:-%. 
 
 
 
 ■>.^^- ^ 
 
 </> .■^' 
 
 ' .A 
 
 S <> 
 
 aX^' ^V. 
 
 
 
 
 "^^y- v^"^ 
 
 
 ■^o rx\ 
 
 X^ '^ ^ 
 
 o 0' 
 
 
 
 
 ,.^'' 
 
 
 
 
 '■ t . •^. 
 
 
 A^ , ' ' « * -'o 
 
 o 
 
 ,0o 
 
 "bo^ 
 
 ,\" --<. 
 
 
 ,0 o 
 
 
 V 
 
 A^^' ^>-- 
 
 
 .^^ 
 
 
 ^' * 
 
 . '^V:n, 
 
 >^'^':. 
 
 
 ■1 
 
 ^^. ^»'A* A^ 
 o 0^ 
 
 
 
 A^"" '^-^. 
 
 1 "o 
 
 5,^ ^ O o v 
 O 
 
 
 . V^ 
 
 ^OO^ 
 
 Oo 
 
 
 ,.--•" '»=5 X -A 
 
 
 ^ v^ .-^^-^-^. 
 
 
 
 '->r 
 
 
 .■?- 
 

 
 ^^ 
 
 ..XV-.,, 
 
 
 .X"^ ^^ ^:-''-'%. ■- 
 
 s>\ ■':. 
 
 
 
 0' > 
 
 '^ & « -^ 
 
 ''^ /' 
 
 '. -^v^ C^ 
 
 . ^ /\ 
 
 \ 
 
 m.'- 
 
 xOo. 
 
 g I \ " ^N. 
 
 ,0 
 
 
 'O, 
 
 ci-. 
 
 '2- 
 
 
 
 -.^^ »,. 
 
 ^- 
 
 .^^ 
 
 -0- 
 
 
 ^^ 0^ 
 
 
 ., '^^ 
 
 
 
 
 ■^■ 
 
 '^ .0-' 
 
 >'-. l^mWr 
 
 ■r % -^jmrn 
 
 
 o 0' 
 
 
 
 ^^ -^c. 
 
 
 '"^-.^^■' :mm\ %^' 
 
 
 
 V <j> 
 
 
 -V 
 
 v^ 
 
 ci-, i- 
 
 v 
 
 
 
 xV\^ 
 
 o, ■' >, 
 
 -9^ 
 
 ,^' ^. 
 
 % ,^' :^/\ 
 
 
 TS- 
 
 
 x^^ 
 
 
 '^ r^ ..^ A 
 
 <^. 
 
 
 
 '--^ ,^ .v.«, -^^ 
 
 ■■■3;' 
 
 ^^ '^ 
 
 Oo, 
 
 ^^^ "^^. =. <^ 
 
 '/, 'c^ 
 
 
 . ,,^^^ 
 
 
 V> ,<;t 
 
 ^^ 
 
 ^. 
 
PUBLICATIONS 
 
 OF THE POLLAK FOUNDATION FOR 
 
 ECONOMIC RESEARCH 
 
 NUMBER ONE 
 THE MAKING OF INDEX NUMBERS 
 
THE MAKING 
 OF INDEX NUMBERS 
 
 A Study of Their 
 Varieties^ Tests^ and Reliability 
 
 BY 
 IRVING FISHER 
 
 PR0PE8S0E OF POLITICAL ECONOMT, YALE UNIVERSmr 
 
 BOSTON AND NEW YORK 
 
 HOUGHTON MIFFLIN COMPANY 
 1922 
 
':^y 
 
 V 
 
 .6 
 
 vA^^^ 
 
 COPYRIGHT, 1922, BY THE POLLAK FOUNDATION FOR ECONOMIC RESEARCH 
 ALL RIGHTS RESERVED 
 
 ?.^o 
 
 Wat jRibcTSftie ^ti* 
 
 CAMBRIDGE • MASSACHUSETTS 
 FKINTED IN THE U.S.A. 
 
 DEC 21 72 
 
 C;lAG92526 
 
TO 
 
 F. Y,_EDGEWORTH 
 
 AND 
 
 CORREA MOYLAN WALSH 
 
 PIONEERS IN THE 
 EXPLORATION OF INDEX NUMBERS 
 
PREFATORY NOTE 
 
 All sciences are characterized by a close approach to exact 
 measurement. How many of them could have made much 
 progress without units of measurement, generally under- 
 stood and accepted, it is difficult to imagine. In order to 
 determine the pressure of steam, we do not take a popular 
 vote : we consult a gauge. Concerning a patient's tempera- 
 ture, we do not ask for anybody's opinion: we read a ther- 
 mometer. In economics, however, as in education, though 
 the need for quantitative measurement is as great as in 
 physics or in medicine, we have been guided in the past 
 largely by opinions and guesses. In the future, we must 
 substitute measurement for guesswork. Toward this end, 
 we must first agree upon instruments of measurement. To 
 the Pollak Foundation for Economic Research it seems 
 fitting, therefore, that its first publication should be The 
 Making of Index Numbers. 
 
 In this book, the author tests by every useful method, 
 not only all the f ormulse for index mnnbers that have been 
 used, but as well all that reasonably could be used; and he 
 tests them by means of actual calculations, extensive and 
 painstaking, based on actual statistical records. He proves 
 that several of the methods of constructing index numbers 
 now in common use are grossly inaccurate; he makes clear 
 why some formulae are precise and others far from it; he 
 points out how to save time in the work of calculation; and 
 he shows how to test the results. Thus he provides us with 
 methods of measuring such illusive things as fluctuations 
 in real wages, in exchange rates, in volume of trade, in the 
 cost of living, and in the purchasing power of the dollar. 
 Finally, he points out that, once a good method of con- 
 
viii PREFATORY NOTE 
 
 structing index numbers has been generally accepted, the 
 usefulness of the instrument will be vastly increased, and 
 will then be extended to many other fields where precise 
 measurement is greatly needed. 
 
 But, after all, is it possible to devise a means of measure- 
 ment that is sufficiently precise to be used as a basis for 
 determining matters of such concern to all human beings 
 as contracts, currency measures, price adjustments, and 
 wage schedules? The doubts on this question that have 
 hitherto stood in the way of the universal use of index 
 numbers must vanish before Professor Fisher's demonstra- 
 tions. He shows that an index number may be so precise 
 an instrument that the error " probably seldom reaches one 
 part in 800, or a hand's breadth on the top of Washington 
 Monument, or less than three ounces on a man's weight, or 
 a cent added to an $8 expense." He shows, further, that all 
 the forms of index numbers that satisfy his few, simple 
 tests give results so nearly alike that it matters little or 
 nothing, for most practical purposes, which form is em- 
 ployed. Any one of these forms is comparable, in point of 
 accuracy, with many of the instruments that are univer- 
 sally and unquestioningly employed in other scientific 
 fields. 
 
 The use of yardsticks of forty different lengths would be 
 a source of endless confusion: the use of forty different 
 kinds of index numbers is no less confusing. If experts fail 
 to clear up this confusion because they disagree on non- 
 essentials, it will seem to the many thousands of people, 
 to whom the mathematics of the subject is a mystery, as 
 though the experts were widely at variance on fundamen- 
 tals. And so, without due cause, index numbers in general 
 will be discredited and the scientific study of economics 
 impeded. For this reason, it is to be hoped that all those 
 who are capable of understanding the subject will see their 
 
PREFATORY NOTE ix 
 
 way clear to agreeing upon the Ideal Formula as the best 
 in point of accuracy. It is to be hoped, furthermore, that 
 they will agree in adopting and advocating for general use 
 the closely similar Formula No. 2153 
 
 ( 
 
 S(go+ei)Po' 
 
 since it is the one which best combines speed of calculation 
 with as high a degree of accuracy as is ever needed for prac- 
 tical purposes. In any event, the Pollak Foundation will 
 have achieved its purpose in publishing this volume, if it 
 has a part in bringing about the abandonment of faulty 
 methods of constructing index numbers, the general adop- 
 tion of any dependable method, and the consequent pro- 
 gressive substitution, wherever precise measurement is 
 possible, of scientific method for personal opinion. 
 
 William Trufant Foster 
 
 Editor of the Pollak Pvhlications 
 Newton, Massachusetts 
 " December 1, 1922 
 
PREFACE 
 
 This book amplifies a paper read in December, 1920, at 
 the Atlantic City meeting of the American Statistical 
 Association. An abstract of that paper was printed in 1921 
 in the March number of the Association's Quarterly Pub- 
 lication. The same paper, somewhat elaborated, was also 
 read before the American Academy of Arts and Sciences at 
 Boston, in April, 1921. 
 
 One of the main conclusions of these papers was accepted 
 at once, namely, that the formula here called the "ideal" 
 is the best form of index number for general purposes. 
 The further contention that this formula is the best for all 
 purposes was stoutly denied by most critics, with the not- 
 able exception of Mr. CM. Walsh, who had reached the 
 same conclusion independently and from a different start- 
 ing point. 
 
 Out of this partial disagreement, a number of writings 
 on index numbers have appeared, such as Professor War- 
 ren M. Persons' article in the Review of Economic Statis- 
 tics for May, 1921, on ''Fisher's Formula for Index Num- 
 bers." Professor Allyn A. Young in his article "The Meas- 
 urement of Changes of the General Price Level," in the 
 Quarterly Journal of Economics for August, 1921, reaches 
 the same formula as "the best single index number of the 
 general level of prices," although he apparently reserves 
 judgment as to its limitations. Professor Wesley C. Mit- 
 chell, in the revision of his monograph on "Index Num- 
 bers of Wholesale Prices in the United States and Foreign 
 Countries" (published as Bulletin No. 28 J,, of the United 
 States Bureau of Labor Statistics, October, 1921), takes 
 a somewhat similar position. 
 
xii PREFACE 
 
 In order to help resolve the questions remaining at issue, 
 a large number of calculations have been made for this 
 book in addition to the large number which had already 
 been made. Any one who has not himself attempted a 
 like task can scarcely realize the amount of time, labor, 
 and expense involved. Some of the work must have been 
 abandoned had not the Pollak Foundation for Economic 
 Research come to the rescue. 
 
 The result has been a much more complete survey of 
 possible formulae than any hitherto attempted. Although, 
 in a subject like this, absolute completeness is out of the 
 question, since the number of possible formulae is infinite, 
 nevertheless, the whole field has been so mapped out as to 
 leave no large gaps. The aim has been to settle decisively, 
 if possible, the questions of how widely the various results 
 reached by different possible methods diverge from each 
 other, and why. Each of more than a hundred formulae 
 has been examined and calculated in four series. Each of 
 these series has its role to play in this study, even formulae 
 which are found, in the end, to have no practical use. 
 
 This book is, therefore, primarily an inductive rather 
 than a deductive study. In this respect it differs from 
 the Appendix to Chapter X of the Purchasing Power of 
 Money, in which I sought deductively to compare the 
 merits of 44 different formulae. The present book had its 
 origin in the desire to put these deductive conclusions to 
 an inductive test by means of calculations from actual his- 
 torical data. But before I had gone far in such testing of my 
 original conclusions, I found, to my great surprise, that the 
 results of actual calculation constantly suggested further 
 deduction until, in the end, I had completely revised both 
 my conclusions and my theoretical foundations. Not that 
 I needed to discard as untrue many of the conclusions 
 reached in the Purchasing Power of Money; for the only 
 
PREFACE xiii 
 
 definite error which I have found among my former con- 
 clusions has to do with the so-called ''circular test" which 
 I originally, with other writers, accepted as sound, but 
 which, in this book, I reject as theoretically unsound. But 
 some of the other tests given in the Purchasing Power of 
 Money, while perfectly legitimate, are of little value as 
 quantitative criteria for a good index number. The most 
 fundamentally important test among those treated in the 
 earlier study is the ''time reversal" test. This and a new 
 test, the "factor reversal" test, are here constituted the 
 two legs on which index numbers can be made to walk. 
 
 In the algebraic analysis, relegated almost wholly to 
 the Appendices, I have refrained, as far as possible, from 
 developing the many possible mathematical transforma- 
 tions and discussions of index numbers, in the belief that, 
 fascinating as these are, the mathematics of index numbers, 
 except as they serve practical ends, would not interest 
 many readers. For the same reason such mathematical 
 analysis as is included has, in some cases, been greatly 
 condensed, the results alone being given. Without such 
 condensation of unimportant details a hundred or more 
 additional pages in the Appendices would have been 
 necessary. 
 
 One incidental result of this study is to show that many 
 precise and interesting relations or laws exist connecting 
 the various magnitudes studied — index numbers, dis- 
 persions, bias, correlation coefficients, etc. Thus this field 
 of study, almost alone in the domain of the social sciences, 
 may truly be called an exact science — if it be permissible 
 to designate as a science the theoretical foundations of a 
 useful art. 
 
 The subject has seemed elusive because it is partly em- 
 pirical and partly rational, and these two aspects of it have 
 not been coordinated. But, although the present volume 
 
xiv PREFACE 
 
 is a combination of theoretical and practical discussion, the 
 theoretical is entirely in the interest of the practical. Most 
 writers on index numbers have been either exclusively the- 
 oretical or exclusively practical, and each of these two 
 classes of writers has been very little acquainted with the 
 other. By bringing these two worlds into closer contact 
 I hope that, in some measure, I may have helped forward, 
 both the science and the art of index numbers. The im- 
 portance of this new art in our economic life is already 
 great and is rapidly increasing. 
 
 While the book includes, I hope, all the chief results of 
 former studies in index numbers, its main purpose is not 
 so much to summarize previous work as to add to our 
 knowledge of index numbers and, as a consequence, to set 
 up demonstrable standards of the accuracy of index num- 
 bers and their suitability under various circumstances. 
 Many of the results reached turned out to be quite different 
 from what I had been led by the previous studies of others 
 as well as of myself to expect. 
 
 I am greatly indebted to the many persons who have 
 helped me in the preparation of the book — especially to 
 Mr. R. H. Coats, Dominion Statistician of Canada; Dr. 
 Royal Meeker, Chief of the Scientific Division of the Inter- 
 national Labour Office; Professor Warren M. Persons, of 
 the Harvard Committee on Economic Research; Pro- 
 fessor Allyn A. Young, of Harvard University; Professor 
 Wesley C. Mitchell, of Columbia University; Director 
 William T. Foster and Professor Hudson Hastings, of 
 the Pollak Foundation for Economic Research; Professor 
 Frederick R. Macaulay, of the National Bureau of Eco- 
 nomic Research; Mr. Correa Moylan Walsh; Mr. H. B. 
 Meek, instructor at Yale in mathematics; Mr. V. I. Caprin, 
 Mr. L. B. Haddad and Mr. M. H. Wilson, Yale students; 
 Miss Else H. Dietel, my research secretary, and my brother, 
 
PREFACE XV 
 
 Mr. Herbert W. Fisher. Professor Hastings, Mr. Meek, 
 Mr Walsh, Miss Dietel, and my brother have read the 
 entire manuscript. Their very valuable criticisms and sug- 
 gestions have been both detailed and general. 
 
 Irving Fisher 
 
 Yale Universitt 
 October, 1922 
 
 Addendum to pp. 240-242. 
 
 As this book goes to press. Professor Allyn A. Young, of 
 Harvard University, writes, calling attention to the fact 
 that what I call the "ideal formula" is mentioned by Ar- 
 thur L. Bowley in Palgrave's Dictionary of Political Econ- 
 omy, vol. Ill, p. 641. This reference, in 1899, antedates any 
 of those mentioned on p. 241 in this book. 
 
 Referring to measuring the ''aisance relative," or the 
 relative well-being of labor, Bowley says: "The best 
 method theoretically for measuring 'aisance relative' ap- 
 pears to be as follows: calculate the quantity by method 
 (ii) twice, taking first a budget typical of the earlier, then 
 of the later year, valuing them at the prices of both years 
 and obtaining two ratios. The average (possibly the geo- 
 metric rather than the arithmetic) of these ratios measures 
 the relative 'aisance.'" He then gives, among others, the 
 "ideal formula." 
 
SUGGESTIONS TO READERS 
 
 IN GENERAL 
 
 After finishing one of his long and much worked-over 
 novels, Robert Louis Stevenson expressed a fear that no- 
 body would read it. In these days of many books and 
 little time for reading them, when even a novel must be 
 short to be much read, a book on Index Numbers can 
 hardly expect to become a "best seller." This book has 
 grown to three times the length originally contemplated, 
 and the very effort to make it readable for the general 
 reader has increased its length. 
 
 It aims to meet the needs of several quite different 
 classes of readers: the specialist in index numbers who is 
 mathematical; the specialist who is not mathematical; the 
 university student who would become a specialist; the 
 student who wants merely to understand the fundamental 
 principles of the subject; the practical user of index num- 
 bers; and, finally, the general reader — he who merely 
 wants to know something about index numbers. The book 
 is also intended to serve as a reference book to be consulted 
 by economists, and statisticians, including business statis- 
 ticians. It is also designed to serve as a textbook for 
 classes in statistics. While it aims primarily to add to our 
 knowledge of index numbers and to set new standards for 
 judging them, most of the time and effort expended upon 
 the writing have been directed toward the reader who has 
 had no previous acquaintance with the subject. In other 
 words, the book tries to popularize the exposition even of 
 the somewhat intricate parts which are placed in the appen- 
 dix. The body of the text, especially if the fine print be 
 omitted, ought to be intelligible to any intelligent reader. 
 
xviii SUGGESTIONS TO READERS 
 
 Every important point has been illustrated graphically. 
 There are 123 charts. I believe that the chief reason why, 
 hitherto, the making of index numbers has been a mys- 
 tery to most people is the absence of just such graphic 
 charts. 
 
 IN PARTICULAR 
 
 1. Only the specialist in the field of Index Numbers is 
 expected to read every word. Appendix 1, consisting of 
 notes on the text, is best read in conjunction with the pas- 
 sages to which the notes relate. 
 
 2. The non-mathematical reader will doubtless omit the 
 mathematical parts of the Appendix. He need not omit 
 the few simple algebraic expressions in the text, nor all of 
 the appendix material. 
 
 3. The non-specialist may well omit all the appendix 
 material, although the non-mathematical parts of the 
 Appendix are almost as easy to read as the text, having 
 been placed in the Appendix merely because concerned 
 with details or side issues. 
 
 4. The general reader who would still further shorten 
 the time of reading may omit the fine print, reducing the 
 printed pages to be read to 216 exclusive of the 71 pages 
 of diagrams and the 33 pages of tables. He may also omit 
 the system of paragraphs occurring in almost every section 
 after the italicized words, "numerically" and "algebrai- 
 cally," totalling 20 pages, unless, after reading a paragraph 
 beginning with the word "graphically," he feels the need 
 of supplementing such paragraphs by the numerical or al- 
 gebraic expressions. These three methods of exposition, 
 numerical, graphic, and algebraic, run parallel through- 
 out the book. This reduces the number of pages to 196. 
 
 5. The mere skimmer will find the main conclusions in 
 the last chapter, XVII. 
 
SUGGESTIONS TO READERS xix 
 
 6. The use of the book as one of reference will be facili- 
 tated by the list of tables and charts, the "keys" re- 
 ferred to below, the section headings, the italicized words 
 ^^ numerically,^^ "graphically,^^ "algebraically,'^ and the ar- 
 rangement of the charts which, when they occur in pairs, 
 places the 'price chart always on the left page and the 
 quantity chart on the right. 
 
 7. Appendix V may be referred to with profit whenever 
 the reader finds mention of a formula merely by its iden- 
 tification number. Attention is especially called to § 1 
 and § 2 of Appendix V, the ''Key to the Principal Alge- 
 braic Notations," and the "Key to Numbering of For- 
 mulse." A few minutes' inspection of this easy mnemonic 
 system of numbering will enable the reader to recognize 
 at sight "Formula 1," "Formula 21," "Formula 53," 
 "Formula 353," or any other number, and mentally 
 locate it in the system. 
 
 8. The reader seeking directions for computing index 
 numbers by the nine most practical formulae will find 
 them in Appendix VI, § 2. 
 
TABLE OF CONTENTS 
 
 CHAPTER PAGE 
 
 I. Introduction 1 
 
 ^ II. Six Types of Index Numbers Compared . . , .11 
 
 ^ III. Four Methods of Weighting .43 
 
 IV. Two Great Reversal Tests 62 
 
 V. Erratic, Biased, and Freakish Index Numbers . . 83 
 
 Vl. The Two Reversal Tests AS Finders OF Formulae . .118 
 
 VII. Rectifying Formula BY "Crossing" them . . . 136 
 
 VIII. Rectifying Formula by Crossing their Weights . . 184 
 
 IX. The Enlarged Series of FormuLjE 197 
 
 > X. What Simple Index Number is best? 206 
 
 ^ XI. What is the best Index Number? 213 
 
 XII. Comparing all the Index Numbers with the "Ideal" 
 
 (Formula 353) 243 
 
 XIII. The So-called Circular Test 270 
 
 XIV. Blending the Apparently Inconsistent Results . . 297 
 XV. Speed of Calculation 321 
 
 XVI. Other Practical Considerations ...... 330 
 
 "^ XVII. Summary and Outlook 350 
 
 Appendix I. Notes to Text 371 
 
 Appendix II. The Influence of Weighting .... 439 
 
 Appendix III. An Index Number an Average of Ratios 
 
 Rather than a Ratio of Averages . . . 451 
 
xxii TABLE OF CONTENTS 
 
 PAGE 
 
 Appendix IV. Landmarks in the History of Index Numbers 458 
 
 Appendix V. List of Formula for Index Numbers , . 461 
 
 Appendix VI. Numerical Data and Examples . . . 489 
 
 Appendix VII. Index Numbers by 134 Formula for Prices 
 BY THE Fixed Base System and (in note- 
 worthy cases) the Chain System . . . 498 
 
 Appendix VIII. Selected Bibliography 519 
 
 Index 521 
 
LIST OF TABLES 
 CHAPTER II 
 
 TABLE PAGE 
 
 1. The Simple Arithmetic Index Number for 1914 as Calculated 
 from the 36 Prices for 1913 and 1914 ...... 16 
 
 2. Simple Arithmetic Index Number (Formula 1) for Prices . 23 
 
 CHAPTER III 
 
 3. Values (in Millions of Dollars) of Certain Commodities . . 47 
 
 4. Hybrid Values (in Millions of Dollars) of Certain Commodi- 
 ties 55 
 
 5. Identification Numbers of Primary Formulae . . , .61 
 
 CHAPTER IV 
 
 6. Value Ratios for 36 Commodities, 1913-1918 .... 74 
 
 CHAPTER V 
 
 7. Joint Errors of the Forward and Backward Applications of 
 Each Formula (that is, under Test 1) in Per Cents ... 84 
 
 8. Joint Errors of the Price and Quantity Applications of Each 
 Formula (that is, under Test 2) in Per Cents .... 85 
 
 9. The Four Systems of Weighting the Price Relatives for 1917, 
 
 — > -— ) etc 97 
 
 10. Dispersion of 36 Price Relatives, (1) before the World War, 
 and, (2) during it 109 
 
 11. Dispersion of 36 and of 12 Quantity Relatives .... 110 
 
 CHAPTER VII 
 
 12. Identification N'lmbers of Formulae of Main Series . . . 182 
 
 CHAPTER VIII 
 
 13. Derivation of Cross Formulae and Cross Weight Formulae . 186 
 
 14. Index Numbers by Cross Weight Formulae (1123, 1133, 1143, 
 
xxiv LIST OF TABLES 
 
 TABLE PAGE 
 
 1153) compared with Index Numbers by Corresponding 
 Cross Formulae (123, 133, 143, 153) 189 
 
 15. Index Numbers by Cross Weight Formula (1003, 1004, 1013, 
 1014) compared with Corresponding Cross Formulae . . . 190 
 
 16. Doubly Rectified Formulae derived from Primary Weighted 
 Formulae 194 
 
 CHAPTER IX 
 
 17. Enlarged Arithmetic-Harmonic Group . . . . . .199 
 
 18. Enlarged Geometric Group . 200 
 
 19. Enlarged Median Group 200 
 
 20. Enlarged Mode Group 201 
 
 21. Enlarged Aggregative Group 201 
 
 22. The Five-Tined Fork 204 
 
 CHAPTER XI 
 
 23. Excess or Deficiency of Simple Mode of Price Relatives (in Per 
 Cents of Simple Geometric) 216 
 
 24. Excess or Deficiency of Simple Median of Price Relatives (in 
 Per Cents of Simple Geometric) 217 
 
 25. Two Best Index Numbers 224 
 
 26. Selected Index Numbers 226 
 
 27. Probable Errors of an Index Number of Prices or Quantities 
 worked out by anyone of the 13 Formulae considered as Equally 
 Good Independent Observations 227 
 
 CHAPTER XII 
 
 28. Index Numbers by 134 Formulae arranged in the Order of Re- 
 moteness from the Ideal (353) (as shown by the Fixed Base 
 Figures for the Price Indexes) 244 
 
 29. Averages of Each of the Various Classes of Index Numbers . 257 
 
 30. Accuracy of Simple Geometric and Simple Median, judged by 
 the Standard of Formula 353 for 36 Commodities, Prices . . 261 
 
 31. Accuracy of Simple Geometric and Simple Median, judged by 
 the Standard of Formula 353 for 36 Commodities, Quantities . 261 
 
 32. Accuracy of Simple Geometric and Simple Median, judged by 
 the Standard of Formula 53 for 1437 Commodities . . .262 
 
LIST OF TABLES xxv 
 
 CHAPTER XIII 
 
 TABLE PAGE 
 
 33. The "Circular Gap," or Deviation from fulfilling the So-called 
 "Circular Test" of Various Formulae 278 
 
 34. The "Circular Gap," or Deviation from fulfilling the So-called 
 "Circular Test," of Formula 353 (in all possible 3- Around 
 Comparisons of Price Indexes) . 280 
 
 35 The "Circular Gap," or Deviation from fulfilling the So-called 
 "Circular Test," of Formula 353 (in all possible 4- Around 
 Comparisons of Price Indexes) 281 
 
 36. The "Circular Gap," or Deviation from fulfilling the So-called 
 "Circular Test," of Formula 353 (in all possible 5- Around 
 Comparisons of Price Indexes) 282 
 
 37. The "Circular Gap," or Deviation from fulfilling the So-called 
 "Circular Test," of Formula 353 (in all possible 6- Around 
 Comparisons of Price Indexes) 283 
 
 38. "Circular Gaps "for Formula 353 ....... 284 
 
 39. List of Formulae in (Inverse) Order of Conformity to So-called 
 Circular Test 289 
 
 CHAPTER XIV 
 
 40. Formula 353 on Bases 1913, 1914, 1915, 1916, 1917, 1918; also 
 Formula 7053, the Average of the Six Preceding, and 7053 re- 
 duced to make the 1913 Figure 100 301 
 
 41. Four Single Series of Six Index Numbers as Makeshifts for the 
 Complete Set of Table 40, Prices 307 
 
 42. Four Single Series of Six Index Numbers as Makeshifts for the 
 Complete Set of Table 40, Quantities 307 
 
 43. The Influence of Broadening the Base in reducing Bias . .315 
 
 CHAPTER XV 
 
 44. Rank in Speed of Computation of Formulae ..... 322 
 
 CHAPTER XVI . 
 
 45. Deviations from 200 Commodities Index 337 
 
 46. Comparison of the Aggregate Value of the 100, 50, 25, 12, 6, 
 and 3 Commodities with the Aggregate Value of the 200 Com- 
 modities 339 
 
 47. (Inverse) Order of Rank of Formulae 348 
 
xxvi LIST OF TABLES 
 
 APPENDIX I (Note to Chapter V, § 11) 
 
 TABLE PAGE 
 
 48. For Finding the Bias corresponding to any given Dispersion 390 
 
 49. Standard Deviations (for Prices) 391 
 
 50. Special Dispersion Index compared with Standard Deviation 
 (logarithmically calculated) for the 36 Commodities (Simple) . 392 
 
 51. Special Dispersion Index compared with Standard Devia- 
 tion (logarithmically calculated) for the 36 Commodities 
 (Weighted) 393 
 
 APPENDIX I (Note to Chapter XI, § 13) 
 
 52. Price and Quantity Movements of Rubber and Skins and 
 their Average by Formulae 53 and 54 414 
 
 53. Price and Quantity Movements of Lumber and Wool and their 
 Average by Formulae 53 and 54 414 
 
 54. Index Numbers by Formulae 353 and 2153 for Rubber and 
 Skins 415 
 
 55. Index Numbers by Formulae 353 and 2153 for Lumber and 
 Wool 415 
 
 APPENDIX I (Note to Chapter XIII, § 9) 
 
 56. Cross References between the Numbers for Formulae tabulated 
 in the Purchasing Power of Money and the Numbering used in 
 this Book 419 
 
 57. Showing the Formulae which fulfill and do not fulfill Three 
 Supplementary Tests 422 
 
 APPENDIX T (Note to Chapter XV, § 2) 
 
 58. Formula 2153P as a Percentage of Formula 353P (according to . 
 
 54 
 Various Values of — and 353Q, both expressed in per cents) 430 
 53 
 
 APPENDIX II (§ 6) 
 
 59. Comparative Effects on the Index Number for 1917 (by 
 Formula 3) of increasing Tenfold the Weights of Half of the 36 
 Commodities according as the Commodities are taken at Ran- 
 dom, or selected to make the Largest Effect 446 
 
LIST OF TABLES xxvii 
 
 TABLE PAGE 
 
 60. Index Numbers computed by using Different Weights for 
 Skins 446 
 
 61. Index Numbers computed by using Different Weights for Hay 447 
 
 APPENDIX V (§3) 
 
 62. Formulae for Index Numbers 466 
 
 APPENDIX VI (§ 6) 
 
 63. Prices of the 36 Commodities, 1913-1918 489 
 
 64. Quantities Marketed of the 36 Commodities, 1913-1918 . 490 
 
 APPENDIX VII 
 
 65. Index Numbers by 134 Formulae for Prices by the Fixed Base 
 System and (in noteworthy cases) the Chain System . . 498 
 
LIST OF CHARTS 
 
 (In all cases the charts are plotted on the "ratio chart" in which the vertical scale is so 
 arranged that the same slope always represents the same percentage rise. For a full descrip- 
 tion of the advantages of this method the reader is referred to "The Ratio Chart," Irving 
 Fisher, Quarterly Publication of the American Statistical Association, June, 1917.) 
 
 CHAHT PAGE 
 
 1. Averaging Two 5 
 
 2. Averaging Three 6 
 
 3. Individual Prices and Quantities Dispersing from 1913 . . 12 
 
 4. Year to Year Dispersion of Prices and Quantities ... 20 
 
 5. Simple Arithmetic Index Number of Prices of 36 Commodities 
 compared with "No. 353" 24 
 
 6. Index No. 353 of Prices contrasted (1) with dotted lines above, 
 each diverging 5% in a year; (2) with dotted lines below, each 
 diverging 1% in a year 25 
 
 7. Uniform Slopes = Uniform Ratios 26 
 
 8. Simple Index Numbers of Prices and Quantities ... 32 
 
 9. The Five-Tine Fork of 6 Curves 50 
 
 10. The Two Extreme Methods of Weighting the Median . . 52 
 
 11. Forward ('17-'18) and Backward ('18-17) Simple Arithmetics 
 contrasted 68 
 
 12. P(by Formula 353) x Q(by Formula 353) = V 178% x 125% 
 
 = 223% 76 
 
 13. P(by Formula 9) x Q(by Formula 9) not = to F 187% x 132% 
 not = 223%, 77 
 
 14. Type Bias of Formula 1 88 
 
 15. Three Types of Index Numbers: Arithmetic, Geometric, Har- 
 monic 92 
 
 16. Four Methods of Weighting compared: By base year values, 
 
 by mixed values (in two ways), and by given year values . . 98 
 
 17. Four Methods of Weighting compared for Medians . . .100 
 
 18. Double Bias (Weight Bias and Type Bias) of Formula 9 . .102 
 
 19. Weight Bias of Formula 29 104 
 
 20. The Five-Tine Fork of 18 Curves 106 
 
 21. Insensitiveness of Median and Mode to Number of Commodi- 
 ties 114 
 
XXX LIST OF CHARTS 
 
 CHART 
 
 PAGE 
 
 22. The Harmonic Forward is Parallel to the Arithmetic Backward 120 
 
 23. Three Types of Index Numbers: Factor Antitheses of Har- 
 monic, Geometric, Arithmetic 126 
 
 24. Four Methods of Weighting compared 128 
 
 25. The Simple Geometric : compared with the Simple Arithmetic 
 and Harmonic and their Rectification by Test 1 ... 138 
 
 26. Rectified Arithmetic and Harmonic, Simple .... 146 
 
 27. Rectified Geometric, Simple 150 
 
 28. Rectified Median, Simple 152 
 
 29. Rectified Mode, Simple . . 154 
 
 30. Rectified Aggregative, Simple 156 
 
 31. Simple Index Numbers and their Antitheses and. Derivatives: 
 Satisfying neither Test; satisfying Test 1 only; satisfying 
 Test 2 only; satisfying both Tests (Modes omitted) . . .158 
 
 32. Rectified Arithmetic and Harmonic, Weighted (by Values in 
 One Year) 160 
 
 33. Rectified Arithmetic and Harmonic, Weighted (by "Mixed" 
 Values) 162 
 
 34. Rectified Geometric, Weighted (by Values in One Year) . 164 
 
 35. Rectified Geometric, Weighted (by "Mixed" Values) . . 166 
 
 36. Rectified Median, Weighted (by Values in One Year) . .168 
 
 37. Rectified Median, Weighted (by "Mixed "Values) . . .170 
 
 38. Rectified Mode, Weighted 172 
 
 39. Rectified Aggregative, Weighted 174 
 
 40. Weighted Index Numbers and their Antitheses and Deriva- 
 tives: Satisfying neither Test; satisfying Test 1 only; satisfying 
 Test 2 only; satisfying both Tests (Modes omitted) . . . 176 
 
 41. Range of Prices and Quantities and of Three Types of Index 
 Numbers; Weighted: Satisfying neither Test; satisfying only 1 
 
 or only 2; satisfying both 1 and 2 (Modes and Medians omitted) 178 
 
 42. Weighted Index Numbers Doubly Rectified (Modes omitted) 180 
 
 43. Close Agreement of Cross Formulae and Cross Weight Formulae 188 
 
 44. Close Agreement of Cross Formulae and Cross Weight Formulae 
 (fully rectified) 192 
 
 45. Weighted Aggregatives for 90 Raw Materials: War Industries 
 Board Statistics 232 
 
 46. Formulae 53 and 54 applied to Stock Market .... 234 
 
 47. Formulae 53 and 54 applied to 12 Leading Crops (after W. M. 
 Persons), 1880-1920 236 
 
LIST OF CHARTS xxxi 
 
 CHART PAGE 
 
 48. Formulae 53 and 54 applied to 12 Leading Crops (after W. M. 
 Persons), 1910-1919 238 
 
 49. Ranking as to Accuracy of All Index Numbers: 
 
 1. Worthless Index Numbers 249 
 
 2. Poor Index Numbers 250 
 
 3. Fair Index Numbers 252 
 
 4. Good Index Numbers 254 
 
 5. Very Good, (6) Excellent, and (7) Superlative Index 
 Numbers 255 
 
 50. Simple Geometric and Simple Median compared ^vith Ideal for 
 Different Numbers of Commodities 263 
 
 51. Circular Test: Gaps for Years 0-4-5 of Formula? 1, 9, 23, 141, 
 
 151 279 
 
 52. Circular Test: Largest Gaps for 3-Around, 4-Around, 5- 
 Around, 6-Around Comparisons 286 
 
 53. Dispersion (measured by Standard Deviations) (Fixed Base) . 290 
 
 54. Dispersion (measured by Standard Deviations) (Chain) . . 292 
 
 55. Dispersion (measured by Standard Deviations) (Fixed Base) 
 (Sauerbeck's Figures) 294 
 
 56. Comparison for Six Bases of Formulae 53, 54, 353 . . . 304 
 
 57. Optional Varieties of 353 308 
 
 58. Simple Median and Quartiles drawn from Origin . . . 310 
 
 59. 353 and 6023 compared for 12 Leading Crops, 1880-1919 (Day 
 and Persons) 314 
 
 60. 353 and 6023 compared for 12 Leading Crops, 1910-1919 
 (Day and Persons) 316 
 
 61. Effect of Number of Commodities on Index Numbers . . 338 
 
 62. Finding the Simple Mode 372 
 
 63. Different Cross Weightings of 53 and 54 400 
 
 64. 353 compared with its Cross Weight Rivals .... 404 
 
 65. Distribution of 1437 Price Relatives, Forward and Backward 409 
 
 66. Simple vs. Cross Weighted Index Numbers 440 
 
 67. Simple vs. Cross Weighted Index Numbers, Factor Antitheses 442 
 
 68. Weighting is relatively unimportant 448 
 
THE MAKIN^a 
 OF INDEX I^UMBERS 
 
 CHAPTER I 
 
 INTRODUCTION 
 
 § 1. Objects of the Book 
 
 For those who have made any attempt to penetrate 
 their mysteries, index numbers seem to have a perennial 
 fascination. Because of recent upheavals of prices, the 
 interest in this method of measuring such upheavals is 
 rapidly spreading. During the last generation index 
 numbers have gradually come into general use among 
 economists, statisticians, and business men. The skepti- 
 cism with which they were once regarded has steadily 
 diminished. In 1896, in the Economic Journal, the Dutch 
 economist, N. G. Pierson, after pointing out some ap- 
 parently absurd results of index numbers, said: ''The 
 only possible conclusion seems to be that all attempts 
 to calculate and represent average movements of prices, 
 either by index numbers or otherwise, ought to be aban- 
 doned." No economist would today express such an 
 extreme view. And yet there lingers a doubt as to the 
 accuracy and reliability of index numbers as a means of 
 measuring price movements. 
 
 It is perfectly true that different formulae for calculat- 
 ing index numbers do yield different results. But the 
 important question, never hitherto answered in a com- 
 prehensive way, is : How different are the results, and 
 
 1 
 
2 THE MAKING OF INDEX NUMBERS 
 
 can we find reasons for accepting some and rejecting 
 others ? 
 
 To answer this general question as to the trustworthi- 
 ness of index numbers is one of the two chief purposes of 
 the present book. In order to make the answer conclu- 
 sive, all the formulae for index numbers which have been 
 or could reasonably be constructed, have been investi- 
 gated and tested in actual calculations based on actual 
 statistical records. We shall find that some of the for- 
 mulae in general use and unhesitatingly accepted by un- 
 critical users are really very inaccurate, while others have 
 an extraordinary degree of precision. The reasons for 
 these differences will be investigated as well as the attri- 
 butes essential to precision. 
 
 The second chief purpose of this book is to help make 
 the calculation of index numbers rapid and easy. To this 
 end we shall show what formulae are best in theory and 
 practice, and shall indicate certain short cuts for their 
 calculation. 
 
 § 2. An Index Number Defined 
 
 Most people have at least a rudimentary idea of a "high 
 cost of living" or of a "low level of prices," but usually 
 very little idea of how the height of the high cost or the 
 lowness of the low level is to be measured. It is to meas- 
 ure such magnitudes that "index numbers" were in- 
 vented. 
 
 There would be no difficulty in such measurement, 
 and hence no need of index numbers, if all prices moved 
 up in perfect unison or down in perfect unison. But 
 since, in actual fact, the prices of different articles move 
 very differently, we must employ some sort of compro- 
 mise or average of their divergent movements. 
 
 If we look at prices as starting at any time from the 
 
INTRODUCTION 3 
 
 same point, they seem to scatter or disperse like the 
 fragments of a bm-sting shell. But, just as there is 
 a definite center of gravity of the shell fragments, 
 as they move, so is there a definite average move- 
 ment of the scattering prices. This average is the ''index 
 number." Moreover, just as the center of gravity is 
 often convenient to use in physics instead of a list of the 
 individual shell fragments, so the average of the price 
 movements, called their index number, is often convenient 
 to use in economics. 
 
 An index number of prices, then, shows the average 
 percentage change of prices from one point of time to an- 
 other. The percentage change in the price of a single 
 commodity from one time to another is, of course, found 
 by dividing its price at the second time by its price at 
 the first time. The ratio between these two prices is 
 called the price relative of that one particular commodity 
 in relation to those two particular times. An index num- 
 ber of the prices of a number of commodities is an average 
 of their price relatives. 
 
 This definition has, for concreteness, been expressed 
 in terms of prices. But in like manner, an index number 
 can be calculated for wages, for quantities of goods im- 
 ported or exported, and, in fact, for any subject matter 
 involving divergent changes of a group of magnitudes. 
 
 Again, this definition has been expressed in terms of 
 time. But an index number can be applied with equal 
 propriety to comparisons between two places or, in fact, 
 to comparisons between the magnitudes of a group of 
 elements under any one set of circumstances and their 
 magnitudes under another set of circumstances. But 
 in the great majority of cases index numbers are actually 
 used to indicate price movements in time. 
 
4 THE MAKING OF INDEX NUMBERS 
 
 § 3. Illustrations — Numerical, Graphic, Algebraic 
 
 An index number is an average. There are many kinds 
 of averages — the arithmetic, the geometric, etc., of which 
 only the arithmetic is known to most people. In these 
 preliminary illustrations, therefore, we shall employ the 
 arithmetic average, but always specify "arithmetic" in 
 order not to lose sight of the fact that this is but one kind 
 of average. 
 
 Numerically, if wheat has risen 4 per cent since some 
 specified date, say January 1, 1920 (say from $1. a 
 bushel to $1.04), and beef has risen 10 per cent in the same 
 time (say from 10 cents per pound to 11), the simple 
 arithmetic average percentage rise of wheat and beef is 
 midway between 4 per cent and 10 per cent, or 7 per; 
 
 cent (that is, ^^-^^ = 7). Then 107 per cent is the ''in- 
 
 dex number" for the present prices of these two articles 
 as compared with those of the original date, called the 
 ''base" and taken, for convenience, as 100 per cent. Or : 
 
 Commodity 
 
 January 1, 1920 
 
 Present Time 
 
 Wheat 
 
 100 per cent 
 100 per cent 
 
 104 per cent 
 110 per cent 
 
 Beef 
 
 
 Simple arithmetic average 
 
 100 per cent 
 
 107 per cent 
 
 Thus 107 per cent is an index number based on the twc 
 price ratios, or "price relatives," 104 per cent and 110 
 per cent. 
 
 Graphically, Chart 1 pictures the numerical results given 
 above. 
 
 Algebraically, if the price of one commodity in 1920 
 (January 1) is po and, in 1921, pi, and the price of another 
 commodity in 1920 is p'o and, in 1921, p'l, then their 
 
INTRODUCTION 
 
 Averaging Two 
 
 e nf both 
 
 '^^ heaf 
 
 W4^^ 
 
 Jan. L /9Z0 JQn. i. i9Zi 
 
 Chakt 1. Percentage changes in price of two commodities and the average 
 percentage change. 
 
 price ratios or "price relatives" are — and ^, and the 
 
 2?o V 
 
 simple arithmetic average of the two, that is, the simple 
 
 arithmetic index number, is 2l-_2_2, it is convenient to 
 
 multiply the result by 100 to express it in percentages. 
 
 The same method applies, of course, to more than two 
 
 prices. Thus, if three prices, say sugar, wheat, and beef, 
 
 rise respectively 4 per cent, 4 per cent, and 10 per cent, 
 
 4 4-4 4- 10 
 their average rise is — or 6 per cent, and the "in- 
 
 o 
 
 dex number" is 106 as compared with the original price 
 
 level of 100 taken as a base of comparison. 
 
 Graphically, Chart 2 shows the simple arithmetic aver- 
 age just described. 
 
 Algebraically, the simple arithmetic index number of 
 three commodities is evidently 
 
 . Po V'o p"o 
 
THE MAKING OF INDEX NUMBERS 
 
 Averaging Three 
 
 
 /06 ( ^V 
 
 el 
 
 
 Jan I. I9Z0 Jan. / I9ii 
 
 Chart 2. Percentage changes in price of three commodities and the 
 average percentage change. 
 
 § 4. Weighting 
 
 The preceding calculation treats all the commodities 
 as equally important ; consequently, the average was 
 called ''simple." If one commodity is more important 
 than another, we may treat the more important as though 
 it were two or three commodities, thus giving it two or 
 three times as much ''weight" as the other commodity. 
 
 Thus, suppose that wheat is taken to be twice as im- 
 portant as beef. Then the average rise of wheat and 
 
 beef, instead of being —^ — = 7, as it was when the two 
 commodities were regarded as equally important, becomes 
 
 ^ ^^ — = 6, just as though there were three commodi- 
 o 
 
 ties, thus making the index number 106 instead of 107. 
 
 In this average, wheat is weighted twice as heavily as 
 
 beef. If, reversely, beef is given twice as much weight 
 
 in determining the index number as wheat, the average 
 
 rise is ^ "^ — <■ = 8 and the index number is 108 in- 
 o 
 
 stead of 107. 
 
INTRODUCTION 7 
 
 Algebraically, if the wheat is weighted twice as heavily 
 as the beef — that is, if their weights are as 2 to 1 — the 
 formula for this weighted arithmetic index numberbe- 
 comes 
 
 3 
 
 It makes no difference to the result whether the weights 
 be 2 and 1 as above, or 4 and 2, or 20 and 10, or any other 
 two numbers of which one is double the other, since the 
 denominator increases proportionally. Thus, if the 
 weights were 14 and 7 the formula would be 
 
 Pl) + 7(P» 
 
 14(Si 
 
 21 
 which could evidently be reduced to the first formula 
 simply by canceling ''7" in the numerator and denomi- 
 nator. 
 
 Thus ''weighting" is clearly relative only. If we weight 
 wheat and beef evenly, say 10 and 10, evidently the re- 
 sult is the simple average. So a simple average may be 
 said to be a weighted average in which the weights are all 
 equal. Strictly speaking, therefore, there is no such thing 
 as an itnweighted average. 
 
 In general algebraic terms, if the weight for wheat is 
 w and that for beef is w', the weighted arithmetic average 
 is 
 
 w 
 
 (2i) + „'(P>) 
 
 w + w' 
 Graphically, the effect of weighting wheat heavily is 
 evidently to bring the index number line of Chart 1 down 
 nearer to the wheat line as in Chart 2, while weighting 
 heef more heavily swings it up toward the beef line. 
 
8 THE MAKING OF INDEX NUMBERS 
 
 We have illustrated the two most common varieties of 
 index numbers, the simple arithmetic and the weighted 
 arithmetic, or, as they might in strict accuracy be called, 
 the evenly weighted and the unevenly weighted arith- 
 metic index numbers. But, as already noted, there are 
 many kinds of index number formulae other than the 
 arithmetic. In fact, there are as many possible varieties 
 of formulae as there are different varieties of averages, 
 and these are infinite. 
 
 § 5. Attributes of an Index Number 
 
 Moreover, index numbers differ from each other not 
 only as to the kinds of formulae used in calculating them, 
 but also in several other respects, or "attributes." 
 Briefly, all the attributes of an index number, twelve in 
 number, may be enumerated under three groups as fol- 
 lows: 
 
 I. As TO THE Construction of the Index Number 
 
 (1) The general character of the data included, e.g. ' ' whole- 
 sale prices" or "retail prices" of commodities, or "prices 
 of stocks," or "wages," or "volume of production," etc. 
 
 (2) The specific character of data included, e.g. "foods," 
 still further specified as "butter," "beef," etc. 
 
 (3) Their assortment, e.g. a larger proportion of quo- 
 tations of meats than of vegetables. 
 
 (4) The number of quotations used, e.g. "22 commodi- 
 ties" as in the case of the Economist index number (until 
 recently) as contrasted with "1474 commodities" as in 
 the case of the War Industries Board. 
 
 (5) The kind of mathematical formula employed for 
 calculating the index number, e.g. the "simple arithmetic 
 average" or the "weighted geometric average," etc. 
 
INTRODUCTION 9 
 
 II. As TO THE Particular Times or Places :ro Which 
 THE Index Number Applies 
 
 (1) The period covered, e.g. ''1913-1918," or the terri- 
 tory covered, e.g. certain specified cities of which the 
 price levels are to be compared. 
 
 (2) The base, e.g. the year 1913. 
 
 (3) The interval between successive indexes, e.g. "yearly " 
 or "monthly." 
 
 Ill, As TO THE Sources and Authorities 
 
 (1) The agency which collects, calculates, and publishes 
 the index number, e.g. "Bradstreet's" or the "United 
 States Bureau of Labor Statistics." 
 
 (2) The markets used, e.g. the "Stock" or "Produce" 
 Exchanges of "New York" or the "primary markets of 
 the United States." 
 
 (3) The sources of quotations, e.g. the "leading trade 
 journals" or the books of business houses. 
 
 (4) The publication containing the index number, e.g. 
 the Bulletin of the United States Bureau of Labor Sta- 
 tistics. 
 
 Of these 12 attributes characterizing an index number, 
 I shall deal in detail with one only, namely, the formula. 
 The other 11 attributes, previous writers have covered 
 to a large extent, and I shall content myself with a very 
 brief summary of their conclusions, which will be given 
 at the end of this book. 
 
 § 6. Fairness of Index Nmnbers 
 
 The multipUcity of formulae for computing index num- 
 bers has given the impression that there must be a corre- 
 sponding multipucity in the results of these computations, 
 with no clear choice between them. But this impression 
 
10 THE MAKING OF INDEX NUMBERS 
 
 is due to a failure to discriminate between index numbers 
 which are good, bad, and indifferent. By means of cer- 
 tain tests we can make this discrimination. 
 
 The most important tests are all embraced under the 
 single head of fairness. The fundamental purpose of an 
 index number is that it shall fairly represent, so far as one 
 single figure can, the general trend of the many diverging 
 ratios from which it is calculated. It should be the **just 
 compromise" among conflicting elements, the "fair aver- 
 age," the ''golden mean." Without some kind of fair 
 spUtting of the differences involved, an index number is 
 apt to be unsatisfactory, if not absurd. How we are to 
 test the fairness of an average will be shown in Chapter 
 IV. 
 
 Meanwhile it will be advisable first, to describe the 
 various types of index numbers ; for, thus far, we have dis- 
 cussed only the arithmetic type. 
 
CHAPTER II 
 
 SIX TYPES OF INDEX NUMBERS COMPARED 
 
 § 1. The Dispersion of Individual Prices and Quantities 
 
 As a preliminary to calculating various kinds of index 
 numbers we may picture the movements of the 36 in- 
 dividual commodities which will be used for the compari- 
 sons in this book. 
 
 Graphically, Chart ^ 3P shows the movements of the 
 prices of these 36 commodities considered as diverging 
 from a common starting-point in 1913, and Chart 3Q 
 shows, in like manner, the movements of the quantities 
 marketed of these same 36 commodities. 
 
 A casual observer, looking at the diverging and tangled 
 course of prices and quantities, would be tempted to give 
 up in advance, not only any attempt to find index numbers 
 which can truly represent changes in the ''general trend" 
 of these widely scattering figures, but also to wonder 
 whether the words ''general trend" corresponded to any 
 real and clear idea. He would note that at the close of 
 the period, in 1918, the price of rubber was 32 per cent 
 below its starting-point, in 1913, while the price of wool 
 was 182 per cent above its starting-point. Thus, their 
 price relatives, in 1918 relatively to 1913, are as 68.02 to 
 100 and as 282.17 to 100, the latter being 4 times the 
 former, with the other 34 price relatives widely scattered 
 between. As to quantities, he would find that the quan- 
 tity of rubber in 1918 stood at 303.54 and that of skins 
 
 * All charts in this book are "ratio charts," as explained in detail later 
 in this chapter. 
 
 11 
 
12 
 
 THE MAKING OF INDEX NUMBERS 
 
 Individual 
 Prices 
 
 Dispersing from 
 1913 
 
 wool 
 
 colion. pig Iron 
 
 wheat '^^ 
 
 lard, barky 
 k'ron bars, coke 
 ,l[n plate.skins 
 ^bacon.t,ogs 
 
 /pigfin.eggs.bay 
 -'^&ad''''^''bifCCOt 
 
 ■lime, silk 
 [beaf, had 
 pork, cement 
 butter, silver 
 ,copper 
 kattle 
 
 petroleum 
 lumber 
 anth. coal 
 hides 
 
 coffee 
 
 rubber 
 
 IZ 
 
 74 
 
 /5 
 
 16 
 
 17 
 
 16 
 
 Chart 3P. Showing the enormously wide dispersion of the price move- 
 ments of the 36 commodities. (The eye is enabled to judge the relative 
 vertical positions of the curves in this and other charts by means of the 
 little dark vertical line marked "5 %" inserted to serve as a measuring rod. 
 Thus in 1917 coffee is about 5 per cent higher than rubber while petroleum 
 is about 20 per cent higher than coffee and anthracite coal 10 per cent 
 higher than petroleum.) 
 
SIX TYPES OF INDEX NUMBERS 
 
 13 
 
 Quantities 
 
 Dispersing from 
 
 rubbtr 
 
 copper 
 
 tinplafa 
 
 .cattle 
 (wool 
 won bars 
 /beef 
 pig tin 
 "^petroleum 
 
 -barley.siik 
 ^bacon.^ats 
 rcorree 
 ^tead 
 
 - ^^^^-butter 
 
 .wheat lard 
 
 ■hides 
 
 silver 
 
 dumber 
 lime 
 
 cement 
 
 white lead 
 
 mutton 
 steel 
 
 73 14 75 76 17 78 
 
 Chakt ZQ. Showing the enormously wide dispersion of movements of 
 the quantities marketed of the 36 commodities. 
 
14 THE MAKING OF INDEX NUMBERS 
 
 at 10.45 (too low to get on the chart) so that the former 
 was 29 times the latter, with the other 34 quantity rela- 
 tives widely scattered between. 
 
 How is it possible to find a common trend for such 
 widely scattered price relatives or quantity relatives? 
 Will not there be as many answers to such a question as 
 there are methods of calculation ? Will not these answers 
 vary among themselves 50 per cent or 100 per cent? The 
 present investigation will show how mistaken is such a 
 first impression. 
 
 § 2. Uniform Data Used for Comparisons 
 
 The 36 price movements and the 36 quantity move- 
 ments just pictured will constitute the raw material for 
 calculating the many kinds of index numbers which we 
 shall consider. Thus the very same data will be used 
 for calculating different kinds of index numbers by 134 
 different formulae. These data are a part of the mass of 
 statistics, collected by Wesley C. Mitchell for the War 
 Industries Board, for wholesale prices and quantities 
 marketed of 1474 commodities in the United States. The 
 list of these 36 commodities and the figures for the prices 
 and quantities of each are given in Appendix VI, § 1. 
 
 One chief reason for employing data from the records 
 of the War Industries Board is that they are based on the 
 only ^ collection of data which includes figures for quanti- 
 ties as well as for the prices of each commodity. This 
 same set of data is used for all of the comparisons under 
 the various formulae. We may be sure that our tests 
 are severe and conclusive because the period covered, 
 1913 to 1918, is (as will be shown statistically, later) a 
 period of extraordinary dispersion in the movements 
 both of prices and quantities. 
 
 ' Since the present work was begun there have appeared the studies 
 by Professors Day and Persons of 12 commodities cited later. 
 
SIX TYPES OF INDEX NUMBERS 15 
 
 In view of this fact we may be confident that the close- 
 ness of agreement, which the following calculations show 
 among those index numbers which are not demonstrably 
 unfair in their construction, does not exaggerate but 
 actually understates the closeness which will be encoun- 
 tered in ordinary practice. ^ 
 
 § 3. The Simple Arithmetic Average of Relative Prices 
 by the Fixed Base System 
 
 Although we shall calculate index numbers by 134 
 different formulae, they all fall under six types : the arith- 
 metic, harmonic, geometric, median, mode, and aggrega- 
 tive.^ These are the only types of average ever considered 
 for index numbers, or ever likely to be considered, and one 
 of them, the mode, might almost have been omitted as 
 never having been seriously proposed for actual use, al- 
 though often referred to in connection with the subject. 
 
 None of the six, except the simple arithmetic average 
 of relative prices, are familiar to most people. In fact 
 the very word ''average" means, to most people, only the 
 simple arithmetic average. Let us, therefore, begin by 
 defining this kind of average in order to differentiate it 
 from others. 
 
 The simple arithmetic average of a number of terms 
 is their sum divided by the number of the terms. Thus 
 to average 3 and 4 we divide their sum, (7) by their num- 
 ber (2) and obtain 3^ as the simple arithmetic average of 
 3 and 4. Again, averaging likewise 5, 6, and 7 we get 
 
 ^ + ^-^'^ = 6, and averaging 8, 8.5, 9, 9.7 we get 
 
 8 + 8.5 + 9 + 9.7 _g3^ 
 4 
 
 *As to the word "aggregative" see Appendix I (Note A to Chapter 
 11, § 3). 
 
16 
 
 THE MAKING OF INDEX NUMBERS 
 
 To apply this sort of calculation to index numbers, 
 let us take the following skeleton table of the prices of our 
 36 commodities for the two years, 1913 and 1914 : 
 
 TABLE 1 THE SIMPLE ARITHMETIC INDEX NUMBER 
 FOR 1914 AS CALCULATED FROM THE 36 PRICES FOR 
 1913 AND 1914 
 
 
 COMMODITT 
 
 Prices 
 IN Cents 
 
 Price 
 Relatives 
 
 No. 
 
 1913 
 
 1914 
 
 mxlllt 
 
 1 
 
 Bacon, per lb 
 
 12.36 
 62.63 
 
 12.95 
 62.04 
 
 104.77 
 
 2 
 
 Rqrlpv npr bu. 
 
 99.06 
 
 
 
 
 
 
 
 
 
 36 
 
 Oats, per bu 
 
 37.58 
 
 41.91 
 
 111.52 
 
 
 
 36 ) 3467.36 
 96.32 
 
 The first two columns of figures give the actual prices, 
 the last column gives the relative prices, found by calling 
 each price in 1913 100 per cent, while the average of these 
 is the index number sought. 
 
 Thus, to obtain the index number of these commodities 
 for 1914, relatively to 1913 as the base, two steps are in- 
 volved : first, to get the relation between each commodi- 
 ty's 1914 price and its 1913, or base, price. This is a 
 ratio. It is expressed in percentages and is called a rela- 
 tive price or ''price relative." There is, thus, a price 
 relative for bacon, another price relative for barley, and 
 so on — a price relative for each separate commodity. 
 To obtain these price relatives is the first step to an index 
 number and may be called "percentaging." The second 
 step is to average these relatives — and may be called 
 "averaging the percentages." 
 
 The first item on the list is bacon, the price of which 
 
SIX TYPES OF INDEX NUMBERS 17 
 
 in 1913 was 12.36 cents per pound and, in 1914, 12.95 
 cents per pound, which is 4.77 per cent higher. That is, 
 percentaging the prices of bacon we find the price in 1914, 
 relatively to 1913, to be 100 X (12.95 -^ 12.36), or 104.77 
 per cent. Likewise, barley fell from 62.63 cents per 
 bushel to 62.04, the latter being 99.06 per cent of the 
 former. Thus, 99.06 per cent is the price relative of bar- 
 ley (for 1914 relatively to 1913 taken as 100), and so on 
 to the end, where oats rose in 1914 to 111.52, as compared 
 with 100 taken as its price in 1913. 
 
 Having thus percentaged the prices into price relatives, 
 we proceed to average the percentages. The simple 
 arithmetic average of these price relatives, namely, of 
 104.77, 99.06, . . . , 111.52, is found by first taking their 
 sum (3467.36) and then dividing this sum by their 
 number (36). The result is 96.32 per cent, the desired 
 simple arithmetical index number, giving the price level 
 of 1914 as a percentage relatively to 100 in 1913 as the 
 base of comparison. The base is the year for which each 
 price is taken as 100 per cent (or any other common 
 figure) } 
 
 In the same way, the simple arithmetic index number 
 for 1915 relatively to 100 in 1913 as a base is 98.03, or 
 1.97 per cent below 1913 ; and for the next three years, 
 1916, 1917, and 1918 respectively, the simple arithmetic 
 index numbers are 123.68, 175.79, 186.70 — all relatively 
 to 100 in 1913 as base — or higher than 1913 by 23.68 
 per cent, 75.79 per cent, and 86.70 per cent respectively. 
 
 Sometimes it is convenient to make some other year 
 than the base 100 per cent. Thus, we might wish to 
 translate the above series (100.00, 96.32, 98.03, 123.68, 
 175.79, 186.70, all calculated on 1913 as a base) into pro- 
 portional numbers with 100 in place of 186.70 for 1918. 
 1 See Appendix I (Note B to Chapter II, § 3). 
 
18 THE MAKING OF INDEX NUMBERS 
 
 The series then becomes 53.56, 51.59, 52.51, 66.25, 94.16, 
 100.00. 
 
 But this replacement of the awkward number 186.70 
 in 1918 by the more convenient number 100, and the 
 proportionate reduction of the original 100 in 1913 to 
 53.56, does not really change the base from 1913 to 1918. 
 1913 is still the base, but the base number is changed from 
 100 to 53.56 ; for the base number is the number common 
 to all the commodities. Evidently to change an index 
 number for 1918 from 186.70 to 100 does not make 
 each separate commodity 100. The commodities having 
 before had 36 different numbers, the average of which 
 was 186.70 will now have 36 different numbers, the aver- 
 age of which is 100. On the other hand, 1913, which be- 
 fore had every commodity 100, will now have every 
 commodity 53.56 ; therefore, 1913 is still the base. Thus, 
 we must sometimes distinguish between the true base year 
 and the year for which the index number is taken as 100. 
 After a series of index numbers has been computed it 
 is very easy so to reduce or magnify all the figures in 
 proportion, or to make any year 100 which we choose. 
 
 §4. The Simple Arithmetic Average of Relative Prices 
 by the ''Chain" System 
 
 In the preceding discussion all the index numbers were 
 calculated relatively to 1913 as a common base. The 
 price of every one of the 36 commodities was taken as 
 100 per cent in 1913, pnd then, by percentaging, the price 
 relatives of the other year were found, and then averaged. 
 But, of course, any other year could be used as the base. 
 Thus we might take 1918 as the base and calculate any 
 other year relatively to 1918. Or we could use one base 
 for one comparison and another base for another com- 
 parison. If every one of our six years were used as the 
 
SIX TYPES OF INDEX NUMBERS 19 
 
 base for every other year, we would have 30 index num- 
 bers in all, and these would all be discordant among 
 themselves. 
 
 The usual practice is to keep to one year or period — 
 the earhest year of the series, or sometimes an average of 
 several years — as the base for the calculation of the 
 price relatives. This ' ' fixed base ' ' method gives us a series 
 of figures which, in practice, are used not only for compar- 
 ing each year with 1913, but for comparing each year with 
 the one before or after. Thus, the last two figures, 175.79 
 and 186.70, are regarded as showing the price levels of 
 1917 and 1918 relatively not only to 1913, but to each 
 other. But properly to measure the price movement be- 
 tween the two years 1917 and 1918, we ought not to be 
 obliged to take some third year, like 1913, as a base. We 
 should be able to compare 1917 and 1918 directly with 
 each other. By the ''chain of bases system" each year 
 is taken as the base for calculating the index number 
 of the next, and the resulting figures are then linked to- 
 gether to form a "chain" of figures. This will be clear 
 if we take one link at a time. 
 
 First, we calculate the index number of 1914 relatively 
 to 1913 as a base. In this case the calculation is identical 
 with that of the fixed base system when 1913 is the base. 
 We have, then, the first link, which is 96.32 per cent. 
 Next, we calculate the index number of 1915 relatively, 
 not to 1913, but to 1914 as the base. That is, we per- 
 centage the prices of 1915 by taking each price of 1914 as 
 100 per cent, thus obtaining 36 price relatives quite dif- 
 ferent from any previously calculated under the fixed 
 (1913) base system; and then average these 36 price 
 relatives. We now have the second hnk. This is 101.69 
 per cent, the index number of 1915 relatively to 1914 as 
 100 per cent. 
 
20 
 
 THE MAKING OF INDEX NUMBERS 
 
 But this index (of 1915 to 1914) is only a link in the 
 chain. We must still join it to the preceding hnk to ob- 
 
 Year to Year Dispenion 
 of Prices 
 
 13 M '15 16 17 78 
 
 Chakt 4P. The lines from 1913 to 1914 are the same as in Chart 3P; 
 the Hnes for subsequent years are parallel to their positions in Chart 3P, 
 but are shifted so as to start over again from a new common point in each 
 successive year. 
 
 tain the index of 1915 to 1913 via 1914. This requires 
 a third step, namely, multiplying this second link (1915 
 to 1914) by the first (1914 to 1913), thus : 101.69 per cent X 
 96.32 per cent = 97.94 per cent. 
 
SIX TYPES OF INDEX NUMBERS 
 
 21 
 
 In the same way we calculate the third link, the index 
 number for 1916 relatively to 1915 as a base (that is, by 
 percentaging relatively to 1915 and averaging the re- 
 
 Year io Year Dispersion 
 of Quaniiiies 
 
 13 '14 'IS '16 
 
 Chart 4Q. Analogous to Chart 4P. 
 
 suiting price relatives). We then join this third hnk 
 (127.97 per cent) on to the chain by multiplying it by the 
 two previous (127.97 per cent X JPi.69 per cent X 96.32 
 per cent), obtaining 125.33 per cent as the chain figure 
 for 1916 relative, indirectly, to 1913. That is, this is the 
 index number for 1916 relative to 1913 as 100 per cent, 
 but via the intermediate bases, 1914 and 1915. 
 
22 THE MAKING OF INDEX NUMBERS 
 
 In short, by this chain system, or step by step system, 
 each year's index number is first calculated as a separate 
 link relatively to the preceding year as the base. But 
 after these separate year-to-year, or link index numbers, 
 are thus calculated by the usual two processes of per- 
 centaging and averaging, they are joined together by the 
 third process of Unking, or successive multiplications to 
 form ''chain" figures. Consequently, for the final series 
 only the initial base, 1913, stays at 100 per cent. This 
 third process, finking, is added because it is much more 
 convenient to have only one 100 per cent year in the final 
 series than to use the year-to-year links in which each 
 year is 100 per cent for the next. 
 
 § 5. Charts Dlustrating the Chain System 
 
 Graphically, the averaging of the separate links is 
 shown in Charts 4P and 4Q, where the prices and quan- 
 tities are pictured as dispersing, first, from 1913 to the 
 next year, and then from 1914 to the next, and so on by 
 successive steps. Each new point of departure is taken 
 as an average of the preceding set of lines so that all these 
 points constitute the chain series of index numbers. 
 
 The two methods, fixed base and chain, may, of course, 
 be applied to every formula. For some formulae the two 
 methods give identical results ; for others, not. In the 
 case of the simple arithmetic index number they do not. 
 
 § 6. The Simple Arithmetic, Both Fixed Base and Chain. 
 Illustrated Numerically and Graphically 
 
 Table 2 shows the simple arithmetic index numbers by 
 both methods — - fixed base method and chain method — 
 together with the individual links of the chain.^ 
 
 ^ Appendix I (Note B to Chapter II, § 3) might profitably be consulted 
 here. 
 
SIX TYPES OF INDEX NUMBERS 
 
 23 
 
 TABLE 2. SIMPLE ARITHMETIC INDEX NUMBER 
 (FORMULA 1)1 FOR PRICES 
 
 (By fixed base method and by chain method) 
 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 1913 as fixed base 
 
 1914 as base for 1915 
 
 1915 as base for 1916 
 
 1916 as base for 1917 
 
 1917 as base for 1918 
 
 By chain of above bases 
 (product 2 of above 
 links successively) 
 
 100. 
 100. 
 
 96.32 
 100.00 
 
 96.32 
 
 98.03 
 101.69 
 100.00 
 
 97.94 
 
 123.68 
 
 127.97 
 100.00 
 
 125.33 
 
 175.79 
 
 140.15 
 100.00 
 
 175.65 
 
 186.70 
 
 110.11 
 193.42 
 
 1 Complete tables of the index numbers reckoned by all of the 134 formulae are given 
 in Appendix VII. The formula themselves are given in Appendix V. 
 
 » 97.94 is obtained by multiplying 96.32 X 101.69 ; 125.33 by multiplying 96.32 X 101.69 
 X 127.97, etc. In multiplying, we must remember that all the figures are per cents and 
 that 100 per cent is unity or 1.00, while 96.32 per cent is .9632, etc. That is, before mul- 
 tiplying percentages, we must shift the decimal point two places to the left; and, of 
 course, after obtaining the result (e.g. 1.9342 for 1918), we must shift the decimal point 
 back again (i.e. for 1918, 193.42). 
 
 Graphically, in charting price movements, each index 
 number is represented by a point high or low in the dia- 
 gram according as the index number is large or small. 
 The whole series of points for different dates, whether 
 each point is obtained by the fixed base method or by the 
 chain of bases method, may be joined together, forming a 
 curve. The picture of the simple arithmetic index num- 
 ber relative to 1913 as a fixed base is given in Chart 5P 
 (curve labeled "1"). The ''chain" figures, relative, in- 
 directly, to 1913, are indicated by small balls which come 
 sometimes above and sometimes below the original curve 
 calculated by the ''fixed base" method. There are no 
 balls for the year 1914 as the two numbers for that year 
 are, of course, identical. 
 
 This graphic system of distinguishing the results of the 
 "fixed base" and "chain base" methods of working an 
 
24 THE MAKING OF INDEX NUMBERS 
 
 ^^^ t 
 Simple Arithmetic Index Number of Prices /-'-^^"^'^sj 
 
 of 36 Commodities / ^^^ 
 
 / ^ 
 Compared with 'No. 353' / / 
 
 15 14 15 16 17 18 
 
 Chart 5P. Comparison of two index numbers of prices of the 36 com- 
 modities, by Formula No. 1 (simple arithmetic) and Formula No. 353 (the 
 " ideal" as later explained). Each of the points joined by lines is relative 
 directly to the fixed base (1913), and each small ball is relative indirectly 
 to 1913 via intermediate years {i.e. relative directly to the preceding small 
 ball as base, which in turn is likewise relative to its preceding ball, and 
 so on back to 1913). 
 
 index formula will be used throughout the following in- 
 vestigation so that the "fixed base" and "chain base" 
 results may be compared on various charts for all the 
 six types of formulae — arithmetic, harmonic, geometric, 
 median, mode, aggregative. In the case of the simple 
 arithmetic index number there is evidently an appreciable 
 discrepancy between the fixed base and the chain figures. 
 
 § 7. Aids to Interpreting the Charts 
 
 To interpret such curves as the foregoing and those 
 which follow, it will help the reader to note carefully the 
 heights representing an increase of one per cent, five per 
 cent, etc. In Chart 6P, for instance, the length of the 
 dark vertical line marked "5 %" (as noted under Chart 
 3P) affords a visual measuring rod by which it is possible 
 to get a clear idea of the percentage by which any given 
 
SIX TYPES OF INDEX NUMBERS 
 
 25 
 
 point in any diagram in this book is higher than any other 
 point, all the diagrams being drawn on the same scale. 
 In Chart 6P the application of this measuring rod to the 
 slopes of the Unes is indicated in another way. Each of 
 the short lines lying above the curve ascends in a year 
 
 Index No. 353 of Prices 
 
 contrasted 
 (i) with doffed lines above each diyerqing 
 s% in a year, 
 
 (2) wifh doffed lines belm. each dlrerginq 
 1% in Q (jeor 
 
 .^-'- ^'' 
 
 353 
 
 '/3 M V5 '16 '17 18 
 
 , Chabt 6P. An aid to the eye for judging contrasts in subsequent charts. 
 
 five per cent more than the corresponding line in the 
 curve, while each of the short lines below the curve as- 
 cends one per cent more than the corresponding line in 
 the curve. 
 
 Chart 7 will also help in future interpretations of 
 curves. By the method of plotting here used (called the 
 ratio chart method ^), the line representing a uniform per- 
 centage of change, say ten per cent per year, will simply 
 go on being straight. Thus, if an index number increases 
 in the first year ten per cent, that is, from 100 to 110, and 
 
 * For a full discussion of the advantages of this method see Irving Fisher, 
 "The Ratio Chart," Quarterly Publications of the American Statistical As- 
 sociation, vol. XV (1917), p. 677. The method is also called "logarithmic." 
 
26 
 
 THE MAKING OF INDEX NUMBERS 
 
 Uniform Slopes ^ Uniform Rdflos 
 
 Chart 7. Showing the fundamental feature of the "ratio chart" method 
 used throughout this book, namely, the uniform significance of direction. 
 The upper Une representing a continuous series of equal -percentage increases 
 (100 to 110 is 10 per cent; 110 to 121 is 10 per cent; 200 to 220 is 10 per 
 cent) is straight. The three lower lines are parallel to each other, one rep- 
 resenting the actual prices $1.20 and $1.80, one representing the price rela- 
 tives starting with 100 per cent, and the other the price relatives ending 
 with 100 per cent. 
 
SIX TYPES OF INDEX NUMBERS 27 
 
 likewise ten per cent in the second year, that is, from 110 
 to 121, and so^ on, increasing each year ten per cent (in 
 the last, from 200 to 220), it will simply continue its 
 straight course, the rises of 10, 11, ... , and 20 all being 
 equal 'percentage rises (though not equal differences). 
 
 It further follows that any two lines representing equal 
 percentage rates of change will be parallel. Thus, if a 
 commodity changes in price from $1.20 per bushel to 
 $1.80 per bushel, or 50 per cent, the line representing 
 this change in actual prices will be parallel to a line 
 representing merely their relative changes from 100 per 
 cent to 150 per cent and parallel also to a line repre- 
 senting the reverse relative changes from 100 per cent 
 backward to 66 1 per cent. 
 
 The central curve (Chart 6P) might have been any curve. 
 As a matter of fact it is the curve obtained by using the 
 formula called 353 in this book (the calculations being 
 relative to 1913 as a fixed base). Since Formula 353 is 
 the one which we shall find to be the best, — the ''ideal" 
 one — the reader may care, for the sake of future com- 
 parisons, to establish at the outset a mental picture of 
 this curve. 
 
 § 8. The Algebraic Formula for the Simple Arithmetic 
 Index Number 
 
 Algebraically, the formula for the simple arithmetic 
 average was previously given for two and for three com- 
 modities. For 36 commodities the formula for 1914 as 
 year "1" (relatively to 1913 as the base year, or year 
 ^'0") is evidently 
 
 Po 7> P V 
 
 36 
 
28 THE MAKING OF INDEX NUMBERS 
 
 In order to avoid writing so many terms the best usage 
 is to call the numerator ^ — ] where the symbol "S" 
 
 is the Greek letter Sigma or "S," the initial letter of 
 "Sum." It does not denote a quantity, but is an abbrevi- 
 ation for the words 'Hhe sum of terms like the following 
 sample" so that the above expression, written with this 
 convenient abbreviation for summation, is 
 
 ,„ \PoJ 
 36 
 
 or, more generally. 
 
 
 2. 
 <Po 
 
 n 
 
 where n stands for the number of commodities, whether 
 this be 36 or any other number. 
 
 Just as p stands for price, so we may let q stand for 
 quantity (bushels, etc.). The simple arithmetic index 
 number for the quantities of the 36 commodities would, 
 therefore, be 
 
 \qo/ 
 
 2, 
 
 n 
 Similarly the formula for 1915 (year "2") relatively 
 to 1913 (year "0") is, for prices. 
 
 
 n 
 
 and, for quantities. 
 
 © 
 
 S 
 n 
 
SIX TYPES OF INDEX NUMBERS 29 
 
 Again, by replacing "2" with "3" we have the formulae 
 for 1916. Likewise those for 1917 and 1918 are obtained 
 by similarly substituting "4" and '*5." 
 
 Turning from these fixed base formulae to the chain 
 system, we first note that the formula for the simple 
 arithmetic index number of prices for 1915 relatively to 
 1914 — that is, the second link in the chain system — is 
 evidently 
 
 <-) 
 
 n 
 Since the formula for 1914 relatively to 1913 is 
 
 (B) 
 
 2. 
 
 Kp 
 
 n 
 
 the formula for 1915 relatively to 1913 via 1914 is the 
 product of the two preceding expressions; likewise, the 
 chain formula for 1916 is the product of three such ex- 
 pressions, and so on for any number of hnks. 
 
 § 9. The Simple Arithmetic — Usage and Utility 
 
 The simple arithmetic average is perhaps still the fa- 
 vorite one in use. It was used as early as 1766 by Carli.^ 
 It is used by the London Economist, the London Statist 
 (continuing Sauerbeck's index number) and many other 
 makers of index numbers. 
 
 In the present exposition, the simple arithmetic average 
 is put first merely because it naturally comes first to the 
 reader's mind, being the most common form of average. 
 In fields other than index numbers it is often the best 
 form of average to use. But we shall see that the simple 
 arithmetic average produces one of the very worst of 
 
 ^ See C. M. Walsh, The Measurement of General Exchange Value, p. 534. 
 
30 THE MAKING OF INDEX NUMBERS 
 
 index numbers. And if this book has no other effect 
 than to lead to the total abandonment of the simple arith- 
 metic type of index number, it will have served a useful 
 purpose. 
 
 The simple arithmetic index number just described is 
 listed in the Appendix as Formula 1 and will often be 
 referred to by that identification number. 
 
 § 10. The Simple Harmonic 
 
 The next simple index number to be explained is the 
 harmonic, the identification number of which in this 
 book is 11. (The numbers between ''1" and ''11" will 
 be assigned to other formulae to be introduced later.) 
 
 The process of calculating the simple harmonic average 
 is somewhat like that of calculating the simple arithmetic, 
 differing merely in that reciprocals are employed. The 
 term "reciprocal" is here used in the mathematical sense, 
 the reciprocal of any number being the quotient ob- 
 tained by dividing unity by that number. If the number 
 is expressed in fractional form, its reciprocal is found 
 by turning the fraction upside down. Thus the reciprocal 
 of 2 {i.e. f) is i ; the reciprocal of 3 (i.e. f ) is i ; of f is f , etc. 
 
 There are three steps in calculating the simple harmonic 
 average of any given set of ratios : 
 
 (1) turn the ratios upside down ; 
 
 (2) take the simple arithmetic average of these in- 
 verted figures ; 
 
 (3) turn the average thus obtained right side up again. 
 Thus to take the simple harmonic average of | and ^: 
 
 (1) their upside down ratios, or "reciprocals," are | 
 and^; 
 
 (2) the simple arithmetic average of the last two is V ; 
 
 (3) the reciprocal of the last is i^ or if, which is the 
 desired simple harmonic average. 
 
SIX TYPES OF INDEX NUMBERS 
 
 31 
 
 This harmonic average of f and -f- (which is if) is less than 
 the simple arithmetic average of f and f , which is H. 
 
 Let us apply this process to index numbers. It is the 
 second process — averaging, the percentaging being al- 
 ready done. Taking, then, the 36 price relatives indicated 
 in Table 1 above, viz., 104.77 per cent, i.e. 1.0477, .9906, 
 . . . , to 1.1152 (the 36th), then inverting them into .9545, 
 1.0095,. . . , to .8967; then taking the simple arithmetic 
 average of these, which is 1.0506 ; then inverting the latter, 
 we get finally .9519 or 95.19 per cent, which is the simple 
 harmonic index number. This is less than the simple 
 arithmetic index number for the same year (96.32 per 
 cent) already found. 
 
 The complete series of simple harmonic index numbers 
 of prices, both by the fixed base and the chain system, 
 are given below and also, for comparison, the simple 
 arithmetic by the same two methods. 
 
 Formula No. 
 
 Type 
 
 Base 
 
 1913 
 
 1914 
 
 1913 
 
 1916 
 
 1917 
 
 1918 
 
 1 
 11 
 
 Simple arithmetic 
 Simple harmonic 
 
 Fixed 
 Chain 
 
 Fixed 
 Chain 
 
 100. 
 100. 
 
 100. 
 100. 
 
 96.32 
 96.32 
 
 95.19 
 95.19 
 
 98.03 
 97.94 
 
 95.58 
 95.64 
 
 123.68 
 125.33 
 
 119.12 
 117.71 
 
 175.79 
 175.65 
 
 157.88 
 158.47 
 
 186.70 
 193.42 
 
 171.79 
 167.76 
 
 It will be noted that there are great differences here 
 among the results of the four methods, especially in 1917 
 and 1918 ; and that the harmonic is always less than the 
 arithmetic. The reason for this need not be considered here. 
 
 Graphically, the harmonic index number (fixed base) 
 is given in Charts SP and 8Q (Curve 11) with all the five 
 other simple index numbers, — arithmetic, geometric, 
 median, mode, aggregative. As in the case of the chain 
 arithmetics and fixed base arithmetics, the chain har- 
 monics do not agree with the fixed base harmonics. 
 
32 THE MAKING OF INDEX NUMBERS 
 
 Simple Index Numbers of Prices 
 
 14 
 
 '15 
 
 16 
 
 17 
 
 18 
 
 7J 
 
 Chakt 8P. In the upper group, the simple geometric (21) necessarily 
 lies between the simple arithmetic (1) above it and the simple harmonic 
 (11) below it. Of the lower group, the simple median (31) most resembles 
 the upper group, while the simple mode (41) and simple aggregative (51) 
 are each sui generis. The two groups are separated to save confusion, 
 really forming two distinct diagrams. 
 
 Algebraically, the simple harmonic average of the two 
 
 price ratios, ^ and 2^, is the reciprocal of the arithmetic 
 Po p 
 
 average of their reciprocals ; that is, 
 
 2 
 
 Pi V'l 
 For three terms the formula is 
 
 3 
 
 Pi V 
 For n terms the formula is 
 
 Po , p 0_i_P 
 
 V I 
 
 n 
 Pi 
 
SIX TYPES OF INDEX NUMBERS 
 
 Simple Index Numbers of Quantities 
 
 33 
 
 7J 
 
 74 75 IS 17 
 
 Chabt 8Q. Analogous to Chart 8P. 
 
 73 
 
 The harmonic index number has found few champions. 
 One of these is F. Coggeshall.^ We shall find, however, 
 that the simple harmonic is a sort of ''antithesis" of the 
 simple arithmetic ; and when we arrive at their faults 
 we shall find the two equally at fault but in opposite 
 directions. 
 
 § 11. The Simple Geometric 
 
 We now come to the simple geometric index number. 
 The reader whose conception of an average has been 
 limited to the arithmetic is referred to Appendix I (Note 
 A to Chapter II, § 15) for a general definition of average 
 which will include the harmonic and the others used 
 below. Suffice it here to define the geometric average. 
 
 Given the price relatives, in order to get the average 
 of them (that is, the index number) by the simple geo- 
 
 *F. Coggeshall, "The Arithmetic, Geometric, and Harmonic Means," 
 Quarterly Journal of Economics, vol. 1 (1886-87), pp. 83-86. 
 
34 THE MAKING OF INDEX NUMBERS 
 
 metric formula (21 in our series), instead of adding to- 
 gether the price relatives of the listed commodities and 
 then dividing their sum by the number of terms (n) we 
 multiply the price relatives together and then extract the 
 nth root. 
 
 Thus to get the simple geometric average of 2 and 8, 
 we take thei r prod uct (16) and extract its square root, 
 obtaining V2 X 8 = 4. To get the simple geometric 
 average of the three numbers 4, 6, and 9, we take their 
 product (216) and extract the cube root, obtaining 6 as 
 the simple geometric average. To get the simple geo- 
 metric average of the four numbers 3, 4, 6, and 18, we 
 take their product (1296) and extract the fourth root, 
 obtaining 6. 
 
 Numerically, to apply the geometric process to index 
 numbers, we multiply all the 36 price relatives, 104.77 
 per cent, 99.06 per cent, . . . , 111.52 per cent, together 
 and extract the 36th root, a process made easy by means 
 of logarithmic tables.^ The result is 95.77 per cent for 
 1914 relatively to 1913, whereas the simple arithmetic 
 method gave 96.32, and the simple harmonic, 95.19 per 
 cent. The geometric will be found to he between the 
 arithmetic (which is always above it) and the harmonic 
 (which is always below it). 
 
 Graphically, the geometric index number is given in 
 Chart 8 (Curve 21) with all the five other simple index 
 numbers, — arithmetic, harmonic, median, mode, aggre- 
 gative. 
 
 Algebraically, the simple geometric average of two price 
 ratios is given by the formula 
 
 4 
 
 Po P 
 
 * For model examples to aid in the practical calculation of this as well as 
 of eight other sorts of index numbers, see Appendix VI, § 2. 
 
SIX TYPES OF INDEX NUMBERS 35 
 
 For three, the formula is 
 
 PlyP lyP 1 
 
 Po Po p 
 For any number, n, it is 
 
 ^PiX^X^X. . . (n terms). 
 Po Po p 
 
 In the case of the simple geometric average the "chain" 
 figures are always identical with those calculated rela- 
 tively to a fixed base.^ 
 
 Jevons,^ in 1863, used and advocated the simple geo- 
 metric. It still finds some favor among statisticians 
 and, as we shall see, really deserves a high place among 
 the simple averages, when simple averages are called for. 
 But whether the fact that the chain figures agree with 
 the fixed base figures is a virtue will be discussed in Chap- 
 ter XIII. 
 
 § 12. The Simple Median 
 
 The simple median (Formula 31) is calculated, not by 
 the processes of adding and dividing, or of multiplying 
 and extracting a root, but merely by selecting the middle- 
 most term. Thus, the median of 3, 4, and 5 is evidently 4, 
 the middle term. The median of 1, 3, 3, 4, 4, 4, 5, 6, 6, 6, 
 6, 7, 7 is 5, since 5 stands in the middle of the list, there 
 being six items smaller and six items larger. The median 
 height of a line of 51 soldiers standing in the order of their 
 heights is the height of the middlemost soldier, i.e. the 
 26th from either end. 
 
 When the number of terms is even, there are two middle- 
 most terms instead of one. If these two are alike either 
 of them may, of course, be called the median. If the two 
 
 ^ For proof, see Appendix I (Note to Chapter II, § 11). 
 
 ^ See Walsh, The Measurement of General Exchange Value, p. 557. 
 
36 
 
 THE MAKING OF INDEX NUMBERS 
 
 middle terms differ, then the median lies between them 
 and cannot be definitely determined without recourse to 
 some other process of averaging such, for example, as tak- 
 ing the simple arithmetic or simple geometric average of 
 the two middle terms. 
 
 By the fixed base method (recurring to our 36 com- 
 modities), the median of the price relatives of 1914 (rela- 
 tively to 1913) is 99.45, and that of 1915 (also relatively 
 to 1913) is 98.57.^ By the chain method, the median 
 for 1915 (relatively to 1913 via 1914) becomes 99.33. The 
 two methods under the median are compared below : 
 
 FORMUIiA 
 
 No. 
 
 Base 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 31 
 31 
 
 Fixed 
 Chain 
 
 100. 
 100. 
 
 99.45 
 99.45 
 
 98.57 
 99.33 
 
 118.81 
 117.50 
 
 163.81 
 155.86 
 
 190.92 
 180.07 
 
 Graphically, Chart 8 (Curve 31) shows the median as 
 well as the five other simple index numbers. 
 
 Professor Edgeworth (1896) recommended the simple 
 median. Since this advocacy several statisticians have 
 used it, including A. L. Bowley and Wesley C. Mitchell. 
 
 § 13. The Simple Mode 
 
 The simple mode (Formula 41) is found by arranging 
 the items in order of size, just as in the case of the median, 
 and then selecting, not the middlemost term, but the com- 
 monest term. ; hence the word "mode" indicating ''most 
 in vogue." Thus the mode of 1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 
 6, 6, 7 is 5 ; for 5 occurs four times while no other number 
 occurs oftener than three times. 
 
 But, even more than the median, the mode is ambig- 
 
 * For model examples to aid in the practical calculation of this as well as 
 eight other sorts of index numbers, see Appendix VI, § 2. 
 
SIX TYPES OF INDEX NUMBERS 37 
 
 uous and vague and needs to be helped out by other pro- 
 cesses than that given in its own definition. Ordinarily, 
 few of the items are just alike so that, to make the mode 
 a workable average, we do not really count the repetition 
 of precisely equal terms but the repetition of terms f alhng 
 within haiUng distance of each other, or, more precisely, 
 within certain arbitrarily chosen hmits. 
 
 Thus, for the fine of soldiers, we should probably not 
 find any two of exactly the same height ; but we could 
 easily classify them in groups differing by inches. At 
 one end of the fine are the short men between, say, 5 feet 
 6 inches and 5 feet 7 inches of which there may be only, 
 say, two soldiers. Let us, for convenience of thought, 
 imagine these to be set apart in a group by themselves, 
 separated a Httle from the next taller group contain- 
 ing, say, five soldiers between the heights 5 feet 7 inches 
 and 5 feet 8 inches, and these in turn separated from those 
 within the next inch (5 feet 8 inches to 5 feet 9 inches) 
 numbering, say, 20 soldiers. Within the next inch of 
 height (5 feet 9 inches to 5 feet 10 inches) are, say, 30 
 soldiers; the next (5 feet 10 inches to 5 feet 11 inches), 
 25 ; the next (5 feet 11 inches to 6 feet), 10. 
 
 Evidently here the commonest height is that of the 
 group (of 30 soldiers) between 5 feet 9 inches and 5 feet 
 10 inches, which is, therefore, the mode. To put any 
 finer point on it, i.e. to find the mode any more closely 
 than within an inch would require either subdividing by 
 haK-inch intervals, or else mathematically or graphically 
 adjusting the figures so as to make a ''smooth" curve of 
 frequency and then taking the maximum on this ideal 
 curve to represent the mode, or resorting to some other 
 extraneous aid. 
 
 The best example of the mode as apphed to index 
 numbers is that afforded by the "Summary of the History 
 
38 THE MAKING OF INDEX NUMBERS 
 
 of Prices during the War" (Bulletin No. 1, of the War 
 Industries Board) hj Wesley C. Mitchell. Thus, to 
 take the mode of 1918: Out of 1437 commodities, the 
 prices of which were reckoned relatively to the pre-war 
 year as a base, there were two commodities the prices of 
 which were between 30 and 49 per cent of the pre-war 
 prices; four between 50 and 69 per cent; 17 between 
 70 and 89 per cent, and for the succeeding similar in- 
 tervals of 20 each, the following successive numbers of 
 items, namely : 61 items, 64 items, 130, 212, 219, 164, 135, 
 104, 76, 54, 42, 30, 31, 16, 13, 7, 7, 8, 4, 4, 4, 5, 3, 4, 1, 
 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 1, 1, 0, 0, 1, etc. The price of 
 the last named soUtary commodity was between 890 and 
 909 per cent of its pre-war price. The mode here Hes in 
 the compartment having the largest number (219). This 
 compartment is that between 170 and 189. The mode 
 lies, therefore, somewhere between 170 per cent and 189 
 per cent. The exact location of the mode is always 
 more or less mythical. In this case 173 is obtained by a 
 graphical method as the value of the mode. To put any 
 finer point on it would be almost meaningless. 
 
 Thus the median and the mode are both somewhat 
 indeterminate, the mode especially so in case of wide and 
 irregular dispersion of price relatives unless the number 
 of items runs into the hundreds or thousands. 
 
 Numerically, in the case of the 36 commodities the 
 mode for 1914 (relatively to 1913) graphically obtained, 
 was 98.^ Another method (calculating it indirectly from 
 the simple arithmetic and the simple median) makes it 
 106 and another similar method, applied in the reverse 
 direction, makes it 109. But when dealing with so few 
 commodities as 36 the mode is so indeterminate that it is 
 not worth while to employ it. For completeness, how- 
 » See Appendix I (Note to Chapter II, § 13). 
 
SIX TYPES OF INDEX NUMBERS 39 
 
 ever, the mode (as calculated by a graphic method) has 
 been entered in the tables, although omitted from most 
 of the charts. 
 
 By the chain method, the mode for 1915, relatively 
 to 1913, via 1914, is roughly 95 ; by the fixed base method 
 it is 98. The figures for all the years are given in Appen- 
 dix VII. 
 
 Graphically, the simple mode (Curve 41) with the five 
 other simples is given in Chart 8. 
 
 The mode has never been either used or proposed for 
 use in index numbers. But Wesley C. Mitchell, in Bulle- 
 tin 173, and its revised edition 284, o^ the United States 
 Bureau of Labor Statistics, and (as has just been noted) 
 in his War Industries Board ''History of Prices," has 
 presented some figures to illustrate the mode, as have 
 some other writers. Mr. C. M. Walsh has suggested 
 that the position of the mode in relation to the arithmetic 
 and other averages may help us select the best average 
 to use. But even this supposed utility of the mode will 
 be found illusory. 
 
 § 14. The Simple Aggregative 
 
 Last of the simple index numbers is the simple aggre- 
 gative (Formula 51). This is the percentage obtained 
 by taking the aggregate, or sum, of all the actual prices 
 for a given year and dividing this by the sum of the prices 
 for the base year. Thus, while the arithmetic starts off 
 by adding relative prices, the aggregative starts off by 
 adding actual prices. 
 
 Numerically, the sum of all the prices in 1913 (i.e. 12.36 
 -h 62.63 + . . . + 37.58) is 23889.48 and the sum of 
 all the prices in 1914 {i.e. 12.95 + 62.04 + . . . -F 41.91) 
 is 22905.24, so that the simple aggregative index number is 
 
 22905 *^4 
 
 oooonTo ' or 95.88 per cent. Under the simple aggregative 
 
Pi + p'i4-p"i+ . 
 
 . . 
 
 Po + p'o + p"o+ . 
 
 • • 
 
 > 
 
 2pi 
 
 
 40 THE MAKING OF INDEX NUMBERS 
 
 formula, the chain figures and the fixed base figures are 
 identical, as is evident.^ 
 
 Graphically, Chart 8 gives the simple aggregative (Curve 
 51) with the five other simples. 
 
 Algebraically, the formula for the aggregative index 
 number is 
 
 or, more briefly, 
 
 2po 
 
 We have seen that to get the simple aggregative index 
 number we do not first calculate price relatives at all; 
 we use the original prices. In fact, unlike the other types 
 of index numbers, the simple aggregative, being the ratio 
 of sums or aggregates of prices, cannot be calculated from 
 the price relatives alone. It requires the actual prices 
 themselves. It would not be enough to know that the price 
 of, say, sugar was twice what it was at the base date in 
 order to be able to calculate the aggregative index num- 
 ber. We would need to go back to the actual prices of 
 sugar at the two dates — whether, for instance, 6 cents 
 and 12 cents respectively, or some other pair of figures in 
 the same proportion, such as 8 cents and 16 cents. 
 
 The simple aggregative index number is usually re- 
 garded as almost worthless ; and so it is, unless the units 
 of measurement are discreetly chosen. 
 
 The aggregative form of an index number was used as 
 early as 1738 by Dutot.^ The only conspicuous instance 
 of its actual use is in Bradstreet's index number where the 
 prices are first reduced to prices per pound for every item. 
 
 ' See Appendix I (Note to Chapter II, § 14). 
 
 2 See C. M. Walsh, The Measurement of General Exchange Value, p. 534. 
 
SIX TYPES OF INDEX NUMBERS 41 
 
 § 15. The Six Simple Index Numbers Compared 
 
 In our tables the simple index numbers, namely, the 
 simple arithmetic, simple harmonic, simple geometric, 
 simple median, simple mode, and simple aggregative 
 index numbers, have, as already stated, as their identi- 
 fication numbers 1, 11, 21, 31, 41, 51, respectively. 
 
 These six types represent six different processes of cal- 
 culation, namely: for (1), adding the price relatives to- 
 gether and dividing by their number; for (11), adding 
 their reciprocals together and dividing into their number ; 
 for (21), multiplying the price relatives together and ex- 
 tracting the root indicated by their number; for (31), 
 arranging the price relatives in order of size and selecting 
 the middlemost; for (41), so arranging but selecting the 
 commonest; for (51), adding together the actual prices 
 of each year and taking the ratio of these sums.^ 
 
 Graphically, Chart 8 gives all the six simple index num- 
 bers, both of prices and quantities, corresponding to 
 Formulae 1, 11, 21, 31, 41, 51. Curves 1, 11, 21 are drawn 
 from a common origin, separately from the others, be- 
 cause they are interrelated. No. 21 always lying between 
 1 and 11.2 
 
 As to the other three, the median lies, with one trifling 
 exception, above the mode. This is not a law but is apt 
 to be the case when, as in the present example, the items 
 averaged are more widely dispersed upward than down- 
 ward, for downward dispersion is limited by the existence 
 of a zero below which prices and quantities cannot sink. 
 
 The simple aggregative is a law unto itself, by reason 
 of its peculiar and haphazard weighting. I have called 
 
 ^ For a general definition of average covering these six and others see 
 Appendix I (Note A to Chapter II, § 15). 
 
 2 See Appendix I (Note B to Chapter II, § 15). 
 
42 THE MAKING OF INDEX NUMBERS 
 
 it "simple," but it is not simple in quite the same sense 
 as are the other five. As Walsh says, it is ''haphazard," 
 being dependent on the accident of what measures or 
 units are used in pricing the commodities in the hst. If 
 silver, instead of being quoted per ounce (as it was in 
 computing this average because the ounce is the usual 
 unit used in pubUshed silver quotations) , had been quoted 
 in tons, and if coal had been quoted in ounces, instead 
 of tons, the result would be entirely different. Silver 
 would dominate, and the average curve would nearly 
 coincide with the silver curve (in Chart 3), while coal 
 would have a negligible influence. 
 
 It must be admitted that this first view of the six differ- 
 ent types of index numbers is not reassuring. If one of 
 these indexes were as good as another, then certainly 
 they would all be almost good for nothing ; for they dis- 
 agree with each other very widely indeed, both when com- 
 puted for a fixed base and when computed through a 
 chain of bases. The lowest index number for 1917 (that 
 by Formula 41) is 135, and the highest (that by Formula 
 1) is 175.79, which latter is 30 per cent above the former. 
 While this range is much less than the divergence of the 
 individual price relatives themselves, it is altogether too 
 great to make possible any statistics worthy of the name. 
 All that could be claimed is that, where there is not so 
 wild a dance of prices as in the war years, the six types 
 of averages will themselves be less discordant. But, 
 fortunately for the science of index numbers, the six types 
 do not, as we shall see, have equal claims. 
 
CHAPTEE III 
 
 FOUR METHODS OF WEIGHTING 
 
 § 1. Weighting in General 
 
 It has aheady been observed that the purpose of any 
 index number is to strike a ''fair average " of the price move- 
 ments — or movements of other groups of magnitudes.^ 
 At first a simple average] seemed fair, just because it 
 treated all terms alike. And, in the absence of any knowl- 
 edge of the relative importance of the various commodi- 
 ties included in the average, the simple average is fair. 
 But it was early recognized that there are enormous dif- 
 ferences in importance. Everyone knows that pork is 
 more important than coffee and wheat than quinine. 
 Thus the quest for fairness led to the introduction of 
 weighting. At first the weighting was rough and ready, 
 being based on guesswork. Arthur Young called barley 
 twice as important as wool, coal, or iron, while he 
 called "provisions" four times as important, and wheat 
 and day labor each five times as important. 
 
 But what is the just basis for assigning weights? Arbi- 
 trary weighting may be an improvement over a simple 
 index number ; but, if abused, it may aggravate the un- 
 fairness. If we were deliberately to seek the most 
 unfair weighting, we could give any one commodity so 
 preponderate a weight as to make the resulting index 
 number practically follow the course of that particular 
 conmiodity. 
 
 1 "Purchasing power" included, although not explicitly treated in 
 this book. See Appendix I (Note to Chapter III, § 1). 
 
 43 
 
44 THE MAKING OF INDEX NUMBERS 
 
 To cite an extreme example, take the 1366 commodities 
 in the carefully weighted index number of the War In- 
 dustries Board. According to this excellent index num- 
 ber, prices rose between the pre-war year {i.e. the year 
 from July 1, 1913, to July 1, 1914), and the calendar 
 year 1917 in the ratio of 100 to 175. This figure is a very 
 fair representative of the 1366 figures from which it was 
 calculated, although these range all the way from a price 
 relative of 35 for oil of lemon to a price relative of 3910 
 for potassium permanganate. But if we deUberately 
 chose to weight potassium permanganate as a billion 
 times as important as every other commodity, the re- 
 sulting index number for the 1366 commodities would 
 practically coincide with the price movement of potas- 
 sium permanganate. Likewise, if we were, instead, to 
 weight oil of lemon a billion times as important as every 
 other, the index number would become practically identi- 
 cal therewith. Obviously, in either case, we should be 
 grossly unfair. In the one case our index number would 
 yield the absurd conclusion that the prices of 1917 aver- 
 aged 39 times as high as pre-war prices. In the other 
 case it would yield the equally absurd conclusion that 
 the prices of 1917 averaged only a little over a third of 
 the pre-war prices. In each case the trouble would be 
 that a commodity, really very unimportant as compared 
 to wheat, steel, flour, cotton, and hundreds of other com- 
 modities, would be treated as though it were enormously 
 more important. 
 
 We are not yet ready to say what system of weighting 
 is the fairest, nor shall we be ready until we have set up 
 certain tests of fairness. We shall then reach the curious 
 conclusion that, contrary to common opinion, no system 
 of weighting is universally the fairest; that the fairest 
 weighting for the arithmetic, harmonic, and geometric 
 
FOUR METHODS OF WEIGHTING 45 
 
 types, for instance, are all different. Here we must be 
 content to lay the foundations, by describing the four 
 primary systems of weighting which have been or might 
 'he set up. 
 
 As we have seen, weighting any term in an index num- 
 ber is virtually counting it as though it were two or three, 
 or some other multiple, as compared with another term 
 counted only once. This apphes to any of the six types 
 of averages. 
 
 But on what principle shall we weight the terms? 
 Arthur Young's guess and other guesses at weighting 
 represent, consciously or unconsciously, the idea that 
 relative money values of the various commodities should 
 determine their weights. A value is, of course, the prod- 
 uct of a price per unit, multiplied by the number of 
 units taken. Such values afford the only common meas- 
 ure for comparing the streams of commodities produced, 
 exchanged, or consumed, and afford almost the only 
 basis of weighting which has ever been seriously proposed. 
 If sugar is marketed to the extent of ten billion dollars' 
 value a year, while salt is marketed at only five billion 
 dollars' value a year, there is clearly ground for regarding 
 sugar as twice as important as salt. 
 
 § 2. Weighting by Base Year Values or by Given 
 V Year Values 
 
 But any index number implies two dates, and the values 
 by which we are to weight the price ratios for those two 
 dates will themselves be different at the two dates. 
 
 Constant weighting (the same weight for the same item 
 in different years) is, therefore, a mere makeshift, never 
 theoretically correct, and not even practically admissible 
 when values change widely. In Revolutionary days, 
 
46 THE MAKING OF INDEX xNUMBERS 
 
 candles were important, but today the total money value 
 of the candle trade is negligible. Rubber tire values are 
 important today, but were unimportant two decades 
 ago. In comparing the price levels of today and many 
 years ago what weight shall we give to rubber tires or 
 candles? We have two evident choices. We may take 
 as our money values either those in the earUer year or 
 those in the later year. 
 
 § 3. Numerical Illustration 
 
 It often makes a great deal of difference which of these 
 two systems of weights is used. Between 1913 and 1917 
 some commodities rose greatly, not only in price, but in 
 money value marketed ; others scarcely at all. In gen- 
 eral, the 36 commodities rose in money value between 
 1913 and 1917 about 100 per cent (their total value rising 
 from $13,105,000,000 to $25,191,000,000). If every com- 
 modity had thus doubled in money value, their relative 
 weights would remain unchanged so that it would make 
 no difference to the index number which year's weights 
 were used. 
 
 But, as a matter of fact, values of the different com- 
 modities rose very unequally. Some rose much more 
 and others much less than 100 per cent. Bituminous 
 coal had a value of $1.27 per ton X 477,000,000 tons, or 
 $606,000,000 in 1913, and in 1917, $1,976,000,000, or 
 more than three times as much. Anthracite coal, on 
 the other hand, had a value of $35,000,000 in 1913, and 
 in 1917, $44,000,000, or only about 25 per cent more. 
 Clearly, then, bituminous coal has a relatively greater 
 weight when the 1917 money values are used as weights 
 than when the 1913 values are used. 
 
 Table 3, assuming 1913 for ''base" year and 1917 for 
 "given" year, shows the comparative effect of weighting 
 
FOUR METHODS OF WEIGHTING 
 
 47 
 
 according to 1913 values marketed and weighting accord- 
 ing to 1917 values marketed. The figures given in the 
 table are, of course, the multipUers to be used in weight- 
 ing the various price relatives. 
 
 TABLE 3. VALUES (IN MILLIONS OF DOLLARS) OF 
 CERTAIN COMMODITIES 
 
 COMMODITT 
 
 1913 
 
 (Base Year) 
 
 1917 
 
 (Given Yeak) 
 
 Bituminous coal 
 
 606 
 140 
 462 
 422 
 
 1976 
 
 Coke 
 
 Pig iron 
 
 604 
 1502 
 
 Oats 
 
 1011 
 
 Anthracite coal 
 
 35 
 1282 
 
 96 
 1971 
 
 44 
 
 Petroleum 
 
 Coffee 
 
 Lumber 
 
 1848 
 
 123 
 
 2227 
 
 All of the first four items in the table are examples of 
 commodities the prices and values of which rose extraor- 
 dinarily from 1913 to 1917, so that their weights, if taken 
 by 1917 figures, would be very great. Contrariwise, the 
 last four items are examples of commodities the prices 
 and values of which rose very little. 
 
 A glance at the table will show the preponderance in 
 1913 of the last four commodities taken as a whole, rela- 
 tively to the first four, and the preponderance in 1917 of 
 the first four relatively to the last four. The reason 
 for this change of relative weights is, of course, that the 
 upper group rose more in price than the lower. 
 
 Let us calculate an index number by both the two con- 
 trasted methods of weighting. Let the type of index 
 number be, say, the arithmetic and let us calculate it 
 
48 THE MAKING OF INDEX NUMBERS 
 
 for 1917 relatively to 1913 as a base, from the usual data 
 for the 36 commodities. We begin by using the base 
 year values as weights. This kind of index number has 
 as its formula, No. 3. For bacon in 1917, the price rela- 
 tive (as previously calculated) is 192.72 per cent and the 
 base year value (12.36 cents per pound X 1077 milUon 
 pounds) is 133.117 million dollars. For barley, the price 
 relative is 211.27 per cent and the base year value (62.63 
 cents per bushel X 178.2 miUion bushels) is 111.607 
 million dollars, etc. According to the arithmetic method, 
 we first multiply each price relative by its weight and 
 then divide by the sum of the weights. The results are : 
 
 For bacon 192.72 per cent X 133.117 million dollars = 256.54 million 
 dollars. 
 
 For barley 211.27 per cent X 111.607 million dollars = 235.79 million 
 dollars, etc. 
 
 The sum of all 36 such results is 21238.49 million dol- 
 lars, which divided by the sum of the weights, 13104.818 
 million dollars, gives 1.6207, or 162.07 per cent as the 
 desired index number. 
 
 Thus, the arithmetic index number for 1917, when the 
 base year (1913) system of weighting is used (Formula 
 3), is 
 
 (1^)133.117 + f-llf)ui.607 ^_^^ 
 
 13104.818 100 
 
 In other words, from 1913 to 1917, the price level rose 
 (according to Formula 3) from 100 to 162.07. But, by 
 using the given year values as weights (Formula 9), the 
 resulting index number is 180.72 per cent, exceeding the 
 former (162.07) by 11.51 per cent. 
 
 Likewise, the harmonic index number for 1917, 
 when the base year system of weighting is used (Formula 
 13), is 
 
FOUR METHODS OF WEIGHTING 49 
 
 13104.818 147.19 
 
 In other words, from 1913 to 1917, the price level rose 
 (according to Formula 13) from 100 to 147.19. But, by- 
 using the given year values as weights (Formula 19), 
 the resulting index number is 161.05 per cent, exceeding 
 the former (147.19) by 9.42 per cent. 
 
 Likewise, the geometric index number, for 1917, 
 when the base year system of weighting is used (Formula 
 23), is 
 
 154.08 
 
 V 100 y -^ V 100 J X • • • ^QQ ■ 
 
 In other words, from 1913 to 1917, the price level rose 
 (according to Formula 23) from 100 to 154.08. But by 
 using the given year values as weights (Formula 29), the 
 resulting index number is 170.44, exceeding the former 
 (154.08) by 10.62 per cent. 
 
 Here is a new source of differences ! Not only does it 
 make a considerable difference what type of average is 
 used, — whether arithmetic, or harmonic, or geometric, 
 — but it also makes a great difference what the weighting 
 is, — whether base year weighting or given year weight- 
 ing, or simple {i.e. even) weighting. 
 
 Were we to stop at this point, we should be even more 
 inclined to join N. G. Pierson and give up index numbers 
 in disgust as a delusion and a snare. 
 
 § 4. Graphic and Algebraic 
 
 Graphically, Chart 9 shows the contrast between the 
 two weighted arithmetic index numbers as well as the 
 corresponding contrasts between the two weighted har- 
 monic and the two weighted geometric index numbers. 
 
 It will be observed that the upper harmonic (Formula 
 
50 THE MAKING OF INDEX NUMBERS 
 
 19) is almost coincident with the lower arithmetic (For- 
 mula 3), the other arithmetic and harmonic (Formulae 
 9 and 13) diverging about equally on opposite sides of 
 these central hues. The two geometries (Formulae 23 
 and 29) He about midway — one between the two har- 
 
 The Five 'Tine Fork 
 
 of 6 Curves 
 
 (Prices) 
 
 13 M 15 16 17 18 
 
 Chaet 9P. Three types of index numbers, the arithmetic (3 and 9), 
 the harmonic (13 and 19), and the geometric (23 and 29), each type being 
 weighted in two ways, namely, by the values of the base year (3, 13, 23) 
 and by the values of the given year (9, 19, 29), forming five nearly equidis- 
 tant tines of the fork. In each case, the given year weighting makes for 
 a higher position of the curve than the base year weighting. (This holds 
 true whether prices are rising or falling.) 
 
 monies in the lower half of the chart and the other be- 
 tween the two arithmetics in the upper half. Each of 
 the three types (arithmetic, harmonic, geometric) thus 
 has its two curves forking about equally; but their re- 
 spective forks are placed in three substantially equidis- 
 tant positions, the lower tine of the uppermost fork 
 (arithmetic) almost coinciding with the upper tine of the 
 lowermost fork (harmonic), while the remaining pair of 
 tines (geometric) split the other two pairs. 
 
FOUR METHODS OF WEIGHTING 51 
 
 Chart 10 shows the similar, but much smaller, contrast 
 for the weighted medians (Formulae 33 and 39). The 
 "mode," were it charted, would show even less contrast ; 
 in fact, in the rough approximation here used it shows 
 none at all, although, strictly, for it as for every other 
 type, the given year weighting always makes for a higher 
 index number than does the base year weighting.^ 
 
 The Five-Tine Fork 
 
 of 6 Curves 
 
 (Quantities) 
 
 ij 
 
 M 15 16 77 
 
 Chart 9Q. Analogous to Chart 9 P. 
 
 18 
 
 Algebraically, the arithmetic index number weighted 
 by base year values (Formula 3) and written for any given 
 year (as year 1) relatively to the base year (year 0) is 
 evidently 
 
 ?>ogo + p'o g'o H- . . . 
 or, by the shorter method of writing, 
 
 ^PoQc 
 
 Po 
 
 Spogo 
 In like manner, the arithmetic index number weighted 
 by given year values (Formula 9) is 
 
 1 As to calculating the weighted median and mode see Appendix I 
 (Note to Chapter III, § 4). 
 
52 
 
 THE MAKING OF INDEX NUMBERS 
 
 ^--Ci) 
 
 The weighted formulae for other types are given in Ap- 
 pendix V. 
 
 § 6. Weighting by Base Year Values Easiest 
 
 The weighting by base year values has been employed 
 by statisticians more frequently than the weighting by 
 
 The Two Extreme Methods oF 
 
 Weighting Median 
 (Prices) 
 
 '/} 
 
 'K 
 
 15 
 
 17 
 
 18 
 
 Chart lOP. Showing the median type of index number, weighted by 
 the values of the base year (33) and by the values of the given year (39), 
 the latter weighting resulting, as before, in a higher curve than the former. 
 The difference between the two weightings is not so great as in the case of 
 the arithmetic, harmonic, and geometric types, indicating that the matter 
 of weighting makes less difference to the median than it does to those types. 
 
 given year values because, with a fixed base, only one set 
 of values needs to be worked out for a whole series of index 
 numbers. Calculating only one set of values saves labor 
 as compared with calculating a separate set for each given 
 year. Another reason why weighting by base values 
 has so often been employed is that often only one set of 
 weights can be worked out. For instance, a census year 
 
FOUR METHODS OF WEIGHTING 53 
 
 may give the data required for starting off an index num- 
 ber with that census year as a base while similar data for 
 the succeeding years may be unavailable for want of a 
 yearly census. 
 
 The United States Bureau of Labor Statistics has used 
 base weighting with an arithmetic type of index number. 
 The Harvard Committee on Economic Research, in the 
 Day index number of production, employs it with a geo- 
 metric type. Weighting by given year values (as in 
 Formula 9) has been proposed by Palgrave for arithmetic 
 index numbers. 
 
 The Two Extreme Methods of 
 
 Weighting Median 
 (Quantities) 
 
 13 M 15 '16 77 73 
 
 Chart lOQ. Analogous to Chart lOP. 
 
 § 6. Two Intermediate Systems of Weighting 
 
 Besides the two systems of weighting which have just 
 been described there are two other analogous systems, 
 making four in all. Of these four, the system of base 
 value weights will be called "weighting 7" and the sys- 
 tem of given value weights will be called "weighting 7F." 
 The other two systems {II and 7/J) still to be described 
 fall logically between these extremes. In Systems II 
 and III each commodity is weighted by a hybrid value, 
 relating not to the base year alone nor the given year 
 alone but partly to one and partly to the other. In sys- 
 
54 THE MAKING OF INDEX NUMBERS 
 
 tern II the value is made by multiplying the -price of each 
 commodity in the base year by the quantity of that com- 
 modity in the given year. In system III each commodity 
 is weighted by the other hybrid value formed by multi- 
 plying its price in the given year by its quantity in the 
 base year. That is : 
 
 /, each weight = base year price X base year quantity 
 
 II, each weight = base year price X given year quantity 
 
 III, each weight = given year price X base year quantity 
 
 IV, each weight = given year price X given year quantity 
 
 Algebraically, the weights used in the four systems of 
 weighting are, respectively : 
 
 /. po^o, p'og'o, etc. 
 
 II. poqi, p'oq'i, etc. 
 
 III. piqo, p\q'o, etc. 
 
 IV- PiQh p'lQ'h etc. 
 
 In the following Table 4, of weights, if we take the same 
 eight commodities previously cited (§3 above) and apply 
 the weight systems II and III, we find that, while every 
 figure has changed, there are still the same marked tend- 
 encies as in Table 3. In the first column, the lower group 
 of four articles preponderates over the upper group of four 
 articles ; and in the second column vice versa. 
 
 Thus, in both tables, the relative importance of the 
 two groups of commodities changes greatly between one 
 column and the other. The reason is that the two groups 
 of commodities are purposely contrasted as to price 
 change (but not as to quantity change) . It follows that, 
 if the second column weights are used, the upper four 
 commodities which rise the more in price will be the 
 more heavily weighted, while the opposite is true if the 
 first column weights are used. These points will be 
 elaborated in Chapter V. 
 
FOUR METHODS OF WEIGHTING 
 
 55 
 
 TABLE 4. HYBRID VALUES (IN MILLIONS OF DOLLARS) OF 
 CERTAIN COMMODITIES 
 
 Commodity 
 
 1913 Prices Mttltiplied 
 BY 1917 Quantities 
 
 1917 Prices Multiplied 
 BY 1913 Quantities 
 
 Bituminous coal . . 
 Coke 
 
 701 
 172 
 
 577 
 596 
 
 1708 
 494 
 
 Pig iron 
 
 1203 
 
 Oats 
 
 715 
 
 
 
 Anthracite coal .... 
 
 Petroleum .... 
 
 Coffee 
 
 40 
 
 1835 
 
 147 
 
 1916 
 
 39 
 
 1292 
 
 80 
 
 Lumber 
 
 2290 
 
 
 
 Let us trace specifically the effects of all four systems 
 of weighting. 
 
 The arithmetic formulse give the following index num- 
 bers for 1917, relatively to 1913 as the base : 
 
 Arithmetic by weight system / (Formula 3) 162.07 per cent 
 
 II ( " 5) 161.05 per cent 
 " " " " III ( " 7) 180.53 per cent 
 
 " " " " IV ( " 9) 180.72 per cent 
 
 The harmonic formulae give : 
 
 Harmonic by weight system I (Formula 13) 147.19 per cent 
 " " " " II ( " 15) 144.97 per cent 
 
 " " " " III ( " 17) 162.07 per cent 
 
 " " " " IV ( " 19) 161.05 per cent 
 
 The geometric formulae give : 
 
 Geometric by weight system / (Formula 23) 154.08 per cent 
 " " " " II ( " 25) 152.45 per cent 
 
 " " " " III ( " 27) 170.82 per cent 
 
 " " " " IV ( " 29) 170.44 per cent 
 
 Numerically, the above calculations show that weight 
 system II gives results almost identical with weight sys- 
 tem J, while likewise weight systems III and IV are al- 
 
56 THE MAKING OF INDEX NUMBERS 
 
 most identical, there being a wide gap, however, between 
 these two pairs. 
 
 This disparity, as indicated, is due to the fact that, in 
 deriving the weights I and //, base year prices are used, 
 while in III and IV, given year prices are used, the prices 
 in both cases out-influencing the quantities. 
 
 The same contrasts (of 7, II as against III, IV), though 
 less pronounced, are found in the weighted medians; 
 but in the modes these contrasts, while present, are im- 
 perceptible. 
 
 We have now cited not only a simple arithmetic for- 
 mula, but four weighted arithmetic formulae, and hke- 
 wise a simple and four weighted harmonic formulae, a 
 simple and four weighted geometric formulae, a simple 
 and four weighted median formulae, and a simple and four 
 weighted mode formulae, for obtaining index numbers. 
 
 § 7. Only Two Systems of Weighting for the Aggregative 
 
 Up to this point, therefore, we have considered foui 
 forms of weighting for each of the five types of index 
 numbers. The sixth, or aggregative, type of index num- 
 ber has as yet been considered only in its ''simple" form. 
 Because of its peculiar construction it is capable of only 
 two systems of weighting at all analogous to those we have 
 been considering. As we have seen, the simple aggre- 
 gative is a very peculiar average of price ratios (price 
 relatives) being a ratio of the sums of the prices themselves. 
 Thus the simple aggregative gives : 
 
 Fum of all prices of 1917 _ index number for 1917 rela- 
 sum of all prices of 1913 lively to 1913. 
 
 Consequently the weighting cannot be applied to the 
 price ratios as such, but must be apphed direcily to the 
 prices themselves — both in numerator and denominator. 
 
FOUR METHODS OF WEIGHTING 57 
 
 Of course, the same weight is to be appHed, in this way, 
 to the prices of the same items in both numerator and 
 denominator. 
 
 Now, in the previous formulae, the weights were values. 
 But value is price multiplied by quantity. In the aggre- 
 gative formulae, however, the price part is already there 
 as the only thing to be weighted. 
 
 It would be absurd to multiply price by value 
 (which already contains price). Consequently, in the 
 aggregative formulae, the weights must be just quantities 
 and these quantities must be either the quantities of the 
 base year (1913) or the quantities of the given year (1917). 
 If we wish to keep up the analogy with the four kinds of 
 weighting, used for all the other types, we may consider 
 the weighting of the aggregative by base year quantities 
 as weighting I (Formula 53), and the weighting by given 
 year quantities as weighting IV (Formula 59), omitting 
 II and III entirely.^ 
 
 § 8. Numerical Calculation of Weighted Aggregative 
 
 Numerically, to illustrate by our 36 commodities, let 
 us outline the calculation of the aggregative by base 
 weighting (Formula 53) for the index number of 1914 
 (relatively to 1913 as base). ^ This is defined as the ratio 
 of the sum of the hybrid values for 1914 (because reckoned 
 with the quantities of 1913) to the true values for 1913. 
 
 The denominator of this fraction, i.e. the true value 
 in 1913, is, as above, $13,104,818,000. The numerator 
 is derived in a similar way. Beginning with bacon, we 
 obtain its (hybrid) value by multiplying its price in 1914 
 (12.95 cents per pound) by its quantity, not in 1914 but 
 
 * See Appendix I (Note to Chapter III, § 7). 
 
 2 For model examples to aid in the practical calculation of this as well as 
 of eight other sorts of index numbers, see Appendix VI, § 2. 
 
58 THE MAKING OF INDEX NUMBERS 
 
 in 1913 (1077 million pounds), obtaining $0,129^ X 1077, 
 or 139.47 million dollars. Similarly, the barley value is 
 62.04 cents per bushel X 178.2 milhon bushels, or 
 110.56 milhon dollars, and so on, the total of the 36 
 such values being $13,095,780,000, the desired numerator. 
 
 The ratio of this numerator to the above denominator 
 comes out 99.93 per cent, the index number sought. This 
 is by Formula 53, weighting /. 
 
 For Formula 59, using "given year" weighting IV, the 
 calculation is similar.^ The numerator is 13033.034, the 
 sum of the true values in 1914, and the denominator is 
 12991.81, the sum of the hybrid values for 1913 (found by 
 using the prices of 1913 and the quantities of 1914). The 
 ratio of the numerator to the denominator is 100.32 per 
 cent, almost the same as the 99.93 per cent by the other 
 formula (53). 
 
 These two index numbers (Formulae 53 and 59), con- 
 trasted merely as to whether base year quantities or 
 given year quantities are used, show no tendency to the 
 wide contrast between base year and given year weighting 
 found in the arithmetic, harmonic, and geometric index 
 numbers. There is no tendency for Formula 53, in which 
 base year quantities are used, to be less than Formula 59, 
 in which given year quantities are used. The two curves 
 are very close together and even cross each other. As 
 the reader may suspect, the reason for this close similarity 
 is that the price element which, in previous weighting 
 systems was the distuibing element, is here missing, the 
 weights being mere quantities. 
 
 § 9. The Algebraic Formulae 
 
 Algebraically, the aggregative index number for prices 
 with base year weighting or weighting I (Formula 53) is 
 1 See also Appendix VI, § 2. 
 
FOUR METHODS OF WEIGHTING 59 
 
 Pigo + p'lq'o + p'\q''o + . . . 
 Poqo + p'q q'o + p"oq"o + • • . 
 or, 
 
 Spigo ^ 
 
 while with the given year weighting, or weighting IV, the 
 aggregative (Formula 59) is 
 
 2pogi 
 The corresponding index numbers of quantities 
 (weighted by prices) are 
 
 / (base year prices) - 
 
 IV (given " " ) 
 
 
 § 10. Historical 
 
 The first of the two weighted aggregative formulae, 
 
 (Formula 53 for prices), is the form used by the 
 
 ^PoQ'o 
 
 United States Bureau of Labor Statistics. It is a 
 return to an old idea, since this method was expUcitly 
 formulated and advocated by Laspeyres in 1864, and 
 Walsh gives it the name of Laspeyres' method.^ 
 
 The present vogue of this method is largely due to the 
 vigorous advocacy of it and strong arguments for it made 
 by G. H. Knibbs, the Government Statistician for Aus- 
 traha. It has been formally recommended by a vote of a 
 recent conference of the statisticians of the British Em- 
 pire. 
 
 The second of the two formulae, -~^ (59), was ad- 
 
 * See Walsh, The Measurement of General Exchange Value, p. 558. 
 
60 THE MAKING OF INDEX NUMBERS 
 
 vocated and employed by Paasche in 1874. Walsh calls 
 it Paasche's method.^ 
 
 These two names will recur: Laspeyres' formula, 53 
 (aggregative weighted I) and Paasche's formula, 59 
 (aggregative weighted IV). 
 
 § 11. Relation of Weighted Aggregative to Weighted 
 Arithmetic and Weighted Harmonic 
 
 It is of much interest to note that the arithmetic aver- 
 age weighted by base values (7, or Formula 3) necessarily 
 reduces, when simplified, to Laspeyres' formula (53) — 
 that is, the aggregative average weighted by base quanti- 
 ties ; while the harmonic average weighted by given year 
 values {IV, or Formula 19), when simplified, likewise 
 reduces to Paasche's formula (59) — that is, the aggre- 
 gative average, weighted by given year quantities ; and 
 that furthermore the arithmetic average weighted by 
 weight method II (Formula 5) reduces to Paasche's; 
 and the harmonic average weighted by weight method 
 III (Formula 17) reduces to Laspeyres'.^ 
 
 Algebraically, the proof of these propositions is simple.' 
 Graphically, from what has been said it follows that 
 each of the two central curves of Chart 9 has a triple 
 meaning. Each represents an arithmetic, harmonic, 
 and aggregative index number. What is labeled 3 might 
 be labeled also 17 and 53 and what is labeled 19 might 
 be labeled also 5 and 59. 
 
 § 12. Formulae thus far Available 
 
 We see, then, that there are four primary methods of 
 weighting (7, 77, 777, IV) appUcable to five of the six 
 
 ^ The Measurement of General Exchange Value, p. 559. 
 
 ' Ibid., pp. 306-7, 350, 352, 511. Walsh was the first to point out these 
 identities excepting that which he refers to as having been first pointed out 
 by me {i.e. 3 and 53). » See Appendix I (Note to Chapter III, § 11). 
 
FOUR METHODS OF WEIGHTING 
 
 61 
 
 types of index numbers, namely, arithmetic, harmonic, 
 geometric, median, mode, and two analogous methods 
 (7, IV) applicable to the sixth (aggregative). Let us 
 now 'Hake account of stock" and see what index num- 
 bers we have thus far obtained all together. We have 
 the following : 
 
 TABLE 5. 
 
 IDENTIFICATION NUMBERS OF PRIMARY 
 FORMULAE 
 
 Weighting 
 
 Abith. 
 
 Harm. 
 
 Geom. 
 
 Median 
 
 Mode 
 
 Aggkbg. 
 
 Simple (or even) . 
 
 1 
 
 11 
 
 21 
 
 31 
 
 41 
 
 51 
 
 /. Base year only . . . 
 
 3 
 
 13 
 
 23 
 
 33 
 
 43 
 
 53 
 
 //. Base year prices X 
 
 
 
 
 
 
 
 given year quan- 
 
 
 
 
 
 
 
 tities 
 
 5 
 
 15 
 
 25 
 
 35 
 
 45 
 
 
 ///. Given year prices X 
 
 
 
 
 
 
 
 base year quan- 
 
 
 
 
 
 
 
 tities 
 
 7 
 
 17 
 
 27 
 
 37 
 
 47 
 
 
 IV. Given year only . . 
 
 9 
 
 19 
 
 29 
 
 39 
 
 49 
 
 59 
 
 This variety may seem at first merely to increase our 
 sense of bewilderment and distrust of index numbers. 
 But we shall find grounds for discriminating between the 
 various formulae. Moreover, as has been noted, and as 
 is evident from inspecting the formulae in Appendix V, 
 there are four duplications in the table (53 = 3 = 17; 
 59 = 19 = 5). 
 
 These various weighting systems are, of course, not 
 the only possible ones. In Chapter VIII we shall consider 
 systems formed by taking averages or means of the above 
 varieties of weights. There seem to be no others pro- 
 posed worth very serious attention.^ 
 
 » See Appendix I (Note to Chapter III, § 12). 
 
CHAPTER IV 
 
 TWO GREAT REVERSAL TESTS 
 
 § 1. Reversal Tests in General 
 
 As indicated at the close of the last chapter, not all 
 index numbers have equal claims to be considered as 
 truly representative of price movements. They may be 
 good, bad, or indifferent, and our next task is to set up 
 certain criteria for distinguishing them as such. 
 
 The fundamental question, mentioned in Chapter I, 
 § 6, is that of fairness. The requirement of fairness is 
 often expressed by the demand, *' put yourself in his place." 
 Fairness is not fair which takes account of whose ox is 
 gored. In short, ''It is a poor rule that won't work both 
 ways." This kind of test, ''the golden rule" of fair 
 dealing among men is, in a sense, the golden rule in the 
 domain of index numbers also. 
 
 Index numbers to be fair ought to work both ways — 
 both ways as regards any two commodities to be averaged, 
 or as regards the two times to be compared, or as regards 
 the two sets of associated elements for which index 
 numbers may be calculated — that is, prices and quan- 
 tities. The rule of changing places applies separately 
 to each of the three following sets of magnitudes : 
 first, the several commodities ; second, the two times ; 
 third, the two factors — prices and quantities. To be 
 specific, this rule of changing places means three separate 
 things : interchanging any two commodities, inter- 
 changing the two times, interchanging prices and 
 
 62 
 
TWO GREAT REVERSAL TESTS 63 
 
 quantities. In short, we must, in some sense, treat 
 alike : (a) any two commodities ; (6) the two times ; 
 (c) the two factors. 
 
 The first test is seldom if ever violated. It is men- 
 tioned here for completeness and to afford a basis for a bet- 
 ter appreciation of the two less obvious tests which follow. 
 
 In order to avoid 'confusion the three tests will be dis- 
 tinguished as: 
 
 *'PreUniinary" — The commodity reversal test 
 Test 1 — The time reversal test 
 
 Test 2 — The factor reversal test 
 
 Any formula to be fair should satisfy all three tests. 
 The requirement as to commodities is that the order of the 
 commodities ought to make no difference — that, to be 
 specific, any two commodities could be interchanged, i.e. 
 their order reversed, without affecting the resulting index 
 number. This is so simple as never to have been for- 
 mulated. It is merely taken for granted and observed 
 instinctively. Any rule for averaging the commodities 
 must be so general as to apply interchangeably to 
 all of the terms averaged. It would not be fair, for 
 instance, arbitrarily to average the first half of the com- 
 modities by the arithmetic method and the other half 
 by the geometric, nor fancifully to weight the seventh 
 commodity by 7 and the tenth commodity by 10 so that 
 if the seventh and tenth commodities were interchanged 
 the result would be affected.^ 
 
 ^ It may be worth while, for contrast, to note an example of an average, 
 in another field of thought, for which the order of the terms is not inter- 
 changeable. If the German Reparation Debt were represented by bonds 
 of 100 billion marks drawing 10 per cent interest for the first 15 years, 
 6 per cent for the next 15 years, and 3 per cent for a third period of 15 years, 
 the "average" rate of interest for all three periods will not be independ- 
 ent of the order. It would be different if, for instance, the first period 
 were at 3 per cent and the last at 10 per cent. (See Irving Fisher's The 
 Bate of Interest, New York, 1907, p. 372.) 
 
 \ 
 
64 THE MAKING OF INDEX NUMBERS 
 
 The other two tests mentioned (which will be referred 
 to as Test 1 and Test 2), although thoroughly analogous 
 to the Preliminary Test, have not been so well observed. 
 On the contrary many index numbers in actual use fail 
 to observe either of them, and none at all observe the 
 second ! 
 
 § 2. The Time Reversal Test 
 
 Just as the very idea of an index number impUes a set 
 of commodities, so it implies two (and only two) times 
 (or places). Either one of the two times may be taken 
 as the ''base." Will it make a difference which is chosen ? 
 Certainly it ought not and our Test 1 demands that it 
 shall not. More fully expressed, the test is that the for- 
 mula for calculating an index number should be such that 
 it will give the same ratio between one point of comparison 
 and the other point, no matter which of the two is taken as 
 the base. 
 
 Or, putting it another way, the index number reckoned 
 forward should be the reciprocal of that reckoned back- 
 ward. Thus, if taking 1913 as a base and going forward 
 to 1918, we find that, on the average, prices have doubled, 
 then, by proceeding in the reverse direction, we ought 
 to find the 1913 price level to be half that of 1918, from 
 which we started as a base. Putting it in still another 
 way, more useful for practical purposes, the forward and 
 backward index number multiplied together should give 
 unity. 
 
 The justification for making this rule is twofold : (1) no 
 reason can be assigned for choosing to reckon in one 
 direction which does not also apply to the opposite, and 
 (2) such reversibiUty does apply to any individual com- 
 modity. If sugar costs twice as much in 1918 as in 1913, 
 then necessarily it costs half as much in 1913 as in 1918. 
 
TWO GREAT REVERSAL TESTS 65 
 
 By analogy we demand that any formula for an index 
 number, by which we find the price level of 1918 is double 
 that of 1913, ought to tell us that the price level of 1913 
 is half that of 1918. 
 
 This requirement is still more appealing to our sense 
 of fairness if we take not two times, but two places ; we 
 might be confused by the fact that succession in time is 
 different, forward from backward, and wonder for a 
 moment whether there might not be some hidden but logi- 
 cal reason for using the earlier of the two dates as the base 
 rather than the later. But in comparisons between places 
 there is not even this semblance of a reason for regarding 
 one of the two points of comparison as the base rather 
 than the other. 
 
 § 3. The Time Reversal Test Illustrated Numerically 
 
 Yet most forms of index numbers in use do not con- 
 form to this reversal test ! For instance, the simple arith- 
 metic average does not. 
 
 Numerically, the following illustrations show this. 
 Suppose the price of bread is twice as high in Philadelphia 
 as in New York (20 cents a loaf as against 10 cents) and, 
 reversely, the price of butter is twice as high in New York 
 as Philadelphia (60 cents a pound instead of 30 cents). 
 In price relatives or percentages, taking New York prices 
 as 100 per cent, the figures are : 
 
 Bread : New York 100 per cent Philadelphia 200 per cent 
 Butter: " " 100 " " " 50 " " 
 
 The simple arithmetic index number for Philadelphia 
 is — — , or 125 per cent, and would make it appear 
 
 that bread and butter were on the average 25 per cent 
 higher in Philadelphia than in New York. But if we 
 take Philadelphia as 100 per cent, the figures are : 
 
66 THE MAKING OF INDEX NUMBERS 
 
 Bread : Philadelphia 100 per cent New York 50 per cent 
 Butter: " 100 " " " " 200 " " 
 
 This gives ^Q+ ^^^ = 125 per cent, or 25 per cent higher 
 
 in New York than in Philadelphia. Since each city can- 
 not be 25 per cent above the other, something must be 
 wrong with the formula which yields such a preposterous 
 result. No reason can be assigned why the formula should 
 be appHed with New York as a base, which will not equally 
 justify making Philadelphia the base ; and no more reason 
 can be assigned for making one of any two years compared 
 the base which will not equally justify making the other 
 year the base. - • 
 
 Again, suppose bread rose in price between 1913 and 
 1918 from 10 cents to 15 cents a loaf, i.e. in price relatives 
 or percentages from 100 to 150, and butter from 20 cents 
 per pound to 50 cents per pound, or from 100 to 250. The 
 index number for 1918 relatively to 1913 as a base is then 
 
 — = 200 per cent. But, reversing the comparison 
 
 and taking 1918 as the base, we find the price ratios for 
 1913 to be, for bread, 66| per cent and, for butter, 40 per 
 cent. The average of these is not the required 50 per 
 cent but 53^ per cent. Consequently, the product of 
 the two opposite index numbers is not, as it should be, 
 unity, or 100 per cent, but 200 X 53^ = 106f per cent, 
 or 6| per cent too great. 
 
 Again, taking the simple arithmetic average of the 
 36 price relatives for 1917 relatively to 1913, or 175.79 
 per cent, and reversely, taking the simple arithmetic 
 average of the same prices for 1913 relatively to 1917, 
 or 63.34 per cent, and multiplying these two together 
 we get, not unity or 100 per cent, but 111.35 per cent. 
 Evidently there is here an error of 11.35 per cent. 
 
TWO GREAT REVERSAL TESTS 67 
 
 That is, the simple arithmetic average, checked up by 
 itself forward and backward in time, stultifies itself by 
 exactly 11.35 per cent. The error of 11.35 per cent 
 must rest somewhere. It may be that the 175.79 per 
 cent for 1917 relatively to 1913 is too high by 11.35 per 
 cent, or it may be that the 63.34 per cent for 1913 rela- 
 tively to 1917 is too high by 11.35 per cent, or it may be 
 that the two figures (175.79 and 63.34) share the error, 
 equally or unequally. We cannot say. What we can say 
 is that both the 175.79 and the 63.34 cannot be true at 
 once and that between them there is a total, or net joint 
 error of exactly 11.35 per cent. 
 
 Again, we find that the simple arithmetic index number 
 of the 36 commodities makes out the price level of 1915 
 to be If per cent higher than that of 1914 with 1914 as 
 the base while, reversely, it makes out the price level of 
 1914 to be i per cent higher than that of 1915 with 1915 
 as the base. In other words here is an actual case where 
 each of two years is represented by the arithmetic index 
 number as being higher priced than the other ! 
 
 The simple harmonic index number also fails to meet 
 Test 1. 
 
 The simple geometric index number, on the other hand, 
 conforms to Test 1. It gives 166.65 per cent for 1917 
 relatively to 1913 and 60.01 per cent for 1913 relatively 
 to 1917, the product of which is exactly 100 per cent. 
 The general proof of this is deferred to Chapter VI. 
 
 This conformity to Test 1 (time reversal) does not, 
 of course, prove that the geometric index number is 
 exactly correct. It means simply what it says, that the 
 simple geometric is self-consistent when applied reversely 
 in time. There may be errors in both figures which offset 
 each other when they are multiplied but there is no net 
 or joint error in the product. All we can say is that we 
 
68 
 
 THE MAKING OF INDEX NUMBERS 
 
 know the simple arithmetic index number, for instance, 
 has failed to tell the truth and that we have not yet caught 
 
 Forward (17-18) and 
 
 Backward (18-17) 
 
 Simple Arithmetics 
 contrasted 
 
 (Prices) 
 
 17 16 
 
 <.'^= 
 
 Chart IIP. Each line forward, representing the changing price of a 
 commodity between 1917 and 1918, is prolonged backward to represent the 
 reciprocal change from 1918 to 1917. Yet the simple arithmetic averages 
 of these two fans of lines are not prolongations of each other. 
 
 the simple geometric in a lie. We must wait till we apply 
 to it Test 2. 
 
 The simple median, mode, and aggregative all fulfill 
 Test 1. The general proof is deferred to Chapter VI. 
 
TWO GREAT REVERSAL TESTS 
 
 69 
 
 Forward (V-Wand 
 
 Backward CId' 17) 
 
 Simple Arithmetics 
 contrasted 
 
 (Quantities) 
 
 simp le 
 arithmetic 
 
 Chart IIQ. Analogous to Chart UP. 
 
70 THE MAKING OF INDEX NUMBERS 
 
 But none of the weighted index numbers yet described 
 conforms to Test 1. Thus, only four out of the 28 kinds 
 of index numbers so far encountered fulfill Test 1. 
 
 § 4. The Time Reversal Test Illustrated Graphically 
 
 We have seen that an index number calculated forward 
 should be the reciprocal of the index number calculated 
 backward. Such harmonious results would be repre- 
 sented by parallel lines in our charts. But in the case 
 of the arithmetic average the two lines will not be parallel ; 
 that is, the arithmetic backward is not the reciprocal of 
 the arithmetic forward. 
 
 Graphically, this is illustrated in Charts IIP and 11 Q 
 which repeat from Charts 3P and 3Q the dispersion of 
 the 36 individual prices (and the 36 quantities) from 1917 
 to 1918. To represent the reverse dispersion from 1918 
 to 1917, in order not to let the two sets of radiating Hues 
 interfere with one another, and, for simplicity, they have 
 been radiated from the same point, simply to the left in- 
 stead of to the right. We thus really have two separate 
 charts; the one common point representing 1917 for the 
 right hand chart but representing 1918 for the left hand 
 chart. 
 
 Now the line for any individual commodity drawn 
 backward must, in our ratio method of charting, take the 
 same direction as that for the same commodity drawn 
 forward, so that the left set of radiating lines are simply 
 the backward prolongations of the right set. 
 
 But (and this is the point to be noted) while each of 
 these 36 price Unes individually is the prolongation of 
 its mate, yet the two opposite lines for their average (the 
 arithmetic index number) are not the prolongations each 
 of the other. The two longer and darker hues represent 
 these arithmetic index numbers forward and backward : 
 
TWO GREAT REVERSAL TESTS 71 
 
 and, while the arithmetic forward shows a rise from 100 
 to 110.11, the arithmetic backward shows a fall only 
 from 100 to 94.46. The two form a bend at the origin 
 and one or both ends must be too high. This tendency 
 to go higher than it should is characteristic of the arith- 
 metic index number. 
 
 § 5. The Time Reversal Test Expressed Algebraically 
 
 Algebraically, Test 1 (time reversal) may be stated in 
 general terms as follows. Let the two dates (or two 
 places) be distinguished as and 1 and let Foi be the for- 
 ward index number of prices, i.e. that for date 1, relatively 
 to date taken as a base. Then Pio will be the hack- 
 ward index number, i.e. that for date relatively to date 1 
 taken as a base. With this notation we may express Test 
 1 in algebraic terms as follows : Poi X Fio should = 1. 
 This is the same as saying that Foi must be such a formula 
 that if the subscripts and 1 be interchanged, the new 
 formula resulting will become the reciprocal of the old. 
 
 The failure of the simple arithmetic index number to 
 conform to Test 1 is clearly seen if we examine its alge- 
 braic expression. If we take the year designated by "0'* 
 as the base, the simple arithmetic index number for year 
 "l"is 
 
 
 2 
 
 n 
 whereas, if we reverse the comparison by taking year "1" 
 as the base (that is, interchange the subscripts) the sim- 
 ple arithmetic index number for year ''0" is 
 
 _w_. 
 
 n 
 These two expressions are inconsistent with Test 1^ not 
 
72 THE MAKING OF INDEX NUMBERS 
 
 being reciprocals of each other. That is, they are not of 
 such a form that their product will necessarily be unity. 
 
 § 6. The Factor Reversal Test 
 
 The factor reversal test is analogous to the time reversal 
 test. Just as our formula should permit the interchange 
 of the two times without giving inconsistent results, so 
 it ought to permit interchanging the prices and quan- 
 tities without giving inconsistent results — i.e. the two 
 results multiplied together should give the true value 
 ratio. 
 
 Whenever there is a price of anything exchanged, there 
 is implied a quantity of it exchanged, or produced, or con- 
 sumed, or otherwise involved, so that the problem of an 
 index number of the prices implies the twin problem of the 
 index number of the quantities. Thus the index number 
 of the prices at which certain commodities are sold at 
 wholesale goes hand in hand with the index number of 
 the quantities of these commodities sold at wholesale. 
 Likewise we find paired the index numbers of the prices 
 and quantities of industrial stocks sold on the New York 
 Stock Exchange, or the index numbers of rates of wages 
 and of the quantities of labor sold at those rates of wages, 
 or the index numbers of the rates of discount for loans and 
 the volume of loans made at those rates of discount. 
 
 § 7. The Simple Arithmetic Index Number Tested 
 by Factor Reversal 
 
 Of the 28 formulae thus far reached, not a single one 
 conforms to Test 2 ! 
 
 Numerically, take Formula 1, the simple arithmetic, 
 and apply it to an example which is simple enough to fol- 
 low through in detail. Suppose the price of bacon is 
 twice as high in 1918 as in 1913 while the price of rubber 
 
TWO GREAT REVERSAL TESTS 73 
 
 is exactly the same in 1918 as in 1913 ; and suppose that 
 the quantity of bacon sold in 1918 is half as much as the 
 quantity sold in 1913 while the quantity of rubber is the 
 same in both years. Evidently the value of bacon sold in 
 1918 is the same as the value of that used in 1913 (since 
 half the quantity of bacon is sold at twice the price) 
 and hkewise the value of the rubber remains unchanged 
 (since both its price and quantity remain unchanged). 
 Consequently, the total value of both together remains 
 unchanged also. A good index number of these prices 
 multiplied by the corresponding index number of these 
 quantities ought, therefore, to give (in this case) 100 per 
 cent. 
 
 With these figures in mind let us test the mettle of the 
 simple arithmetical average by applying it alike to the 
 above prices and quantities. By this formula the index 
 number of prices in 1918 as compared with 1913 is 
 
 200il00 ^ ^^^^^^^^^^ 
 
 and the index number of quantities is 
 — it — 75 per cent. 
 
 Multiplied together these results give 112| per cent in- 
 stead of the true 100 per cent. Here is an error of 12§ 
 per cent either in the index number of prices, or in that 
 of quantities, or shared jointly between them. 
 
 Again, suppose bread doubles in price and triples in 
 
 quantity so that its value sextuples, and butter triples 
 
 in price and doubles in quantity so that its value also 
 
 sextuples ; then their combined value certainly sextuples. 
 
 But the simple arithmetic index number would make it 
 
 2 + 3 
 appear that bread and butter had increased in price —~—» 
 
74 THE MAKING OF INDEX NUMBERS 
 
 3 4-2 
 or 2 1 fold, and that their quantity had increased ^ > or 
 
 2 1 fold, according to which their values are represented 
 to have increased 2\ X 2|, or 6j fold instead of sixfold, 
 the true figure. 
 
 The value ratio, unlike an index number of prices or quan- 
 tities, is not an estimate but a fact. There can be no am- 
 biguity about it or any question of reckoning it by dif- 
 ferent methods as in the case of index numbers. Thus, 
 in 1913, the value of the bacon sold was its price, 12.36 
 cents per pound multiplied by its quantity, 1077 milUon 
 pounds, or 133 million dollars. In the same way the value 
 of the barley sold was 62.63 cents per bushel X 178.2 mil- 
 Uon bushels = 112 million dollars, and so on for each of the 
 other 34 commodities. The sum total of these 36 prod- 
 ucts, or the value aggregate in 1913 (SpoQ'o) is 13104.818 
 million dollars and can be nothing else. Likewise for the 
 last year, 1918, the value aggregate (Spagj) is 29186.105 
 and can be nothing else. Thus the ratio of the total value 
 
 of 1918 to the total value of 1913 is ?^|^, or 222.71 per 
 
 13105. 
 
 cent and can be nothing else. The complete table of 
 value ratios follows : 
 
 TABLE 6. VALUE RATIOS FOR 36 COMMODITIES 
 1913-1918 
 
 Year 
 
 Value Ratio 
 
 1913 
 
 100.00 
 
 1914 
 
 99.45 
 
 1915 
 
 108.98 
 
 1916 
 
 135.75 
 
 1917 
 
 192.23 
 
 1918 
 
 222.71 
 
 
 
 These are, if we choose to call them so, ''index numbers " 
 of the total or aggregate value. But, whereas the index 
 
TWO GREAT REVERSAL TESTS 75 
 
 numbers of prices or of quantities may be calculated by 
 many different methods, the comparative merits of which 
 are debated in this book, the ''index numbers" of value are 
 indubitable and undebatable. They, therefore, afford a 
 fixed rock of truth, by which we may reckon the drifting 
 courses of the various index numbers of prices and quan- 
 tities. The problem then is to find a form of index 
 number such that, applied ahke to prices and quantities, 
 it shall correctly ''factor" any such value ratio. 
 
 Thus we can say with absolute certainty that the total 
 value in 1918 was 223 per cent of the total value in 1913. 
 But when we ask how far this increase from 100 to 223 rep- 
 resents increased prices and how far it represents in- 
 creased quantities, we enter the quagmire of index 
 numbers. We are searching for a formula which, applied 
 to prices, will really measure the increase of the prices, 
 and, applied to quantities, will really measure the in- 
 crease of the quantities ; and such that to make these 
 two results consistent, their product should give the re- 
 quired 223 per cent. 
 
 The justification for Test 2 is twofold : (1) no reason 
 can be given for employing a given formula for one of the 
 two factors which does not apply to the other, and, 
 (2) such reversibility already applies to each pair of indi- 
 vidual price and quantity ratios, and should, in all logic, 
 apply to the index numbers which aim to represent them 
 in the mass. 
 
 We know that if the price of bread in 1918 was double 
 its price in 1913 and if the quantity marketed in 1918 was 
 triple that in 1913 then the total value of bread marketed 
 in 1918 was six times that marketed in 1913. By anal- 
 ogy we have a right to expect of our index numbers, if 
 they show prices, on the average, to have doubled, and 
 quantities to have tripled, that sixfold correctly repre- 
 sents the increase in total value. 
 
76 
 
 THE MAKING OF INDEX NUMBERS 
 
 Algebraically, Test 2 is 
 
 PoiXQoi=|^i^=F, 
 
 01 
 
 PCby formula 353) X Q(by formula 353) - y 
 
 223% 
 
 175% 
 
 \/2S% 
 
 ' Uoo% 
 
 yj 
 
 7-^ 
 
 75 
 
 7tf 
 
 77 
 
 75 
 
 Chart 12. The product of the price index, P, times the quantity index, 
 Q, both calculated by the same formula (No. 353) equals the correct "value 
 ratio," V. (In this ratio chart, therefore, the total height above the 
 origin, 100 per cent, of the point in the chart labeled 223 equals the sum 
 of the heights of the two points labeled 125 and 178, above the same 
 origin.) 
 
 § 8. The Factor Reversal Test Illustrated Graphically 
 
 Graphically, Chart 12 shows the relation of index 
 numbers which correctly conform to Test 2. It shows 
 how one of the index numbers, to be explained later 
 (No. 353), fulfills Test 2. This formula, when applied 
 to our 36 prices, yields 178 per cent for the index number 
 in 1918, relatively to 1913 as a base, and when applied 
 to quantities, yields 125 per cent ; and these two figures 
 
TWO GREAT REVERSAL TESTS 77 
 
 multiplied together give correctly the true value ratio, 
 223 per cent, as given in Table 6. 
 
 Chart 13 shows how incorrect, on the other hand, are 
 the index numbers calculated by Formula 9, the weighted 
 arithmetic average in which the weights are the values 
 in the given or current year. 
 
 P (by formula 9) X Q (by formula 9) not sz to Y / 
 tQ>77oXl327o noi^to223% / 
 
 f/ 
 
 / 
 
 \/52% 
 
 \'00% 
 
 7J 74 75 7e 77 76 
 
 Chart 13. Analogous to Chart 12 except that the product of the two 
 index numbers is not, as it should be, equal to the value ratio. The dotted 
 line, representing the product, lies above the true value by a percentage 
 expressing the joint error of the two indexes (for prices and for quantities). 
 
 § 9. The Factor Reversal Test Reveals a Joint Error 
 
 Just as when studying Test 1, we checked up any type 
 of index number by noting how far the product of the 
 index number reckoned forward by the index number 
 
78 THE MAKING OF INDEX NUMBERS 
 
 reckoned backward departed from unity, so, through 
 Test 2, we check up by noting how far the product of 
 the price ratio (index number for prices) by the corre- 
 sponding quantity ratio (index number for quantities) 
 departs from the value ratio. 
 
 To illustrate how great this error may be; we recur to 
 our 36 commodities. We know that the total value of 
 the 36 commodities in 1917 was $25,191,000,000 and in 
 1913 it was $13,105,000,000 so that the true value ratio 
 was the ratio of these numbers, or 192.23 per cent. But 
 the simple arithmetic index number (No. 1) for prices for 
 1917 relatively to 1913 is 175.79 per cent, and the cor- 
 responding index number for the same dates for quan- 
 tities is 125.84 per cent. The product of these two is 221.21 
 per cent, which is larger than the truth (192.23 per cent) 
 by 15.08 per cent. 
 
 This is an exact measure of the inconsistency of the two 
 arithmetic index numbers with each other as checked up 
 by the truth. Thus again does the simple arithmetic 
 stultify itself. There is a joint error here of 15.08 per cent 
 somewhere, just as, in checking up by Test 1, we found 
 that there was an error of 11.35 per cent somewhere. 
 And, just as before, we cannot say exactly where the error 
 lies. The 15.08 per cent error may be in the price index, 
 or in the quantity index, or it may be shared between them. 
 
 As to the simple geometric, it will be remembered that 
 we could not convict it of error by using Test 1 ; but, by 
 using Test 2, we can now convict it of error. The simple 
 geometric index number for 1917, relatively to 1913, for 
 prices, is 166.65 per cent and, for quantities, 118.75 per 
 cent; the product of these two (instead of being 192.23 
 as it should) is 197.90, which is 2.95 per cent too high. 
 
 In this way, by means of Test 2, we can convict every 
 fair of index numbers for prices and quantities in our 
 
TWO GREAT REVERSAL TESTS 79 
 
 list, as thus far constituted, of some degree of error. Some 
 formulse, of course, come much nearer than others to con- 
 forming to Test 2. The least joint error among the for- 
 mulse thus far hsted is 53's. For prices for 1917 relatively 
 to 1913 this gives 162.07, and for quantities, 119.36, the 
 product of which is 193.45 per cent which is only 0.6 per 
 cent higher than the required 192.23. Incidentally it 
 may be noted that this joint error of 53P and 53Q is the 
 same as the joint error we found by Test 1 for 53P and 
 59P and is the same as the joint error of 53Q and 59Q. 
 
 , § 10. The Factor Reversal Test Analogous to 
 the Other Reversal Tests 
 
 Algebraically, the various sorts of reversibility can best 
 be seen by taking some particular formula as an example. 
 Let us take Formula 53 (Laspeyres'). For prices for- 
 ward, Formula 53 is 
 
 2pogo* 
 For prices backward this same Formula 53 becomes 
 
 the "0" and "1" being reversed, or interchanged. The 
 two above apphcations of Formula 53 are exactly alike 
 except that one is forward in time and the other is back- 
 ward. Each is an index number of prices. 
 Starting again with 
 
 Spogo 
 
 for prices forward, let us this time interchange or reverse, 
 not the ''0" and " 1," but the "p's" and "g's." We then get 
 
 Sgipo 
 
 XqoPq 
 
80 THE MAKING OF INDEX NUMBERS 
 
 The last two applications of Formula 53 are exactly 
 alike except that one is for prices and the other is for quan- 
 tities. Each is a forward index number. 
 
 Thus the only difference between the two tests is 
 that, starting, say, with the Formula 53 for prices for- 
 ward, 
 
 for Test 1 we erase *'0" wherever it occurs and write 
 ''!" in its place, and vice versa; whereas, for Test 2, we 
 erase "p" wherever it occurs and write "g" in its place, 
 and vice versa. 
 
 Test 1 tells us that after the specified reversal of sym- 
 bols, the new formula, multiplied by the old, should 
 give unity, i.e. 
 
 Xpoqo Spigi 
 Test 2 tells us that after the specified reversal of sym- 
 bols the new formula multiplied by the old should give 
 the value ratio, i.e. 
 
 XpoQa Xqopo ~ Spogo * 
 
 In the case of this particular formula (53) neither of 
 these equations holds true, so that neither test is fulfilled. 
 
 While we are noting the algebraic interpretation of 
 Tests 1 and 2, we may as well recur to the ''Preliminary 
 Test" regarding the interchange or reversal of any two 
 commodities. We start again with 
 
 ^PiQo „_ PiQo + p'lq'o + p"iq"o + . ■ • 
 Zpoqo poQa + p oq Q + p oq o+ • • • 
 
 but now reverse the places, not of "0" and "1" nor of 
 "p" and "5," but of " ' " and " " " (or of any other 
 two accents representing two different commodities). 
 
TWO GREAT REVERSAL TESTS 81 
 
 That is, we erase " ' " wherever it occurs and write " " " 
 in its place, and vice versa. The result is : 
 
 PiQo + v"\q"Q + p'lg'o + . . . 
 
 Pogo + p"og"o + p'og'o + . • .' 
 which new formula is (except in form) the same as the 
 old — as the "Preliminary Test" or conunodity reversal 
 test requires.^ 
 
 Thus the commodity reversal test, the time reversal test, 
 and the factor reversal test alike require that the formula 
 be such that we can, with impunity, interchange symbols. 
 For the commodity reversal test the reversible symbols are 
 the commodity symbols, any two superscripts such as '"" 
 and " "" ; for the time reversal test the reversible sym- 
 bols are the two time symbols, the two subscripts ''0" and 
 '' 1" ; and for the factor reversal test the reversible sym- 
 bols are the two factor symbols, the two letters "p" and 
 ^'g." Reversibility "with impunity" means that the 
 results of such reversal shall be appropriate to the case. 
 For commodity reversal, the new and old forms of the for- 
 mula ought to be equal ; for time reversal, they ought to 
 be reciprocals ; for factor reversal, they ought correctly 
 to "factor" the value ratio. 
 
 These three tests are the only reversal tests possible, 
 because any formula for an index number contains just 
 three sets of symbols, the letters, the subscripts, the super- 
 scripts. The three reversal tests (Preliminary, Test 1, 
 and Test 2) merely require that the formula shall allow 
 each of the three kinds of symbols of which it is com- 
 posed to shift about with impunity. 
 
 As these requirements of reversibility are purely formal 
 and mathematical, they evidently have a very wide range 
 
 ^ This test is met by all the formulae in this book. If the reader wishes 
 to picture a case where this test would not be fulfilled, let him suppose a 
 minus sign in place of one of the plus signs. Also see § 1 above. 
 
82 THE MAKING OF INDEX NUMBERS 
 
 of application. They apply to any index number — 
 wholesale prices, retail prices, wages, interest, production, 
 and many others — where we have several items dis- 
 tinguishable by superscripts such as '"," ""," *""," 
 etc., two times, or places, or other groupings distinguish- 
 able by two subscripts, such as "0" and "1," and two 
 magnitudes distinguishable by two letters such as "p" 
 and "g" after the analogy of the case we just took.* 
 
 § 11. Historical 
 
 Test 1, the time reversal test, seems first to have been 
 used by Professor N. G. Pierson in 1896. ^ Its great im- 
 portance was recognized by C. M. Walsh in 1901,^ and 
 by myself in 1911,^ as well as by other writers. 
 
 UnHke Test 1, Test 2 has hitherto^ been entirely over- 
 looked, presumably because index numbers of quantities 
 have so seldom been computed and, almost never, side 
 by side with the index number of the prices to which they 
 relate. Moreover, the analogy between the three kinds 
 of reversal naturally escaped attention since most users 
 of index numbers have thought in concrete terms not 
 algebraic ; they formed a mental image of time reversal 
 only from the calendar, and saw no advantage in pic- 
 turing it sjnnboUcally as an interchange of "0" and "1" 
 in a formula. 
 
 1 See Appendix I (Note to Chapter IV, § 10). 
 
 * Economic Journal, Vol. vi, March, 1896, p. 128. 
 
 8 Measurement of General Exchange Value, pp. 324-32, 368-69, 389-90. 
 
 * Purchasing Power of Money, p. 401. 
 
 ' It was first formulated in the paper, of which this book is an expan- 
 sion, read December, 1920, and abstracted in "The Best Form of Index 
 Number," Quarterly Publication of the American Statistical Association, 
 March, 1921. 
 
CHAPTER V 
 
 ERRATIC, BIASED, AND FREAKISH INDEX NUMBERS 
 
 § 1. Joint Errors between Index Numbers 
 
 We have seen that there are two great reversal tests : 
 (1) that the product of forward and backward indexes 
 should equal unity, and (2) that the product of price and 
 quantity indexes should equal the value ratio. If the 
 former product is not equal to unity, the deviation from 
 unity is a joint error of the forward and backward in- 
 dexes ; and, hkewise, if the latter product is not equal to the 
 value ratio, the deviation from that figure is a joint error 
 of the price and quantity indexes. 
 
 Tables 7 and 8 — one for each test — show the joint 
 errors of each of the 28 formulae. Take, for instance, 
 the index numbers for prices and quantities as between 
 1913 and 1917. Under Test 1, the error lies jointly be- 
 tween the index /or 1917 relatively to 1913, and the index 
 for 1913 relatively to 1917, when both are reckoned by any 
 given formula ; while under Test 2 the error hes jointly 
 between the price index and the quantity index, when both 
 are reckoned by any given formula. 
 
 It will be seen that the joint errors vary from zero to 
 nearly 30 per cent (for Formula 11, 1918, Test 2) ; and that 
 Formulae 7, 9, 13, 15 show very large joint errors, while 
 those of 3, 5, 17, 19, 53, 59 are among the smallest. Not 
 a single one of the 28 formulae is entirely free from one 
 or the other of the joint errors, and only four (21, 31, 41, 
 51) are free from either error. These four conform to 
 Test 1. (Each of the weighted modes, 43, 45, 47, 49, 
 
 83 
 
84 
 
 THE MAKING OF INDEX NUMBERS 
 
 has too small a joint error under Test 1 to be measured 
 by the rough method used for calculating them.) In 
 other words, every one of these formulae is certainly 
 erratic, as revealed by the two tests. It may, of course, 
 
 TABLE 7. JOINT ERRORS OF THE FORWARD AND BACK- 
 WARD APPLICATIONS OF EACH FORMULA (THAT IS, 
 UNDER TEST 1) IN PER CENTS 
 
 (Price Indexes) 
 
 Example: The first figure, + 1.19, is found as follows: The index num- 
 ber forward X the index number backward (both by Formula 1) = 96.32 
 per cent X 105.06 per cent = 101.19 per cent as compared with the truth, 
 100 per cent — an error of +1.19 per cent. 
 
 Formula 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 No. 
 
 (Per Cents) 
 
 (Per Cents) 
 
 (Per Cents) 
 
 (Per Cents) 
 
 (Per Cents) 
 
 1 
 
 + 1.19 
 
 +2.56 
 
 +3.83 
 
 + 11.34 
 
 + 8.68 
 
 3 
 
 -0.39 
 
 -0.43 
 
 -0.24 
 
 + 0.63 
 
 + 0.25 
 
 5 
 
 +0.39 
 
 +0.43 
 
 +0.24 
 
 - 0.63 
 
 - 0.25 
 
 7 
 
 +0.90 
 
 +3.73 
 
 +6.08 
 
 +24.53 
 
 + 12.07 
 
 9 
 
 + 1.68 
 
 +4.59 
 
 +6.56 
 
 +22.78 
 
 + 11.03 
 
 11 
 
 -1.17 
 
 -2.50 
 
 -3.69 
 
 -10.19 
 
 - 7.99 
 
 13 
 
 -1.65 
 
 -4.39 
 
 -6.15 
 
 -18.55 
 
 - 9.93 
 
 15 
 
 -0.90 
 
 -3.60 
 
 -5.73 
 
 -19.70 
 
 -10.77 
 
 17 
 
 -0.39 
 
 -0.43 
 
 -0.24 
 
 + 0.63 
 
 + 0.25 
 
 19 
 
 +0.39 
 
 +0.43 
 
 +0.24 
 
 - 0.63 
 
 - 0.25 
 
 21 
 
 0. 
 
 0. 
 
 0. 
 
 0. 
 
 0. 
 
 23 
 
 -1.01 
 
 -2.42 
 
 -4.14 
 
 - 9.60 
 
 - 4.99 
 
 25 
 
 -0.26 
 
 -1.59 
 
 -2.80 
 
 -10.75 
 
 - 5.53 
 
 27 
 
 +0.26 
 
 + 1.62 
 
 +2.88 
 
 + 12.05 
 
 + 5.85 
 
 29 
 
 + 1.02 
 
 +2.48 
 
 +4.32 
 
 + 10.62 
 
 + 5.26 
 
 31 
 
 0. 
 
 0. 
 
 0. 
 
 0. 
 
 0. 
 
 33 
 
 -0.41 
 
 -0.58 
 
 -1.75 
 
 - 4.71 
 
 - 5.04 
 
 35 
 
 -0.13 
 
 -0.24 
 
 -1.29 
 
 - 2.23 
 
 -10.15 
 
 37 
 
 +0.13 
 
 +0.24 
 
 + 1.30 
 
 + 2.29 
 
 + 11.30 
 
 39 
 
 +0.41 
 
 +0.58 
 
 + 1.78 
 
 + 4.95 
 
 + 5.31 
 
 41 
 
 0. 
 
 0. 
 
 0. 
 
 0. 
 
 0. 
 
 43 
 
 0.± 
 
 0.± 
 
 o.± 
 
 0.=fc 
 
 0.=fc 
 
 45 
 
 0. =fc 
 
 0. =i= 
 
 0. =t 
 
 o.± 
 
 0. =b 
 
 47 
 
 0.± 
 
 0. ± 
 
 o.± 
 
 0. ± 
 
 0.=fc 
 
 49 
 
 0.=*= 
 
 o.± 
 
 o.± 
 
 0.=t 
 
 o.± 
 
 51 
 
 0. 
 
 0. 
 
 0. 
 
 0. 
 
 0. 
 
 53 
 
 -0.39 
 
 -0.43 
 
 -0.24 
 
 + 0.63 
 
 + 0.25 
 
 59 
 
 +0.39 
 
 +0.43 
 
 +0.24 
 
 - 0.63 
 
 - 0.25 
 
BIASED INDEX NUMBERS 
 
 85 
 
 TABLE 8. JOINT ERRORS OF THE PRICE AND QUANTITY 
 APPLICATIONS OF EACH FORMULA (THAT IS, UNDER 
 TEST 2) IN PER CENTS 
 
 (Forward Indexes) 
 
 Example: The first figure, —3.85, is found as follows : The index num- 
 ber for price X the index number for quantity (both by Formula 1) =96.32 
 per cent X 99.27 per cent = 95.617 per cent, as compared with the true 
 value ratio, 99.45 per cent — an error of —3.85 per cent of the true 99.45. 
 
 Formula 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 No. 
 
 (Per Cents) 
 
 (Per Cents) 
 
 (Per Cents) 
 
 (Per Cents) 
 
 (Per Cents) 
 
 1 
 
 -3.85 
 
 +2.19 
 
 + 12.73 
 
 + 15.08 
 
 + 5.40 
 
 3 
 
 -0.39 
 
 -0.43 
 
 - 0.24 
 
 + 0.63 
 
 + 0.25 
 
 6 
 
 +0.39 
 
 +0.43 
 
 + 0.24 
 
 - 0.63 
 
 - 0.25 
 
 7 
 
 + 1.55 
 
 +4.53 
 
 + 5.67 
 
 + 18.44 
 
 + 11.92 
 
 9 
 
 +2.26 
 
 +5.53 
 
 + 6.47 
 
 +16.62 
 
 + 10.58 
 
 11 
 
 -8.01 
 
 -5.66 
 
 + 3.27 
 
 - 8.27 
 
 -29.50 
 
 13 
 
 -2.51 
 
 -4.46 
 
 - 4.96 
 
 -12.58 
 
 -11.18 
 
 15 
 
 -1.67 
 
 -3.80 
 
 - 4.81 
 
 -14.02 
 
 -11.90 
 
 17 
 
 -0.39 
 
 -0.43 
 
 - 0.24 
 
 + 0.63 
 
 + 0.25 
 
 19 
 
 +0.39 
 
 +0.43 
 
 + 0.24 
 
 - 0.63 
 
 - 0.25 
 
 21 
 
 -5.84 
 
 -1.86 
 
 + 7.79 
 
 + 2.95 
 
 - 7.22 
 
 23 
 
 -1.40 
 
 -2.57 
 
 - 3.62 
 
 - 6.53 
 
 - 5.22 
 
 25 
 
 -0.61 
 
 -1.79 
 
 - 2.46 
 
 - 7.81 
 
 - 5.61 
 
 27 
 
 +0.60 
 
 + 1.87 
 
 + 2.51 
 
 + 8.74 
 
 + 5.91 
 
 29 
 
 + 1.35 
 
 +2.81 
 
 + 3.19 
 
 + 7.40 
 
 + 5.08 
 
 31 
 
 -0.66 
 
 -3.55 
 
 + 2.41 
 
 + 1.02 
 
 + 4.02 
 
 33 
 
 -0.85 
 
 -5.04 
 
 - 8.85 
 
 - 5.69 
 
 - 7.05 
 
 35 
 
 -0.49 
 
 -4.42 
 
 - 8.72 
 
 - 3.21 
 
 - 7.15 
 
 37 
 
 +0.04 
 
 -2.37 
 
 - 6.83 
 
 + 3.80 
 
 + 4.74 
 
 39 
 
 +0.23 
 
 -1.78 
 
 - 6.65 
 
 + 2.46 
 
 - 1.23 
 
 41 
 
 -5.77 
 
 -9.26 
 
 -13.60 
 
 -19.16 
 
 + 3.83 
 
 43 
 
 -1.94 
 
 -5.66 
 
 -18.18 
 
 -16.33 
 
 - 6.67 
 
 45 
 
 -1.94 
 
 -5.66 
 
 -18.18 
 
 -16.33 
 
 - 6.67 
 
 47 
 
 -1.94 
 
 -5.66 
 
 -18.18 
 
 -16.33 
 
 - 6.67 
 
 49 
 
 -1.94 
 
 -5.66 
 
 -18.18 
 
 -16.33 
 
 - 6.67 
 
 61 
 
 -1.28 
 
 -0.92 
 
 - 5.98 
 
 - 7.41 
 
 + 4.61 
 
 53 
 
 -0.39 
 
 -0.43 
 
 - 0.24 
 
 + 0.63 
 
 + 0.25 
 
 69 
 
 +0.39 
 
 +0.43 
 
 + 0.24 
 
 - 0.63 
 
 - 0.25 
 
 be erratic beyond these revelations, as a small joint error 
 may be the net effect of large but offsetting errors in the 
 two index numbers for which that joint error is re^?ealed. 
 
86 THE MAKING OF INDEX NUMBERS 
 
 We shall find reasons for believing this to be true of the 
 modes particularly. 
 
 § 2. Bias, under Test 1, Inherent in Arithmetic and 
 Harmonic Types of Formulae 
 
 But, in many cases, we can convict a formula not only 
 of being erratic when tested by Test 1, but also, under 
 that test, of being distinctly biased, i.e. subject to a fore- 
 seeable tendency to err in one particular direction. Under 
 Test 1, four formulae conform (21, 31, 41, 51) ; six (which 
 reduce to two when duplicates are excluded) are merely 
 erratic (3, 5, 17, 19, 53, 59) ; and 18 are biased. Of these 
 18, the following nine have an upward bias : 1, 7, 9, 27, 
 29, 37, 39, 47, 49, while the following nine have a down- 
 ward bias: 11, 13, 15, 23, 25, 33, 35, 43, 45. 
 
 All cases of provable bias are under Test 1. Let us 
 begin with Formula 1 . It can be proved that the prod- 
 uct of this formula, applied forward and backward, 
 instead of being unity, as required by Test 1, always nec- 
 essarily exceeds unity. 
 
 Numerically, that this is true in any given case, can 
 readily be seen by trial. Thus, suppose two commodities 
 of which the forward price ratios are 100 and 200 per 
 cent, and the backward, therefore, 100 per cent and 50 
 per cent. We are to show that 
 
 (100 + 200) (100 + 50) 
 2^2 
 
 exceeds unity. This is 150 per cent X 75 per cent, or 113 
 per cent, which exceeds unity by 13 per cent. 
 
 Algebraically, the proof that the product of the arith- 
 metic forward by the arithmetic backward always and 
 necessarily exceeds unity is given in the Appendix.^ 
 * See Appendix I (Note to Chapter V, § 2). 
 
BIASED INDEX NUMBERS 87 
 
 Thus, Formula 1, the simple arithmetic average, has nec- 
 essarily a positive joint error. While we cannot go further 
 and say, in any given case, how much of this error lies 
 in its forward form and how much in its backward form, 
 in the absence of any reason to accuse the one more than 
 the other, we are justified in accusing both equally. The 
 proportionate share of the total necessary error thus pre- 
 sumed to belong to each is called its ''bias." In Table 7, 
 the bias of the index number of prices, by Formula 1, for 
 the 36 commodities, is, for 1917, one-half of 11.34 per 
 cent, or about 5f per cent.^ That is, the arithmetic 
 average exhibits an inherent tendency to exaggeration, 
 a ''bias," such that, in the instance cited, it yields a re- 
 sult probably too high by about 5^ per cent. 
 
 This inherent tendency in the arithmetic type always 
 exists irrespective of the method of weighting used. 
 So long as the same weights are used forward and back- 
 ward, the product of the arithmetic forward and backward 
 will exceed unity. The reasoning in the Appendix, above 
 cited,^ applies to the arithmetic index number as such, 
 whether simple or weighted. By similar reasoning, it 
 may be shown that the harmonic index number, with or 
 without any given weighting, has an inherent bias down- 
 ward. That is, its forward and backward forms, multi- 
 plied together, give a result always and necessarily less 
 than unity. The joint error is the difference between 
 unity and the product of the harmonic forward by the 
 harmonic backward. 
 
 Graphically, the intimate relationship between the arith- 
 metic and harmonic bias (which are, at bottom, the same) 
 
 ^ The mathematical reader will prefer to reckon the equal s hares m ore 
 precisely, i.e. in equal proportions instead of equal parts (i.e. \/l.ll34- 1). 
 But the result is, of course, approximately the same. 
 
 2 Appendix I (Note to Chapter V, § 2). 
 
88 
 
 THE MAKING OF INDEX NUMBERS 
 
 is clearly seen in Charts 14P and 14Q made from Chart 11. 
 By reversing the direction of the dotted line representing 
 
 the simple arith- 
 
 TypeBias of Formula Nal 
 (Prices) 
 
 77 
 
 simple arithmetic 
 forward 
 
 78 
 
 simple arithmetic 
 backward ^^—-' 
 
 simple harmonic 
 forward 
 
 \5% 
 
 77 
 
 78 
 
 metic backward, 
 we represent its 
 reciprocal. But 
 this reciprocal 
 turns out to be 
 the simple har- 
 monic forward. 
 Thus the chart 
 shows that the 
 failure, previously 
 pointed out, of the 
 arithmetics for- 
 ward and back- 
 ward to be each 
 the prolongation of the other is precisely the same thing 
 as the failure of the arithmetic forward and the harmonic 
 forward to coincide with each other. The use of the har- 
 monic enables us to get rid of all backward lines and merely 
 contrast forward lii;ies. Consequently, the joint error of 
 the arithmetic {i.e. the deviation from unity of the product 
 of the forward and backward arithmetics) , previously pic- 
 tured as the bend between two lines which ought to be pro- 
 longations of each other, is now pictured as the angle be- 
 tween two forward lines which ought to coincide. Half of 
 this divergence represents the upward bias of the arith- 
 metic and half the downward bias of the harmonic. 
 
 Chart 14P. The simple harmonic forward the 
 same reversed, as the simple arithmetic back- 
 ward. The bias is half the gap at the right. 
 
 § 3. Joint Error Expressible by Product or Quotient 
 
 Thus, the joint error, either of the arithmetic or of the 
 harmonic, may be written in two ways. The old way 
 was as the difference between unity and the 'product of the 
 
BIASED INDEX NUMBERS 89 
 
 arithmetic forward by the arithmetic backward. The new 
 way is as the difference between unity and the quotient 
 of the arithmetic _ _. t ai i 
 
 forward by the T/pe dios oF rormula Na I 
 
 harmonic forward (Quont'tties) 
 
 which, as stated, 
 
 is the reciprocal ^ _ 
 
 of the arithmetic ^^simple arithmetic 
 
 backward. These -^^J)achvard simple aritlimefic 
 
 two alternative ^^-^^ ^ Forward 
 
 ways of exhibit- ' ^-.^^mple ttarmorjic 
 
 ing the joint er- \5% ^-^^^^rward 
 
 ror are important // '^8 
 
 enough to be for- Chart UQ. Analogous to Chart 14P. 
 
 mulated mathe- 
 matically. That is, the old way is 
 
 arithmetic forward X arithmetic backward exceeds unity. 
 But we may substitute for "arithmetic backward" its 
 equal, "the reciprocal of the harmonic forward," giving 
 
 arithmetic forward X -i ^—7 3 exceeds unity, 
 
 harmonic forward 
 
 or, more briefly, 
 
 arithmetic forward ■, ., 
 
 : — J — exceeds unity. 
 
 harmonic forward 
 
 Similarly the joint error of the harmonic may be written 
 either as 
 
 harmonic forward X harmonic backward is less than unity 
 
 or, as follows, harmonic forward -^ ^^^^ ^^^^ ^^ .^^^ 
 
 arithmetic forward 
 The new, or quotient, form is in each case the more con- 
 venient and obviates the need of using any backward 
 index numbers. 
 
90 THE MAKING OF INDEX NUMBERS 
 
 But, while the quotient form is the easier to handle and 
 much the more convenient to use in computations and 
 charts, the product form affords the more convincing 
 proof of bias. If only the quotient form were mentioned, 
 it might be hastily inferred that our only reason for ascrib- 
 ing an upward bias to the arithmetic and a downward 
 bias to the harmonic is that the former exceeds the latter. 
 But the argument goes much deeper. The argument is 
 not merely that one of two index numbers exceeds an- 
 other. The point is that the harmonic essentially repre- 
 • sents an arithmetic backward. We ascribe an upward 
 bias to the arithmetic solely on the showing of the arith- 
 metic itself — because the arithmetic forward multiplied by 
 the arithmetic backward is always greater than unity. 
 In this product form the reasoning does not require the 
 introduction of the harmonic, or any other type of aver- 
 age than the arithmetic. Even if we had never heard of 
 any other average than the arithmetic, it would stand 
 convicted on its own testimony. The same argument, 
 of course, apphes to the harmonic, without invoking the 
 arithmetic average. In short, the harmonic is, as it were, 
 a concealed arithmetic, and so either may be made to dis- 
 appear and give place to the other. 
 
 Graphic Resume of Type Bias 
 
 Graphically, Charts 15P and 15Q show three principal 
 types of index numbers compared. There are five groups, 
 each from a separate origin : one group (at the top) 
 representing the simple index numbers and four groups 
 representing the index numbers having the four weight- 
 ings respectively. Each group contains all three types, 
 so that there are 15 formulae in all. We observe that, in 
 each group, the geometric always lies about midway be- 
 tween the arithmetic and the harmonic, and that this 
 
BIASED INDEX NUMBERS 91 
 
 is true of the index numbers in the chain systems (shown 
 by the balls) as truly as in the fixed base systems (shown 
 by the curves themselves) . The upward bias of the arith- 
 metic and the downward bias of the harmonic manifest 
 themselves in every case. In each group the three curves 
 have the same weighting: 1, 11, 21, — simple; 3, 13, 
 23, —weighting / ; 5, 15, 25, —weighting II ; 7, 17, 27, — 
 weighting III] 9, 19, 29, — weighting IV. The three 
 differ only in type, and, in each case, the arithmetic type 
 is the highest and the harmonic, the lowest. The wide 
 gap between the arithmetic and harmonic in each case 
 represents their joint error (by the quotient method), and 
 so measures the upward bias of the arithmetic and down- 
 ward bias of the harmonic. 
 
 § 5. Bias in the Weighting 
 
 The kind of bias just described inheres in the arith- 
 metic and harmonic types of average. But there is another 
 kind of bias inhering in the system of weighting used and 
 affecting all the weighted formulae thus far described, 
 except the aggregatives. That is, weight bias applies to 
 any type of index number susceptible of value weighting. 
 The weights of the aggregative are, of course, not values 
 but mere quantities, as has been explained. 
 
 To illustrate weight bias, take, for example, the geo- 
 metric index number. We know that the geometric type, 
 as such, has no bias, and it will be remembered that the 
 simple geometric obeys Test 1 (being merely erratic under 
 Test 2). But when we weight the geometric under, for 
 instance, system IV, we, at once, impart an upward bias. 
 Empirically, this is proved by the fact that, if we take 
 this geometric IV, both forward and backward, the prod- 
 uct is invariably found to exceed unity. 
 
 Again, as we have seen, the arithmetic type, as such, 
 
92 
 
 THE MAKING OF INDEX NUMBERS 
 
 Tliree Types of Index Numbers 
 of Prices 
 
 Arithmetic 
 
 Geometric 
 
 Harmonic 
 
 7J 
 
 74^ 
 
 75 
 
 7S 
 
 77 
 
 76 
 
 Chart 15 P. The geometric always lies about midway between the 
 arithmetic and harmonic, whether fixed base or chain. The five groups are 
 separated to save confusion, really forming five distinct diagrams. The 
 gap of each arithmetic and harmonic from the middle is its type bias. 
 
 does have a bias. But when we weight the arithmetic 
 under system /7 we impart an additional bias ; its bias 
 
BIASED INDEX NUMBERS 
 
 Three Types of index Numbers 
 of Quantities 
 
 Arithmpfic 
 
 Geometric 
 
 Harmonic 
 
 93 
 
 75 7-^ 75 VS 77 
 
 . Chart 15Q. Analogous to Chart 15P. 
 
 78 
 
 is approximately doubled thereby. Empirically, this is 
 proved by the fact that, if we take this arithmetic IV j 
 both forward and backward, the product is invariably 
 found to exceed unity by about twice the bias of the 
 
94 THE MAKING OF INDEX NUMBERS 
 
 simple arithmetic. In this way, by actual trial, we can 
 convince ourselves of the truth of the proposition that the 
 weighting systems / or // impart a downward bias to 
 any index number, while /// and IV impart an upward 
 bias. 
 
 § 6. Outline ^ of Argument as to Geometric, 
 Median, and Mode 
 
 Besides such empirical evidence, good logical reasons 
 for this weight bias exist ; but they are not so simple to 
 set forth as were the reasons for the arithmetic and har- 
 monic type bias, chiefly because the weight bias, with 
 which we now have to deal, unhke type bias, with which 
 we dealt in previous sections, is partly a matter of mere 
 probability. In studying weight bias it will be more con- 
 venient to take up the quotient method first. We shall 
 see: 
 
 (1) For any given type of formula having value 
 weights, the index numbers with weightings I or II are, 
 in general, smaller than the index numbers with weight- 
 ings III or IV ; 
 
 (2) These inequalities are partly necessary, partly 
 probable. That the index number weighted / is less than 
 III and that II is less than IV are mathematically neces- 
 sary. But that / is less than IV and // than /// can- 
 not be proved to be absolutely necessary but only to be 
 highly probable ; 
 
 (3) Since, then, IV exceeds 7, and III exceeds II, 
 the quotient of ZF divided by / exceeds unity, as does III 
 divided by //. These excesses may provisionally be 
 called joint errors. Such a joint error allotted in equal 
 proportions to index numbers weighted I and IV, or to 
 
 * For details of the argument in this and the following section see Ap- 
 pendix I (Note to Chapter V, § 6), which may best be read after reading 
 the text. 
 
BIASED INDEX NUMBERS 95 
 
 index numbers weighted II and III, gives each index 
 number its bias ; 
 
 (4) Weight bias is most simply seen in the case of the 
 weighted geometries, medians, and modes because these 
 have weight bias only, uncompUcated by type bias. In 
 these cases the quotient form of weight bias is easily de- 
 rived from the product form, and vice versa. Let us take, 
 for instance, the geometries I and IV, or Formulae 23 
 and 29, and express the weight bias of 29. The quotient 
 form of this bias is half the excess above unity of the 
 quotient ff (both indexes being forward or both back- 
 ward). This excess will be found to be identical with 
 half the excess above unity of the product of 29 forward 
 X 29 backward. Likewise, if we take 25 and 27, the 
 weight bias of 27 is half the excess above unity of |^, which 
 is the same as half the excess above unity of 27 forward 
 X 27 backward. 
 
 As previously stated, the product form is the preferable 
 one to use in our logic because it employs only one for- 
 mula. Thus it makes 29 convict itself of error by con- 
 fronting it, as it were, by its own reversed image in the 
 looking glass. 
 
 I ~ The foregoing relate to the four systems of weighting 
 as apphed to the geometries, medians, or modes. The 
 weight biases of the geometric compare closely in magni- 
 tude with those found for the type bias of the arithmetic 
 and harmonic. 
 
 § 7. Supplementary Argument as to Arithmetic 
 and Harmonic 
 
 With the weighted arithmetic and weighted harmonic, the case is more 
 complex. Take arithmetic IV, Formula 9, or Palgrave's formula. The 
 (forward) arithmetic IV, divided by (forward) arithmetic I, Formula 3, 
 is here not identical with the arithmetic IV forward multiplied by 
 arithmetic IV backward, because type bias complicates the situation. 
 
96 THE MAKING OF INDEX NUMBERS 
 
 The product mentioned (arithmetic IV forward by arithmetic IV back- 
 ward) is identical with the quotient of arithmetic IV forward divided by 
 the harmonic I, Formula 13 forward. That is, A IV for. X A IV back. 
 
 A IV A IV 
 
 (or 9 for. X 9 back.) is not identical with (or f) but with - 
 
 A. I xl I 
 
 (or ■^). But we know, from our study of type bias, that the harmonic 
 / lies below arithmetic / and, in fact, that their joint error is the excess 
 above unity of arithmetic / divided by harmonic /. Hence we find, from 
 our present study of weights, that arithmetic IV exceeds arithmetic /; 
 and, from our former study of types, that arithmetic I exceeds harmonic 
 /. It follows that arithmetic IV doubly exceeds harmonic /. Conse- 
 quently, it is doubly true that arithmetic IV divided by harmonic 7 ex- 
 ceeds unity. But this is the same thing as saying that it is doubly true 
 that arithmetic IV forward multiplied by arithmetic IV backward ex- 
 ceeds unity. 
 
 Thus, we convict arithmetic IV hy itself, although as a step in our 
 reasoning we included the type joint error of arithmetic I and harmonic I. 
 That is, of the decreasing series : arithmetic IV, arithmetic I, harmonic 7, 
 the first exceeds the third by a joint error, not only in the quotient sense 
 but also in the product sense; likewise, the second exceeds the third by 
 a joint error, not only in the quotient sense but also in the product sense ; 
 but the first exceeds the second, by a joint error, only in the quotient 
 sense. That is, the total excess of arithmetic IV over harmonic 7 is 
 type and weight bias, in the product sense ; part of this total excess, 
 namely, that of arithmetic 7 over harmonic 7, is type bias in the product 
 sense; hence, indirectly, the remaining excess, i.e. that of arithmetic 7 F 
 over arithmetic 7, is weight bias in the product sense. Thus, the arith- 
 metic IV has a double dose of upward bias, part of its bias being due to 
 its being of the arithmetic type and part being due to its having weighting 
 7 V. The same is true of arithmetic 777 ; while the harmonics 7 and 77 
 have a double dose of downward bias. 
 
 § 8. The Argument, Numerically, Algebraically, 
 and Graphically 
 
 I have outlined these steps of reasoning, partly to 
 help the reader who chooses to follow the argument in 
 the Appendix ^ in detail, and partly to make it unneces- 
 sary to do so for readers who do not so choose. Here, for 
 brevity, I will merely indicate the results by actual figures. 
 
 Numerically, then, we can see how the matter works 
 out by repeating here the weights of selected commodi- 
 ties under the four systems of weighting. 
 
 * See Appendix I (Note to Chapter V, § 6). 
 
BIASED INDEX NUMBERS 
 
 97 
 
 TABLE 9. THE FOUR SYSTEMS OF WEIGHTING THE 
 
 PRICE RELATIVES FOR 1917, ^, ^, etc. 
 
 Po Po 
 
 COMMODITT 
 
 Weighting 
 
 System / 
 
 Poqa 
 
 Weigihting 
 
 System // 
 
 pm 
 
 Weighting 
 System /// 
 
 PiQO 
 
 Weighting 
 System IV 
 
 PiQi 
 
 /////, also 
 IV/II, also 
 
 Pi 
 
 Pa 
 
 
 (in milliona of dollars) 
 
 (in per cents) 
 
 Bituminous coal . . 
 Coke 
 
 606 
 140 
 
 462 
 422 
 
 701 
 172 
 
 577 
 596 
 
 1708 
 494 
 
 1203 
 715 
 
 1976 
 
 604 
 
 1502 
 
 1011 
 
 282 
 
 352 
 
 Pig iron 
 
 260 
 
 Oats 
 
 170 
 
 
 
 Anthracite coal . . . 
 
 Petroleum 
 
 Coffee 
 
 35 
 1282 
 
 98 
 1971 
 
 40 
 
 1835 
 
 147 
 
 1916 
 
 39 
 1292 
 
 80 
 2290 
 
 44 
 
 1848 
 
 123 
 
 2227 
 
 111 
 101 
 
 83 
 
 Lumber 
 
 116 
 
 
 
 Thus, bituminous coal rose in price from 100 to 282, and 
 the weights under systems I and III are 606 and 1708, 
 which are also exactly in the ratio of 100 to 282 (or, again, 
 the weights under II and IV are 701 and 1976 — also ex- 
 actly as 100 to 282). Thus, the last column not only 
 gives, in each case, the price relative, or price rise, but also 
 the weight rise {i.e. the ratio of III to I and of IV to II). 
 Algebraically, the reason for this last-named result is 
 clear. As the headings of the columns indicate, the 
 weights under III and / are ^4^0, etc., and po?o, etc., and 
 
 the ratio of these weights, -2i2P, reduces, by cancellation, 
 
 Pogo 
 
 to — , which is identical with the price relative. Thus 
 
 the greater the price relative, the more heavily is it 
 weighted under system III (as compared with system /) 
 and in exact proportion. Under system III (as compared 
 with system 7) the rule is "to him that hath shall be 
 
98 THE MAKING OF INDEX NUMBERS 
 
 given" — that is, the high price relatives draw relatively- 
 high weights and the low, low. Consequently, the high 
 
 Four Methods of Weighting Compared 
 
 By baje year ya/ues (p,(^. etc) 
 
 mixed - Ip/f. " ) 
 
 " ip.q, ") 
 
 •. g/yen year - (p.^ ••) 
 
 (Prices) 
 
 'IJ 
 
 H 
 
 IS 
 
 'iff 
 
 17 
 
 *» 
 
 Chart 16P. When the 'price elements in the weights are changed, the 
 index number is greatly changed, and in a foreseeable direction. When 
 the quantity elements are changed, the index number is scarcely altered, 
 and in no foreseeable direction. The weight bias is half of a gap. 
 (Changing the price elementd is as between curves 3 and 7, 13 and 17, 23 
 and 27, or between 5 and 9, 15 and 19, 25 and 29. Changing the quantity 
 elements is as between 3 and 5, 13 and 15, 23 and 25, or between 7 and 9, 17 
 and 19, 27 and 29. There are three distinct diagrams. Hereafter, the reader 
 will be expected to distinguish for himself between separate diagrams on 
 the same chart, by the fact that they have separate origins.) 
 
 price relatives have more influence on the resulting index 
 number (which is an average of all the price relatives) 
 than under system J, and, therefore, make the resulting 
 
BIASED INDEX NUMBERS 
 
 99 
 
 index number larger than that resulting under system /. 
 Likewise, IV has exactly the same contrast with II. 
 
 It is clear, then, that under systems /// and IV the 
 high price relatives are heavily weighted and so dominate 
 their average (the index number), i.e. raise it; or, if we 
 prefer to say so, under systems / and // the low price 
 
 Four Methods or Weighting Compared 
 (Quantifies) 
 
 73 '14 'm 'le 77 18 
 
 Chart 16Q. Analogous to Chart 16P (interchanging " price " and 
 "quantity")- 
 
 relatives are heavily weighted and so dominate their 
 average, i.e. lower it. The cards in the weighting are 
 stacked so that weighting / or II pulls the index number 
 down, or weighting /// or IV pushes it up, or both. The 
 former weighting has a bearish, as the latter has a bullish, 
 influence, or both, and in the absence of any other data 
 and with no reason to believe the error all one way, we 
 can best describe the tendency as a ''bias" in the weight- 
 
100 THE MAKING OF INDEX NUMBERS 
 
 ing ; an upward bias for III and IV and a downward for 
 I and 11. 
 
 Thus, in an index number of prices the 'price element 
 in the weight has far greater influence on the result than 
 the quantity element. We need not trouble much as to 
 the quantity element, but we must take great pains to 
 see that the price element is what it should be. Instead 
 
 Four Methods of Weighting Compared 
 (Prices) 
 
 Median 
 
 factor 
 
 Anmeses ofj^edm 
 
 '/3 M 75 'le *I7 78 
 
 Chart 17P. The effects on the index number of changing the weight- 
 ing are, in the case of the median, similar to, but smaller and more erratic 
 than, the effects in the cases of the arithmetic, harmonic, and geometric. 
 In some years the agreement is closer than is the case in the arithmetic, 
 harmonic, and geometric, but when there is a difference it is apt to be 
 much more pronounced. 
 
 of having to "mind our p's and ^f's" we need only mind 
 our "p's"! But for the quantity indexes the opposite 
 holds. 
 
 Graphically, weight bias manifests itself in Charts 16P 
 and 16Q in each of the three groups of curves. In each 
 group the four curves are of the same type and differ 
 
BIASED INDEX NUMBERS 
 
 101 
 
 only in weighting. It will be noted that the curves, end- 
 ing in 3 or 5 (weightings I and //), always practically 
 coincide, as do the curves ending in 7 or 9 (weightings 
 III and IV), although there is, in all three cases, a wide 
 gap between the former pair on the one hand and the 
 latter pair on the other. The mystery of this persist- 
 ently recurring gap representing the joint error (by the 
 quotient method) is to be solved by the existence of a 
 
 Four Methods of Weighting Compared 
 {Quantities) 
 
 VJ y-f 75 7S 77 
 
 Chart 17Q. Analogous to Chart VIP. 
 
 73 
 
 distinct upward bias of III and IV and downward bias of 
 / and II. 
 
 Charts 17P and 17Q (upper diagrams) show the medi- 
 ans, which exhibit the same sort of biases, though less than 
 in Chart 16, and resemble the two medians of Chart 10. 
 But we notice a curious and important difference between 
 these charts and Charts 16P and 16Q for the arithmetic, 
 harmonic, and geometric. In all these preceding cases 
 the curves ending in 7 and 9, for example, nearly, but 
 not quite, coincided with each other, according as slight 
 changes in the incidence of weighting produced corre- 
 
102 THE MAKING OF INDEX NUMBERS 
 
 spondingly slight effects. But, in the case of the median, 
 the effects of changed weighting go by fits and starts. In 
 most instances curves 37 and 39, for instance, stick even 
 closer together than 7 and 9, or 17 and 19, or 27 and 29. 
 But when the cleavage between them is broken at all they 
 are apt to be torn wide apart. This characteristic of the 
 median, its insensitiveness, as contrasted with the arith- 
 metic, harmonic, and geometric, has already been referred 
 to. The four modes (not charted) are indistinguishable. 
 
 § 9. Double Bias Illustrated Numerically and 
 Graphically 
 
 Double BiasiWeightBias and Numerically, our 
 
 TypeBias)cfromul.Na9 ^^0"" 
 
 (Prices) haps that of Pal- 
 
 grave's index num- 
 *.. . ber (Formula 9 in 
 5(Ati) our series), the 
 13 (Hi) arithmetic weighted 
 
 tF 
 
 \5% 
 
 '17 78 
 
 IV, weighted by 
 
 given year values, 
 
 PiQi, etc. This 
 
 ,nT, CI. • ,- X. J- index has a very 
 
 Chakt 18P. Showing, by the divergence 
 
 between 9 forward and 9 backward, their large jomt error 
 
 joint error, half of which is the upward bias under Test 1 which 
 of Formula 9. Tliis divergence, or joint error, . 
 
 is also shown by the divergence between curves ^6 ^^^ ^^ analyze. 
 
 9 and 13. In this form it is easily subdi- In Table 7 We find 
 
 vided into three parts of which the middle is . r • • , <• 
 
 negligible. Of the rest, half is upward bias ^'^f J^^^^ ®^^^^ ^^^ 
 
 of 9, comprising two parts, weight bias and this Palgrave for- 
 type bias (the weight bias being half of the , annliPrl in 
 
 divergence between 9 and 5, and the type bias "lU^'** '^^ appiieu io 
 
 being half of the divergence between 3 and 1917 relatively tO 
 
 13). The other two quarters of the whole IQIQ f^ Ua 22 78 
 constitute the similar, but downward, double ' 
 
 bias of 13. p e r c e n t. F o r 
 
BIASED INDEX NUMBERS 103 
 
 1918 relatively to 1917, it is 6.99 per cent. That is, Pal- 
 grave's index number taken forward multiplied by Pal- 
 
 rrtlt^baT OoutleBlasm^^^^^^^ 
 
 ward is 1 + .0699. T/pedias) of Formula No. 9 
 
 About half of this (Quantities) 
 
 error of 6.99 per y^ 
 
 cent, or 3.5 per g^^j^) 
 
 cent, may be as- ^^^^%m)^^^^^ 
 
 signed to each of ^ j 
 
 the two forms (for- ^^ *" 
 
 ward and back- ^ . ^ . , . ^, . ,„« 
 
 ,v „T , ,, Chart 18Q. Analogous to Chart 18P. 
 
 ward). We shall 
 
 find that this 3.5 per cent error is in turn made up of 
 
 three parts. 
 
 Graphically, Charts 18P and 18Q show the whole joint 
 error and the three parts into which it may be divided. 
 The hues are numbered with the identification numbers 
 and also lettered ("A" for arithmetic, "H" for har- 
 monic, with the Roman numerals attached to indicate 
 the system of weighting). Beginning with Palgrave's 
 formula (9, or arithmetic IV) taken forward, let us also 
 take it backward, as shown by the dotted line, likewise 
 labeled "9" or ''A 7F." These two applications of Pal- 
 grave's formula, forward and backward, multiplied to- 
 gether do not give unity. In other words, the forward 
 and backward lines are not prolongations of each other. 
 The prolongation forward of the backward line gives 
 us 13 or H I,^ and the divergence between 9 and 13 (i.e. the 
 vertical distance between the right-hand ends of Hues 9 
 and 13 in the chart) represents the percentage joint error 
 of 9 forward and backward, that is, .0699. 
 
 This joint error consists of three parts. Practically 
 1 For proof see Appendix I (Note to Chapter V, § 9). 
 
104 
 
 THE MAKING OF INDEX NUMBERS 
 
 the upper half, that is, the divergence between 9 and 5, 
 is due to changing the weighting of the arithmetic from 
 IV (as used in 9) to // (as used in 5), i.e. from piQi, 
 etc., to poQi, etc., i.e. by changing the price element in 
 the weighting. The next part is very small and due to 
 changing further the weighting system from II (as used 
 in 5) to 7 (as used in 3), i.e. from poQi, etc., to po3o. 
 etc., i.e. by changing the quantity element in the weights. 
 Finally, the third part, practically the lower half, is due 
 to changing from the arithmetic type (A I or 3) to the 
 harmonic type (H I or 13), while retaining the same 
 weighting system (7). 
 
 Recapitulating, we note three shifts : (1) a shift of the 
 price element in the weights, (2) a shift of the quantity 
 element in the weights, and (3) a shift of the type of aver- 
 age. The middle shift is always almost negligible and 
 
 may be either up 
 
 Weight Bias of Formula Na29 
 (Prices) 
 
 77 
 
 29(em) 
 25 (Gu) 
 25(GJ) 
 
 \5% 
 
 Chart 19 P. 
 
 ya 
 
 or down. Both 
 the other shifts 
 are necessarily 
 down in the or- 
 der we have read 
 them. The first 
 shift represents 
 a joint error of 
 arithmetic IV 
 and 77 (9 and 5), 
 being the 
 
 Showing by the divergence be- lialf 
 
 tween 29 forward and 29 backward their joint /J u* f 
 
 error, half of which is the upward bias of For- UpWard ^ Dias 01 
 
 mula 29. This divergence or joint error is also weighting IV 
 
 shown by the divergence between curves 29 and i y.r,'\e +Via 
 
 23. In this form it is easily subdivided into two ^"^ ^^^^' V 
 
 parts, of which the lower is negligible, and half downward biaS 
 
 of the upper is the upward bias of 29 with r jj T'Vip last 
 weighting IV, and the other half the downward 
 
 bias of 23 with weighting /. shift represents a 
 
BIASED INDEX NUMBERS 105 
 
 Weight Bias or Formula Na29 
 (Quantities) 
 
 joint error of the arithmetic and harmonic types, half be- 
 ing the upward bias of the arithmetic, and half, the down- 
 ward bias of the 
 harmonic. 
 
 By a different 
 choice of Unes the 77 '/8 
 
 analysis may be ^^..^-i^^i'f^i^ar^rf 
 
 presented some- ^.^-^-"^''''''^ y ^ 
 
 what differently, 29(gi^*-^'' 1 '*' 
 
 but the essential ^ '^ 
 
 fact will always Chabt IQQ. Analogous to Chart 19P. 
 
 appear that 9, or A lY , has a double dose of up- 
 ward bias, first, because it is of the arithmetic iy'pe and, 
 secondly, because its system of weighting is IV, while 
 13, or H /, has a double dose of downward bias, being 
 both harmonic and weighted by system /. 
 
 The example chosen illustrates both kinds of bias, 
 weight bias and type bias. Only the arithmetic and har- 
 monic formulae have type bias, consequently the corre- 
 sponding diagrams for the geometric, median, and mode 
 are simpler, as there is no type bias. Charts 19P and 
 19Q show the contrast between weightings I (23) and IV 
 (29) for the geometric. 
 
 We see, then, that the joint errors shown in Tables 
 7 and 8 are not altogether unaccountable or, as we may 
 say, accidental ; but are, in two instances, due to clearly 
 discernible causes. First, the arithmetic and harmonic 
 index numbers have a definite bias, upward and downward 
 respectively, and, secondly, the methods of weighting III, 
 IV, on the one hand, and J, II, on the other, have like- 
 wise an upward and downward bias respectively.^ 
 
 * The reader should not forget that all these results are general ; they 
 hold good whether prices are rising or falling ; they are not due to any se- 
 lection of commodities (other than that self-selection by which, e.g. xmder 
 given year weighting, high price relatives draw high weights). 
 
106 THE MAKING OF INDEX NUMBERS 
 
 § 10. The Five-tined Fork 
 
 Charts 20P and 20Q (upper) give a bird's-eye view of 
 how the four methods of weighting affect the three princi- 
 pal types of index numbers, arithmetic, harmonic, and geo- 
 
 77?^ Five 'Tine Fork 
 
 of 18 Curves 
 
 (Prices) 
 
 106. 
 
 no. 
 
 124. 
 
 j2e 
 
 
 13 
 
 '14 
 
 '15 
 
 te 
 
 77 
 
 75 
 
 Chakt 20P. The five-tined fork given in Chart 9P, with additional 
 curves, and their factor antitheses (lower dotted diagram), which arrange 
 themselves in the inverse order of the originals. The four gaps are biases. 
 
 metric, exhibiting both single and double bias. We can 
 see substantially the same five-tined fork as in Charts 9P 
 and 9Q, where only weights I and IV were used. But in 
 Charts 20P and 20Q No. 5 is added to No. 3, and almost 
 coincides with it, 7 almost coincides with 9, 15 with 13, 
 
BIASED INDEX NUMBERS 
 
 107 
 
 17 with 19, 25 with 23, 27 with 29 ; also 3 and 5 coincide 
 absolutely with 17 and 19 respectively.^ 
 
 The middle tine is the bottom of the arithmetic index 
 numbers (weighting /, or curve 3, and II, or curve 5), 
 and, at the same time, it is the top of the harmonic 
 
 77?^ Five 'Tine Forlc 
 of 18 Curves 
 (Quantities) 
 
 •— — ■•■«BS#' 
 
 73 'M 15 '/e '17 Vd 
 
 Chakt 20Q. Analogous to Chart 20P. The spacing of the upper dia- 
 gram is equal and opposite to that of the lower part of 20P; that of the 
 lower is equal and opposite to that of the upper part of 20P. 
 
 (weighting III, or 17, and 7 y, or 19), while the other two 
 arithmetics (weighting III, or 7, and IV, or 9) are at 
 the extreme top, and the other two harmonics (weighting 
 /, or 13, and II, or 15) are at the extreme bottom. The 
 extreme upper and lower tines represent doubly biased 
 index numbers. The geometries, as in Charts 9P and 
 9Q, having single bias, he astride of the central tine. 
 
 ^ The reader, at this point, may disregard the curves numbered 103 and 
 upward, and also ail the even-numbered curves. These will be referred 
 to later. 
 
108 THE MAKING OF INDEX NUMBERS 
 
 That is, the geometries with weightings III and IV (27 and 
 29) lie substantially midway within the arithmetic two 
 tined fork, while those with weightings / and II (23 and 
 25), likewise midway within the harmonic two-tined fork. 
 
 § 11. Bias Depends on Dispersion 
 
 All the various formulae for any year would, of course, 
 agree in their results if all the price relatives for that year 
 happened to agree. The more nearly the price relatives 
 coincide, the more nearly the averages will coincide, and 
 the more the price relatives scatter or disperse, the more 
 the formulse can be expected to disagree. It is interest- 
 ing, therefore, to trace the effect which the dispersion 
 of the original data has on the disagreement between 
 index formulae, and, especially, on the disagreement be- 
 tween the biased formulae. 
 
 The relation between bias and dispersion is not a re- 
 lation of simple proportion. Thus, in the period 1914- 
 1917, the bias increased quite out of proportion to the 
 dispersion. Nevertheless a definite formula can be given 
 connecting any bias with the dispersion of the price rela- 
 tives.i When the dispersion is small, the bias is very 
 small indeed. This explains why the bias of the arith- 
 metic type has not been clearly discerned by users of index 
 numbers. As shown in the table given in the Appendix, 
 the average dispersion of the price relatives above and be- 
 low their mean must reach about 20 per cent to make the 
 bias as much as 1.67 per cent. But if the prices disperse 
 30 per cent, or half as much again, the bias doubles. And 
 when the average dispersion is 50 per cent the bias reaches 
 8.34 per cent. When the dispersion reaches 100 per 
 cent, i.e. when the high price relatives are, on the average, 
 
 ^ See Appendix I (Note to Chapter V, § 11), where methods of measur- 
 ing dispersion are given, with formulae and tables. 
 
BIASED INDEX NUMBERS 
 
 109 
 
 double their mean and the low price relatives are, on the 
 average, half of their mean, the bias reaches 25 per cent. 
 (Any "mean " will do.) Thus, if we know the dispersion, 
 we can tell how biased an arithmetic index number may- 
 be in any given case and approximately correct it. 
 
 § 12. Our 36 Commodities Disperse Unusually Widely 
 
 It is in time of war, crises, or other disturbance that 
 the dispersion of prices is hkely to be great. Consequently, 
 the arithmetic index number is the most untrustworthy 
 for such periods, e.g. through 1861-1875 and 1914-1922. 
 It is chiefly from the last-named period that our data for 
 the prices and quantities of the 36 commodities were 
 taken. They disperse very widely, therefore, as com- 
 pared with the dispersion we find in any peace time 
 period of the same length. Table 10 shows the average 
 dispersion of our 36 price relatives and of 36 of Sauer- 
 beck's price relatives (the 36 commodities most nearly 
 comparable to our 36) : 
 
 TABLE 10. DISPERSION 1 OF 36 PRICE RELATIVES, 
 (1) BEFORE THE WORLD WAR, AND (2) DURING IT 
 
 (In per cents) 
 
 Yeak 
 
 1846 
 1856 
 1866 
 1876 
 1886 
 1896 
 1906 
 1913 
 
 Satjbbbeck's 
 
 base 
 20 
 44 
 29 
 25 
 28 
 35 
 42 
 
 Year 
 
 1913 
 1914 
 1915 
 1916 
 1917 
 1918 
 
 This Book's 
 
 base 
 10 
 16 
 24 
 
 58 
 45 
 
 ^Measured by the (arithmetically) calculated "standard deviation," as 
 explained in Appendix I (Note to Chapter V, § 11). 
 
110 
 
 THE MAKING OF INDEX NUMBERS 
 
 It will be noted: (1) that in the four years, 1913 to 
 1917, the dispersion reached 58 per cent, which was more 
 than any figure reached in the entire period of 67 years, 
 1846 to 1913 ; (2) that, in both series, war increases dis- 
 persion, the Civil War year, 1866, having the highest 
 figure in the first column ; (3) that the return of peace re- 
 duces the dispersion, as witness the figures for 1876 and 
 1918 ; and (4) that, in general, there is a progressive in- 
 crease in dispersion with the lapse of time, as witness the 
 figures for 1886 to 1913. These same points are always 
 in evidence whatever period is examined. 
 
 The preceding table relates to prices only. Unfortu- 
 nately there are no quantity relatives associated with 
 Sauerbeck's price figures. But Professors Day and Per- 
 sons have worked out quantity figures for 12 crops. 
 Table 11 shows the dispersion of the quantities of the 36 
 commodities and of the 12 crops studied by Professors 
 Day and Persons : * 
 
 TABLE 11. DISPERSION 2 OF 36 AND OF 12: 
 
 QUANTITY RELATIVES 
 
 (In per cents) 
 
 Year 
 
 36 Commodities 
 
 Year 
 
 12 Crops 
 
 1913 
 
 base 
 
 1880 
 
 38 
 
 1914 
 
 12 
 
 1885 
 
 25 
 
 1915 
 
 17 
 
 1890 
 
 26 
 
 1916 
 
 17 
 
 1895 
 
 25 
 
 1917 
 
 24 
 
 1900 
 
 18 
 
 1918 
 
 27 
 
 1905 
 
 16 
 
 
 
 1910 
 
 base 
 
 
 
 1915 
 
 18 
 
 
 
 1920 
 
 10 
 
 * See Edmund E. Day, "An Index of the Physical Volume of Produc- 
 tion," The Review of Economic Statistics, pp. 246-59, September, 1920. 
 
 * Measured by the (geometrically) calculated "standard deviation," 
 weighted as explained in Appendix I (Note to Chapter V, § 11). 
 
BIASED INDEX NUMBERS 111 
 
 It will be seen that the dispersion of the 36 quantities 
 reaches, in only five years, a figure higher than that 
 reached in a span of 25 years for the 12 crops. The only 
 instance in which the 12 crop dispersion reaches a higher 
 figure is in 1880, 30 years away from the base. 
 
 It is because of the unusually great dispersion of our 
 36 prices and quantities that these data afford a very 
 severe test of accuracy of the conclusions reached in this 
 book. The index numbers which we have calculated 
 and shall calculate, whether biased, freakish, or merely 
 slightly erratic, differ among themselves much more than 
 they would during six years of peace. Thus in Table 7 
 the biased Formula 1 has a joint error of 11.34 per cent, 
 calculated forward and backward between 1913 and 1917, 
 only four years apart. But Professor A. W. Flux ^ shows 
 that Sauerbeck's index number calculated forward and 
 backward between two periods, ten years apart (one of the 
 two being the period 1904-1913 and the other the year 
 1919), gives a discrepancy of only eight per cent; he also 
 shows that the Board of Trade index number calculated 
 forward and backward between 1871 and 1900, a span of 
 29 years, gives a discrepancy of 13 per cent, which is only 
 a little more than the 11.34 per cent we find here, although 
 covering seven times as long a period of time. 
 
 § 13. Formul© may be Erratic without being Biased 
 
 In the case of Palgrave's index number (Formula 9 
 above discussed) the two kinds of bias — type bias and 
 weight bias — conspire, as we have seen, to raise the 
 index number and the same is true of Formula 7. Like- 
 wise for Formulae 13 and 15 the two conspire downward. 
 
 For Formulae 3 and 5 ( the same as Formulae 17 and 19), 
 on the other hand, the two types of bias almost exactly 
 ^ Journal Royal Statistical Society, March, 1921, p. 174. 
 
112 THE MAKING OF INDEX NUMBERS 
 
 offset each other. Thus, Formula 3, by virtue of being 
 arithmetic, has an upward bias, but, by virtue of having 
 weighting /, has also a downward bias; likewise as to 
 Formula 5. As there is no way of telling which of 
 the two opposing tendencies will be the greater, the net re- 
 sult may be said to be unbiased, though still erratic, 
 for bias is a foreseeable tendency to err in one direction. 
 
 Again, taking the same formulae considered as harmon- 
 ics, we may say that Formula 19, being harmonic, is 
 biased downward, but, being weighted by system 77 is 
 also biased upward ; and hkewise as to Formula 17. Or, 
 taking the same formulae considered as aggregatives (for 
 3 is the same as 53, and 19 as 59), we may say that For- 
 mula 53 has no bias ; for, while it is one-sided in that it 
 contains the quantities for only one of the two years, 
 the other being omitted, we cannot ordinarily foretell 
 whether this fact will raise or lower the index number ; 
 and likewise as to Formula 59. We can, however, say 
 that the Formulae 53 and 59 are slightly erratic; for, 
 taken forward and backward, the product is not unity 
 though very close to it, as Tables 7 and 8 show. Thus 
 the weighted aggregatives, or their equivalent arithmetics 
 and harmonics, are erratic without being biased ; some 
 other random selection of commodities than those here 
 chosen might show a negative error in place of a posi- 
 tive error in our tables, and vice versa. Thus, we must 
 distinguish sharply between index numbers like Formula 
 51, which are simply very erratic, and those like Formula 
 9 or Formula 13, which are very much biased. 
 
 § 14. Erratic and Freakish Index Numbers 
 
 It may be assumed, for the present at least, that all index 
 numbers are erratic to some degree. One of the chief 
 objects of this book is to show to what degree. 
 
BIASED INDEX NUMBERS 113 
 
 Tables 7 and 8 convict every one of the 28 index num- 
 bers so far considered of some error. In the case of 18 
 of the formulae we can show reason for some at least of 
 the errors, the part which has been described and dis- 
 cussed as "bias." In the cases of the other ten formulae, 
 the joint errors shown are ''accidental" in the sense that 
 we can assign no reason beforehand for their being in 
 one direction or the other. Thus as to Formula 7, which 
 shows in Table 7 a joint error in 1917 of +24.53 per cent 
 under Test 1 (and of +18.44 per cent under Test 2 in 
 Table 8) and a positive joint error in the ten columns 
 of the two tables, we can confidently predict ^ that we 
 shall always find a positive joint error whatever the data 
 may be which enter the formula. But, as to Formula 
 21 which, under Test 2, shows a joint error in 1916 of 
 +7.79 per cent and in 1918 of —7.22 per cent, there 
 can be no assurance whether, for any other particular 
 set of data, the joint error will be positive or negative. 
 All we can say is that 21 is certainly erratic. 
 
 Nor can we infer from these tables what the whole 
 error of any formula, whether biased or merely erratic, 
 really is. Thus from Table 7 we find that Formula 43, 
 the mode, with base year weighting, shows an imper- 
 ceptible joint error, and 21, 31, 41, 51, no error at all. 
 But this may be due to the fact that errors forward and 
 backward happen to offset each other. That this is the 
 case is proven by Table 8 which finds errors in all these 
 formulae, that for 43 reaching —18.18 per cent in 1916. 
 Thus, if the real error under Formula 43, in the price 
 index forward, in 1916, is —5.44 per cent and backward 
 the same, while the real error in the quantity index for- 
 ward is —13.34 per cent and backward the same, the 
 
 * From the analysis in Appendix I (Note to Chapter V, § 2) and 
 Appendix I (Note to Chapter V, § 6). 
 
114 THE MAKING OF INDEX NUMBERS 
 
 figures in both tables would be explained. As a matter 
 of fact these errors are the real errors of No. 43 — in- 
 dubitable within a small fraction of one per cent. But 
 we are not yet ready to show this. 
 
 Thus a small joint error, being only a net error between 
 two index numbers, is compatible with large errors in 
 
 Insensitiveness of Median and Mode 
 1o Number of Commodities 
 
 31 
 
 \5% 
 
 3 5 7 $ II 13 15 17 _ 19 21 23 25 27 29 31 33 35 
 
 NUMBER OF COMMODITIES 
 
 Chart 21. Showing that a change in the number of commodities from 
 3 commodities to 5, 7, etc., commodities seldom affects the median (31) and 
 mode, even the weighted mode (43 and 47) . Both median and mode remain the 
 same throughout the sixteen changes in the number of commodities, ex- 
 cept for six changes in the median (two at the *) and two in the mode. 
 When the mode does change, it changes violently. 
 
 both. But how can we ascribe individual error otherwise 
 than by dividing by two the joint errors in the tables? 
 
 While the answer to this question must be given in 
 stages or instalments, we can, at this point, show that 
 the modes, which never have perceptible bias, are never- 
 theless very erratic, and the medians, which seldom have 
 much bias, are moderately erratic. The evidence Ues in 
 the fact that the mode and, to a less extent, the median, 
 
BIASED INDEX NUMBERS 115 
 
 are insensitive to many of the factors of which an index 
 number is expected to be a sensitive barometer. 
 
 The introduction of a new commodity ought, evidently, 
 to change, in some degree, any price index which pretends 
 to be a sensitive expression of the data from which it is 
 computed (unless, of course, the new commodity happens 
 to have a price relative exactly equal to the index 
 number). But the mode and median often remain un- 
 changed, Uke the hands of a clock not rigidly connected 
 with the wheels which are supposed to move it. 
 
 Again, every change in weights will be reflected by a 
 change in any truly sensitive index number. But the 
 mode will often, in fact usually, remain inert even when 
 the weighting is changed radically. 
 
 Graphically, both these points are illustrated by Chart 
 21 which traces median and mode numbers through suc- 
 cessive stages as, one after another, we introduce new 
 commodities, beginning with three (Ume, pig iron, and 
 eggs) taken by lot and adding successively new commodi- 
 ties by lot, two by two, until all 36 are introduced. The 
 median taken is the simple median ; the mode is weighted. 
 The weighted median is not taken because it is somewhat 
 sensitive to weighting and we are illustrating insensitive- 
 ness. 
 
 We see that in all the 17 stages at which there ought to 
 be a change the median changes only six times and the 
 mode only twice! No clock can keep time to the second 
 if it jumps only once in a minute, or once in an hour. 
 Such a clock must invariably be in error most of the time, 
 although, from the clock itself, we cannot say how much. 
 In short, the horizontal hnes in the diagram betray the 
 existence of error, but not how much error. Further- 
 more, as to the mode, the fact that Formulae 43, 45, 47, 
 49 can all be represented by the same curve shows that 
 
116 THE MAKING OF INDEX NUMBERS 
 
 the mode pays no attention to big changes in weighting, 
 thus further betraying error. When an index number 
 is highly erratic we have called it freakish. Evidently the 
 modes, even the weighted modes, are freakish and the 
 median, likewise, though in less degree. 
 
 Formula 51 is freakish for another reason. Instead 
 of being insensitive to influences which ought to affect 
 it, it is sensitive to influences which ought not to affect 
 it. Evidently an index number, to be a true barometer 
 of prices, ought not to be affected by irrelevant circum- 
 stances, such as whether the price of cotton is quoted by 
 the pound or by the bale. Formula 51 alone of all the 28 
 formulae will be so affected and is therefore unrehable, 
 very erratic, or freakish.^ Finally, every other simple index 
 number may be considered somewhat freakish because its 
 weights are arbitrarily equal, in defiance of the obvious 
 inequaUties among the commodities in real importance. 
 
 Thus, out of the 28 formulae, we know that 18 are biased. 
 Of these 18, ten are also freakish {viz, 1, 11, 33, 35, 37, 
 39, 43, 45, 47, 49). Besides these there are four other 
 freakish formulae {viz. 21, 31, 41, 51). This leaves only 
 two formulae not condemned on either score. These two 
 are 53 and 59 (or 3 and 5, or 17 and 19). 
 
 Formulae 53 and 59 are very close together. Thus, al- 
 ready, we find that all of the 28 formulae which differ 
 widely from each other have a discernible reason to differ 
 — bias or freakishness — while those for which we can- 
 not discover any reason for differing do not, in fact, differ 
 very much. 
 
 § 15. Bias and Errors Generally are Relative 
 
 We shall see that the ''ideal" formula, 353, gives an 
 almost absolute standard by which to measure errors. 
 
 * This feature is discussed in detail in Appendix III. 
 
BIASED INDEX NUMBERS 117 
 
 But, for the present, it is better not to try to imagine 
 any absolute standard, however much we may disHke 
 to rest on mere ''relativity." When we say, for instance, 
 that Formula 1 has an upward bias and 11 a downward 
 bias, both of, say, four per cent in 1917, we mean simply 
 that these four per cent errors apply in addition to any 
 other errors there may be. We thus think of each bias as 
 measured relatively to the half-way point between 1 and 
 11, but without assuming necessarily that this half-way 
 point is itself correct. This half-way point may, for aught 
 we yet know, be too high by ten per cent ; in which case 
 the error of 1 is 10 + 4, or 14 per cent, and of 11, 10 — 4, 
 or six per cent. In that case the bias of 11 is still four 
 per cent downward, despite the fact that the net error is 
 six per cent in the opposite direction. Thus we may 
 say, as compared with any other index number without 
 assignable bias. Formula 1 has an upward bias, ''other 
 things being equal." 
 
 § 16. Historical 
 
 The term "bias" has been used by Bowley and other 
 statisticians as applied to errors. The idea of type bias 
 was expressed, in other language, by Walsh.* Also, while 
 he did not recognize weight bias, he did point out that the 
 arithmetic average should be used with the weighting 
 of the base year and the harmonic with the weighting 
 of the given year.^ 
 
 Perhaps, as pointed out to me by Walsh, Sauerbeck 
 had an inkUng of the upward bias of the arithmetic aver- 
 age in a passage quoted by N. G. Pierson,' although Sau- 
 erbeck had no remedy to propose. 
 
 * Measurement of General Exchange Value, pp. 327-28. 
 2 lUd., pp. 307, 349. 
 
 * Economic Journal, March, 1896, p. 128. 
 
CHAPTER VI 
 
 THE TWO REVERSAL TESTS AS FINDERS OF FORMULA 
 
 § 1. The Time Reversal Test as Finder of Formulae 
 
 ' Not only do the two tests reveal joint errors per- 
 taining to each formula, but they afford the means of 
 rectification. But before we can thus rectify any given 
 formula we must first find for it two other formulse re- 
 lated to it. These two other formulse are "antithetical" 
 to the original formulae ; one being its antithesis re- 
 specting Test 1 and the other its antithesis respecting 
 Test 2. These two antitheses of any formula will there- 
 fore be called its time antithesis and its /ador antithesis. 
 To find these two antitheses is our next task and the ob- 
 ject of this chapter. 
 
 The time antithesis of any given formula is found by 
 applying Test 1 to that formula. As we know, Test 1 
 involves two steps : 
 
 (1) Interchanging the two times and thus obtaining 
 the index number reversed in time. 
 
 (2) Dividing the last found expression into unity. 
 The result ought to be the original formula itself in 
 
 order that Test 1 may be fulfilled. If it is not, then the 
 resulting formula, instead of being identical with the 
 original formula, is its time antithesis. That is, the time 
 antithesis of any index number between one time and 
 another is found by applying the very same formula the 
 other way round and then turning it upside down. 
 
 Algebraically, the first step, applying the formula the 
 other way round, consists in interchanging the subscripts 
 
 118 
 
TESTS AS FINDERS OF FORMULAE 119 
 
 (say "0" and ''1"), i-e. erasing "0" wherever it occurs 
 and writing "1" in its place, and vice versa. Thus For- 
 mula 7, viz.f 
 
 -;^; becomes — :^; 
 
 The second step, inverting, i.e. dividing into unity, con- 
 sists of interchanging numerator and denominator. 
 Thereby, the above becomes the required time antithesis, 
 
 ^Poq. 
 
 £)■ 
 
 We have taken a particular case for the sake of illustra- 
 tion. In the most general terms the process is : Let Poi 
 represent any index number for time '^1" relatively to 
 time ''0." The ''other way round" is Pio, and this 
 
 "turned upside down" is ^-, which, therefore, is the 
 
 " 10 
 
 general expression for the time antithesis of Pol 
 
 It may easily be shown that the antithetical relation- 
 ship is necessarily mutual, the original formula being deriv- 
 able by the very same process from its antithesis, so that 
 each of the two is the time antithesis of the other.^ 
 
 § 2. Numerical Illustration of Time Antithesis 
 
 Let us illustrate these two steps by starting once more 
 with the simple arithmetic index number of prices for 
 1918 relatively to 1917 and repeating, in slightly different 
 form, some of what was shown in Chapter V. This 
 is 110.11 per cent. The first of the two steps is to inter- 
 change 1917 and 1918, i.e. to calculate the simple arith- 
 metic index number of prices for 1917 relatively to 1918. 
 ^ See Appendix I (Note to Chapter VI, § 1). 
 
120 THE MAKING OF INDEX NUMBERS 
 
 This is 94.46 per cent. But these two index numbers, 
 forward and backward, are mutually inconsistent, not 
 being reciprocals, i.e. their product not being unity or 
 100 per cent. Test 1 is not fulfilled. 
 
 But, by the second step, we divide one of them, 94.46 
 per cent, into 100 per cent, or unity, obtaining 105.86 
 per cent, which is the time antithesis of 110.11. This 
 105.86 per cent is the figure which multiplied by the arith- 
 metic backward, 94.46 per cent, will give the true required 
 100 per cent. It will be recognized as the simple harmonic. 
 
 § 3. Graphic Illustration of Time Antithesis 
 
 Thus the simple harmonic is the time antithesis of the 
 simple arithmetic. The illustration of this relationship 
 
 is given in Chart 22. 
 The Harmonic forward is parallel Here the original index 
 
 to the Arithmetic backward , number is that of the 36 
 
 prices as they changed 
 
 from 1917 to 1918 and 
 
 ^flrw •, for^ff^^ is represented by the 
 
 i^^ar^"'" arithmetic "forward" 
 
 curve. The effect on 
 
 ^^ari this index number of 
 
 Qr0^^^^ "" interchanging the" O's" 
 
 -, .jQ and "I's," or in this 
 
 Chart 22. As in Charts UP and 14Q case, interchanging the 
 the harmonic may be regarded as the arith- "4's and 5's," 1.6. in- 
 metic backward in disguise, being parallel terchanging the years 
 
 1917 and 1918, is repre- 
 sented by the "backward" pointing curve. This shows 
 how the simple arithmetic would portray the price 
 change in going backward from 1918 to 1917. The result 
 is represented by drawing, parallel to the last named, the 
 dotted fine pointing forward. This is the harmonic. 
 
TESTS AS FINDERS OF FORMULAE 121 
 
 § 4. Algebraic Expression of Arithmetic and Harmonic 
 Time Antitheses 
 
 Beginning with Formula 1 (simple arithmetic) and 
 subjecting it to our twofold procedure we obtain : 
 
 2, 
 Original Formula 1 
 
 
 n 
 
 
 (1) Interchanging the ''O's" and __ 
 
 71/ 
 
 (2) Inverting 
 
 Po\ 
 
 ViJ 
 
 S( 
 
 The result is the time antithesis of the original simple 
 arithmetic. But the formula thus found is evidently 
 Formula 11, the ''simple harmonic." That is, Formulse 1 
 and 11 are, as noted in the last section, time antitheses 
 of each other. The harmonic (forward) is thus the arith- 
 metic backward reversed in direction.^ 
 
 Next take the arithmetic weighted /. 
 
 Original Formula 3 
 
 Vpo/ 
 
 I 
 Pi 
 
 ;pigi' ^° 
 
 (1) Interchanging the "O's" and '' I's"^^— — 
 
 ■^Piyi 
 
 (2) Inverting 
 
 \piJ 
 
 which is Formula 19, harmonic weighted IV. Thus 
 Formulae 3 and 19 are time antitheses. 
 
 *■ Cf. C. M. Walsh, Measurement of General Exchange Value, pp. 327-28. 
 
122 
 
 THE MAKING OF INDEX NUMBERS 
 
 Similarly, Arithmetic Formula 5 and Harmonic Formula 17 are time 
 antitheses. 
 
 Similarly, Arithmetic Formula 7 and Harmonic Formula 15 are time 
 antitheses. 
 
 Similarly, Arithmetic Formula 9 and Harmonic Formula 13 are time 
 antitheses. 
 
 Tabulating more simply, we may indicate the time 
 antitheses by connecting lines as follows, the weighted 
 arithmetic being related to the weighted harmonic in 
 reverse order : 
 
 Weighting 
 
 Arithmetic 
 
 Harmonic 
 
 Formula No. 
 
 Formula No. 
 
 Simple 
 
 1-e 
 
 >ll 
 
 Weighted / 
 
 X 
 
 yrl3 
 
 Weighted // 
 
 5^ 
 
 y^l5 
 
 Weighted III 
 
 7^ 
 
 ^\^;^i7 
 
 Weighted IV 
 
 y 
 
 ^19 
 
 § 5. Time Antithetical Geometries, Medians, and 
 Modes 
 
 The other formulae in our list are also related. 
 Algebraically, applying the same processes to the sim- 
 ple geometric, we have : 
 
 Original Formula 21 
 
 ^1 
 
 xi^Ox^^ 
 
 p 
 
 p:\\ 
 
 X 
 
 (1) Interchanging the »j/p 
 
 ^©<ft) 
 
 X[^-J]X 
 
 (2) Inverting (and sim 
 plifying) 
 
 which is identical with the original formula. Thus Test 1 
 
TESTS AS FINDERS OF FORMULA 
 
 123 
 
 is fulfilled or, if we wish to say so, the simple geometric 
 is its own time antithesis. 
 
 Likewise, taking the geometric weighted I, we find 
 its time antithesis to be geometric weighted IV while, 
 similarly, geometric // and geometric III are the time 
 antitheses of each other so that the antithetical relation- 
 ships of the geometries, as shown by connecting lines, are : 
 
 Simple Geometric 
 
 21) 
 
 Weighted / 
 Weighted II 
 Weighted III 
 Weighted IV 
 
 23 
 
 25] 
 
 27 
 
 29 
 
 
 Likewise the simple median,^ Formula 31, fulfills Test 1 
 and the weighted medians have the same sort of relation- 
 ships as in the case of the geometries, i.e. 
 
 31) 
 
 33 
 35 
 37 
 39 
 
 The same scheme of relationships applies to the mode. 
 
 41) 
 
 43 
 45 
 
 47 
 49 
 
 ^ If the number of terms is even, the test is fulfilled only provided we 
 take the geometric mean of the adjacent middle terms as the median. 
 
124 THE MAKING OF INDEX NUMBERS 
 
 § 6. Antithetical Aggregatives 
 
 The simple aggregative comes next. 
 Original formula 
 
 
 (1) Interchanging the "O's" and 'Ts" |^° 
 
 (2) Inverting ^ 
 
 Po 
 
 which is identical with the original, Formula 51. 
 
 Taking Formula 53, the aggregative weighted /, we have : 
 
 Original Formula 53 ?MP 
 
 ^Poqa 
 
 (1) Interchanging the "O's" and '' I's" ?Mi 
 
 ^Piqi 
 
 (2) Inverting J^-' 
 
 which is not identical with the original (53) but is For- 
 mula 59. 
 
 Thus we have 
 
 51)^ 
 
 531 
 59 
 
 § 7. Review of the 28 Formulae 
 
 We have now paired as time antitheses all our 28 for- 
 mulae relatively to Test 1. Evidently, if we had started 
 with only one member of each pair, we could have dis- 
 covered the other member by means of the twofold pro- 
 cedure. For instance, if we had not included the bar- 
 
TESTS AS FINDERS OF FORMULA 125 
 
 monies in our original list, we should have been led to 
 discover them by applying the twofold procedure to the 
 arithmetics, or vice versa. Again, if we had not included 
 in our original list the weighted formulae whose identi- 
 fication numbers end in 7 and 9, we should have been led to 
 discover them, by applying the twofold procedure to those 
 whose identification numbers end in 3 and 5, or vice versa. 
 
 § 8. The Factor Reversal Test as Finder of Formulae 
 
 Thus far we have considered only Test 1 (time reversal 
 test) as a finder of formulae; and the formulae we found 
 were formulae already discussed — a harmonic for an 
 arithmetic, a weighted geometric for a differently weighted 
 geometric, etc. The test was to reverse the times and 
 then invert the resulting formula {i.e. divide it into unity). 
 This was called the time antithesis, and turned out in 
 every case to be an old formula. 
 
 When we apply Test 2 (the factor reversal test), how- 
 ever, we shall actually be led to new formulae not included 
 in the previous 28. 
 
 The factor antithesis of any given formula for, say, 
 the "price index, is found by applying Test 2 to that for- 
 mula. Test 2 involves two steps : 
 
 (1) Interchanging the prices and quantities, thus ob- 
 taining the index number of quantities. 
 
 (2) Dividing the last found expression into the value ratio . 
 The result ought to be the original formula itself in 
 
 order that Test 2 may be fulfilled. If it is not, then the 
 resulting formula, instead of being identical with the 
 original formula, is its factor antithesis. 
 
 § 9. Numerical Illustration of Factor Antithesis 
 
 Consider the year 1917. The simple arithmetic index 
 number oi prices for 1917 is 175.79 per cent of the 1913 
 
126 THE MAKING OF INDEX NUMBERS 
 
 / 
 
 A'^ 
 
 / 
 / 
 
 Three Types of Index Numbers / y22 
 
 of Prices / y 
 
 / / ' 
 
 Factor Antitheses of / y ^^ 
 
 Harmonic ^ / / ^^14 
 
 Ceometric / / / ^ ^ -^^ 
 
 / y 
 
 ^2 
 
 Arithmetic 
 
 
 
 //V // t^^ ^>?. 
 
 
 j^ /// /// 
 
 ^^ 
 
 28 
 
 a 
 
 /// 
 
 ,^^^ 
 
 J// 
 
 f I5^ 
 
 ZJ 74 75 'le 17 7B 
 
 Chart 23P. In the case of the factor antitheses, the harmonic is above 
 and the arithmetic below the geometric, in an order the reverse of that in 
 which the original three types were arrayed. 
 
 price level, and the corresponding index number of quan- 
 tities for 1917 is 125.84 per cent of the 1913 quantity level. 
 One of these is too big, since their product evidently ex- 
 ceeds the value ratio, which is only 192.23 per cent. By 
 
TESTS AS FINDERS OF FORMULAE, 127 
 
 dividing this value ratio, 192.23, by the second factor 
 (quantity index), 125.84, we get a new price index, 152.76. 
 This we call the factor antithesis of the original price 
 index, 175.79 — that is, the factor antithesis of the simple 
 
 Three Types of Indey Numbers 
 of Quantifies 
 
 Factor Antitheses of 
 
 Harmonic ^n 
 
 Geometric ^^ -— -^ 
 
 Arithmetic _--r — ^-""^^ 
 
 75 t4 7S ye '17 *i8 
 
 Chart 23Q. Analogous to Chart 23P. 
 
 arithmetic index of prices. It is the figure which, when 
 used as a factor and multiplied by the simple arithmetic 
 quantity index, will give the true required 192.23 per 
 cent. 
 Or, reversely, dividing the value ratio, 192.23 per cent, 
 
128 THE MAKING OF INDEX NUMBERS 
 
 by the first factor (the price index), 175.79 per cent, we 
 obtain 109.35 per cent, the factor antithesis of the quan- 
 tity index, 125.84 per cent, i.e. the figure which if used 
 
 Four Methods of Weighting Compared 
 (Prices) 
 
 [ador Antjlhemfjjif^'^^^^^^ 
 
 Factor Ant. of Georru}tjic_ 
 
 factor Ant,£j!sn^°J!E 
 
 
 yj H '/5 'le '\7 % 
 
 Chart 24P. Comparison of methods of weighting applied to factor 
 antitheses of the index numbers given in Chart 16P. The change from 
 weights with base prices (curve numbers ending in 4 and 6) to weights with 
 given year prices (curve numbers ending in 8 and 0) shifts the curve down- 
 ward. Changes in the quantities have little effect one way or the other. 
 
 as the quantity factor and multiphed by the original price 
 factor, 175.79, will give the true value ratio, 192.23 per cent. 
 The factor antithesis of Formula 1 is numbered 2 ; 
 the factor antithesis of Formula 3 is numbered 4, and 
 so on. That is, each odd numbered formula has as its 
 factor antithesis the following even number.^ 
 
 1 The complete system of numbering formulae is given in Appendix V, § 2. 
 
TESTS AS FINDERS OF FORMULA 129 
 
 § 10. Graphic lUustratiGn of Factor Antithesis 
 
 Graphically, Charts 23P and 2SQ show three principal 
 types of factor antitheses arranged in five groups by 
 weights. The order is in each case the reverse of that of 
 the original index number (see Charts 15P and 15Q) . The 
 
 Four Methods of Weighting Compared 
 
 (Quantities) 
 
 f^tS- 
 
 ..A„#58 
 
 , ^fi5^: — »- -;;;t: 
 
 
 .J^— "— =" 
 
 focft 
 
 
 ^^^^2a« 
 
 \5% 
 
 '13 n 75 'le 17 '18 
 
 Chabt 24Q. Analogous to Chart 24P. The shifts are equal (and oppo- 
 site) to those of 16P. Those of 24P are equal and opposite to those of 16Q. 
 
 factor antitheses (even numbered) of the price indexes 
 exhibit the same biases as the original quantity indexes 
 (odd numbered) in the reverse order. 
 
 Charts 24P and 24Q, classifying the opposite way, 
 show the four varieties of factor antitheses correspond- 
 ing to the four systems of weighting, arranged in three 
 groups by types. 
 
130 THE MAKING OF INDEX NUMBERS 
 
 Charts 20P and 20Q, lower part, show the combination 
 diagram for the factor antitheses. It is similar to 
 Charts 20P and 20Q, upper part, in reverse order. 
 
 Charts 17 P and 17Q, lower part, show the factor antith- 
 eses of the median. They differ only slightly from each 
 other and exhibit the same inertness or tendency for 34 
 and 36 (and 38 and 40) to stick close together, except 
 occasionally when they fly apart. 
 
 The factor antitheses of the modes (not charted) would 
 be indistinguishable from each other. 
 
 § 11. Algebraic Expression of Factor Antitheses 
 
 Algebraically, the first step, interchanging prices and 
 quantities, consists merely in interchanging "p" and 
 "g" in any formula, i.e. erasing "p" wherever it occurs, 
 and writing "q" in its place, and vice versa; the second 
 step is dividing the result into the value ratio. 
 
 For example, according to the simple arithmetic, the 
 index number for time 1 relatively to time is 
 
 Original Formula 1 
 
 (1) Interchanging "p's" and "q's 
 
 (2) Dividing into 
 
 (2i) 
 
 n 
 
 'r," 
 
 \qo^ 
 
 n 
 
 
 SpoQ'o 
 
 
 n 
 Thus Formula 1 does not meet Test 2, but the appli- 
 cation of that test leads to a new formula, the factor antith- 
 esis of the former. 
 
TESTS AS FINDERS OF FOEMULE 131 
 
 Again, Formula 7, ms., — ^ — :;;^ — becomes ^^^ — * 
 
 The second step consists in dividing the last found into 
 
 ?Ml giving Formula 8. 
 2pogo 
 
 The above are particular cases for illustration. In 
 the most general terms we may let Poi be any index 
 number for prices. Substituting ''^'s" for ''p's," and 
 vice versa, we get Qoi, and dividing into the value ratio, 
 
 we get— ^-^^-7- Qoi as the general expression for the fac- 
 ^PoQo 
 
 tor antithesis of Poi. 
 
 The (even numbered) weighted antitheses of price 
 
 indexes exhibit the same biases as the (odd numbered) 
 
 original quantity indexes in opposite order. 
 
 § 12. The Various Roles of Laspeyres' and 
 Paasche's Formulae 
 
 In precisely the same way we may obtain all the other 
 factor antitheses. In the case of Formula 3 (or its substi- 
 tutes. Formulae 17 and 53) the result is subject to simphfica- 
 tion. 
 
 As we aheady know, Formula 3 reduces to 53. Let 
 us start therewith and apply the factor reversal test. 
 
 Original Formula 53 |2i£o 
 
 2pogo 
 
 (1) Interchanging "p's" and "g's" |^i^ 
 
 (2) Dividing this into— ^^^ and cancehng SpoQ'o 
 
 Spo3o ^qiPo 
 
 which (according to our identification numbering) is called 
 Formula 54. This is evidently identical with Formula 59. 
 
132 
 
 THE MAKING OF INDEX NUMBERS 
 
 Thus it will be noted that the factor antithesis of For- 
 mula 53 (namely 54) is identical with its time antithesis, 
 59, which we have known as Paasche's formula. Here- 
 after Paasche's formula will be usually referred to as 54 
 rather than as 59. 
 
 For the sake of uniformity of method we designate 
 the factor antithesis of Formula 53 as 54 (and likewise 
 of Formula 3 as 4, and of Formula 17 as 18, all of which 
 [54, 4, 18] are identical with 59). Again, starting with 
 Formula 59, we get as its factor antithesis a formula desig- 
 nated as 60 which, of course, turns out to be identical 
 with 53. Likewise the factor antitheses of Formulae 5 
 (called 6) and of 19 (called 20) are all identical with 60. 
 
 Our table of formulae now has two sets of six identicals 
 (3,6, 17, 20, 53, 60) and (4,5,18, 19,54,59), represent- 
 ing two types, type L (Laspeyres') and type P (Paasche's). 
 
 The following list of weighted arithmetic, harmonic, 
 and aggregative formulae is arranged to show the repeti- 
 tions of L (Laspeyres') and P (Paasche's) formulae (the 
 only repeaters in the entire Ust of formulae). 
 
 Arithmetic 
 
 Aggregative 
 
 Harmonic 
 
 Formula No. 
 
 Formula No. 
 
 Formula No. 
 
 
 
 13 
 
 
 
 14 
 
 
 
 15 
 
 
 
 16 
 
 3L 
 
 53 L 
 
 17 L 
 
 4P 
 
 54 P 
 
 18 P 
 
 6P 
 
 59 P 
 
 19 P 
 
 6L 
 
 60 L 
 
 20 L 
 
 7 
 
 
 
 8 
 
 
 
 9 
 
 
 
 10 
 
 
 
 Thus L and P fall always among these three types. 
 
TESTS AS FINDERS OF FORMULA 
 
 133 
 
 § 13. List of 46 Formulae 
 
 We had 28 formulae which included four identicals. 
 All the even numbered formulae which we have just added 
 to the Ust by applying Test 2 are new, excepting only 
 Formulae 54 and 60 and their identicals. Thus instead 
 of 28 formulae, or 24 after canceling identicals, we now 
 have 56, or 46 after canceUng identicals. They are as 
 follows : 
 
 Aeithmetic 
 
 Harmonic 
 
 Geometric 
 
 Median 
 
 Mode 
 
 Aggregative 
 
 1 
 
 11 
 
 21 
 
 31 
 
 41 
 
 51 
 
 2 
 
 12 
 
 22 ■ 
 
 32 
 
 42 
 
 52 
 
 3L 
 
 13 
 
 23 
 
 33 
 
 43 
 
 53 L 
 
 4P 
 
 14 
 
 24 
 
 34 
 
 44 
 
 54 P 
 
 5P 
 
 15 
 
 25 
 
 35 
 
 45 
 
 
 6L 
 
 16 
 
 26 
 
 36 
 
 46 
 
 
 7 
 
 17 L 
 
 27 
 
 37 
 
 47 
 
 
 8 
 
 18 P 
 
 28 
 
 38 
 
 48 
 
 
 9 
 
 19 P 
 
 29 
 
 39 
 
 ^ 49 
 
 59 P 
 
 10 
 
 20 L 
 
 30 
 
 40 
 
 50 
 
 60 L 
 
 The following list omits duplicates (53 and 54 being 
 retained but their duphcates omitted). 
 
 Simple 
 
 Fac. an. of simple 
 
 Weighted I 
 
 Fac. an. of weighted /. . 
 
 Weighted II 
 
 Fac. an. of weighted // . 
 
 Weighted/// 
 
 Fac. an. of weighted /// 
 
 Weighted IV 
 
 Fac. an. of weighted IV . 
 
 
 u 
 
 S 
 
 S 
 
 
 S 
 
 S 
 
 I 
 
 o 
 
 i 
 
 
 
 
 < 
 
 w 
 
 a 
 
 
 
 1 
 
 11 
 
 21 
 
 31 
 
 41 
 
 2 
 
 12 
 
 22 
 
 32 
 
 42 
 
 _ 
 
 13 
 
 23 
 
 33 
 
 43 
 
 - 
 
 14 
 
 24 
 
 34 
 
 44 
 
 - 
 
 15 
 
 25 
 
 35 
 
 45 
 
 - 
 
 16 
 
 26 
 
 36 
 
 46 
 
 7 
 
 - 
 
 27 
 
 37 
 
 47 
 
 8 
 
 - 
 
 28 
 
 38 
 
 48 
 
 9 
 
 - 
 
 29 
 
 39 
 
 49 
 
 10 
 
 - 
 
 30 
 
 40 
 
 50 
 
 51 
 
 52 
 
 53 
 54 
 
134 THE MAKING OF INDEX NUMBERS 
 
 These 46 formulae may be called the 'primary formulae. 
 The additional ones which follow in subsequent chapters 
 are all derivatives from these 46 primary formula. 
 
 Of these 46 distinct formulae : 
 
 six are simples, viz. 1, H, 21, 31, 41, 51 
 
 sLx are the factor antitheses of the simples, viz. 2, 12, 22, 32, 42, 52 
 
 two are Laspeyres' and Paasche's, 53, 54 
 
 {aggregatives which interchange with some of 
 
 the arithmetics and harmonics) 
 
 two are other weighted arithmetics, 7, 9 
 
 two are the factor antitheses of these, 8, 10 
 
 two are other weighted harmonics, 13, 15 
 
 two are the factor antitheses of these, 14, 16 
 
 four are weighted geometries, 23, 25, 27, 29 
 
 four are the factor antitheses of these, 24, 26, 28, 30 
 
 four are weighted medians, 33, 35, 37, 39 
 
 four are the factor antitheses of these, 34, 36, 38, 40 
 
 four are weighted modes, 43, 45, 47, 49 
 
 four are the factor antitheses of these, 44, 46, 48, 50 
 
 § 14. Historical 
 
 As has been pointed out in previous chapters, the time 
 reversal test has, to all intents and purposes, been used 
 by many previous writers. These same writers, notably 
 Walsh, have likewise observed the essential symmetry 
 of Formulae 23 and 29, of 1 and 11, and of 3 and 19 (or 53 
 and 59). - 
 
 As to factor antitheses. Formula 52 has been used by 
 Drobisch and Sir Rawson-Rawson (who proposed to 
 measure the average price level of imports or exports by 
 dividing values by tonnage). Formula 22 has been pro- 
 posed by Nicholson and Walsh. Among other factor 
 antitheses Formula 2154 (to be described later) was 
 proposed by Walsh while 4154 (also to be described 
 later) has been proposed by Lehr. As these are all factor 
 antitheses of other formulae, the principle of such antith- 
 esis must have been more or less consciously recognized. 
 The other factor antitheses in our list, derived from the 
 
TESTS AS FINDERS OF FORMULAE 135 
 
 general application of the principle, seem not to have 
 been expressed. Nevertheless the general principle may 
 be said to be recognized whenever any series of statistics 
 of money values (such as of imports, exports, production, 
 clearings) are ''deflated" by dividing by an index number 
 of prices to obtain a rough index of the underlying quanti- 
 ties (physical volume of imports, exports, production, 
 trade). 
 
CHAPTER VII 
 
 RECTIFYING FORMULA BY "CROSSING" THEM 
 
 § 1. Crossing Time Antitheses 
 
 We have thus far reached two chief results from the 
 use of the tests. First, we have noted which formulsB 
 meet, and which fail to meet, these tests. Of the 46, 
 only four, the simple geometric, median, mode, and aggre- 
 gative, meet Test 1 and none of the 46 meet Test 2. Sec- 
 ondly, we have, by Test 1, found for each formula its 
 time antithesis (in each case an old odd numbered for- 
 mula) and by Test 2 its factor antithesis (in each case, 
 except for some dupUcations, a new even numbered for- 
 mula). 
 
 We now come to a third use of these tests, namely, to 
 "rectify" formulse, i.e. to derive from any given formula 
 which does not satisfy a test another formula which does 
 satisfy it ; so that now we are about to pass from our 
 46 primary formulse to the region of derivative formulse. 
 
 This is easily done by "crossing," that is, by averaging, 
 antitheses. If a given formula fails to satisfy Test 1 
 its time antithesis will also fail to satisfy it ; but the two 
 will fail, as it were, in opposite ways, so that a cross be- 
 tween them (obtained by geometrical averaging) will 
 give the golden mean which does satisfy. This will be 
 true in all cases, whether the formulse paired and crossed 
 are arithmetic, harmonic, geometric, median, mode, or 
 aggregative (or any other for that matter). As will be 
 shown, the geometric mean of the two antithetical index 
 numbers may always be used for "crossing" them 
 
 136 
 
RECTIFYING FORMULA BY "CROSSING" 137 
 
 whether the two themselves be geometric, median, mode, 
 aggregative, or one arithmetic and the other harmonic 
 (arithmetic-harmonic). If we thus cross the two antithe- 
 ses geometrically, the resulting formula will satisfy 
 the test. But if we cross them arithmetically, or har- 
 monically, it will not. By this simple process of crossing 
 (geometrically) we can ''rectify" any formula whatever 
 so far as securing conformity to either or both of the two 
 tests goes. 
 
 Thus, take the simple arithmetic. Its time antithesis 
 is the simple harmonic. Neither of these fulfills the first 
 or time reversal test. But the failure of each in one di- 
 rection is exactly matched by the failure of the other in 
 the opposite direction, and we shall see that the cross 
 between the two meets the test exactly. 
 
 § 2. Numerical Illustration 
 
 The simple arithmetic index number for 1917 on 1913 
 as base is 175.79 per cent. The simple harmonic (time 
 antithesis) for 1917 on 1913 as base is 157.88 per cent. 
 Neither of these satisfies Test 1 but the cross between 
 them is 166.60 per cent which does satisfy Test 1 since it 
 is the reciprocal of 60.02 per cent, the figure reached by 
 the same process applied the other way round in time. 
 
 Thus 166.60, the rectified arithmetic (and, of course, 
 the rectified harmonic as well), unlike the original un- 
 rectified or simple arithmetic, 175.79 per cent, and the 
 original unrectified or simple harmonic, 157.88, conforms 
 to Test 1, i.e. is such that multipUed by the similarly 
 obtained figure for the reverse direction, 60.02 per cent, 
 it gives exactly 100 per cent, or unity ; in other words, 
 the forward and backward are reciprocals. 
 
 The simple geometric index number, on the other hand, 
 being its own time antithesis, i.e. conforming to Test 1, 
 
138 
 
 THE MAKING OF INDEX NUMBERS 
 
 requires no rectification (so far as that test goes). For 
 1917 this simple geometric index number is 166.65 per 
 cent and, in the reversed direction, it is 60.01 per cent, 
 which is the reciprocal of 166.65 per cent. 
 
 The Simple Geometric 
 
 compared with the simple Arithmetic and 
 Harmonic and their rectification by test I. 
 (Prices) 
 
 13 
 
 U 
 
 75 
 
 16 17 18 
 
 Chakt 25P. The geometric (21) is practically identical with 101, the 
 geometric mean of 1 and 11. 
 
 The entire (price) series of the two, i.e. the recti- 
 fied simple arithmetic-harmonic, 101, and the simple 
 geometric, 21, are : 
 
 
 1913 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Eectified Arithmetic- 
 Harmonic (101) 
 
 100 
 
 95.75 
 
 96.80 
 
 121.38 
 
 166.60 
 
 179.09 
 
 Simple Geometric (21) 
 
 100 
 
 95.77 
 
 96.79 
 
 121.37 
 
 166.65 
 
 180.12 
 
 Comparison between these two index numbers satisfy- 
 ing Test 1 reveals an unexpected result — that is, a re- 
 markably close agreement. Thus, the supposed conflict 
 between the geometric and arithmetic index numbers 
 disappears by "rectification." 
 
RECTIFYING FORMULtE BY "CROSSING" 139 
 
 Hitherto there has been a disposition to think that the 
 arithmetic and geometric stand on an equality, that, while 
 the arithmetic lies above the geometric and the harmonic 
 lies below it, this is Uttle more than an interesting fact. 
 
 Jevons and a few others, on the other hand, have had 
 a disposition to prefer the geometric as one always pre- 
 fers a "golden mean" to extremes, but without assigning 
 any clear reason for the preference. The mere fact that 
 the geometric lies between two others is not a very 
 logical reason for preferring it. 
 
 The Simple Geometric 
 
 compared with the simple Arifhmetic and 
 Harmonic and their rectification by test I 
 (Quantities) 
 
 '/3 '14 15 16 17 18 
 
 Chart 25Q. Analogous to Chart 25P. But 21 and 101 disagree in 1918. 
 
 We did, however, find a very good reason for rejecting 
 the simple arithmetic (and, likewise, the simple harmonic) 
 index number. It will not work both ways in time con- 
 sistently with itself. But when, by ''rectification," this 
 defect is remedied the resulting rectified arithmetic no 
 longer presents any problem arising from results dis- 
 crepant with the geometric mean. 
 
 Test 1 thus serves as a touchstone for (1) convicting 
 the arithmetic (and harmonic) of self -inconsistency ; 
 (2) remedying that inconsistency, reaching another for- 
 mula entirely free of this defect. 
 
140 THE MAKING OF INDEX NUMBERS 
 
 § 3. Graphic Illustration 
 
 Graphically, Charts 25P and 25Q show the rectified index 
 number (Formula 101) by crossing the simple arithmetic (1) 
 and simple harmonic (11) and its practical identity with 
 the simple geometric (21). 
 
 § 4. Algebraic Proof that Rectification Can Always 
 be Accomplished by Crossing Time Antitheses 
 
 Algebraically, the full proof is ridiculously simple. Let 
 Poi be any formula for the index number of prices of date 
 1 relatively to date 0. Its time antithesis, as was shown 
 
 in § 1 of the preceding chapter, is — -. The geometric 
 mean is found by multiplying these two expressions to- 
 gether and extracting the square root, \-^- This is 
 
 ■I 10 
 
 the new formula which we are to prove conforms to 
 Test 1. 
 
 Let us apply Test 1. 
 
 (1) Interchanging the '^O's" and 'Ts," V^' 
 
 -I 01 
 
 (2) Multiplying this by the original, we get unity as 
 the test requires. 
 
 Therefore the cross between any two time antitheses 
 will obey Test 1. 
 
 Thus, as Formulae 1 and 11 are time antitheses, the 
 formula V(l) X (11) must fulfill Test 1, the time re- 
 versal test. We have called this new formula, 101. In 
 all, we may derive the following new formulae fulfilling 
 the time reversal test by virtue of the fact that each is a 
 cross between two time antitheses : 
 
RECTIFYING FORMULA BY "CROSSING" 141 
 
 § 5. List of Rectified Formulae by Crossing Time 
 Antitheses 
 
 FormulsD derived from arithmetics and harmonics. . 
 
 Vl X 11 or Formula 101 
 
 V2 X 12 or Formula 102 
 
 V3^l9 or Formula 103 
 
 V4 X 20 or Formula 104 
 
 . V5 X 17 or Formula 105 
 
 Ve X 18 or Formula 106 
 
 V7 X 15 or Formula 107 
 
 VS X 16 or Formula 108 
 
 V9 X 13 or Formula 109 
 
 VlO X 14 or Formula 110 
 
 Formulae derived from geometries : 
 
 V'23 X 29 or Formula 123 
 V24 X 30 or Formula 124 
 V25 X 27 or Formula 125 
 V26 X 28 or Formula 126 
 Formulae derived from medians : 
 
 , V33 X 39 or Formula 133 
 
 V34 X 40 or Formula 134 
 
 V35 X 37 or Formula 135 
 
 V36 X 38 or Formula 136 
 
 Formulae derived from modes : 
 
 V43 X 49 or Formula 143 
 V44 X 50 or Formula 144 
 V45 X 47 or Formula 145 
 V46 X 48 or Formula 146 
 
142 THE MAKING OF INDEX NUMBERS 
 
 Formulse derived from aggregatives : 
 
 \/53 X 59 or Formula 153 
 \/54 X 60 or Formula 154 
 
 Formula 153 for prices is : 
 
 4 
 
 
 and for quantities : J Sgi??o ^ gg^^ 
 ^S^oPo 2goPi 
 
 This is what we shall call our ''ideal" formula. It is 
 evidently identical with Formula 154, since 53 is identi- 
 cal with 60 and 59 with 54. Likewise the resulting 153 
 and 154 dupUcate Formulae 103, 104, 105, 106 (which 
 result from various identicals of 53 and 54). 
 
 In numbering these rectified formulse for identification, 
 it will be observed that we simply use the number 100, in- 
 creased by the number of the lower numbered of the two 
 time antitheses from which each is derived.^ 
 
 § 6. Crossing Factor Antitheses 
 
 We come now to Test 2, and factor antitheses. Recti- 
 fication relatively to Test 2 is accomplished by taking 
 the geometric mean between any two formulse which are 
 factor antitheses. Again the proof, given in the Appen- 
 dix, is simple.^ 
 
 § 7. List of Rectified Formulae by Crossing Factor 
 
 Antitheses 
 Thus we obtain the following formulse conforming to 
 Test 2 : 
 
 Vl X 2 or Formula 201 
 V3 X 4 or Formula 203 
 
 * The complete system of numbering formulae is given in Appendix V, § 2. 
 
 * See Appendix I (Note to Chapter VII, § 6) for proof and discussion. 
 
RECTIFYING FORMULA BY "CROSSING" 143 
 
 V5 X 6 or Formula 205 
 
 V7 X 8 or Formula 207 
 
 Vq X 10 or Formula 209 
 Vu X 12 or Formula 211 
 Vl3 X 14 or Formula 213 
 Vl5 X 16 or Formula 215 
 Vl7 X 18 or Formula 217 
 Vl9 X 20 or Formula 219 
 V21 X 22 or Formula 221 
 V23 X 24 or Formula 223 
 V25 X 26 or Formula 225 
 V27 X 28 or Formula 227 
 V29 X 30 or Formula 229 
 V31 X 32 or Formula 231 
 V33 X 34 or Formula 233 
 V35 X 36 or Formula 235 
 V37 X 38 or Formula 237 
 V39 X 40 or Formula 239 
 V4I X 42 or Formula 241 
 V43 X 44 or Formula 243 
 V45 X 46 or Formula 245 
 V47 X 48 or Formula 247 
 V49 X 50 or Formula 249 
 V5I X 52 or Formula 251 
 V53 X 54 or Formula 253 
 V59 X 60 or Formula 259 
 
 In numbering these formulae for identification, it will 
 be observed that we simply use the number 200, increased 
 by the number of the lower numbered of the two factor 
 
144 THE MAKING OF INDEX NUMBERS 
 
 antitheses from which each is derived (just as, in reference 
 to Test 1 we used the number 100 plus the number of 
 the lower numbered of the two time antitheses). Of 
 course, the ability of a formula to conform to one of the 
 two tests does not necessarily imply ability to conform 
 to the other (although, as a matter of fact, it tends in 
 that direction). Accordingly, most of the 200-group 
 of formulae are distinct from — even though usually 
 giving results close to — the 100-group of formulae. 
 
 Among the 200-group there are six alike, viz., those 
 crossing Laspeyres' and Paasche's, or the following: 
 Formulae 203, 205, 217, 219, 253, 259 ; and these are not 
 only identical with each other but, as will be seen by 
 inspection, identical with six of the 100-group, namely 
 with Formulae 153 and 154, and the duplicates of the 
 latter, namely Formulae 103, 104, 105, 106. This formula, 
 
 4 
 
 2£i9o ^ Spig i 
 
 SpoQ-o 2pogi 
 
 mentioned before as our ideal, is the cross between Las- 
 peyres' and Paasche's. It is the only formula which 
 occurs both in the 100 and the 200 lists. 
 
 § 8. Fourfold Relationship of Antitheses 
 
 It may be easily shown ^ that,"if any two index numbers 
 are time antitheses of each other, then their respective 
 factor antitheses are also time antitheses of each other. 
 Thus, Formulae 1 and 11 being time antitheses of each 
 other, Formulae 2 and 12 (their factor antitheses) are 
 also time antitheses of each other. Likewise Formulae 
 23 and 29 being time antitheses. Formulae 24 and 30 (their 
 respective factor antitheses) are also time antitheses of 
 each other. 
 
 1 Algebraically, the proof of this theorem is simple and is given in Ap- 
 pendix I (Note A to Chapter VII, § 8). 
 
RECTIFYING FORMULAE BY "CROSSING" 145 
 
 Similarly it may be easily shown ^ that if any two in- 
 dex numbers are factor antitheses of each other, then their 
 respective time antitheses are also factor antitheses of 
 each other. 
 
 § 9. Rectifying Simple Arithmetic and Harmonic 
 by Both Tests 
 
 Thus we find our formulae arranging themselves in 
 quartets, which not only form two pairs of time antithe- 
 ses, but also form two pairs of factor antitheses — all of 
 them failing to meet tests, but rectifiable through cross- 
 ing. 
 
 Thus the quartet of formulae : 
 
 1 11 
 
 2 12 
 
 are such that either horizontal pair yields a formula con- 
 forming to Test 1 (i.e. V (1) X (11) is Formula 101, and 
 V (2) X (12) is Formula 102) while the vertical pairs 
 yield formulae conforming to Test 2 (i.e. V (1) X (2) 
 is 201 and V(ll) X (12) is 211). 
 
 It may be shown that, in any such quartet, the crosses 
 of the two pairs of time antitheses are factor antitheses 
 of each other and the crosses of the two pairs of factor 
 antitheses are time antitheses of each other. 
 
 We are now ready to follow through the complete or 
 double rectification of all formulae. This is obtained 
 by crossing the crosses and gives the same result in which- 
 ever order it is done, — whether first crossing the time 
 antitheses and then crossing the results, or first crossing 
 the factor antitheses and then crossing the results, and 
 the result is the same as the fourth root of the product 
 
 ^Algebraically, the proof is given in Appendix I (Note B to Chapter 
 VII, § 8). 
 
146 THE MAKING OF INDEX NUMBERS 
 
 Recti fled Arifhmetic and Harmonic, Simple / 
 
 (Prices) 
 
 /2 
 
 73 
 
 W 
 
 75 
 
 V5 
 
 V 
 
 'la 
 
 Chart 26P, The upper tier are the curves of a quartet of related for- 
 mulae. The next tier are formed by welding, or crossing geometrically, 
 each pair of time antitheses (1 and 11 yielding 101; 2 and 12 yielding 
 102) ; the next, by welding each pair of factor antitheses (1 and 2 yielding 
 201; 11 and 12 yielding 211); and the last, by welding all four in the 
 upper tier (or both in the second or both in the third). No one of the 
 upper tier fulfills either test ; those of the second fulfill Test 1 but not Test 
 2 ; those of the third fulfill Test 2 but not Test 1 ; those of the last ful- 
 fill both tests. 
 
RECTIFYING FORMULA BY "CROSSING" 147 
 
 of the entire quartet. ^ Thus this fourth root, the double 
 rectification of any of the quartet of formulae, must satisfy 
 both tests. 
 
 Recti fled Arithmetic and Harmnic, Simple 
 
 (Quantities) 
 
 W 
 
 n 75 iS 17 
 
 Chaet 26Q. Analogous to Chart 26P. 
 
 7a 
 
 A doubly rectified formula is numbered 300, increased 
 by the number of the lowest numbered of the quartet 
 of formulae from which it is derived. All the relation- 
 ships may be illustrated by the following scheme for the 
 quartet Formulae 1, 11, 2, 12, above cited. 
 
 1 See Appendix I (Note to Chapter VII, § 9). 
 
148 
 
 THE MAKING OF INDEX NUMBERS 
 
 1 
 
 crossed with 
 
 11 
 
 gives 101 
 
 crossed 
 with 
 
 
 crossed 
 with 
 
 crossed 
 with 
 
 2 
 
 crossed with 
 
 12 
 
 gives 102 
 
 gives 
 201 
 
 crossed with 
 
 gives 
 211 
 
 gives 
 gives 301 
 
 § 10. Numerical Illustration 
 
 We may illustrate all three rectifications by taking the 
 figures for 1917 for the quartet of FormulsB 1, 11, 2, 12. 
 These are for the index numbers of prices : 
 
 (1) = 175.79 
 
 (2) = 152.75 
 
 (11) = 157.88 
 
 (12) = 172.11 
 
 The geometric means or rectifications of the time an- 
 titheses are 
 
 (101) = Vl X 11 = V175.79 X 157.88 = 166.60 
 
 (102) = \/2 X 12 = V152.75 X 172.11 = 162.14 
 
 It is interesting to observe that these results conform- 
 ing to Test 1 are not so far apart as the original figures 
 which do not so conform. 
 
 Similarly the rectifications respecting Test 2 are 
 
 (201) = Vl X 2 = V175.79 X 152.75 = 163.87 
 (211) = Vll X 12 = V157.88 X 172.11 = 164.84 
 
 It is interesting to observe that these figures conform- 
 ing to Test 2 are closer together than the original figures 
 which do not so conform. 
 
RECTIFYING FORMULAE BY "CROSSING" 149 
 Finally, the complete rectification gives 
 
 (301) = VlOl X 102 = \/l66.60 X 162.14 = 164.35 
 
 = V2OI X 211 = \/l63.87 X 164.84 = 164.35 
 = v^ 1 X 2 X 11 X 12 = 
 
 v^ 175.79 X 152.75 X 157.88 X 172.11 = 164.35 
 
 § 11. Graphic Illustration 
 
 Charts 26P and 26Q give the rectification of the simple 
 arithmetic and harmonic, i.e. of Formulae 1, 11, 2, 12 (in 
 which quartet 1 and 1 1 are time antitheses of each other, 
 as are 2 and 12, while 1 and 2 are factor antitheses of 
 each other, as are 11 and 12). These four are drawn 
 from the same origin (upper part of the figure), the factor 
 antitheses, or even numbered, being dotted lines. 
 
 Their rectifications by Test 1 are drawn immediately 
 below, the dark curve 101 being the rectification of For- 
 mulae 1 and 11 ; and the dotted curve 102 being the rec- 
 tification of 2 and 12. These two rectified formulae agree 
 with each other better than the original formulae. 
 
 The third tier, on the other hand, gives the rectifica- 
 tions by Test 2, 201 being the rectification of Formulae 
 1 and 2, and 211 of 11 and 12. These two also are closer 
 than the first four. 
 
 Finally, the lowest tier gives 301, the completely rectified 
 index number. It may be considered as the rectification 
 by Test 2 of the pair rectified by Test 1, or it may be 
 considered as the rectification by Test 1 of the pair recti- 
 fied by Test 2, or it may be considered as the rectifica- 
 tion of the whole original quartet by both tests at once. 
 
 Thus each rectification splits a difference and each 
 index number represented by the final curve is the geo- 
 metric average of the four from which it is derived. These 
 methods of rectification by crossing apply generally. 
 
150 
 
 THE MAKING OF INDEX NUMBERS 
 
 § 12. Rectifying Simple Geometric, Median, Mode, and 
 Aggregative by Both Tests 
 
 Graphically, Charts 27P and 27Q show the recti- 
 fication of the simple geometric. This is a shorter 
 
 X 22,22 
 
 Rectified Geometric, Simple 
 
 (Prices) 
 
 73 74 75 7tf V7 73 
 
 Chabt 27P. Analogous to Chart 26P ; but the quartet 21, 21, 22, 22 
 contains two duplicates, so that the upper tier of four curves reduce to 
 two; the two of the second tier simply repeat those last named and the 
 two curves in the third tier reduce to one. The one in the lower tier merely 
 repeats the last named. 
 
 process than that shown in the last section as the sim- 
 ple geometric already meets Test 1 and only needs recti- 
 fication by Test 2. But, for uniformity, we put in all four 
 steps, the first "rectification" being, in this case, merely 
 a repetition of the formulse ; for we may regard Formula 
 21 as its own time antithesis, and 22 as its own. 
 
RECTIFYING FORMULA BY "CROSSING" 151 
 
 That is, the first tier gives the quartet, 
 
 21 21 
 
 22 22 
 
 (Formula 21 being the time antithesis of 21 and Formula 
 22 of 22, while one of the 22's is the factor antithesis of 
 one of the 21's and the other 22 of the other 21). In 
 
 Recfified Geomeiric, Simple 
 
 (Quaniities) 
 
 ^^^22.22 
 
 13 74 75 16 17 W 
 
 Chaet 27Q. Analogous to Chart 27P. 
 
 the second tier. Formula 121 is the ''horizontal" recti- 
 fication of Formulae 21 and 21, i.e. is identical with 21, 
 and likewise. Formula 122 is the "horizontal" rectifica- 
 tion of Formulae 22 and 22, i.e. is identical with 22. The 
 third tier, 221, is supposed to represent two coincident 
 formulae, one the ''vertical" rectification of one pair, 
 Formulae 21 and 22, and the other of the other pair, 21 
 and 22. The fourth tier is evidently identical with the 
 third, being the rectification of Formulae 221 and 221 (as 
 well as of 121 and 122). 
 
 Were it not for the fact that usually we have four 
 really distinct formulae to rectify we would omit two of 
 
152 
 
 THE MAKING OF INDEX NUMBERS 
 
 these tiers (the second and last) ; for the only real recti- 
 fication is by Test 2. 
 
 Charts 28P and 28Q show in exactly the same way 
 the rectification of the simple median, and Charts 29P 
 and 29Q that of the simple mode, and Charts SOP and 
 30Q that of the simple aggregative. 
 
 Rectified Median. Simple 
 
 {Prices ) 
 
 '13 74 75 'le U7 '18 
 
 Chart 28P. Analogous to Chart 27P as to duplications. 
 
 § 13. Results of Doubly Rectifying Simples 
 
 Graphically, Charts 31P and 31Q show at a glance 
 the rectification of all simples (modes omitted). The 
 top tier simples are only 1 and 11 because the mode (41) 
 is omitted; and because 21, 31, 51, already conforming 
 to Test 1, are postponed to the second tier, where 
 they occur as 121, 131, 151, along with those rectified 
 by Test 1. All rectified by Test 2 are in the third tier; 
 
RECTIFYING FORMULiE BY "CROSSING" 153 
 
 while the last gives those rectified by both tests. It 
 will be seen that Curves 301 and 321 are practically paral- 
 lel everywhere except 1917-1918, where Curve 301 (fixed 
 base) still bears evidence of the original distortion due 
 to one commodity, skins. These two are fairly similar to 
 331, while 351 stands alone. Formula 341 (omitted) has 
 comparatively little resemblance to the rest. 
 
 Rectified Median, Simple 
 
 (Quanfiiies) 
 
 '13 74 75 '16 17 '18 
 
 Chart 28Q. Analogous to Chart 28P. 
 
 Thus we may say, in general, that by rectifying simple 
 index numbers we secure a moderate, but only a moderate, 
 degree of agreement among the three principal formulae. 
 That this agreement is not better is because the simples 
 involve such outlandish weighting that they are almost 
 incorrigible. This is especially true of the aggregative 
 Formula 51 with its "haphazard" weighting, which has no 
 relation to the weighting employed by the others. 
 
 Moreover, the rectification of the simples by Test 2 
 
154 
 
 THE MAKING OF INDEX NUMBERS 
 
 involves a practical absurdity. Simple index numbers 
 of prices have an excuse for existing only when we have 
 no knowledge of what weights could be used, that is, no 
 
 Rectified Mode, Simple 
 
 (Prices) 
 
 \5% 
 
 •13 '14 '15 '16 '17 18 
 
 Chart 29P. Analogous to Chart 27P as to duplications. 
 
 knowledge of the "g's" and so no knowledge of the values, 
 Poqo, etc. But rectifying a simple index number of 
 prices by Test 2, on the other hand, requires its factor 
 antithesis obtained by dividing the corresponding simple 
 
RECTIFYING FORMULA BY "CROSSING" 155 
 
 index number of quantities into the value ratio. This 
 implies that we do know the quantities and values. But 
 if we had all this knowledge we would, in practice, use 
 it at the start, and employ a better system of weighting 
 than the simple weighting. 
 
 Rectified Mode, Simple 
 
 
 (Quantities) 
 
 --^-O^ 
 
 4l.4t 
 
 . .-^ .^.^--'^^ 
 
 ) 42,42 
 
 V5 
 
 7-* 'IS 
 
 Chaet 29Q. 
 
 'te '17 
 
 us to Chart 29 P. 
 
 18 
 
 Nevertheless, for completeness, I have included in 
 this book the rectification of simples. It serves to show 
 how, even starting with the handicap of absurd weight- 
 ing, we can achieve a very considerable rectification, though 
 we can never completely overcome the handicap. 
 
156 
 
 THE MAKING OF INDEX NUMBERS 
 
 § 14. Rectifying the Weighted Arithmetic and 
 Harmonic by Both Tests 
 
 Far more important, therefore, are the rectifications 
 of the weighted index numbers. 
 
 The consideration of the first two quartets on the hst, 
 consisting of Formulae 3, 19, 4, 20, and 5, 17, 6, 18, is 
 
 Rectified Aggregative, Simple 
 
 (Prices) 
 
 tj '/'f 75 ye 77 'id 
 
 Chart SOP. Analogous to Chart 27P as to duplications. 
 
 postponed. The reason is that, in each case, their recti- 
 fication is identical with that of Formulae 53 and 54. 
 
 Graphically, Charts 32P and 32Q show the rectification 
 of the arithmetic-harmonic quartet, Formulae 9, 13, 10, 
 14. By Test 1 in Charts 32P and 32Q we roll or weld 
 Curves 9 and 13 into 109, and 10 and 14 into 110. Again, 
 by Test 2 Curves 9 and 10 are compressed into 209, and 13 
 and 14 into 213. Finally, by putting all four through 
 both roUing mills (in either order, or both at once), we 
 
RECTIFYING FORMULAE BY "CROSSING" 157 
 
 roll them together into the single curve 309 at the bottom 
 of the diagram. 
 
 Charts 33P and 33Q show the same process by which 
 the quartet, Formulae 7, 15, 8, 16, are passed through 
 our rolling mills to be welded into the fully rectified 307. 
 
 Rectified Aggregative, Sinyoie 
 
 (Quantities) 
 
 '13 
 
 74 */5 
 
 Chart 30Q. 
 
 ie 77 
 
 Analogous to Chart SOP. 
 
 78 
 
 Chart 33 resembles Chart 32 very closely in every 
 detail. 
 
 § 15. Rectifying the "Weighted Geometric, Median, 
 Mode, and Aggregative by Both Tests 
 
 Graphically, Charts 34P and 34Q show the rectified 
 quartet of geometries. Formulae 23, 29, 24, 30. These 
 charts resemble Charts 32 and 33 except that the four 
 formulae to start with are only about half as far apart. 
 
 Charts 35P and 35Q show the rectification quartet of 
 the geometric Formulae 25, 27, 26, 28, and resemble 
 closely Chart 34 in every detail. 
 
 Charts 36P, 36Q and 37P, 37Q give the rectified 
 
158 
 
 THE MAKING OF INDEX NUMBERS 
 
 dimple Index Numben of Prices 
 
 and Their Antitheses and DenVof/re'S. 
 
 Satisfying neither test 
 test I only. 
 
 both tests 
 (modes omitted) 
 
 73 
 
 14 
 
 15 
 
 16 
 
 17 
 
 ys 
 
 Chart 3 IP. This double rectification of all the simple index numbers 
 (of which 21, 31, 51 and their factor antitheses 22, 32, 52 are omitted in 
 the first tier, being inserted in the second tier as 121, 131, 151 and 122, 
 132, 152) results in only a moderate degree of agreement, as the lowest tier 
 of curves indicates. 
 
RECTIFYING FORMULiE BY "CROSSING" 159 
 
 Simple Index Numbers of Quantifies 
 
 ond Their Antitheses and Deriyotives. 
 
 Satisfying neither test, 
 test I ortiy. 
 
 « doth tests, 
 (modes omitted) 
 
 13 
 
 1^ '15 '16 '17 
 
 Chart 31Q. Analogous to Chart ZIP. 
 
 'IS 
 
160 
 
 THE MAKING OF INDEX NUMBERS 
 
 medians as indicated. They are like the figures for the 
 preceding except that they are usually still closer to each 
 other than the geometric but less consistently so. 
 
 Rectified Arithmetic and Harmonic, Weighted 
 
 (By Values in One Year) 
 (Prices ) 
 
 21^209 
 
 73 
 
 '14 
 
 '15 
 
 16 
 
 '17 
 
 'Id 
 
 Chart 32 P. A quartet of widely differing weighted index numbers, 
 with scattered chain figures, combined by rectification into 309, which is 
 practically identical with the other rectified index numbers that follow, and 
 the chain figures of which practically coincide with the fixed base figures. 
 
 Charts 38P and 38Q give the rectified mode for the 
 quartet of Formulae 43, 49, 44, 50 as well as of 45, 47, 46, 
 48 ; for these two do not need separate charts, being iden- 
 tical up to the limit of our calculations. That is, 43 is 
 practically identical with 45, 44 with 46, 49 with 47, and 
 50 with 48. Thus the mode does not respond appreciably 
 to changing weights. 
 
RECTIFYING FORMULAE BY "CROSSING" 161 
 
 Charts 39P and 39Q show the rectification of the two 
 weighted aggregatives, the formulsB of Laspeyres' (53) 
 and Paasche's (54) which recur again and again in our 
 system of formulae. It may be considered as the recti- 
 fication, not only of the quartet, Formulae 53, 59, 54, 60, 
 
 Recfitied Arithmetic ancf Harmonic, Weighted 
 (dy Values in One Year) 
 (Quantities) 
 
 '13 74 V5 Vtf 77 
 
 Chart 32Q. Analogous to Chart 32P. 
 
 but also of the quartets, 3, 19, 4, 20, and 5, 17, 6, 18, all 
 three quartets being identical. 
 
 This rectification, like some of the preceding, is not 
 really of four but of two only. Also, unlike the case of 
 the others, there is only one real rectification; that is, 
 the first rectification and second are identical with each 
 other as well as, of course, with the two together. Hence 
 there is only the one identical curve for each of the three 
 lower tiers. 
 
162 
 
 THE MAKING OF INDEX NUMBERS 
 
 § 16. Results of Double Rectifications of Weighted 
 Index Numbers 
 
 Graphically, Charts 40P and 40Q show at a glance the 
 rectification of all the weighted index numbers (modes 
 
 Recfified Arifhmeiic and Harmonic, Weighfed I 
 
 (By ••Mixed' Values ) ___^ 7 
 
 /e 
 
 (Prices) 
 
 207 
 
 '13 '14- -IS 'le 'I? Vd 
 
 Chart 33P. Analogous to Chart 32P except that the weighting is by 
 mixed or "hybrid" values. 
 
 omitted). The agreement thus brought about among 
 the weighted index numbers is far greater than that 
 brought about among the simples. In fact, all these 
 rectified weighted index numbers agree perfectly for 
 practical purposes. If the medians were excluded the 
 eye could scarcely detect any discordance. Only the 
 
RECTIFYING FORMULAE BY "CROSSING" 163 
 
 rectified weighted modes (omitted from chart) really 
 disagree with the rest. 
 
 Charts 41 P and 41 Q outUne the hmits of the various 
 weighted formulae (omitting modes and medians), showing 
 that the hmits contract as the tests are fulfilled. This 
 diagram shows that all weighted index numbers (omitting 
 
 Rectified Arithmetic and Harmonic, Weighted 
 
 (By "Mixed" Values ) 
 (Quantifies) 
 
 '13 V4 'IS 16 '17 
 
 Chart 33Q. Analogous to Chart 33 P. 
 
 75 
 
 modes and medians) he within limits far closer together than 
 the original price relatives or quantity relatives averaged. 
 What is more important, it shows that this range is 
 greatly reduced when at least one of the two tests is met. 
 Finally, it shows that those which satisfy both tests 
 lie within an amazingly small range, so small as, for prac- 
 tical purposes, to be entirely neghgible. 
 Charts 42P and 42Q give individually the doubly recti- 
 
164 THE MAKING OF INDEX NUMBERS 
 
 fied weighted index numbers (modes omitted). It will 
 be noted that the eye can scarcely detect any lack of 
 paralleUsm, except sUghtly in the case of the median. 
 
 Rectified Geomefric , Weighted 
 
 (By Values in One Year) 
 (Prices) 
 
 '13 
 
 74 
 
 75 
 
 7^ 
 
 '17 
 
 'IB 
 
 Chart 34P. Analogous to Chart 32P except that all of the quartet are 
 of geometric derivation instead of arithmetic and harmonic. 
 
 § 17. List of Quartets 
 
 Let us now again "take account of stock," and list, 
 first, all the quartets, and then all the formulae. The 
 following is a complete list, omitting duphcates, of all 
 the quartets which may be formed from the 46 primary 
 formulae by matching each formula with its antitheses. 
 
RECTIFYING FORMULAE BY "CROSSING" 
 Arithmetic and Harmonic 
 
 giving Formula 301 
 giving Formula 307 
 giving Formula 309 
 
 Recfitied Geometric, Weighted 
 
 (By Values in One Year) 
 (Quantities) 
 
 165 
 
 1 
 
 11 
 
 2 
 
 12 
 
 7 
 
 15 
 
 8 
 
 16 
 
 9 
 
 13 
 
 10 
 
 14 
 
 13 'M 'IS '16 '17 
 
 Chakt 34Q. Analogous to Chart 34P. 
 
 75 
 
 
 Geometric 
 
 21 
 22 
 
 21 
 22 
 
 giving Formula 321 
 
 23 
 24 
 
 29 
 30 
 
 giving Formula 323 
 
 25 
 26 
 
 27 
 28 
 
 giving Formula 325 
 
166 THE MAKING OF INDEX NUMBERS 
 
 Median 
 
 giving Formula 331 
 giving Formula 333 
 giving Formula 335 
 
 Recti fied Geometric, Weighted 
 
 (By 'Mixed" Values) 
 (Prices) 
 
 31 
 32 
 
 31 
 32 
 
 33 
 34 
 
 39 
 40 
 
 35 
 36 
 
 37 
 
 38 
 
 Chart 35 P. Analogous to Chart ^34P except that the weighting is by 
 mixed or "hybrid" values. 
 
 Mode 
 
 41 
 42 
 
 41 
 42 
 
 43 
 44 
 
 49 
 50 
 
 45 
 46 
 
 47 
 
 48 
 
 giving Formula 341 
 giving Formula 343 
 giving Formula 345 
 
RECTIFYING FORMULAE BY "CROSSING" 167 
 
 Aggregative 
 
 51 
 
 52 
 
 51 
 52 
 
 53 
 54 
 
 59 
 60 
 
 giving Formula 351 
 giving Formula 353 
 
 RecfiFied Geomeiric, Weighted 
 
 (By 'Mixed" Values) 
 
 (Quantities) 
 
 13 'M IS 16 17 
 
 Chakt 35Q. Analogous to Cloart 35F. 
 
 ^8 
 
 The omitted duplicates are 
 
 3 
 4 
 
 
 19 
 20 
 
 and 
 
 5 
 6 
 
 
 17 
 
 18 
 
168 
 
 THE MAKING OF INDEX NUMBERS 
 
 Rectified Median, Weighted 
 (By Values in One Year) 
 (Prices) 
 
 U3 7-f 'IS W 17 
 
 Chakt 36P. Analogous to Chart 32P. 
 
 which are identical with 
 
 53 
 54 
 
 59 
 60 
 
 U8 
 
 all of which identical quartets merely contain Laspeyres' 
 and Paasche's formulae in their various r61es. Each of 
 the last three might be written 
 
 L 
 P 
 
 P 
 
 L 
 
RECTIFYING FORMULA BY "CROSSING" 169 
 
 All these identical quartets remind us again that 
 Laspeyres' and Paasche's formulae are time antitheses of 
 each other and also factor antitheses of each other, as 
 well as that they are arithmetic, harmonic, and aggre- 
 gative. 
 
 Rectified Median, Vfeighted 
 
 (By \/alues in On* Year) 
 (Quantifies) ^^ 34 
 
 '13 y-f '15 W 1(7. 
 
 Chaet 36Q. Analogous to Chart 36P. 
 
 W 
 
 Of the quartets it has doubtless been observed by the 
 reader that some really reduce to duets, namely, those 
 quartets resulting in Formulse 321, 331, 341, 351 (in which 
 cases the two numbers in the same horizontal line are iden- 
 tical) ; and also the quartet resulting in Formula 353 
 (in which case the diagonals are identical, being Las- 
 peyres' and Paasche's formulse). Formula 353, 
 
 V 
 
 
170 
 
 THE MAKING OF INDEX NUMBERS 
 
 which is identical with the 12 formulse indicated in § 7, 
 will hereafter be referred to only as 353. 
 
 Rectified Median, Weighted 
 
 (By 'Mixed' Values) 
 (Prices) 
 
 73 
 
 74 
 
 75 
 
 'le 
 
 '17 
 
 '18 
 
 Chaht 37P. Analogous to Chart 36P except that the weighting is by 
 mixed or "hybrid" values. 
 
 § 18. List of Formtilae thus far Obtained 
 
 The complete list of formulae, including the primary, 
 those fulfilUng Test 1, those fulfilling Test 2, and those 
 fulfiUing both tests, are given in Table 12, in which 
 duplications are omitted (being indicated only by a dash) . 
 
 In this table any formula (like 21) which already fulfills 
 
RECTIFYING FORMULiE BY "CROSSING" 171 
 
 Test 1 before crossing is pushed forward and appears 
 later (as 121) ; and likewise any (like 221) which fulfills 
 both tests after only one kind of crossing is pushed for- 
 ward and appears later (as 321). That is, in this table 
 the numbers with "300" comprise those and those only 
 
 RecfiFied Median, Weighted 
 (By 'Miffed" Values) 
 (Quanfiiies) 
 
 j5e 
 
 73 74 75 'IS '17 'IB 
 
 Chart 37Q. Analogous to Chart 37P. 
 
 which fulfill both tests; the numbers with ''200" com- 
 prise those, and those only, which fulfill only Test 2 ; the 
 numbers with ''100" comprise those, and those only, 
 which fulfill only Test 1, while the numbers less than 100 
 include those, and those only, which fulfill neither test. 
 
 Thus far we have assembled for examination the follow- 
 ing number of formulae : 
 
 46 primary formulae, including eight (21, 22, 31, 32, 41, 
 42, 51, 52) which conform to Test 1 and so are pushed 
 
172 THE MAKING OF INDEX NUMBERS 
 Reciitied Mode, Weighted 
 
 (Prictsi 
 
 44.50 (or 46.46) 
 43,43 (er45.^7i 
 
 243, 249(er245,i47i 
 
 343(or34S) 
 
 13 74 75 7ff '/7 '18 
 
 Chart 38P. Analogous to Charts 32P and 33P. 
 
 forward (to 121, 122, 131, 132, 141, 142, 151, 152) in 
 the last table of formulse ; 
 
 19 new derivative formulae (derived by crossing time 
 antitheses among the primary) conforming to Test 1 
 and including one (153) which conforms to Test 2 as 
 well and so is pushed forward (to 353) in the last table ; 
 
 22 new derivative formulse (derived by crossing factor 
 antitheses among the primary) conforming to Test 2 ; 
 9 new derivative formulse conforming to both tests. 
 
RECTIFYING FORMULAE BY "CROSSING" 173 
 
 This makes 96 separate formulae, of which 38 conform 
 to neither test, 26 conform to Test 1 only, 18 conform to 
 Test 2 only, and 14 conform to both tests. This list of 
 96 formulae constitutes our main series of formulae and 
 
 Rectified Mode, Weighted 
 (Quanfities) 
 
 ilJ.19(oM5,47t 
 
 M4fer/4e) 
 
 Y43(«rH5} 
 
 243.24B(or245,247l 
 
 343(6r345} 
 
 13 74 'S le '17 78 
 
 Chart 38Q. Analogous to Chart 38P. 
 
 includes most of the important kinds. Certain other 
 formulae which will be considered later are, in each case, 
 closely similar to some of these 96 varieties. 
 
 In a later chapter all these and other forms of index 
 numbers will be systematically compared. But already 
 one important conclusion forces itself upon us. It is 
 
174 
 
 THE MAKING OF INDEX NUMBERS 
 
 one which has already been noted, namely, that, after 
 rectification, the great discrepancies which we first noticed 
 among index numbers constructed by different formulae 
 tend to disappear; and that excepting the modes and the 
 index numbers derived from simples, all the index numbers 
 thus far foundwhich obey both tests agree closely witheach other. 
 
 Rectified Aggregative, Weighted 
 
 (Prices) 
 
 13 14 '15 '16 17 '18 
 
 Chart 39P. Analogous to Chart 32P, but the quartet 53, 59, 54, and 
 60 contains two duplicates ; consequently the four curves in the upper 
 tier reduce to two ; the two of the second tier reduce to one ; and the two 
 lower tiers merely repeat the preceding. 
 
 § 19. Other Methods of Crossing 
 
 In this chapter the "cross" between any two formulae 
 has always been the geometric mean between those two 
 formulae. And we have seen that this geometric mean 
 satisfied the test in question. That is, the geometric 
 mean of two time antitheses satisfies the time test, and 
 the geometric mean of two factor antitheses satisfies the 
 factor test. If we try the arithmetic mean or the harmonic 
 
RECTIFYING FORMULA BY "CROSSING" 175 
 
 mean of the two antitheses, it will fail to satisfy the re- 
 quired test. 
 
 Algebraically, this is readily proven by applying the 
 usual twofold routine by which we have tested any for- 
 mula. 
 
 Take, for example, Formulae 53 and 54 or 59. If we cross 
 these two formulae arithmetically instead of crossing them 
 
 Rectified Aggregative, Weighted 
 (Quantifies) 
 
 15 
 
 14 15 16 17 
 
 Chaet 39Q. Analogous to Chart 39P. 
 
 76 
 
 geometrically, we obtain 
 
 2 
 
 Starting with this formula, let us apply to it Test 1, by 
 means of the usual twofold procedure : 
 
 Interchanging the ''O's" and the "I's," 
 SpogiiSpo^o 
 
 2 
 Inverting, 2 
 
 Spogi , 2po£o 
 Spigi Spigo 
 
176 
 
 THE MAKING OF INDEX NUMBERS 
 
 Weighted Index Numbers cf Prices 
 
 ana Their Ant/theses and Derivatives 
 
 Satisfying neither test 
 test I onltf 
 
 .. Z - 
 both tests 
 [modes omitted) 
 
 \ior.»>a.to9.im. 
 
 l23.l24.l2S.l2e, 
 133.134,135.133 
 
 Z33.2SS.23Z239 
 
 \30Z 309.3?3. 
 )^2^3S3.3J^SS9i 
 
 f/S 
 
 74 
 
 75 
 
 W 
 
 77 
 
 vd 
 
 Chart 40P. Analogous to Chart 3 IP, except that the double rectifi- 
 cation of these weighted index numbers results in a much closer agreement 
 than was the case with the simples. 
 
 The resulting formula is not the original arithmetic but 
 the harmonic. Therefore, the original formula fails to 
 conform to Test 1, and the resulting (harmonic) formula 
 is its time antithesis. 
 
 The reader can readily prove that the same formula 
 also fails to satisfy Test 2. In this case the twofold pro- 
 cedure consists, as we know, in interchanging the "p's" 
 
RECTIFYING FORMULAE BY "CROSSING" 177 
 
 Weighted Index Numbers of Quantifies 
 
 and Their Antitheses and Derivatives. 
 
 Saliafijinq neither test 
 
 " test I only. 
 
 » Z " . 
 
 bottt tests. 
 
 (modes omittedl 
 
 107. ma, io9.no. 
 
 I2S.I24.I2S.I2S. 
 l33./3t.l3S.I3e 
 
 {207.209.213.2/S. 
 \223.22S.227.2^Sl 
 
 '30ZS09,323. 
 
 yj 74 'i5 'iS 17 
 
 Chakt 40Q. Analogous to Chart 40P. 
 
 and the "g's" 
 
 ratio 
 
 Spo^o* 
 
 and dividing the result into the value 
 The formula resulting from this twofold pro- 
 
 cedure for Test 2 will, it may surprise the reader to find, 
 be, in this case, the same as the above formula resulting 
 from the twofold procedure for Test 1. That is, the orig- 
 inal 'and final formulae (arithmetic and harmonic crosses 
 of 53 and 54) are not only time antitheses of each other 
 but also factor antitheses of each other. 
 
 In passing, we may note that these two formulae, the re- 
 sulting and the original, are listed in the Appendix as For- 
 mulae 8053 and 8054, being factor antitheses of each other. 
 
 If we should test the harmonic crossing we would sim- 
 ply reverse the above process. We would start with 8054 
 and reach 8053.^ 
 
 » See Appendix I (Note A to Chapter VII, § 19). 
 
178 THE MAKING OF INDEX NUMBERS 
 
 Rcfnge of Prices 
 
 and of 
 
 Three Tkjpes of Index Numbers 
 
 Weighted, Satisfijinq neither tejt. 
 - - only I or onli^ Z. 
 
 •• » both I and ^ 
 
 Imodej and fnedionj omtlted) 
 
 75 
 
 14 
 
 15 
 
 16 
 
 17 
 
 78 
 
 Chakt 41 p. The limits of the weighted index numbers contract markedly 
 as the tests are fulfilled. 
 
 All of the above examples are of the aggregative type. 
 What we have found is that if two aggregatives are 
 crossed arithmetically (or harmonically) the resulting 
 cross will not satisfy either Test 1 or Test 2. 
 
 By like testing of the other types of index numbers, — 
 
RECTIFYING FORMULAE BY 
 Range of Quantities 
 
 and of 
 
 Three Types of Index Numbers 
 
 CROSSING" 179 
 
 Weighted, Satisfqm) neither test. 
 ~ " onli/ I or onlii 2. 
 
 m - both I and 2. 
 
 (/nodes and medians omitted) 
 
 
 yj 14 '15 '16 '17 76 
 
 Chabt 41Q. Analogous to CJiart 41P. 
 
180 THE MAKING OF INDEX NUMBERS 
 
 Weighted Index Numbers Doubly Rectified 
 (Modes Omilted) 
 
 (Prices) 
 
 '/S 
 
 74 
 
 '15 
 
 /6 
 
 17 
 
 78 
 
 Chart 42P. This chart shows separately the seven resultant curves 
 (lowest tier) of Chart 40P, with the chain figures added. 
 
 arithmetic, harmonic, geometric, median, and mode, — 
 we find that crossing arithmetically or harmonically any 
 two time antitheses {i.e. Formulae 3 and 19, 5 and 17, 4 and 
 20, 6 and 18, 13 and 9, 15 and 7, 14 and 10, 16 and 8, 
 23 and 29, 25 and 27, 24 and 30, 26 and 28, 33 and 39, 
 35 and 37, 34 and 40, 36 and 38, 43 and 49, 45 and 47, 
 44 and 50, 46 and 48) will yield formulae which likewise 
 fail to satisfy either test. 
 
 We can thus convince ourselves that not a single one 
 of the 46 primary index numbers can be arithmetically 
 
RECTIFYING FORMULA BY "CROSSING" 181 
 
 Weighted Index Numbers Doubly Rectified 
 (tlodes Omitted) 
 (Quantifies) 
 
 353 
 
 7J 
 
 'H 'J5 "16 '17 
 
 Chart 42Q. Analogous to Chart 42P. 
 
 78 
 
 (or harmonically) crossed with its antithesis (whether 
 time or factor) and yield a result which will satisfy either 
 test. The only question remaining is, may any among 
 them be successfully crossed by any other method than 
 geometrically ? 
 
 We need scarcely consider any other methods of cross- 
 ing index numbers than the six types of averages which 
 we have considered for index numbers themselves. Of 
 these six we have already considered three. The remain- 
 ing three are the median, mode, and aggregative. 
 
 As to using the median or modal method for crossing 
 two formulae, obviously this is impossible. No such 
 averages exist when, as in the present problem, only two 
 terms are to be averaged. 
 
 There remains the aggregative method of crossing 
 two index numbers. This method is inappUcable as a 
 
182 
 
 THE MAKING OF INDEX NUMBERS 
 
 I— I 
 
 Pi 
 
 CO 
 
 O 
 
 o 
 o 
 
 CO 
 
 w 
 m 
 
 o 
 
 H-l 
 
 H 
 < 
 O 
 I— I 
 P^ 
 I— I 
 
 o 
 
 PQ 
 H 
 
 
 1—1 
 
 
 M 
 
 
 
 
 
 
 
 
 
 lO 
 
 1 
 
 iO 
 
 
 
 
 
 
 
 
 H 
 
 CO 
 
 
 CO 
 
 
 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 
 
 
 I 
 
 
 O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 c: 
 
 ym^ 
 
 (M 
 
 
 
 
 
 
 
 
 
 
 lO 
 
 »C 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <; 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CO 
 
 ■* 
 
 
 
 
 
 
 
 
 
 
 lO 
 
 IC 
 
 
 
 
 
 
 
 
 ^ 
 
 
 CO 
 
 
 lO 
 
 
 
 
 
 
 
 'Tt< 
 
 
 rt< 
 
 
 ^ 
 
 
 
 
 
 
 
 CO 
 
 
 CO 
 
 
 CO 
 
 
 
 
 
 
 
 
 CO 
 
 
 lO 
 
 
 t^ 
 
 
 05 
 
 
 o 
 
 1 
 
 
 -^ 
 
 
 ^ 
 
 
 -^ 
 
 
 ■^ 
 
 
 
 
 ca 
 
 
 a 
 
 
 (N 
 
 
 IM 
 
 
 .— 1 
 
 (M 
 
 CO 
 
 •* 
 
 iO 
 
 o 
 
 
 
 
 
 ^ 
 
 
 l-H 
 
 I— 1 
 
 -* 
 
 ^ 
 
 
 
 
 
 
 
 CO 
 
 Tt< 
 
 lO 
 
 CO 
 
 r^ 
 
 00 
 
 Oi 
 
 o 
 
 
 
 
 ':*< 
 
 ^ 
 
 Tf< 
 
 ^ 
 
 rH 
 
 Tt< 
 
 tH 
 
 »o 
 
 
 ^ 
 
 
 CO 
 
 
 lO 
 
 
 
 
 
 
 
 CO 
 
 
 CO 
 
 
 CO 
 
 
 
 
 
 
 a 
 
 CO 
 
 
 CO 
 
 
 CO 
 
 
 
 
 
 
 
 
 CO 
 
 
 lO 
 
 
 1^ 
 
 
 cr> 
 
 
 1 
 
 
 CO 
 
 
 CO 
 
 
 CO 
 
 
 CO 
 
 
 
 
 <M 
 
 
 Cl 
 
 
 (M 
 
 
 (M 
 
 
 l-H 
 
 (N 
 
 CO 
 
 ^ 
 
 lO 
 
 o 
 
 
 
 
 
 
 CO 
 
 CO 
 
 CO 
 
 T-H 
 
 CO 
 
 CO 
 
 CO 
 
 
 
 
 
 
 
 CO 
 
 ■* 
 
 to 
 
 o 
 
 t^ 
 
 00 
 
 o 
 
 o 
 
 
 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 
 ^ 
 
 
 tH 
 
 
 CO 
 
 
 VO 
 
 
 
 
 
 
 
 (M 
 
 
 (N 
 
 
 <N 
 
 
 
 
 
 
 o 
 
 CO 
 
 
 CO 
 
 
 CO 
 
 
 
 
 
 
 
 
 CO 
 
 
 LO 
 
 
 r^ 
 
 
 C5 
 
 
 g 
 
 1 
 
 
 (M 
 
 
 (N 
 
 
 (M 
 
 
 (M 
 
 
 
 
 (N 
 
 
 <N 
 
 
 (M 
 
 
 (N 
 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 Ed 
 
 ,_, 
 
 (M 
 
 CO 
 
 •* 
 
 IC 
 
 o 
 
 
 
 
 
 (M 
 
 iM 
 
 <M 
 
 (M 
 
 (M 
 
 C) 
 
 
 
 
 
 a 
 
 
 '"' 
 
 
 
 
 
 
 
 
 
 
 
 00 
 
 ^ 
 
 lO 
 
 «3 
 
 1^ 
 
 00 
 
 05 
 
 o 
 
 
 
 
 (M 
 
 (M 
 
 (N 
 
 <N 
 
 (N 
 
 (M 
 
 (N 
 
 CO 
 
 
 l-H 
 
 
 
 
 
 
 t^ 
 
 
 a> 
 
 
 
 o 
 
 
 1 
 
 
 1 
 
 
 O 
 
 
 o 
 
 
 O 
 
 CO 
 
 
 
 
 
 
 CO 
 
 
 CO 
 
 
 l-H 
 
 
 CO 
 
 
 lO 
 
 
 
 
 
 
 Z 
 
 l-H 
 
 
 
 
 1—1 
 
 
 1 
 
 
 1 
 
 
 
 IM 
 
 
 (M 
 
 
 (M 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 < 
 
 1— < 
 
 
 
 
 
 
 t^ 
 
 
 a> 
 
 
 ffi 
 
 O 
 
 
 1 
 
 
 1 
 
 
 o 
 
 
 o 
 
 
 6 
 
 (N 
 
 
 
 
 
 
 (M 
 
 
 c^ 
 
 
 1— 1 
 
 CI 
 
 
 
 
 
 t^ 
 
 CO 
 
 05 
 
 o 
 
 
 O 
 
 o 
 
 1 
 
 1 
 
 1 
 
 1 
 
 o 
 
 o 
 
 o 
 
 
 l-H 
 
 I— ( 
 
 
 
 
 
 1—1 
 
 '-' 
 
 
 '-' 
 
 
 
 
 
 
 
 
 
 
 
 PS 
 
 < 
 
 r-l 
 
 .—1 
 
 CO 
 
 1— i 
 
 »o 
 
 O 
 
 1 
 
 1 
 
 1 
 
 1 
 
 tH 
 
 (N 
 
 1 
 
 1 
 
 1 
 
 1 
 
 t> 
 
 00 
 
 Ol 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 tH 
 
RECTIFYING FORMULAE BY "CROSSING" 183 
 
 method of averaging Formulas 3 and 19, or any of the other 
 pairs of antitheses, except the geometric and aggregative 
 index numbers ; because, except in these two cases, there 
 are no appropriate numerators and denominators of the 
 terms to be averaged such as are required to fit into an 
 aggregative formula. 
 
 The reader who is interested will find these two cases 
 discussed in the Appendix.^ 
 
 § 20. Historical 
 
 Except in the case of Formula 353, the history of which 
 will be especially noted later, no crosses of formulae, such 
 as those set forth in this chapter, seem to have been pre- 
 viously pointed out. 
 
 Instead of crossing the formulae themselves, previous 
 students of index numbers have crossed their weights, as 
 will be shown in the next chapter. 
 
 1 See Appendix I (Note B to Chapter VII, § 19). 
 
CHAPTER VIII 
 RECTIFYING FORMULA BY CROSSING THEIR WEIGHTS 
 
 § 1. Introduction 
 
 The foregoing list of 96 formulae thus far obtained, and 
 ending with Formula 353, constitutes a complete system 
 of formulae, primary and derivative, which I shall call 
 the ''main series." The additions to it in this chapter 
 are, in essence, only sHght variations of this main series. 
 These additions are included in deference to the wishes 
 of other students of index numbers, and in order that the 
 list shall cover all formulae previously suggested by others 
 and all points of view. They may be called the "sup- 
 plementary series." 
 
 Each of these additional formulae is weighted and each 
 weight is a cross between two other weights. This cross- 
 ing of two weights is merely an alternative method of 
 combining two kinds of weighted index numbers. To 
 illustrate, if we start with the two formulae, 23 and 29, 
 namely the geometric index numbers, — one weighted 
 according to the values in the base year and the other 
 weighted according to the values in the given year, and 
 which are time antitheses of each other — we can combine 
 these two formulae in either of two ways. One way is 
 that already described in the main series, and con- 
 sists simply in crossing the two index numbers themselves, 
 i.e. multiplying them together and extracting the square 
 root. The result is Formula 123 of the main series. The 
 other way, about to be discussed, is to construct a new 
 formula on the same model as 23 and 29, such that each 
 
 184 
 
RECTIFYING BY CROSSING WEIGHTS 185 
 
 individual weight is a cross between corresponding weights 
 in 23 and 29. This resulting formula is called 1123 and 
 gives, as we shall see, virtually the same result as 123. 
 The result of the first kind of crossing, such as For- 
 mula 123, may be called a cross formula ; and that of the 
 second, such as 1123, a cross weight formula. 
 
 Numerically, Formula 23 (for prices for 1917 rel- 
 atively to 1913) gives 154.08, while 29 gives 170.44. 
 The ir cross by the g eometric mean, as per Formula 123, 
 is V154.08 X 170.44, or 162.05. So much for Formula 
 123, the cross formula between 23 and 29. 
 
 The cross weight formula involves more detail, for 
 we must first cross each of the 36 pairs of weights. For 
 bacon, the weight under Formula 23, that is, the value of 
 bacon in the base year, 1913, is 133.117 while the weight 
 under 29, that is, its value in the given year, 1917, is 282.743. 
 The cross of these weights (by the geometric mean) is 
 V133.117 X 282.743 = 193.86, which is the weight we 
 were seeking for bacon. Similarly, the weight for barley 
 is the cross between 111.607 and 276.549, which is 175.68 ; 
 similarly, the weight sought for beef is 1097.04, and so on. 
 Next we calculate a new index number based on these 
 36 new weights but otherwise precisely analogous to For- 
 mulge 23 and 29. The result is found to be 161.62. This 
 is by Formula 1123. 
 
 Algebraically, Formula 123, the cross formula, is 
 V23 X 29. (The reader who chooses can substitute the 
 algebraic expressions for Formulas 23 and 29 as given 
 in Appendix V.) On the other hand. Formula 1123 — 
 the cross weight formula — is itself given fully in 
 Appendix V. The reader will observe that it is exactly 
 analogous to Formulae 23 and 29, the only difference being, 
 that, instead of the weights poqo, p'oQ'q, etc., as per For- 
 mula 23, or instead of the weights piQi, p'iq\, etc., as per 
 
186 
 
 THE MAKING OF INDEX NUMBERS 
 
 F ormula 29 , we now have the weights VpoqoPiqi , 
 ^p'oq'op'iq'i, etc. 
 
 § 2. The Cross Weight Geometries, Medians, and Modes 
 
 We have taken Formula 1123 as the first illustration of a cross weight 
 formula. It was derived by crossing the weights in Formulae 23 and 29 
 and, on their model, writing a new formula. The same method may be 
 used for combining any two formulte of the same model differing only in 
 their weights. But, it is interesting to observe, if we thus combine For- 
 mulse 25 and 27 we get identically the same result as we have just obtained 
 by combining 23 and 29; for the cross weights in the first case are 
 V'(pogo) X ipiqi), etc., and in the second, V(po5i) X (pi^o), etc., which 
 are evidently the same. Thus Formula 1123 may be just as truly said to 
 come from 25 and 27 as from 23 and 29. On the other hand, the cross 
 formula, 123, is made only from 23 and 29 ; that from Formulae 25 and 27 
 is 125, which is slightly different. 
 
 Likewise we designate by 1133 the formula derived by crossing the 
 weights of the medians, 33 and 39, or of 33 and 37; and by 1143 that by 
 crossing the weights of the modes, 43 and 49, or 45 and 47. Formula 
 1133 agrees closely with 133 and 1143 with 143. 
 
 The preceding formulae, i.e. the cross weight geometries, medians, and 
 modes have been given first because they resemble each other so closely 
 and are the simplest of the six types. 
 
 Table 13 contains the identification numbers for the geometries, medi- 
 ans, and modes, (1) of primary formulae, and (2) and (3) of the two kinds 
 of derivatives from them — the cross formulae and the cross weight 
 formulae. 
 
 TABLE 13. 
 
 DERIVATION OF CROSS FORMULA AND 
 CROSS WEIGHT FORMULAE 
 
 Ttpe 
 
 (1) 
 
 Primary 
 
 Formula 
 
 TO 3E Combined 
 
 Combined 
 (2) (3) 
 
 
 By Crossing 
 the Two Formute 
 
 By Crossing 
 Their Weights 
 
 
 23 and 29 
 25 and 27 
 
 33 and 39 
 35 and 37 
 
 43 and 49 
 
 45 and 47 
 
 123 
 125 
 
 133 
 135 
 
 143 
 
 145 
 
 1123 
 
 Median 
 
 1133 
 
 Mode 
 
 1143 
 
 
 
RECTIFYING BY CROSSING WEIGHTS 187 
 
 § 3. The Cross Weight Aggregatives 
 
 The process of deriving a price index by crossing the 
 weights of the two weighted aggregatives (which we nnay 
 here refer to as Formulse 53 and 59) is sHghtly different, 
 since the weights are not values (Uke poQ'o and piqi), but 
 only quantities (like go and gi). The resulting formula 
 is 1153 of the same model as 53 and 59, but with weights 
 (Vgogi, etc.) which are the crosses of their weights. It 
 agrees closely with Formula 153. 
 
 We have now considered the cross weight geometries, 
 medians, modes, and aggregatives. 
 
 There remain only the arithmetics and harmonics, 
 which will be considered shortly. 
 
 § 4. Comparisons of the Cross Weight FormulaB 
 thus far Obtained 
 
 All the cross weight types just given satisfy Test 1. 
 This may readily be proved in the usual manner by inter- 
 changing the ''O's" and the "I's" in the formulae of Ap- 
 pendix V. Furthermore, each cross weight formula 
 agrees almost exactly with the corresponding cross for- 
 mula, except the median. That is, Formula 1123 is vir- 
 tually the same as 123 or 125, 1143 as 143 or 145, 1153 
 as 153 ( = 353) . Table 14 shows some of these similarities. 
 
 Graphically, the curves representing cross formulae 
 and the curves representing cross weight formulas are 
 indistinguishable, except in the case of the median, as 
 shown in Charts 43P and 43Q. 
 
 § 5. Cross Weight Arithmetics and Harmonics 
 
 There remain to be described the cross weight arithmetic and harmonic 
 formulse. These are numbered 1003 and 1013. In the above table 
 they are not represented, as there were no corresponding cross for- 
 mulae in our previous tables, for the reason, of course, that arithmetic for- 
 
188 
 
 THE MAKING OF INDEX NUMBERS 
 
 mulse are crossed not with other arithmetics, but with harmonics, and, 
 vice versa, harmonics with arithmetics. 
 
 Thus Formula 103 was a cross between 3 and, not 9, but 19 ; Formula 
 104 was a cross between 4 and 20; Formula 107 was a cross between 7 
 and 15 ; etc. But, while we can thus cross two formulw, one of which is 
 arithmetic and the other harmonic, crossing weights of two formulae im- 
 plies that they are both of the same model, differing only in their weights. 
 If the models of two formulae differ we would not know which model to 
 
 Close Agreement oF 
 Cross Formulae and Cross Weight Formulae 
 (Prices) 
 
 1103. m 
 
 1104.104 
 
 13 M 15 16 17 18 
 
 Chart 43P. The pairs indicated practically coincide (1103 with 103, 
 1104 with 104, etc.) except in the case of the medians. All the 16 formulae, 
 shown in pairs in these eight separate diagrams, obey Test 1 but not Test 2. 
 
 use in building the proposed cross weight formula. Thus, if we should 
 cross the weights of an arithmetic and a geometric formula we would not 
 know what to do with the weights after we had them. It is equally mean- 
 ingless to cross the weights of an arithmetic and harmonic. 
 
 In short, crossing weights is meaningless except as applied to two of a 
 kind, such as to two arithmetics or to two harmonics — not one of each ; 
 and when it is applied to two arithmetics or to two harmonics the result- 
 ing cross weight formulae (unlike the other four tj^jes of cross weight for- 
 mulae considered hitherto) will fail to satisfy Test 1. This is another 
 interesting result of the one-sidedness of the arithmetic and of the har- 
 monic. 
 
RECTIFYING BY CROSSING WEIGHTS 189 
 
 Close Agreement of 
 
 Cross^Formulae and Cross-Weight Formulae 
 (Quantifies) 
 
 .103.1103 
 124,1,24 
 
 
 /J lA 15 '16 17 
 
 Chart 43Q. Analogous to Chart 43P. 
 
 7d 
 
 TABLE 14. INDEX NUMBERS BY CROSS WEIGHT FORMULA 
 (1123, 1133, 1143, 1153) COMPARED WITH INDEX NUMBERS 
 BY CORRESPONDING CROSS FORMULAE (123, 133, 143, 153) ^ 
 
 
 Pbices 
 
 FOBMUIiA No. 
 
 
 
 
 
 
 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 123 
 
 100. 
 
 100.12 
 
 99.94 
 
 113.83 
 
 162.05 
 
 177.80 
 
 1123 
 
 100. 
 
 100.14 
 
 99.89 
 
 114.17 
 
 161.62 
 
 177.87 
 
 133 
 
 100. 
 
 100.54 
 
 99.68 
 
 108.12 
 
 159.93 
 
 173.57 
 
 1133 
 
 100. 
 
 100.52 
 
 99.57 
 
 108.39 
 
 162.63 
 
 170.85 
 
 143 
 
 100. 
 
 101. 
 
 100. 
 
 108. 
 
 164. 
 
 168. 
 
 1143 
 
 100. 
 
 101. 
 
 100. 
 
 108. 
 
 164. 
 
 168. 
 
 153 
 
 100. 
 
 100.12 
 
 99.89 
 
 114.21 
 
 161.56 
 
 177.65 
 
 1153 
 
 100. 
 
 100.13 
 
 99.89 
 
 114.20 
 
 161.70 
 
 177.83 
 
 * Omitting, for brevity, the cross formulsB 125 and 145, which agree closely with 123 
 and 143 respectively ; and 135, which also agrees closely with 133 except in the years 1917 
 and 1918 when the former is 162.00 and 178.44 as contrasted with the 159.93 and 173.57 
 for 133 as given in the table. 
 
190 
 
 THE MAKING OF INDEX NUMBERS 
 
 
 Quantities 
 
 FormuijA No. 
 
 
 
 
 
 
 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 123 
 
 100. 
 
 99.30 
 
 109.14 
 
 118.92 
 
 118.85 
 
 125.01 
 
 1123 
 
 100. 
 
 99.34 
 
 109.07 
 
 118.79 
 
 118.82 
 
 125.31 
 
 133 
 
 100. 
 
 98.60 
 
 105.58 
 
 115.82 
 
 118.16 
 
 122.94 
 
 1133 
 
 100. 
 
 98.71 
 
 105.46 
 
 115.50 
 
 118.23 
 
 122.27 
 
 143 
 
 100. 
 
 97. 
 
 103. 
 
 103. 
 
 98. 
 
 124. 
 
 1143 
 
 100. 
 
 97. 
 
 103. 
 
 103. 
 
 98. 
 
 124. 
 
 153 
 
 100. 
 
 99.33 
 
 109.10 
 
 118.85 
 
 118.98 
 
 125.37 
 
 1153 
 
 100. 
 
 99.33 
 
 109.08 
 
 118.82 
 
 118.86 
 
 125.29 
 
 Thus, Formulae 1003, 1004, in which cross weights are used to unite 
 pairs of arithmetics, and Formulae 1013, 1014, in which they are likewise 
 applied to harmonics, correspond to no cross formulae given in our main 
 series. This is why we have numbered them 1003, etc., and not 1103, 
 etc. If we wish to construct cross formulae which correspond to the new 
 cross weight 1003, 1004, 1013, 1014, we need to cross 3 and 9; 4 and 10; 
 13 and 19; 14 and 20. This is done in Table 15 for the purpose of 
 comparison. 
 
 TABLE 15. INDEX NUMBERS BY CROSS WEIGHT FORMULA 
 (1003, 1004, 1013, 1014) COMPARED WITH CORRESPONDING 
 CROSS FORMULAE 
 
 (1913 = 100) 
 
 Formula No. 
 
 Prices 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 
 100.43 
 100.45 
 
 100.99 
 100.93 
 
 116.17 
 116.02 
 
 171.14 
 170.81 
 
 
 v^3 X 9 
 1003 
 
 182.46 
 182.54 
 
 
 99.51 
 
 99.47 
 
 98.52 
 98.60 
 
 112.71 
 
 112.84 
 
 157.98 
 158.01 
 
 
 V4 X 10 
 1004 
 
 173.30 
 173.03 
 
 
 99.79 
 99.81 
 
 98.96 
 98.91 
 
 112.67 
 112.53 
 
 153.96 
 153.51 
 
 
 Vi3 X 19 
 1013 
 
 172.95 
 173.02 
 
 
 100.87 
 100.83 
 
 101.03 
 101.10 
 
 115.43 
 115.54 
 
 165.19 
 165.24 
 
 
 Vl4 X20 
 1014 
 
 183.74 
 182.94 
 
 Here, as before, the cross weight formulae and the cross formulae 
 
RECTIFYING BY CROSSING WEIGHTS 191 
 
 agree almost perfectly. They represent, essentially, two different routes 
 toward the same result. Neither satisfies Test 1 (nor Test 2 for that 
 matter). 
 
 § 6. The Cross Weight Formulas Derived from the 
 Factor Antitheses of the Preceding 
 
 In our weight crossing, in the case of the geometries, medians, and modes, 
 we have taken only the odd numbered formulae. But we may, in like 
 manner, cross the weights employed in Formulae 24 and 30 {i.e. in their 
 denominators) and so build up a new formula on their model. This is 
 called Formula 1124. It is also the cross weight formula from Formulae 
 26 and 28. Likewise we derive Formula 1134 from 34 and 40 (or from 
 36 and 38), and Formula 1144 from 44 and 50 (or from 46 and 48). 
 
 We have now derived, as our complete list of cross weight formulae of 
 odd numbers: Formula 1003, 1013, 1123, 1133, 1143, 1153 and also those 
 to which we have given the corresponding even numbers : Formulae 1004, 
 1014, 1124, 1134, 1144, 1154. But all except Formula 1154 of the latter 
 six even numbered formulae were derived, not as antitheses (although 
 such they are) ^ of the six corresponding odd numbered formulae. They 
 were derived directly by crossing the weights of 4 and 10, 14 and 20, 24 
 and 30, 34 and 40, 44 and 50. 
 
 § 7. Cross Weight Arithmetics and Harmonics are 
 not Truly Rectified 
 
 As stated, the arithmetic Formula 1003 is not analogous to the truly 
 rectified 103; nor is 1013. There is no way whatever, through weight 
 crossing alone, to rectify the arithmetics alone or the harmonics alone 
 relatively to Test 1. To get truly rectified formulae the above results 
 (1003 and 1013) have still to be crossed with each other. That is, in this 
 case, the method of crossing weights must be eked out by the method of 
 crossing formulae. Crossing, then, 1003 and 1013, we obtain a new for- 
 mula, numbered 1103, which does satisfy Test 1 and is the nearest ap- 
 proach to a cross weight formula analogous to the cross formula, 103. 
 Moreover, their results practically coincide. Similarly (by crossing 1004 
 and 1014), we get the new Formula 1104, corresponding to 104. 
 
 Having reached Formulae 1103 and 1104, we insert them in Chart 43 
 as the nearest analogues of 103 and 104. We note that, here again, the 
 results of rectifying by crossing weights on the one hand, and by crossing 
 the formulae themselves on the other, coincide to all intents and purposes. 
 
 We may now add to Table 13 in § 2 the following : 
 
 See Appendix I (Note to Chapter VIII, § 6). 
 
192 THE MAKING OF INDEX NUMBERS 
 
 
 Primary 
 FoRMtri^ 
 
 TO BE 
 
 Combined 
 
 Combined 
 
 Type 
 
 By Crossing the 
 Two Formulse 
 
 By Crossing Their 
 Weights 
 
 By Crossing ihe 
 Two Formulae in 
 the Last Column 
 
 
 3 and 9 
 5 and 7 
 
 13 and 19 
 15 and 17 
 
 3 and 19 
 Sand 17 
 
 omitted ' 1 
 omitted * J 
 
 omitted * 1 
 omitted ' / 
 
 103 
 105 
 
 1003 
 
 1013 
 
 impossible 
 
 1 
 
 
 > 1103 
 
 Arithmetic and Harmonic 
 
 
 ' "Omitted" means that no identification number was given to these crosses as they 
 eerve no purpose in our main series. But the figures for some of these formulse (for prices, 
 fixed base) were calculated and given in § 5 above. 
 
 Close Agreement of Cross-Formulae and 
 
 Cross- Weight Formulae (Fully Rectified) 
 (Prices) 
 
 '303.303 
 323./323 
 J33.,yj3 
 13^3.353 
 
 15 
 
 M 
 
 15 
 
 16 
 
 17 
 
 Id 
 
 Chart 44P. Analogous to Chart 43P, except that here both tests are 
 fulfiUed. 
 
 § 8. List of the Formulae Obeying Test 1 Derived 
 Partly or Wholly by Weight Crossing 
 
 We see that the arithmetic 1003 and the harmonic 1013, and their 
 factor antitheses, 1004 and 1014, all derived by weight crossing, had 
 to be used merely as a preliminary scaffolding for building 1103, derived 
 partly by formula crossing. After discarding the scaffolding our new 
 formulae are 1103, 1123, 1133, 1143, 1153, and these supplementary for- 
 
RECTIFYING BY CROSSING WEIGHTS 193 
 
 mulsB agree almost precisely with their mates (103, 123, 133, 143, 153) in 
 the main series. Their factor antitheses (the next even numbered for- 
 mulae, 1104, 1124, 1134, 1144, 1154) likewise agree closely with their mates 
 (104, 124, 134, 144, 154). 
 
 § 9. Rectifying the New Fomnilae by Test 2 
 
 The new formulae, 1103, 1123, 1133, 1143, 1153 (and their factor an- 
 titheses, the next even numbered formulae) all satisfy Test 1, as do the 
 corresponding formulae in the main series. But not a single one of them 
 satisfies Test 2 (although in the main series one formula, the analogue 
 of 1153, namely, 153, does satisfy Test 2), 
 
 Close Agreement of Cross-Formulae and 
 
 Cross-Weight formulae (Fully RecUfied) 
 (Quanfliies) 
 
 I353.33S 
 
 13 
 
 •M '15 IS 17 
 
 Chaet 44Q. Analogous to Chart 44P. 
 
 7(3 
 
 In order to obtain conformity to Test 2 we must further rectify and, 
 for this purpose, the only process of combining the factor antitheses is by 
 crossing the formulae themselves. Crossing their weights is inapplicable 
 because the two formulae to be combined are, in every instance, of different 
 models. The doubly rectified formulae numbers are given in the last 
 column of Table 16. 
 
 These pairs of corresponding formulae satisfying both tests agree with 
 each other even more perfectly than did the pairs satisfying only Test 1 
 agree with each other. That is, Formula 303 agrees almost exactly with 
 1303, Formulae 323 and 325 with 1323, etc. 
 
 Graphically, Charts 44 show the almost perfect iden- 
 tity of 1303 with 303 and of 1323 with 323, of 1333 with 
 333, 1353 with 353, and 1343 with 343 (or would, were 
 the last two indicated). 
 
 Moreover, we may note in passing, that the entire group 
 
194 THE MAKING OF INDEX NUMBERS 
 
 TABLE 16. DOUBLY RECTIFIED FORMULA DERIVED FROM 
 PRIMARY WEIGHTED FORMULAE 
 
 Wholly bt Formula 
 Crossing 
 
 103 = 104 = 105 = 106 
 (= 153 = 15 4) 
 V' lO/ X 1U8 
 •V^ lOO X 110 
 "^" 123 X 124 
 V l25 X 126 
 V' l33 X 134 
 •V^ 135 X 136 
 Vi43_2<_144 
 Vl45 X 146 
 153 = 154 
 ( = 103 = 104 = 105 = 106) 
 
 Results in Main 
 Series 
 
 = 303 = 305 
 ( = 353) 
 = 307 
 = 309 
 
 = 323 1 
 = 325 J 
 
 = 333 1 
 = 335 / 
 
 = 343 1 
 = 345 / 
 = 353 
 ( = 303 = 305) 
 
 Pahtlt by Weight 
 Crossing 
 
 ■^1103 X 1104 
 
 "^1123 X 1124 
 ^1133 X 1134 
 "^1143 X 1144 
 
 V 
 
 1153 X 1154 
 
 Results in 
 
 Supplementary 
 
 Series 
 
 = 1303 
 
 = 1323 
 = 1333 
 = 1343 
 = 1353 
 
 of rectified formulae, by both methods of crossing, agrees 
 almost absolutely, excepting only those originating from 
 modes and medians, and even the medians agree with the 
 rest well enough for most practical purposes. This re- 
 markable agreement is clear from a study of the figures 
 for all the index numbers given in Appendix VII and will 
 be emphasized later. 
 
 § 10. Several Methods for Crossing Weights as Con- 
 trasted with Only One (in General) for Crossing the 
 Formulae Themselves 
 
 At the close of the last chapter it was shown that to 
 cross formulae the geometric method of crossing is uni- 
 versally appropriate (although in two instances the aggre- 
 gative method would also be applicable). But in weight 
 crossing the geometric method has no such preeminence 
 for we can equally well, in all cases, use the arithmetic 
 method or the harmonic method without prejudicing 
 the conformity of the result to the test v/hich we are 
 
RECTIFYING BY CROSSING WEIGHTS 195 
 
 seeking to meet. In this chapter, I have employed the 
 geometric method of crossing weights chiefly because it 
 is the form of crossing hitherto most in favor. The other 
 methods are discussed in the Appendix.^ They include 
 several interesting and ingenious suggestions which writers 
 on index numbers have made. But only one of them has 
 much practical value. That one (2153) is useful as a 
 short cut approximation to 353. 
 
 § 11. Conclusions 
 
 In this chapter we have obtained the following new 
 formula: 1003, 1004,1013,1014, 1103, 1104, 1123, 1124, 
 1133, 1134, 1143, 1144, 1153, 1154, 1303, 1323, 1333, 1343, 
 1353 ; and, in the Appendix, 2153, 2154, 2358, 3153, 3154, 
 3353, 4153, 4154, 4353. Of these, all coincide approxi- 
 mately with the middle tine of our fork, excepting the arith- 
 metics, Formulae 1003, 1004 (which have an upward 
 and downward bias respectively and which fall on the 
 mid-upper and mid-lower tines) ; and also excepting the 
 harmonics, 1013, 1014 (which have a downward and up- 
 ward bias respectively and which fall on the mid-lower and 
 mid-upper tines) ; and also excepting the modes, 1143, 
 1144, 1343, which are erratic; and, excepting also, 
 possibly, the medians, 1133, 1134, 1333, which ara 
 slightly erratic. 
 
 From what has been said it is now clear that crossing 
 the weights of two formulae of the same model and so 
 forming a new formula of that same model yields almost 
 identically the same numerical result as crossing the for- 
 mulae themselves. It is also clear that formula crossing 
 is a process which can be applied to any two formulae 
 whether the two be of the same model or not, whereas 
 weight crossing cannot be used except where the two 
 1 See Appendix I (Note to Chapter VIII, § 10). 
 
196 THE MAKING OF INDEX NUMBERS 
 
 formulse to be combined are built on exactly the same 
 model, differing only in their weights. 
 • In other words, formula crossing is a universal method 
 of compromising between two formulae, while weight 
 crossing is of restricted application. We found it incapable, 
 for instance, of rectifying any formula by Test 2, and 
 even incapable of rectifying some formulse by Test 1. In 
 short, weight crossing is never necessary and is some- 
 times inapphcable. 
 
 § 12. Historical 
 
 It is rather odd, therefore, that hitherto the simpler 
 and more universally serviceable of the two processes 
 has been almost wholly overlooked. The reason is his- 
 torical tradition. In the history of the index numbers 
 the first stage was to discuss the virtues of the simple 
 index numbers, chiefly the arithmetic and geometric. 
 The next step was to assign weights supposed to be rep- 
 resentative of the conditions prevailing in the periods 
 concerned. Drobisch was, apparently, the first to make 
 use specifically of the quantities of two years compared 
 by an index number. 
 
 Following this line of study, Scrope and Walsh pro- 
 posed the cross weight aggregative formula here num- 
 bered 1153, and Walsh also 1154; Marshall and Edge- 
 worth proposed the cross weight aggregative, Formula 
 2153; Walsh, Formula 2154; Lehr, Formulae 4153 and 
 4154, and Walsh the cross weight geometric, Formula 
 1123. Of the cross formulae, 8053 (see Appendix V) was 
 suggested by Drobisch and Sidgwick. Finally, Formula 
 353, of which more will be said later, was first mentioned, 
 though not at that time advocated, by Walsh. 
 
CHAPTER IX 
 
 THE ENLARGED SERIES OF FORMULA 
 
 § 1. Introduction 
 
 Thus far we have accomplished three chief things. 
 We have shown : 
 
 (1) That there are two important reversibiUty tests 
 of index numbers ; ' 
 
 (2) That certain formulae have a " bias " or constant 
 tendency to err relatively to Test 1 ; 
 
 (3) That any formula whatever can be " rectified " so 
 as to conform to either test or both. 
 
 In the course of this study, we have constantly added 
 to the number of formulae demanding consideration. 
 Before proceeding to compare all these formulae as to their 
 relative accuracy, we may now pause to " take account of 
 stock " and also complete our Hst by the addition of ten 
 more formulae. 
 
 We first set forth the main series of 96 formulae (original 
 and derivative) of which those having identification 
 numbers between 1 and 99 were the primary formulae; 
 those having identification numbers between 100 and 199 
 conformed to Test 1 ; those having identification numbers 
 between 200 and 299 conformed to Test 2; and those 
 having identification numbers between 300 and 399 con- 
 formed to both Test 1 and Test 2. The last and culminat- 
 ing one of these formulae, 96 in number, was Formula 353, 
 
 4 
 
 Xpoqo 2pogi ' 
 
 We shall find this to be theoretically the best formula.^ 
 
 1 For model examples to aid in the practical calculation of this as well 
 as eight other sorts of index numbers, see Appendix VI, § 2. 
 
 197 
 
198 THE MAKING OF INDEX NUMBERS 
 
 To this list of 96 formulae we have just added a supple- 
 mentary list of 28 more formulae, which owe their origin 
 to the process called weight crossing (in place of formula 
 crossing employed in the main series). 
 
 These 28 new formulie are as follows : Those numbered between 1000 
 and 1999, originated in crossing weights geometrically two by two ; those 
 between 2000 and 2999, originated in crossing them arithmetically; those 
 between 3000 and 3999, originated in crossing them harmonically; those 
 between 4000 and 4999, originated in crossing them by means of a special 
 weighted arithmetical average. 
 
 To these 124 formulae we now add ten miscellaneous 
 formulae, which make 38 in the supplementary series, 
 in addition to the 96 in the main series, or 134 in all. 
 
 They are : Those between 5000 and 5999, formed by crossing formulae 
 in the "300" list ; those between 6000 and 6999, formed by using a broader 
 base than one year ; those between 7000 and 7999, formed by averaging the 
 six forms of Formula 353 obtained by using each of the six years as base ; 
 those between 8000 and 8999, formed by crossing formulae arithmetically 
 and harmonically ; those between 9000 and 9999, formed by using round 
 numbers as weights. 
 
 More specifically, these final ten miscellaneous formulae are as follows : 
 As to the 5000's : 
 
 Formula 5307 is the cross between Formulae 307 and 309 ; 
 
 Formula 5323 is the cross between Formulae 323 and 325 : 
 
 Formula 5333 is the cross between Formulae 333 and 335 ; 
 
 Formula 5343 is the cross between Formulae 343 and 345. 
 As to the 6000's : 
 
 Formulae 6023 and 6053 are like 23 and 53 respectively, except that 
 instead of the first year being the base, the base is an average made up of 
 two or more years. 
 
 Formula 7053 is an average of six forms of 353, with six different bases. 
 
 Formula 8053 is the arithmetic average of 53 and 54. Formula 8054 
 is the harmonic average of the same, as well as the factor antithesis of 
 8053. It may be shown that the cross between Formulae 8053 and 8054 is 
 identical with 353. » 
 
 We may classify the 134 formulae which have been 
 noted. They will be classified under five heads, according 
 as they owe their origin to (1) arithmetics and harmonics, 
 (2) geometries, (3) medians, (4) modes, (5) aggrega- 
 tives. 
 
 1 See Appendix I (Note to Chapter IX, § 1). 
 
THE ENLARGED SERIES OF FORMULiE 199 
 
 § 2. List of the Arithmetic and Harmonic Formulae 
 
 The first group, to which Table 17 is devoted, includes the two types, 
 the arithmetics and the harmonics, since in the crossing, which we found 
 necessary, these two could not be kept apart. 
 
 The two upper lines relate to the simples and their derivatives, and the 
 eight lines following relate to the weighted and their derivatives. The 
 first column gives the arithmetic, the second, the harmonic, the 
 third, the derived cross formulce satisfying Test 1, the fourth and fifth, 
 the cross formulae satisfying Test 2, the sixth, the cross formulae satis- 
 fying both tests, thus completing the arithmetic and harmonic formulae 
 in the main series. The remaining columns give the cross weight formulce and 
 their crosses. 
 
 A dash indicates a formula omitted because duplicated elsewhere. 
 These duplications are given below Table 17. In the same way, the du- 
 plications of Tables 18, 19, 20, and 21 are given below them. 
 
 TABLE 17. ENLARGED ARITHMETIC-HARMONIC GROUP 
 
 Primary 
 
 Formulae 
 
 Cross FoRMULuSi 
 
 Cross Weight FoRMULis and Their 
 Crosses 
 
 M 
 'k 
 
 w 
 
 H 
 ^ 
 
 By Test 2 
 
 o 
 
 m 
 
 Arith. 
 
 Harm. 
 
 o d . 
 
 o 
 
 6 
 
 J 
 
 Eh 
 
 11 
 
 Ucoco 
 
 1 
 
 2 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 13 
 14 
 15 
 16 
 
 101 
 102 
 
 107 
 108 
 109 
 110 
 
 201 
 
 207 
 209 
 
 211 
 213 
 215 
 
 301 
 
 307 
 309 
 
 1003 
 1004 
 
 1013 
 1014 
 
 1103 
 1104 
 
 1303 
 
 5307 
 
 Duplicationa (indicated above by dashes) : 
 
 3 = 53 17 = 53 103 = 353 203 = 353 303 = 353 
 
 4 = 54 18 = 54 104 = 353 205 = 353 305 = 353 
 
 5 = 54 19 = 54 105 = 353 217 = 353 
 
 6 = 53 20 = 53 106 = 353 219 = 353 
 
 The above table covers two of our six types. Each of the following 
 four tables covers one. The next three are alike in form. 
 
200 
 
 THE MAKING OF INDEX NUMBERS 
 
 § 3. List of the Geometric, Median, and Mode 
 Groups of Formulae 
 
 The following three tables give lists of all the formulae in the geometric, 
 median, and mode groups, including all derivatives. 
 
 TABLE 18. ENLARGED GEOMETRIC GROUP 
 
 Primary 
 
 Cross Formula 
 
 Cross Weight Formula and 
 Their Crosses 
 
 Cross op 
 
 323 AND 
 
 325 
 
 Formulae: 
 
 By Test 1 
 
 By Test 2 
 
 By Both 
 
 Cross Weight 
 Formulffi 
 
 Their Cross 
 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 
 121 
 122 
 123 
 124 
 125 
 126 
 
 223 
 225 
 
 227 
 229 
 
 321 
 323 
 325 
 
 1123 
 1124 
 
 1323 
 
 5323 
 
 Duplications (indicated above by dashes) : 
 
 21 = 121 221 = 321 
 
 22 = 122 
 
 
 TABLE 
 
 19. ENLARGED ] 
 
 MEDIAN 
 
 GROUP 
 
 
 Peimaht 
 
 FOBMUUE 
 
 Cross Fobmuls! 
 
 Cross Weight FoRMULiB 
 AND Their Crosses 
 
 Cross op 
 
 By Test 1 
 
 By Test 2 
 
 By Both 
 
 Cross 
 Weight 
 Formulae 
 
 Their Cross 
 
 l333 AND 
 
 335 
 
 
 
 131 
 
 
 
 331 
 
 
 
 
 
 
 132 
 
 
 
 
 
 
 33 
 
 133 
 
 233 
 
 333 
 
 1133 
 
 1333 
 
 5333 
 
 34 
 
 134 
 
 
 
 1134 
 
 
 
 35 
 
 135 
 
 235 
 
 335 
 
 
 
 
 36 
 
 136 
 
 
 
 
 
 
 37 
 
 
 237 
 
 
 
 
 
 38 
 
 
 
 
 
 
 
 39 
 
 
 239 
 
 
 
 
 
 40 
 
 
 
 
 
 
 
 Duplications (indicated above by dashes) : 
 
 31 = 131 231 = 331 
 
 32 = 132 
 
THE ENLARGED SERIES OF FORMULA 201 
 
 TABLE 20. ENLARGED MODE GROUP 
 
 Primary 
 
 Formula 
 
 Cross Formula 
 
 Cross Weight Formula 
 AND Their Crosses 
 
 Cross op 
 
 By Test 1 
 
 By Test 2 
 
 By Both 
 
 Cross 
 Weight 
 Formulae 
 
 Their Cross 
 
 343 AND 
 
 343 
 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 
 141 
 142 
 143 
 144 
 145 
 146 
 
 243 
 
 245 
 247 
 249 
 
 341 
 
 343 
 345 
 
 1143 
 1144 
 
 1343 
 
 5343 
 
 Duplications (indicated above by dashes) : 
 
 41 = 141 241 = 341 
 
 42 = 142 
 
 § 4. List of the Aggregative Formulae 
 
 Finally, we have the aggregative group. 
 
 TABLE 21. ENLARGED AGGREGATIVE GROUP 
 
 Primary 
 
 Cross Formula 
 
 Cross Weight Formula and Their 
 Crosses 
 
 Formula 
 
 By Test 1 
 
 By Test 2 
 
 By Both 
 
 Cross Weight 
 Formulse 
 
 Their Cross 
 
 53 
 
 54 
 
 151 
 152 
 
 — 
 
 351 
 
 353 
 
 1153 
 1154 
 
 1353 
 
 Duplications (indicated above — except 59, 60, 259 omitted — by dashes) : 
 51 = 151 153 = 353 251 = 351 
 62 = 152 154 = 353 253 = 353 
 
 59 = 54 259 = 353 
 
 60 = 53 
 
 The preceding lists do not include certain other forms discussed in 
 the Appendix/ namely, Formula 2153, the cross weight by the arithmetic 
 method of crossing ; 3153, the cross weight by the harmonic method; 4153, 
 
 1 See Appendix I (Note to Chapter VIII, § 10). 
 
202 THE MAKING OF INDEX NUMBERS 
 
 the cross weight by Lehr's method of taking a weighted arithmetic average 
 of the weights ; the factor antitheses (2154, 3154, 4154) of these three cross 
 weight formulae and the rectifications (2353, 3353, 4353) by crossing said 
 antitheses (2153 with 2154, etc.). Besides these are a few other mis- 
 ceUaneous forms (6023, 6053, 7053, 8053, 8054, 9051). 
 
 § 5. The Seven Classes 
 
 The 134 formulae constitute the enlarged series of for- 
 mulae embracing all considered in this book. Including 
 duplicates the number is 170 ; besides these there are the 
 five formulae (9001, 9011, 9021, 9031, 9041) given in Appen- 
 dix V, § 3. 
 
 Our problem now is to examine and discriminate between 
 these 134 formulae, — in particular to explain their 
 differences and to select the best. These 134 index num- 
 bers have been classified by type, weighting, and method 
 of crossing. We may also, for convenience in our dis- 
 cussion, classify them under the following seven groups : 
 
 S, the simple index numbers and their derivatives, 
 
 M, the medians and modes and their derivatives, 
 
 2 +, all other weighted index numbers having a double 
 upward bias, 
 
 2 — , all other weighted index numbers having a double 
 downward bias, 
 
 1 +, all other weighted index numbers having a single 
 upward bias, 
 
 1 — , all other weighted index numbers having a single 
 downward bias, 
 
 0, all other weighted index numbers having no bias. 
 
 These seven groups are mutually exclusive, except 
 that the simple modes and the simple medians, and their 
 derivatives, are included under both the first two headings. 
 
 § 6. The Formulae Grouped under the Seven Classes 
 
 The following is a list of the formulse in each of the first two groups : 
 - Group "S": 1, 2, 11, 12, 101, 102, 201, 211, 301, 21, 22, 321, 31, 32, 
 331, 41, 42, 341, 51, 52, 351. 
 
THE ENLARGED SERIES OF FORMULA 203 
 
 Closely associated with the "S" group, though not strictly members of 
 it, are : 9001, 9021, 9031, 9041, 9051.i 
 
 Group "M" : 31-40 inclusive, 
 
 133-136 inclusive, 
 233, 235, 237, 239, 
 331, 333, 335, 
 1133, 1134, 1333, 
 5333. 
 
 41-50 inclusive, 
 143-146 inclusive, 
 243, 245, 247, 249, 
 341, 343, 345, 
 1143, 1144, 1343, 
 5343. 
 (31, 32, 331, 41, 42, 341 are in both the "S" and "M" groups.) 
 
 The other five groups, {i.e. excluding "S" and "M") fall in the five tines 
 of our five-tined fork (or, if we wish to avoid, so far as possible, any blurring 
 of the tines, two such forks, one for the odd, and the other for the even 
 numbered formulae), according to Table 22. 
 
 The formulse hold their approximate positions on the 
 " five-tined fork " wholly according to the following fixed 
 rules : 
 
 Those which have no bias lie approximately in coinci- 
 dence and constitute the middle tine. Those which have 
 only one upward bias, whether type bias or weight bias, 
 likewise agree and form the mid-upper tine. Similarly, 
 those which have only one downward bias, whether type 
 or weight, make the mid-lower tine. Those which have 
 a double upward bias, i.e. a type bias and a weight bias, 
 make the uppermost tine. Likewise those doubly biased 
 downward make the lowermost tine. 
 
 The case of a downward bias of one sort and an upward 
 bias of another being combined is also provided for. Such 
 a curve turns out to have no bias at all, being merely 
 erratic. Therefore, it also lies on the middle tine. For- 
 mula 3 is one of these. As an arithmetic type it has an 
 upward bias, but having weight / it has a downward bias, 
 
 ^ Given in Appendix V, § 3. 
 
204 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 22. THE FIVE-TINED FORK 
 
 Tine 
 
 Arithmetic 
 
 Harmonic 
 
 Geometric 
 
 Aqgbegativi: 
 
 Uppermost (2 +) 
 
 7,9 
 
 14, 16 
 
 
 
 Mid-upper (1 +) 
 
 1003 
 
 1014 
 
 24, 26, 27, 29 
 
 
 
 3=6=(L), 
 4 = 5 = (P) 
 
 17=20=(L), 
 18 = 19=(F) 
 
 
 53=60 = (L), 
 54 = 59 = (P) 
 
 
 107, 108, 109, 110, 
 1103, 1104 
 
 123, 124, 125, 
 126,1123,1124 
 
 1153, 1154 
 
 Middle (0) 
 
 207, 209 
 
 213, 215 
 
 223, 225, 
 227, 229 
 
 2153, 2154, 
 3153, 3154, 
 4153, 4154 
 
 
 203=205 = 
 
 217=219 = 
 
 323, 325, 
 1323 
 
 153 = 154 = 
 253=259 = 
 
 
 
 
 103=104= 
 303 = 305= 
 
 307, 30 
 
 lU5=lUb= 
 
 353=VLXP, 
 
 
 WLXP) 
 9, 1303 
 
 1353, 
 
 2353, 3353, 
 4353, 5307, 
 5323, 6053, 
 7053, 8053, 
 8054 
 
 Mid-lower (1 — ) 
 
 1004 
 
 1013 
 
 23, 25, 28, 30 
 
 
 Lowermost (2 — ) 
 
 8, 10 
 
 13, 15 
 
 
 
 and the two neutralize ; for, after cancellation, 3 reduces 
 to 53, which is of such a type that we cannot accuse it of a 
 proneness to err up rather than down or down rather than 
 up. 
 
 Thus, barring " simples " and " modes " and their 
 derivatives (and possibly medians if we wish to have our 
 results very close), we find that, although we have numer- 
 ous f ormulse, they all fall under only five clearly defined 
 
THE ENLARGED SERIES OF FORMULAE 205 
 
 heads, namely, those without bias, those with single bias 
 up or down, and those with double bias up or down. 
 
 The five tines include all the arithmetic, harmonic, 
 geometric, and aggregative weighted index numbers and 
 their derivatives which we have obtained. 
 
CHAPTER X 
 
 WHAT SIMPLE INDEX NUMBER IS BEST? 
 
 § 1. Introduction 
 
 Our next problem is to compare the numerous formulae 
 which we have found and to select the theoretically best 
 formula or formulae, i.e. the most accurate. This problem 
 may conveniently be subdivided into two parts, viz. : 
 
 1. Assuming that we have no weights available so that 
 we are compelled to use simple averages, which index 
 number then is best ? 
 
 2. Assuming, on the contrary, that we do have the 
 data for assigning unequal weights, which index number 
 then is the best ? 
 
 In this chapter, we shall take up the first of these 
 two problems. The assumption that there are no data 
 for weights at once removes from our list of index numbers 
 of prices all the even numbered ones, and those derived from 
 them; since each of these was obtained by dividing an 
 index number of quantities into a ratio of values, and, 
 therefore, presupposes a knowledge of values and quanti- 
 ties, which are the data for assigning weights. 
 
 Obviously, also, our assumption rules out all the weighted 
 index numbers and their derivatives. The only index 
 numbers now left are : Formulae 1, 11, 21, 31, 41, 51, 101. 
 Our problem, therefore, reduces itself to selecting the 
 best from these seven formulae. 
 
 § 2. Discarding the Two Biased Formulae 
 
 Proceeding by a process of elimination, we may discard 
 Formulae 1 and 11 as they possess an upward and a down- 
 
 206 
 
WHAT SIMPLE INDEX NUMBER IS BEST? 207 
 
 ward bias respectively. This has been proved by Test 1, 
 the time reversal test. Formulae 21, 31, 41, 51, and 101 
 meet successfully Test 1, as has also been proved. Our 
 hypothesis, that no data for quantities or values are 
 available, prevents the appHcation of Test 2, the factor 
 reversal test, since this involves a knowledge of values. 
 
 § 3. Freakishness 
 
 So far as meeting tests is concerned, therefore, all five 
 of the remaining formulae stand on an equality. If we are 
 to discriminate further, it must be on some other basis. 
 Such a basis is what we have called freakishness. All 
 index numbers may be assumed to be somewhat erratic, 
 that is, no one is certain to be absolutely correct. But 
 some can be shown to be more erratic than others, that 
 is, more likely to err. A formula which can be shown to 
 be especially erratic, as compared with other formulae, has 
 been called freakish. 
 
 A biased formula errs in a given direction. An erratic 
 formula or a freakish formula may err in either direction. 
 
 § 4. Discarding Formula 51 as Freakish 
 
 Formula 51 may be discarded as freakish. As has been 
 noted before, while its weighting is called simple, it is not 
 simple in the same sense as the other four formulae. In 
 these four formulae the price relatives have equal weights. 
 But in Formula 51 it is the prices themselves which have 
 equal weights. Consequently, unlike the other four index 
 numbers, Formula 51 is affected by a change in the unit 
 in which any price is quoted. Its simple weighting is 
 thus quite arbitrary, or, as Walsh says, '' haphazard." 
 
 As Formula 51 is applied by Bradstreet, for instance, 
 the unit of each commodity is a pound. The index 
 number is found by taking the sum of the prices per pound 
 
208 THE MAKING OF INDEX NUMBERS 
 
 of a certain bill of goods. A pound of silver and a pound 
 of coal are counted as of equal importance. If the units 
 used in market quotations were employed so that the sum 
 was made up of the price per ounce of silver and the price 
 per ton of coal, the result would be quite different. 
 
 In the case of the aggregative, I doubt even whether the 
 general substitution of the pound for articles usually 
 measured in other units produces any improvement. 
 Most large units, like the ton or bale, are applied to coal 
 and hay merely to lift up the quotation to a figure com- 
 parable to those in which the smaller units are measured. 
 In other words, we avoid quoting hay per pound because 
 the resulting figure would be so small and out of line with 
 quotations of other market figures. Reversely, radium 
 is quoted per milligram and not per ton. 
 
 That is, custom has already unconsciously assigned 
 roughly adjusted weights in hitting upon the units respec- 
 tively applied not only to silver, coal, hay, radium, but 
 probably, to some extent, to almost everything. I am 
 therefore inclined to think that in using Formula 51 it is 
 better simply to add the newspaper quotations in pounds, 
 ounces, tons, yards, etc., indiscriminately rather than to 
 reduce them to one unit. This reduction is based on the 
 misconception that economic weighting is a physical 
 matter. 
 
 Nevertheless, custom has not done its job well. The 
 same substance is very inconsistently quoted according 
 to its various stages of manufacture. Cattle per head 
 and beef per pound give weights for Formula 51 widely 
 different. Iron per ton, copper per pound, pig iron per 
 ton, and tin plates per hundredweight are out of tune. 
 Formula 51, therefore, unless helped out by judicious 
 guessing will be apt to play freakish tricks upon the 
 user. Sometimes, in fact, unless there be some exercise 
 
WHAT SIMPLE INDEX NUMBER IS BEST? 209 
 
 of judgment, it would be difficult to say exactly how 51 
 is to be interpreted ; whether, for instance, cotton is to 
 be entered per bale or per pound, its quotation being 
 expressible both ways. Formula 51 is the only formula 
 among all the 134 where there is any such ambiguity. 
 All other formulae give the same results whether cotton 
 is measured in pounds or bales. 
 
 § 5. Discarding Formula 41 and Possibly Formula 31 
 
 as Freakish 
 
 Quite as appropriate, although in a different way, is 
 the term freakishness as applied to the mode and, in less 
 degree, to the median. While Formula 51 is too respon- 
 sive to changes in the things to be compared, 41 and 31 
 are less responsive than the other formulae to the influence 
 of change in any individual term. 
 
 The fatal weakness of the mode (which is to some 
 extent shared by the median) is that the process by which 
 it is calculated gives undue influence to the few price 
 relatives which happen to lie together in its vicinity, and 
 gives practically no voice at all to the rest of the price 
 relatives. 
 
 Thus, in our regiment of soldiers where we found the 
 modal height to be about 5 feet 9^ inches, this figure 
 would still be the mode even if each of the soldiers taller 
 than, say, 5 feet 10 inches were replaced by a new soldier 
 a foot taller than his predecessor ! Reversely, the 
 shorter men have practically no voice in determining the 
 mode. The modal soldier is thus not a fair representative 
 of the whole regiment because most of the soldiers may 
 be taller or shorter without making any difference to the 
 mode, just as a congressman is not a fair representative 
 of his district when chosen by a clique. 
 
 Where the number of price relatives is small the mode 
 
210 THE MAKING OF INDEX NUMBERS 
 
 is particularly haphazard. With a large number, the dis- 
 tribution assumes some regularity, and the mode becomes 
 more significant. Therefore the mode cannot properly 
 be used unless the number of items is great, and then it 
 should be thought of as only a rough approximation. 
 For this reason it is practically never worth while to use 
 the mode as an index number. It was (with some re- 
 luctance) included in my list because it has been dis- 
 cussed in connection with index numbers, and because 
 it serves as a foil in our comparisons. 
 
 § 6. Freakishness of Simple Median 
 
 The simple median is much more nearly representative 
 of all the price relatives than is the mode, and yet much less 
 representative than the other simple index numbers. 
 Any particular soldier in the regiment could be taken out 
 and replaced elsewhere by another, taller or shorter, 
 without displacing the median, so long as this change in 
 height of the particular soldier did not send him to the 
 other side of the median. All of the soldiers standing on 
 one side of the middle soldier (say the shorter side) could 
 be replaced by still shorter soldiers, even dwarfs, without 
 changing the median in the least. Or they could all be 
 replaced by taller soldiers up to the middle soldier's height 
 without depriving him of his median character as repre- 
 sentative of the regiment. Likewise, on the tall side, 
 all the soldiers could be replaced by giants or all could 
 shrink to the median, without changing the latter. In 
 short, the median, like the mode, is insensitive or unrespon- 
 sive. Every other index number, such as the arithmetic 
 or the geometric, would faithfully register some effect of 
 any change in the regiment, however slight. The extreme 
 end soldiers, exactly like those nearer the center, have some 
 voice and influence in determining the average height. 
 
WHAT SIMPLE INDEX NUMBER IS BEST? 211 
 
 If one of them grows even a quarter of an inch, the average 
 will be affected. The mode and median, on the other hand, 
 are not sensitive barometers but creaky weathervanes 
 which seldom change, and when they do, they change by 
 jumps. 
 
 If, then, we are justified in excluding Formula 51, on 
 account of its freakish weighting, and 41 and 31 on account 
 of their freakish insensitiveness, we have left out of our 
 original seven, only two index numbers, viz., 101 and 
 21, i.e. the arithmetic-harmonic and the geometric. 
 These two agree very closely, so that, so far as accuracy 
 is concerned, there is nothing to choose between them. 
 This close agreement is shown in the following table : 
 
 Formula 
 No. 
 
 Prices — Fixed Base 
 
 1914 
 
 191S 
 
 1916 
 
 1917 
 
 1918 
 
 21 
 101 
 
 95.77 
 95.75 
 
 96.79 
 96.80 
 
 121.37 
 121.38 
 
 166.65 
 166.60 
 
 180.12 
 179.09 
 
 § 7. Doubt as to Formula 31 vs. Formula 21 
 
 The above conclusion, that the geometric (or its equal, 
 Formula 101), is the best, has been reached, however, 
 only on the assumption that simple weighting is proper 
 weighting, an assumption which we know is not correct. 
 In the absence of available weights we are sometimes 
 forced to use simple or equal weighting but we are never 
 justified in assuming that it is really the best weighting. 
 On the contrary, we must assume that this weighting 
 contains unknown errors. It will usually be found, 
 when the true weights are revealed, that the simple 
 weighting was not only erratic but so erratic as to de- 
 serve to be called freakish. In view of this fact, we cannot 
 yet close the argument and give judgment to the geometric 
 as against the median. If the commodities, reckoned 
 
212 THE MAKING OF INDEX NUMBERS 
 
 by simple weighting as though they were of equal impor- 
 tance, are really of very unequal importance, the geometric 
 may, from' its very sensitiveness, be more distorted by the 
 false weighting than the median by its insensitiveness. 
 
 The only way to settle the question whether, in 
 actual fact, the simple geometric or the simple median 
 gives the closer approximation to the result obtained by 
 proper weighting, is actually to compare these three 
 statistically. This will be done in the next chapter with 
 interesting results. 
 
 At this point we are merely justified in concluding that 
 if the simple weighting does not happen to be too erratic, 
 the geometric (or the practically coincident Formula 101) 
 is the best formula of the seven considered in this chapter. 
 
CHAPTER XI 
 WHAT IS THE BEST INDEX NUMBER? 
 
 § 1. Introduction 
 
 At the beginning of the last chapter, we set ourselves 
 two problems : first, to find the best simple index number, 
 which means best on the assumption that we lack the full 
 data needed for weighting, and, second, assuming all 
 needed data to be supplied, to find the very best. In the 
 last chapter we took up the first problem. We are now 
 ready to study the second (and, incidentally to add to 
 our conclusions concerning the first). 
 
 Let us assume, then, that we have accurate and complete 
 data both as to prices and quantities and, therefore, 
 values. The specific question to be answered in this 
 chapter is : What formula for the index number of, say, 
 prices is the most accurate ? 
 
 § 2. Discarding All Simples and Their Derivatives 
 
 We may begin by excluding not only all simple index 
 numbers but all of their derivatives. Such derivatives are 
 mongrels, almost contradictions in terms. As we have 
 seen, a simple index number has as its excuse for existence 
 a supposed lack of available weights. Yet we have 
 rectified our simple index numbers by Test 2, although 
 to use Test 2 presupposes a knowledge of weights. Of 
 course, if we really have a knowledge of these weights we 
 should, as previously pointed out, use that knowledge at the 
 outset, and start off with weighted index numbers. No one 
 
 213 
 
214 THE MAKING OF INDEX NUMBERS 
 
 could argue that we should get the best results by starting 
 with a bad index number, and then trying to reform it 
 by the processes of rectification. 
 
 The rectifications of simple index numbers, therefore, 
 are mere curiosities to show how far the faults of a bad 
 start can be overcome later. The results will be considered 
 at the proper stage ; but, at present, in searching for the 
 most accurate index number possible, we must rule out not 
 only all simples, but all their derivatives, i.e. their antithe- 
 ses and their rectifications, on the principle that we should 
 not expect to " make a silk purse out of a sow's ear." 
 
 § 3. Discarding All Modes and Medians and 
 Their Derivatives 
 
 "We have just ruled out group " S," the simples. We 
 next rule out group " M," the modes and medians (so 
 far as they have not already been ruled out by being in 
 group " S "). Previously, in discussing the mode and 
 median types of index number, we saw that they were 
 freakish in that they were unresponsive to the influence of 
 small changes in the terms averaged. On this account 
 they are clearly less fitted than the other index numbers 
 to provide a refined barometer. All that we need to add 
 here is that this freakishness holds true of the weighted 
 modes and medians as well as of the simple modes and 
 medians. In fact, not only are the mode and median apt, 
 so to speak, to fall accidentally into the clutches of a few 
 of the price relatives instead of being equally in the hands 
 of all, but the weighted mode and weighted median are apt 
 to fall accidentally into the clutches of a single large weight 
 or a very few large weights. If one or two price relatives 
 near the middle of the range of price relatives happen to 
 have large weights they are apt to. control the mode or 
 median absolutely. When the index number is thus 
 
WHAT IS THE BEST INDEX NUMBER? 215 
 
 captured no ordinary change in the price relative can dis- 
 lodge it. It is, so to speak, " stuck." And when a big 
 enough change does dislodge it, it simply jumps into 
 another such situation. The weighted mode is thus almost 
 a one-chance proposition, staking everything, perhaps, on 
 whether or not some one commodity with a monstrous 
 weight happens fairly to represent the rest in its price 
 changes — the chances being, naturally, against it. In 
 using the mode we almost " put all our eggs in one bas- 
 ket." It is doubtful whether a weighted mode (or perhaps 
 even a weighted median) is a better barometer than a 
 simple mode (or simple median), especially where there 
 are only a few commodities involved. 
 
 Because of this characteristic of the mode, its inertness, 
 the modes. Formulae 143 and 145, even though '' rectified " 
 by Test 1 (i.e. by splitting the difference between 43 and 
 49, and between 45 and 47, where there are no observable 
 differences to split), gain no real improvement in accuracy. 
 
 The only real improvement in the modes effected by a 
 " rectification " comes through Test 2. The numerator 
 of the factor antithesis is the value ratio, and in the value 
 ratio every element, p and q, has a voice. But this kind 
 of rectification has power to correct only a small part of 
 the freakishness of the original. And it may be balked 
 in accomplishing even a partial correction ; for the denomi- 
 nator of the factor antithesis (being simply another 
 mode — of quantities instead of prices) also contains 
 freakishness, and this may operate in either direction. 
 The only gain is that, instead of (practically) a one-chance 
 proposition, we now have a two-chance proposition. 
 
 In view of what has been said it is not surprising that the 
 modes (and to some extent the medians) are found to be out 
 of tune with the other index numbers, sometimes far above 
 and sometimes far below, without rhyme or reason. 
 
216 
 
 THE MAKING OF INDEX NUMBERS 
 
 § 4. Possible Improvement by Increasing the 
 Nimiber of Commodities 
 
 This freakishness of the modes (and of the medians) can, 
 of course, be lessened by including a large number of 
 commodities just as any other index number can be 
 improved somewhat in the same way. By taking a very 
 large number of commodities, we could perhaps make the 
 rectified weighted modes and medians approximately 
 coincide with the middle tine of our fork. Unfortunately, 
 we have no data for testing this hypothesis and the simple 
 mode, as given by Wesley C. Mitchell, for the 1437 com- 
 modities studied by the War Industries Board, is as far 
 out of tune with the other types of index numbers, say 
 the simple geometric, as is the simple mode of our 36 
 commodities. 
 
 The mode, in the two cases, Hes above (+) or below (— ) 
 the simple geometric as follows : 
 
 TABLE 23. 
 
 EXCESS OR DEFICIENCY OF SIMPLE MODE OF 
 PRICE RELATIVES 
 
 
 (In 
 
 per cents of 
 
 simple geometric) 
 
 
 No. OF Com- 
 modities 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 36 
 1437 
 
 +2.08 
 + 1.52 
 
 + 1.03 
 -5.93 
 
 -10.74 
 -19.75 
 
 -19.16 
 -14.64 
 
 +5.56 
 -10.53 
 
 Thus, irrespective of the number of commodities, 
 it will be seen that, whereas the arithmetic (as pre- 
 viously shown) always lies above the geometric and the 
 harmonic always below, the mode is above and below about 
 equally often, being above in four of the ten cases and below 
 in six. In the long run we may expect this approximate 
 equality to be more perfect. In fact it is absolutely 
 perfect if we always take into account backward as well as 
 
WHAT IS THE BEST INDEX NUMBER? 217 
 
 forward index numbers, for if the forward (or backward) 
 mode is above the geometric the backward (or forward) 
 must be below it.^ That is, the mode has no inherent 
 tendency to lie either above or below the geometric. 
 Either is equally likely, although there is always a likeli- 
 hood of deviating widely — freakishness. 
 
 Exactly the same discussion applies to the median 
 except that the freakishness is less. For the simple 
 medians we have : 
 
 TABLE 24. 
 
 EXCESS OR DEFICIENCY OF SIMPLE MEDIAN 
 OF PRICE RELATIVES 
 
 
 (In per cents of simple geometric) 
 
 
 No. OP Com- 
 modities 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 36 
 1437 
 
 +3.S4 
 - .01 
 
 + 1.84 
 -5.66 
 
 - 2.11 
 -11.02 
 
 -1.70 
 -6.40 
 
 +6.00 
 - .64 
 
 Comparing Table 24 with Table 23, it will be observed 
 that the median and mode usually jump together, first, 
 on one side of the geometric, and then on the other ; but 
 the median usually jumps less than the mode, thus lying 
 between the mode and geometric. The average ratio of 
 the two deviations (those of the mode and median 
 from the geometric) is 2.5 in the case of the 36 commod- 
 ities and 2.2 in the case of the 1437 commodities. 
 
 It is noteworthy that the mode and median seem to be 
 helow the geometric when prices are rapidly rising. 
 Whether this is usually the case and, if so, why, I do not 
 know. In this particular case, it may be partly accounted 
 for by price fixing preventing many commodities from 
 rising as much as they otherwise would, so that those com- 
 modities which do rise inordinately raise the geometric 
 but scarcely affect the mode or median. 
 
 * See Appendix I (Note to Chapter XI, § 4). 
 
218 THE MAKING OF INDEX NUMBERS 
 
 While the freakishness of the mode and median can prob- 
 ably be reduced by introducing large numbers, it cannot 
 be eliminated altogether. Under all circumstances these 
 index numbers are lame and limping, as compared with 
 the other four types. I have estimated very roughly 
 on the basis of the data above mentioned and the law of 
 distribution of chances that, for a large number of com- 
 modities, say 100, the rectified mode, Formula 343, would 
 keep almost always within two per cent of the middle 
 tine. In the present case of 36 commodities, it is ten per 
 cent off the track for 1917, although for the other years 
 it is usually within three per cent. And the rectified 
 median is within two per cent even in the case of our 36 
 commodities. With 100 commodities it would doubtless 
 agree still more closely. 
 
 § 5. Discarding All " Biased " Index Numbers 
 Leaves Only the Middle Tine (47 Formulas) 
 
 Thus far in our search for the most accurate index 
 number, we have eliminated (1) the " S " group, i.e. all 
 simples and their progeny, and (2) the " M " group, — 
 all medians and modes and their progeny. We have 
 found these index numbers " freakish " or " haphazard," 
 the first group because constructed from badly (that is, 
 evenly) weighted material, and the second because so 
 largely insensitive to changes in the individual prices and 
 quantities. 
 
 As we have seen, the rest of the index numbers do not 
 vary at random but naturally group themselves into the 
 five classes shown by the five-tined fork. That is, they 
 differ from one another not by even gradations but by 
 definite intervals. The causes for this grouping we have 
 already investigated and expressed by the term " bias " 
 to represent a distinct tendency or " list " in a particular 
 
WHAT IS THE BEST INDEX NUMBER? 219 
 
 direction. We now eliminate all biased index numbers 
 (classes 2+, 1+, 1 — , 2 — ), viz., all in the two upper and 
 two lower tines, leaving only the ''0" or unbiased class 
 for further consideration. 
 
 § 6. Selecting from the 47 Formulse, the 13 
 Satisfying Both Tests 
 
 There are 47 distinct formulae represented in this middle 
 tine. Even if we proceeded no further we would have 
 reached an important conclusion — even a startling con- 
 clusion. These 47 formulae agree more closely than the 
 standards of ordinary statistical practice require ! We 
 may say, therefore, that, if we merely exclude formulce 
 obviously freakish or biased, all the rest agree with each 
 other well enough for ordinary practical purposes ! 
 
 But we may go still further in our search for accuracy. 
 Among these 47 approximately agreeing formulae there are 
 two, 53 and 54, which, while free from bias, are not free 
 from joint error. For instance. Formula 53 forward times 
 Formula 53 backward does not give unity but sometimes 
 a little more and sometimes a little less, revealing a slight 
 joint error in the two applications of 53 ; and so also of 54. 
 In short. Formulae 53 and 54 fail to obey Test 1 as also 
 they fail to obey Test 2. The same is true of 6053, 7053, 
 8053, and 8054. Ruling these out, we have left 41 for- 
 mulae all of which obey at least one test. But from these 
 we may eliminate, as not obeying both tests. Formulae 107, 
 108, 109, 110, 1103, 1104, 123, 124, 125, 126, 1123, 1124, 
 1153, 1154, 2153, 2154, 3153, 3154, 4153, 4154, 207, 209, 
 213, 215, 223, 225, 227, 229. We now have left, as 
 obeying both tests, 13 formulae from which to choose the 
 best, namely, 307, 309, 323, 325, 353, 1303, 1323, 1353, 
 2353, 3353, 4353, 5307, 5323. 
 
 The argument here is not that every one of these so far 
 
220 THE MAKING OF INDEX NUMBERS 
 
 surviving formulse is better than all those eHminated, 
 for we shall find that this is not quite true, but that each 
 of the excluded formulse, failing in one or the other of the 
 two tests, is necessarily surpassed by some at least of the 13. 
 Thus Formula 109, failing in Test 2, must be adjudged 
 inferior to its own rectification, 309, which meets both 
 tests, even if it (Formula 109) happens to be superior 
 to some other of the inner circle of 13, say, 1303 ; and we 
 may conclude that 8053 and 8054 are inferior to their own 
 rectification (which is 353) without concluding that they 
 are inferior to some other of the 13, say, to 309. 
 
 In other words, out of the 47 best formulae we are select- 
 ing, not necessarily the best 13, but the 13 which we know 
 must include the best one of all. In still other words, 
 while we have no reason to think that each of these 13 is 
 superior to all the 34 excluded, we do have good reason to 
 believe that each of these 34 is inferior to some one of the 13. 
 
 § 7. Selecting Formula 353 as the " Ideal " 
 
 "We have still to choose from the surviving 13, although 
 their agreement is now close — far closer than practically 
 required. Here the argument changes and becomes much 
 less definite and sure. We can no longer appeal to the two 
 tests as a means of further sifting ; for all the 13 formulse 
 obey both these tests perfectly. But we can still find 
 reasons for preferring one formula to another. We can 
 prefer the crossed formulce to the cross weight formulce 
 and their derivatives (except Formula 2353, reserved 
 for later consideration), thus excluding 1303, 1323, 1353, 
 3353, 4353, and leaving only the eight formulae : 307, 309, 
 323, 325, 353, 2353, 5307, 5323. 
 
 This exclusion is based on the consideration ^ that the 
 cross weight formulse fail to insure a middle course between 
 
 * As shown in Appendix I (Note to Chapter VIII, § 10). 
 
WHAT IS THE BEST INDEX NUMBER? 221 
 
 the original formulae whose weights are crossed. They 
 seem slightly erratic as compared with the rest. Again, 
 on the principle that two equally promising estimates or 
 measures may probably be improved in accuracy by taking 
 their average, 307 and 309 may next be excluded in favor 
 of their cross, 5307 ; and, likewise, 323 and 325, in favor 
 of their cross, 5323. 
 
 This leaves the four formulae, 353, 2353, 5307, 5323. 
 From these four, all practically coinciding, I should be 
 inclined, if forced to choose, first, to drop 2353 in favor of 
 353 on the theory that weight crossing of any kind is 
 probably not as accurate a splitter of differences as formula 
 crossing. This leaves the three formulae, 353, derived 
 from aggregatives ; 5307, derived from arithmetics and 
 harmonics; and 5323, derived from geometries. From 
 these I am inclined next to eliminate 5307 on the ground 
 that it descends from ancestors (7, 8, 9, 10, 13, 14, 15, 16) 
 far wider apart than does 353 or 5323. There seems more 
 chance of error in using figures wide apart than in using 
 those close together. If we must prefer one of the two 
 remaining formulae (353 and 5323) to the other I would 
 drop 5323 for the same reason. 
 
 Thus Formula 353, derived from aggregatives, remains 
 to take the first prize for accuracy. But I should not 
 quarrel with those who would divide the prize with 2353, 
 5307, or 5323, especially the last. 
 
 § 8. Other Argiiments for Formula 353 
 
 Our whole argument has hitherto been on the score of 
 accuracy. If we add the consideration of algebraic 
 simplicity, the superiority of Formula 353 over all its 
 rivals is evident, and very marked. As to ease and 
 rapidity of computation (of which I shall speak more 
 fully later) 353 is immensely superior to all its 12 rivals, 
 
222 THE MAKING OF INDEX NUMBERS 
 
 though some excellent formulae outside of the 13 are still 
 more rapid, as we shall see. 
 
 Hitherto no use has been made of the argument that 
 formulae of widely different nature are likely to be accu- 
 rate if they agree with each other. Every formula was 
 given consideration independently and on its merits. 
 Thus, Formula 9 was condemned, not on the ground that 
 it gave results higher than 353 and the other middle-of- 
 the-roaders, but on the ground that, if twice appUed, once 
 forward and again backward, it gave a result greater 
 than unity so that at least one of its two applications 
 was too large. We could thus prove bias without any 
 comparison with other formulae. Likewise 41 and 43, 
 were condemned as freakish, not because they differ 
 so greatly from other formulae, but because they fail to 
 respond to most of the changes which they aim to average. 
 
 And yet as we have proceeded, step by step, we could 
 not fail to notice that the good formulae give very similar, 
 and the bad formulae, very dissimilar, results, and that 
 the good agree in results despite wide differences of 
 method. And now that we have completed the original 
 line of argument, we may confirm it strikingly by citing, 
 as new and internal evidence, these similarities and dis- 
 similarities. The formulae which we condemned as up- 
 ward biased (on the ground of comparison only with them- 
 selves, reversed in direction), we now find do actually give 
 higher results than 353 and its peers or next bests, 
 the divergence for the doubly biased formulae being about 
 double the divergence of the singly biased formulae; 
 and similarly as to the downward biased. 
 
 The only qualifications to this statement are such as 
 merely further confirm what has been found. Thus the 
 simples, modes, medians, and their derivatives, which, 
 on general grounds, were condemned as very erratic, 
 
WHAT IS THE BEST INDEX NUMBER? 223 
 
 behave peculiarly relative to 353 and the other foremost 
 formulae, and thereby again justify the term " freakish." 
 Thus all the formulae shown to be bad independently 
 are found also to be bad comparatively, — that is, as 
 judged by their departures from the very good formulae. 
 
 Finally, the formulae which we found, by studying each 
 one by itself, to be good, because free from bias and freak- 
 ishness, are also found to be good as judged by each other. 
 That is, they all agree amazingly well, constituting the 
 middle tine of the fork. In fact, I think that anyone 
 who had not followed the former argument but who should 
 merely examine the internal evidence of agreement and 
 disagreement would reach almost exactly the same con- 
 clusions as to which formulae are good and which are bad. 
 At any rate the agreements and disagreements between the 
 134 formulae are, without a single exception, consistent 
 with all the conclusions reached on other grounds. 
 
 § 9. Formulae 353 and 5323 Compared 
 
 We have seen that Formulae 353 and 5323 present 
 almost equal claims to be true barometers of changes in 
 prices and quantities. But their results do not tally 
 absolutely, as Table 25 shows.^ 
 
 In only one instance do the two methods yield precisely 
 the same result and that identity would doubtless dis- 
 appear if we were to carry the computation one decimal 
 further. 
 
 What are we to infer from these disagreements ? Error 
 there must be but we have no warrant for saying one is 
 " absolutely right " and, therefore, all the error is in the 
 
 ^ For the purposes of this comparison Formula 353 might have been 
 called 5353 (although no such number is used in the list), for just as 5323 
 is descended from eight index numbers (23, 24, 25, 26, 27, 28, 29, 30), so 
 may 5353 (i.e. 353) be regarded as derived from eight (3, 4, 5, 6, 17, 18, 19, 
 20). 
 
224 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 25. TWO BEST INDEX NUMBERS 
 (1913 = 100) 
 
 Pbices 
 
 Quantities 
 
 Base 
 
 Formu- 
 la No. 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Fixed 
 
 353 
 5323 
 
 100.12 
 100.13 
 
 99.89 
 99.87 
 
 114.21 
 114.09 
 
 161.56 
 161.59 
 
 177.65 
 177.67 
 
 99.33 
 99.32 
 
 109.10 
 109.11 
 
 118.85 
 118.92 
 
 118.98 
 118.96 
 
 125.37 
 125.35 
 
 Chain 
 
 353 
 
 5323 
 
 100.12 
 100.13 
 
 100.23 
 100.23 
 
 114.32 
 114.45 
 
 162.23 
 162.42 
 
 178.49 
 178.64 
 
 99.33 
 99.32 
 
 108.72 
 108.73 
 
 118.74 
 118.61 
 
 118.49 
 118.36 
 
 124.77 
 124.68 
 
 other. We must infer just what Pierson inferred, that 
 index numbers are not and never can be absolutely pre- 
 cise. There is always a fringe of uncertainty surrounding 
 them. But, while index numbers can never quite pretend 
 to rank with weights and spatial measures in perfection of 
 precision. Table 25 reveals a very high degree of precision, 
 not only far higher than skeptics like Pierson imagined 
 possible, but higher even than believers in index numbers 
 had supposed. 
 
 Table 25 shows that, for prices, the two fixed base fig- 
 ures for 1914 agree within about one part in 10,000 ; 1915 
 within about two parts in 10,000 ; 1916 within about one 
 part in 1000 ; 1917 within about one part in 5000 ; and 
 1918 within about one part in 9000 ; while, for quantities, 
 the corresponding degrees of agreement are substantially 
 the same: one part in 10,000, one in 10,000, one in 1000, 
 one in 6000, one in 6000. Turning to the chain index 
 numbers we find, for prices for 1915, perfect agreement 
 as far as calculated, and for the succeeding years one in 
 1000, one in 1000, one in 800 ; while for quantities the 
 figures are: one in 10,000, one in 10,000, one in 1000, 
 one in 1000, one in 1000. 
 
 When we speak of two magnitudes as agreeing within 
 
WHAT IS THE BEST INDEX NUMBER? 225 
 
 one part in 1000 we are speaking of an extremely high 
 degree of agreement. The agreement is as close as that 
 between two estimates of the height of Washington 
 Monument which differ by a hand's breadth, or two 
 estimates of the height of a man which differ by a four- 
 teenth of an inch, or two estimates of his weight which 
 differ by two ounces. These are higher degrees of preci- 
 sion than those met with in the measures of commodities 
 sold at retail and than most of those met with in whole- 
 sale transactions. They are comparable even with many 
 laboratory measurements. Thus, I learn from the United 
 States Bureau of Standards that measures of volume by 
 glass or brass containers are correct only to one part in 
 5000 to 10,000. The best portable ammeter measures 
 electric current only to one part in 250, and the best port- 
 able voltmeter measures voltage to only one part in 500. 
 
 WTien we consider that these two methods of reckoning 
 an index number by Formulae 353 and 5323 are wholly 
 distinct, that, in one, the processes are adding and divid- 
 ing, and, in the other, they are multiplying and extract- 
 ing roots, it seems truly marvelous that by such widely 
 different routes we should be led to almost absolutely the 
 same goal. It would be absurd to ascribe the agreement 
 wholly to " accident." The coincidences are too numer- 
 ous, even without recourse to the agreements with other 
 index numbers on the middle tine. We cannot escape 
 the conclusion from this comparison that these two index 
 numbers check each other up and prove each other's ac- 
 curacy within an error of usually less than one part in 1000. 
 
 § 10. The " Probable Error " of Formula 353 
 
 We may now cite the close agreement of all the 13 for- 
 mulae which satisfy both tests and are also free of the 
 accusation of freakishness (i.e. are not descended from 
 
226 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 26. SELECTED INDEX NUMBERS 
 (1913 = 100) 
 
 FrxED Base 
 
 FOR- 
 
 Prices 
 
 Quantities 
 
 No. 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 307 
 
 100.13 
 
 99.78 
 
 114.17 
 
 161.04 
 
 177.25 
 
 99.31 
 
 109.20 
 
 118.89 
 
 119.36 
 
 125.65 
 
 309 
 
 100.17 
 
 99.85 
 
 114.25 
 
 162.31 
 
 178.44 
 
 99.29 
 
 109.13 
 
 118.74 
 
 118.43 
 
 124.81 
 
 323 
 
 100.13 
 
 99.89 
 
 113.99 
 
 161.90 
 
 177.98 
 
 99.31 
 
 109.09 
 
 119.09 
 
 118.74 
 
 125.14 
 
 325 
 
 100.12 
 
 99.85 
 
 114.19 
 
 161.28 
 
 177.35 
 
 99.33 
 
 109.13 
 
 118.76 
 
 119.19 
 
 125.57 
 
 353 
 
 100.12 
 
 99.89 
 
 114.21 
 
 161.56 
 
 177.65 
 
 99.33 
 
 109.10 
 
 118.85 
 
 118.98 
 
 125.37 
 
 1303 
 
 100.14 
 
 99.88 
 
 114.22 
 
 161.75 
 
 177.82 
 
 99.32 
 
 109.11 
 
 118.84 
 
 118.84 
 
 125.25 
 
 1323 
 
 100.13 
 
 99.90 
 
 114.23 
 
 161.70 
 
 177.80 
 
 99.32 
 
 109.08 
 
 118.84 
 
 118.88 
 
 125.26 
 
 1353 
 
 100.13 
 
 99.89 
 
 114.22 
 
 161.71 
 
 177.79 
 
 99.33 
 
 109.08 
 
 118.85 
 
 118.87 
 
 125.26 
 
 2353 
 
 100.13 
 
 99.89 
 
 114.22 
 
 161.60 
 
 177.67 
 
 99.32 
 
 109.09 
 
 118.84 
 
 118.94 
 
 125.35 
 
 3353 
 
 100.14 
 
 99.90 
 
 114.35 
 
 161.94 
 
 177.36 
 
 99.31 
 
 109.08 
 
 118.81 
 
 118.70 
 
 125.57 
 
 4353 
 
 100.13 
 
 99.92 
 
 114.26 
 
 161.78 
 
 177.52 
 
 99.32 
 
 109.06 
 
 118.80 
 
 118.82 
 
 125.46 
 
 5307 
 
 100.15 
 
 99.82 
 
 114.21 
 
 161.67 
 
 177.84 
 
 99.30 
 
 109.17 
 
 118.81 
 
 118.90 
 
 125.23 
 
 , 5323 
 
 100.13 
 
 99.87 
 
 114.09 
 
 161.59 
 
 177.67 
 
 99.32 
 
 109.11 
 
 118.92 
 
 118.96 
 
 125.35 
 
 Chain op Bases 
 
 Formu- 
 la No. 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 307 
 
 100.13 
 
 100.22 
 
 114.56 
 
 162.50 
 
 178.42 
 
 99.31 
 
 108.74 
 
 118.49 
 
 118.30 
 
 124.83 
 
 309 
 
 100.17 
 
 100.22 
 
 114.61 
 
 162.76 
 
 179.30 
 
 99.29 
 
 108.74 
 
 118.44 
 
 118.10 
 
 124.21 
 
 323 
 
 100.13 
 
 100.23 
 
 114.45 
 
 162.47 
 
 178.69 
 
 99.31 
 
 108.73 
 
 118.61 
 
 118.32 
 
 124.64 
 
 325 
 
 100.12 
 
 100.23 
 
 114.45 
 
 162.36 
 
 178.58 
 
 99.33 
 
 108.73 
 
 118.62 
 
 118.39 
 
 124.71 
 
 353 
 
 100.12 
 
 100.23 
 
 114.32 
 
 162.23 
 
 178.49 
 
 99.33 
 
 108.72 
 
 118.74 
 
 118.49 
 
 124.77 
 
 1303 
 
 100.14 
 
 100.23 
 
 114.40 
 
 lo2.37 
 
 178.99 
 
 99.32 
 
 108.72 
 
 118.66 
 
 118.39 
 
 124.42 
 
 1323 
 
 100.13 
 
 100.24 
 
 114.65 
 
 162.71 
 
 179.05 
 
 99.32 
 
 108.72 
 
 118.41 
 
 118.14 
 
 124.39 
 
 1353 
 
 100,13 
 
 100.23 
 
 114.33 
 
 162.27 
 
 178.45 
 
 99.33 
 
 108.72 
 
 118.73 
 
 118.46 
 
 124.76 
 
 2353 
 
 100.13 
 
 100.23 
 
 114.32 
 
 162.31 
 
 178.58 
 
 99.32 
 
 108.72 
 
 118.74 
 
 118.43 
 
 124.71 
 
 3353 
 
 100.14 
 
 100.24 
 
 114.28 
 
 162.14 
 
 178.39 
 
 99.31 
 
 108.71 
 
 118.71 
 
 118.48 
 
 124.77 
 
 4353 
 
 100.13 
 
 100.24 
 
 114.38 
 
 162.20 
 
 178.46 
 
 99.32 
 
 108.71 
 
 118.68 
 
 118.51 
 
 124.79 
 
 5307 
 
 100.15 
 
 100.22 
 
 114.59 
 
 162.63 
 
 178.86 
 
 99.30 
 
 108.74 
 
 118.47 
 
 118.20 
 
 124.52 
 
 i 5323 
 
 100.13 
 
 100.23 
 
 114.45 
 
 162.42 
 
 178.64 
 
 99.32 
 
 108.73 
 
 118.61 
 
 118.36 
 
 124.68 
 
WHAT IS THE BEST INDEX NUMBER? 227 
 
 simples, modes, or medians). Table 26 gives these 13 
 index numbers from which we selected Formula 353 as 
 presumably the best. 
 
 By means of Table 26 we can further address our- 
 selves to the problem of measuring the degree of accuracy 
 of Formula 353. Critics hke Pierson have cited the 
 disagreements of index numbers, which they mistakenly 
 assumed to present equal claims to be true barometers of 
 price changes, as evidence that index numbers in general 
 were inaccurate. Though their premises were wrong their 
 logic was right. And we may now apply it, freed from 
 their mistaken premises. We may apply to these 13 
 barometer readings the processes of the theory of proba- 
 bilities, and compute the probable errors. We shall 
 assume, at the outset, that all the 13 have equal claims ; 
 that is, we shall give them equal weights in our probability 
 calculations. This is conservative; that is, it will tend 
 to exaggerate the probable errors of the best. Table 27 
 gives the probable errors as so calculated. 
 
 TABLE 27. PROBABLE ERRORS ^ OF AN INDEX NUMBER 
 OF PRICES OR QUANTITIES WORKED OUT BY ANY ONE 
 OF THE 13 FORMULA CONSIDERED AS EQUALLY GOOD 
 INDEPENDENT OBSERVATIONS 
 
 (In per cents of their average) 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Fixed 
 
 Chain 
 
 .009 
 .009 
 
 .025 
 .006 
 
 .050 
 .069 
 
 .128 
 .079 
 
 .118 
 .104 
 
 > See Appendix I (Note to Chapter XI, § 10). 
 
 Thus we see that the probable error of any of the 
 13 formulae for 1914 was .009 per cent to be added to or 
 subtracted from the index number, say, 100.12, or one 
 part in 10,000. To state this exactly, assuming all of the 
 
228 THE MAKING OF INDEX NUMBERS 
 
 13 to be equally likely to be right, the error of any one of 
 them is as likely as not less than one part in 10,000. Simi- 
 larly, the probable (or as-likely-as-not) error in 1916 of 
 the fixed base figures is .05 per cent of the index number 
 — about one part in 2000. 
 
 The largest error in a single index number is that for 
 1917 relatively to 1913. That is the index number has 
 an error of .128 per cent, or about one part in 800, or about 
 one eighth of one per cent. We may, therefore, be 
 assured that Formula 353, being certainly more accurate, 
 if that be possible, than at least most of the other 12, 
 is able correctly to measure the general trend of the 36 dis- 
 persing price relatives or quantity relatives within less than 
 one eighth of one per cent! That is, the error in, say. For- 
 mula 353, probably seldom reaches one part in 800, or a 
 hand's breadth on the top of Washington Monument, or 
 less than three ounces on a man's weight, or a cent added 
 to an $8 expense. 
 
 The above estimate of one eighth of one per cent is a 
 maximum, for three reasons : (1) it is the maximum of 
 the ten figures in the above table ; (2) the above table is 
 based on an extraordinary war-time dispersion which 
 tends to magnify the disagreements between index num- 
 bers ; and (3) many of the 13 formulae treated as equally 
 reliable are demonstrably less reliable than 353. To re- 
 place the above maximum estimate by a more truly repre- 
 sentative one is not easy and introduces doubtful consider- 
 ations. Without detailing these, I shall merely say that 
 after various other calculations I am convinced that the 
 probable error of Formula 353 seldom reaches one per cent 
 of one per cent. 
 
 Assuming that, for practical purposes, a precision within 
 one per cent of the truth is ample, we see that any first 
 class index number is at least eight times as precise as it 
 
WHAT IS THE BEST INDEX NUMBER? 229 
 
 needs to be. Humanly speaking then, an index number is 
 an absolutely accurate instrument. This does not, of 
 course, have any reference to inaccuracies in the original 
 data, nor to inaccuracies due to the choice of data in- 
 cluded as samples, or representatives of those excluded. 
 It merely means that, given these data, the index 
 number is able to give an unerring figure to express their 
 average movement. As physicists or astronomers would 
 say, the "instrumental error " is negligible. The old idea 
 that among the difficulties in measuring price move- 
 ments is the difficulty of finding a trustworthy mathe- 
 matical method may now be dismissed once and for all. 
 
 § 11. The Purpose to "Which an Index Number Is 
 Put Does Not Affect the Choice of Formula 
 
 It will be noted that Formula 353, or its rivals, has been 
 selected as the best on very general grounds of a formal 
 character. Consequently, the conclusions are as general 
 as the premises from which we started. Whether prices 
 are wholesale or retail, for instance, obviously does not 
 affect the choice of Formula 353 rather than 1, or 31, or 9. 
 For, in either case, there are precisely the same reasons 
 for selecting a formula which is reversible in time or factors 
 and for selecting a formula which will not be freakish, 
 or spasmodic, in its findings. 
 
 But so deeply rooted is the idea that various purposes 
 require various formulae that the general significance of 
 these results is not yet acknowledged by many of the stu- 
 dents of index numbers. I must reserve for a separate 
 article specific answer to those who have rejected the con- 
 clusion, when first briefly stated at the Atlantic City meet- 
 ing of the American Statistical Association, December, 
 1920, that a good formula for one purpose is a good formula 
 
230 THE MAKING OF INDEX NUMBERS 
 
 for all known purposes. But I may note here reasons 
 alleged for rejecting this idea. There seem to be three : 
 
 (1) There is the idea that a conflict exists between 
 measuring the average change of prices and measuring 
 changes in the average level of prices.^ 
 
 (2) There is the idea that changes in the aggregate cost 
 of a specified bill of goods or regimen, as implied in aggre- 
 gative index numbers, is appropriate only for retail 
 trade — despite the fact that Knibbs, the chief protago- 
 nist of this concept, applies this idea of aggregate cost 
 to a specified list of wholesale prices. Of course, it may 
 be applied to any list in any market. What is the custom 
 in the case has nothing to do with the accuracy of the 
 procedure as a mathematical method. 
 
 (3) There is the idea that the character of the dis- 
 tribution of price relatives about the mode or other mean 
 prescribes the choice of, say, the arithmetic or geometric 
 type. This argument defeats itself through the reversal 
 process ; any asymmetry displayed (on the ratio chart, 
 at least) in the distribution of the relatives taken forward 
 is reversed when we have to consider the relatives taken 
 backward. 2 If the arithmetic be adjudged proper for 
 the one it would have to be adjudged improper for the 
 other, thus leading to such an absurd conclusion as that, 
 in calculating the price level of London relatively to New 
 York, the arithmetic index number is appropriate, but 
 in calculating the price level of New York relative to 
 London it would be highly improper ! Moreover, if we 
 count the cases in both directions there are, of course, 
 as many cases of asymmetry in one direction as in the 
 other. It follows that, in the long run, there is no tend- 
 
 1 This is discussed in Appendix III (on ratio of averages vs. average of 
 ratios). 
 
 2 As pictured in Appendix I (Note to Chapter XI, § 11). 
 
WHAT IS THE BEST INDEX NUMBER? 231 
 
 ency to asymmetry in any one direction.^ Also when a 
 large number of relatives are used, there is usually little 
 asymmetry in any case. The opinion to the contrary 
 is based on the wrong method, usually employed, of 
 plotting on an arithmetic scale instead of the ratio chart 
 used in this book. 
 
 But, from a practical standpoint, it is quite unnecessary 
 to discuss the fanciful arguments for using " one formula 
 for one purpose and another for another," in view of the 
 great practical fact that all methods (if free of freakish- 
 ness and bias) agree! Unless someone has the hardihood 
 to espouse bias or freakishness for some '' purpose," 
 whatever formula he advocates will insist on coinciding 
 with whatever formula anyone else advocates. The 
 notion that the aggregative is appropriate for the cost of 
 living, and the geometric for a wholesale price level, and 
 the arithmetic for something else, becomes futile. For 
 if we admit that in each case the rectified forms are to be 
 used, we shall find that the rectified aggregative (Formulse 
 353, 1353, 2353, 3353, 4353), the rectified geometric 
 (323, 325, 1323, 5323), and the rectified arithmetic (307, 
 309, 1303, 5307), all agree ten times as closely as is required 
 for any purpose whatever ! 
 
 The basic reason for misunderstanding on this subject is 
 failure to take into account bias and reversibility in time. 
 So long as the very bad FormulsB 1, 9001, 21, 9021, 31, 
 51, 9051 are used, no wonder writers on index numbers 
 
 ^ Asymmetrical distribution is often characteristic in other fields than 
 index numbers, e.g. human heights or weights. (See Macalister, "Law of 
 the Geometric Mean," Proceedings of the Royal Society, 1879.) But in 
 such cases there is no reversibility. The items averaged are not ratios. 
 In the case of a skull index, on the other hand, the ratio of length to 
 breadth may be reversed as breadth to length and is analogous to index 
 numbers which are ratios of prices to prices or quantities to quantities. 
 Ratios are essentially double ended and produce their own symmetry, by 
 reversal in one form or another. 
 
232 THE MAKING OF INDEX NUMBERS 
 
 seek fanciful reasons for using one of these mutually 
 conflicting formulae for one purpose and another for 
 another. But as soon as it is seen that the weighted index 
 numbers of all these types need rectification, — that there 
 is no more justification for using, for instance, an arithmetic 
 forward than backward, and that, therefore, it should be 
 
 Wighfed Aggregafives for 90 Raw Materials 
 
 V/ar Industries Board Statistics 
 (Prices) 
 
 fS H '/5 'le '17 *f8 
 
 Chart 45P. Showing the same close agreement and absence of bias 
 of Formulae 53 and 54 for the 90 commodities, as was found in the case of 
 the 36 commodities (see Chart 39P, top tier). 
 
 rectified before being used at all, — all these fanciful 
 distinctions and arguments fall to the ground. 
 
 A year ago I issued a friendly challenge to those who 
 object to this conclusion to supply a single case where 
 Formula 353 should not be used. Several have tried to 
 supply such cases but without success. 
 
 It is clear that a considerable part of the disagreement 
 is more apparent than real and due to misunderstandings. 
 Mitchell gives seven purposes requiring, he alleges, dif- 
 
WHAT IS THE BEST INDEX NUMBER? 233 
 
 ferent formulae.^ One of these " purposes " is the com- 
 parison with an existing series of index numbers, in which 
 case the formula used should be identical with that 
 used in the existing series. Naturally ! In a somewhat 
 similar way I, myself, in this book, have found use for 
 134 different formulae for the " purpose " of comparison. 
 Another of Professor Mitchell's purposes is to make an 
 index number which the common man can understand. 
 Of course, we can go on indefinitely enumerating such 
 varieties of purpose. Our purpose may be to secure the 
 
 Weighted Aggregatives for 90 Raw Materials 
 
 War industries Board Statistics 
 (Quantities) 
 
 U3 '/4 '/5 '/e 77 78 
 
 Chart 45Q. Analogous to Chart 45 P. 
 
 cheapest index number. Then Formula 51 is the formula 
 we want. Or our purpose may be to secure the most 
 inaccurate. One of the modes might then be indicated. 
 Formula 353 would not be the best for that purpose ! 
 
 I had assumed, of course, that there was at least this 
 uniformity of '' purpose " : that by the best index number 
 would be understood the index number which was the 
 most accurate measure. If this be taken for granted, 353 
 (or any of the 30 or more others which give the same 
 results) seems the best for all purposes within the domain 
 covered by index numbers. Whether the purpose be an 
 index number of prices, or quantities, or wages, or rail- 
 
 * Bulletin No. 284, United States Bureau of Labor Statistics, pp. 76, 78. 
 
234 THE MAKING OF INDEX NUMBERS 
 
 road traffic, or whether the index number is to measure the 
 value of money, the barometer of trade, the cost of manu- 
 facturing, the volume of manufacture, the same varieties 
 of mathematical processes can be used and will converge 
 to close agreement, — that is, so long as the problem 
 is of the same mathematical form, — as it is in all cases I 
 have yet met with.^ 
 
 In short, an index number formula is merely a statis- 
 tical mechanism like a coefficient of correlation. It is as 
 
 Formulae 55 and 54 Applied To Stock Market 
 
 (Prices) 
 
 \5% 
 
 riAY' 
 te 17 16 19 20 2f 
 
 1921 
 
 Chart 46 P. Showing the same closeness of agreement and absence of 
 bias of 53 and 54 for stock market prices. 
 
 absurd to vary the mechanism with the subject matter to 
 which it is applied as it would be to vary the method of 
 calculating the coefficient of correlation. 
 
 § 12. Comments on Formula 353 and the 
 Aggregatives Generally 
 
 Formula 353 must already have impressed the reader 
 as having noteworthy peculiarities and simplicities. It 
 is formed more simply ^ than any other formula ful- 
 
 1 See Appendix I (Note to Chapter IV, § 10). 
 
 ' To see how much simpler 353 really is than any other formula among 
 the 12 rivals for accuracy we need only compare it with the next in sim- 
 pUcity, 2353, as follows : 
 
 353 = ^fipK^^im 
 
WHAT IS THE BEST INDEX NUMBER? 235 
 
 filling the two tests, being obtained merely from the four 
 magnitudes SpoS'o, Spi^i, SpoQ'i, ^PiQo- The same four are 
 
 Formulae 55 and 54 Applied To Stock Market 
 
 (Quantifies) 
 
 16 
 
 /? 
 
 /S 
 
 HAY- 
 1921 
 
 /9 
 
 20 
 
 21 
 
 Chart 46Q. Analogous to Chart 46P. 
 
 used, simply in different order, for the price index number 
 and the quantity index number.^ 
 
 The formula fulfills both tests, although it is obtained 
 by only one crossing of antecedent formulae. That one 
 
 2353 
 
 -i 
 
 Spig-i X S(9o + ?i) Pi X S(po + Pi) ?o 
 
 2po9o X 2(go + gO po X 2(po + Pi) qx 
 The reader may care, for curiosity, to write out some of the still more 
 complicated formulae such as 5323, the most accurate among the geometries. 
 1 See Chapter VII, § 5, regarding Formula 153 (the same as 353). 
 
236 THE MAKING OF INDEX NUMBERS 
 
 crossing may be the crossing of two time antitheses (53 
 and 59, or 54 and 60, or 3 and 19, or 4 and 20, or 5 and 17, 
 or 6 and 18), or the crossing of two factor antitheses (53 
 and 54, or 59 and 60, or 3 and 4, or 5 and 6, or 17 and 18, 
 or 19 and 20). Thus, it merely needs to conform to one 
 test in order to conform to both. This can be said of no 
 other formula. 
 
 Formulae 5^ and 5^ Applied To 12 Leading Crops 
 (Prices) 
 
 (After WM.Persons) 
 
 54 
 53 
 
 \Sif 
 
 90 
 
 185 
 
 VO 
 
 '95 
 
 •00 
 
 '05 
 
 10 
 
 •15 
 
 '20 
 
 Chakt 47P. Showing almost the same closeness of agreement but the 
 presence of a slight bias, 53 always exceeding 54, except in the one year, 
 1920. The common origin of the two curves is 1910. 
 
 It is derivable from the aggregatives (53, 54, 59, 60), 
 from the arithmetics (3, 4, 5, 6), or from the harmonics (17, 
 18, 19, 20), or from both the arithmetics and the har- 
 monics. Consequently, unlike other formulae, it recurs 
 in its various roles again and again (as 103, 104, 105, 106, 
 153, 154, 203, 205, 217, 219, 253, 259, 303, 305), being en- 
 countered, so to speak, at the many crossroads in our 
 tables. Its constituent formulae, 53 and 54, are likewise 
 frequent repeaters, and are the only pair of formulae which 
 are at once time antitheses and factor antitheses. 
 
 Another interesting fact, as shown in the Appendix,^ 
 
 * See Appendix I (Note to Chapter XIII, §9, "Proportionality Test"). 
 
WHAT IS THE BEST INDEX NUMBER? 237 
 
 is that, while Formula 353 is a perfectly true average, 
 nine of its twelve rivals (all excepting 1353, 2353, 3353 — 
 themselves aggregatives) are not true averages. They 
 fulfill the definition of an average of the price relatives 
 only in case the quantity relatives are all equal. 
 
 Another peculiarity is that the aggregative, alone of 
 all index numbers, does not require calculating price 
 ratios. 
 
 Formulae 53and 54 Applied To 12 Leading Crops 
 (Quantifies) 
 
 (After WMPersons) 
 
 W '85 '90 '95 '00 '05 '10 15 '20 
 
 Chart 47Q. Analogous to Chart 47P. 
 
 § 13. Formulae 53 and 54 Already in Close Agreement 
 
 Last but not least. Formulae 53 and 54 are also in actual 
 fact far closer together than any other of the primary 
 formulae which are crossed. The remarkable closeness 
 between the two index numbers calculated by 53 and 54 
 is not an accident merely happening to be true for the 36 
 commodities here selected. 
 
 Professor Persons has calculated an index of the physi- 
 cal volume of exports for 1920 by Formulae 53 and 54, 
 obtaining 93.3 and 95.1 per cent, differing by two per cent. 
 
 We find the same closeness if we take the 90 commodities 
 
238 
 
 THE MAKING OF INDEX NUMBERS 
 
 (" materials ") for which Professor Wesley C. Mitchell 
 gives the data in the report of the War Industries Board. ^ 
 These are given in Charts 45P and 45Q and show the same 
 closeness of Formulae 53 and 54 for prices and so also the 
 same closeness for quantities. What is equally important 
 to note is that, in both cases, as in the corresponding case 
 
 Formulae 55 and 54 Applied To 12 Leading Crops 
 
 (Prices) 
 (After WhPersons) 
 
 \s% 
 
 (70 
 
 12 73 74 rS K 7/ 
 
 Chakt 48P. Analogous to Chart 47 P. 
 
 '/9 
 
 of 36 commodities, there is no tendency for either of the 
 two curves to be constantly above or constantly below 
 the other. 
 
 Another case from an entirely different field is that of 
 the prices of 100 stocks and the quantities sold on the 
 New York Stock Exchange, from daily quotations. 
 
 'Wesley C. Mitchell, "History of Prices during the War, Summary" 
 (War Industries Board, Bulletin No. 1), p. 45. Mitchell works out only 
 53 (for prices and quantities) but as, fortunately, he gives the data for 
 values, it is easy to calculate 54. Mitchell uses 1913 as 100 per cent al- 
 though the real base of calculation is 1917. Accordingly in the charts I 
 have used 1917 as the common point. The corresponding data for all 
 the 1366 commodities were not published and, although a search was 
 made in my behalf, they cannot be found even in manuscript. Were 
 this possible it would be easy to calculate 53, 54, 353, for the entire 1366. 
 
WHAT IS THE BEST INDEX NUMBER? 239 
 
 Here again the closeness of Formulae 53 and 54 is illus- 
 trated, in Charts 46 P and 46Q. 
 
 Charts 47P, 47Q, and 48P, 48Q are made from the 
 figures of Professor Persons for 12 crops, the first pair 
 by five year intervals, and the second pair by year to 
 year intervals,^ In these cases the divergence is a 
 little greater than in the case of the 36 commodities.^ 
 Charts 48 P and 48Q also give the chain figures, which 
 show a considerable deviation from the fixed base figures. 
 
 Formulae 53 and 54 Applied To 12 Leading Crops 
 
 (Quantities) 
 (After WKPersons) 
 
 Z? 7J 14 15 16 1/ 
 
 Chakt 48Q. Analogous to Chart 48P. 
 
 In all these crop figures there is discernible the effect 
 of an inverse correlation between the price and quantity 
 movements. This is of interest to the student of index 
 numbers in three ways : (1) it signifies a slight modifica- 
 tion of the proposition that Formulae 53 and 54 are not 
 subject to bias ; (2) it confirms the proposition that any 
 bias in the fixed base system is intensified in the chain 
 system; and (3) it shows that such a bias as is here 
 
 ^Warren M. Persons, "Fisher's Formula for Index Numbers," Review 
 of Economic Statistics (Statistical Service of the Harvard University Com- 
 mittee on Economic Research, Cambridge, Mass.), May, 1921, pp. 103-113. 
 
 '^ Somewhat greater, even, than appears at first glance, as the scale of 
 these four charts had to be reduced to get them on the page. The little 
 yardstick in the charts — the "5 %" vertical line — is evidently shorter 
 than that in all preceding charts, indicating that in this chart a given verti- 
 cal distance means a greater per cent increase than in former charts. 
 
240 THE MAKING OF INDEX NUMBERS 
 
 illustrated — a sort of secondary bias, as we shall see — 
 is very small. ^ 
 
 § 14. History of Formula 353 
 
 The constituent Formulae 53 and 54 from which 353 
 is constructed are, as has already been noted, due respec- 
 tively to Laspeyres and Paasche. Formula 53, or Las- 
 peyres, is the most practical of the two when a substitute 
 for 353 has to be used. It (53) was advocated strongly 
 and ably by G. H. Knibbs, Statistician of Australia.^ 
 
 Formula 53 being identical with 3, it has been used 
 sometimes as an arithmetic average with base year 
 weighting and calculated laboriously as such. Apparently 
 I was the first to point out the identity of the two formulae.^ 
 The great service performed by Knibbs was to point out 
 the great saving in time in calculating 53 as an aggregative 
 rather than calculating Formula 3 as an arithmetic index 
 number. Knibbs also points out that 53 (and the same 
 might be said, though less emphatically, of the other ag- 
 gregatives) has the advantage over the geometries and 
 other types of being easily comprehended by the gen- 
 eral public* Formula 53 is used by the United States 
 Bureau of Labor Statistics, having been introduced by 
 Dr. Royal Meeker. The method was recently endorsed 
 in a resolution (No. 81) passed by the British Imperial 
 Statistical Conference in 1920. This resolution reads : 
 
 ^ See Appendix I (Note to Chapter XI, § 13). 
 
 2 See G. H. Knibbs, "Price Indexes, Their Nature and Limitations, 
 the Technique of Computing Them, and Their AppHcation in Ascertaining 
 the Purchasing Power of Money." Commonwealth Bureau of Census and 
 Statistics, Labour and Industrial Branch, Report No. 9, McCarron, Bird 
 & Co., Melbourne, 1918. 
 
 ^ Economic Journal, December, 1897, pp. 517, 520. See also, Pur- 
 chasing Power of Money, table opposite p. 418, heading of Formulae 11 and 
 12 and discussion. 
 
 * Discussed in Chapter XVI, § 8. 
 
WHAT IS THE BEST INDEX NUMBER? 241 
 
 Methods of Constructing Index Numbers 
 
 That the index numbers should be so constructed that their 
 comparison for any two dates should express the proportion 
 of the aggregate expenditure on the selected list of representa- 
 tive commodities, in the quantities selected as appropriate, at 
 the one date, to the aggregate expenditure on the same list of 
 conmiodities, in the same quantities, at the other date. 
 
 This phraseology may perhaps be taken as applicable, 
 not only to Formula 53, but also to 54, 1153, 2153, 
 3153, 4153. Since 2153 is, as we shall see, a short cut 
 method of calculating 353, we may practically include 
 353 also. 
 
 So far as I know, the earliest reference to the formula 
 here numbered 353 is that made by C. M. Walsh inciden- 
 tally in a footnote of The Measurement of General Exchange 
 Value in 1901.^ Walsh's mention of it had escaped 
 my notice until he called my attention to it in correspond- 
 ing with me in 1920. The next mention appears to be in 
 my Purchasing Power of Money, 1911, where it is given 
 as Formula 16 in the table opposite page 418, but I did 
 not then appraise it as the best. 
 
 Apparently the next writer to mention this formula, 
 and with high approval, is Professor A. C. Pigou in his 
 Wealth and Welfare, 1912.^ He regards it as probably 
 the best measure for comparing price levels of two coun- 
 tries. This anticipation by Professor Pigou was called 
 to the attention of Mr. Walsh and myself by Professor 
 Frederick R. Macaulay. 
 
 Next in order comes my preliminary paper on " The 
 Best Form of Index Numbers," read December, 1920, 
 where I advocated 353 as the best or " ideal " for- 
 
 1 p. 429. 
 
 * p. 46. By inadvertence the square root sign is omitted, but is inserted 
 in the later book Economics of Welfare, 1920, p. 84. 
 
242 THE MAKING OF INDEX NUMBERS 
 
 mula.^ Writing contemporaneously, without knowl- 
 edge of my work, C. M. Walsh added this formula to the 
 other index numbers recommended by him as " perhaps 
 the best " in his Problem of Estimation,^ published 
 February, 1921. The same formula was reached from a 
 still different angle by Professor Allyn A. Young ^ as the 
 best for measuring changes of the general price level. 
 
 Several others have accepted 353 (e.g. George R. 
 Davies, Introduction to Economic Statistics, 1922, p. 86) as 
 best for certain purposes. It is a great satisfaction to 
 know that several of us have now reached the conclusion 
 that this formula is the best, even if some still add safe- 
 guarding qualifications. I think we may be confident 
 that the end is being reached of the long controversy 
 over the proper formula for an index number. 
 
 Professor Persons'' refers to the index number which 
 I have called " ideal " as " Fisher's Index Number." 
 This was doubtless pursuant to the too generous sugges- 
 tion of Mr. Walsh at the Atlantic City meeting.^ If the 
 conclusions of this book be accepted, I think my proposed 
 term " ideal " is the most appropriate. But, if my name 
 is to be used, Walsh's, or Walsh's and Pigou's should be 
 used also. 
 
 1 Published, with discussion, in Quarterly Publication of the American 
 Statistical Association, March, 1921. 
 
 » pp. 102-103. 
 
 ' See Qvarterly Journal of Economics, "The Measurement of Changes 
 of the General Price Level," August, 1921, p. 572. 
 
 < "Fisher's Formula for ludex Numbers," Review of Economic Statistics, 
 May, 1921, p. 103. 
 
 * See Quarterly Publication of the American Statistical Association, 
 March, 1921, p. 544. 
 
CHAPTER XII 
 
 COMPARING ALL THE INDEX NUMBERS WITH THE 
 " IDEAL " (FORMULA 353) 
 
 § 1. All Index Numbers Arranged in Order of 
 Their Remoteness from Formula 353 
 
 We have chosen Formula 353 as the most nearly ideal 
 index number, have measured its precision, have found 
 that the 12 others in our list which have the best inde- 
 pendent claims to rival Formula 353 coincide with it for 
 all intents and purposes, and that 34 other index numbers, 
 i.e. those free merely from obvious bias or freakishness, 
 agree with it nearly enough for ordinary requirements. 
 And now we can look back and, by using Formula 353 
 as a standard for comparison (or, if anyone prefers, any 
 other of those deserving honorable mention in our contest), 
 we can compare all the other 133 formulae with that 
 standard. For this comparison I have arranged all the 
 formulae in order of their closeness to Formula 353.^ 
 
 Numerically, Table 28 gives all the 134 ^ index numbers 
 in the order of remoteness from 353, beginning with the 
 remotest and ending with 353 itself. The figures are for 
 prices, not quantities (although the order is substantially 
 the same in both cases) , and for fixed base figures, not chain. 
 In each case the formula number is given (in the first 
 column) for identification. Thus, the first in the list is 
 Formula 12, which is the factor antithesis of the simple 
 harmonic index number. In the second column is given 
 
 ^ For the method used see Appendix I (Note to Chapter XII, § I). 
 
 ^ These form only 119 different ranks because those tied in rank are 
 given the same number. Thus the second number in the list, "118," 
 applies to seven different index numbers. 
 
 243 
 
244 
 
 THE MAKING OF INDEX NUMBERS 
 
 the letter or number of the class, out of the seven classes 
 enumerated in Chapter IX, § 5, to which the index num- 
 ber belongs. Thus 12 belongs to " S," the " simple " group, 
 being a derivative of the simple harmonic index number. 
 In order to simplify the picture, the list of 134 is sep- 
 arated arbitrarily into several classes in increasing order 
 of merit. The first twelve index numbers, constituting 
 the first of these classes, are labeled, rather harshly 
 perhaps, as " worthless " index numbers (to designate 
 the fact that they are the worst). The other six classes 
 are labeled as poor, fair, good, very good, excellent, and 
 superlative. Decimals are omitted (as superfluous for 
 the comparisons) from all classes worse in rank than the 
 *' very good " and these are given but one decimal. Only 
 the " excellent " and " superlatives " are accorded two deci- 
 mals. The reader will quickly form a mental comparison 
 of various formulae by running his eye down the columns, 
 especially 1917, for which the variations are the greatest. 
 
 TABLE 28. INDEX NUMBERS BY 134 FORMULA ARRANGED 
 IN THE ORDER OF REMOTENESS FROM THE IDEAL 
 (353) (AS SHOWN BY THE FIXED BASE FIGURES FOR 
 THE PRICE INDEXES) 
 
 (1913 = 100) 
 
 Identifi- 
 cation 
 
 Number 
 
 Class 
 
 OP 
 
 Formula 
 
 19U 
 
 1916 
 
 1916 
 
 1917 
 
 I (Inverse) 
 Order op 
 Merit 
 
 Worthless Index Numbers 
 
 12 
 
 S 
 
 103 
 
 101 
 
 115 
 
 172 
 
 244 
 
 119 
 
 44 
 
 M 
 
 103 
 
 106 
 
 132 
 
 196 
 
 180 
 
 118 
 
 46 
 
 M 
 
 
 
 " 
 
 
 
 " 
 
 4S 
 
 M 
 
 
 
 " 
 
 
 
 " 
 
 50 
 
 M 
 
 
 
 " 
 
 
 
 " 
 
 144 
 
 M 
 
 
 
 " 
 
 
 
 '• 
 
 146 
 
 M 
 
 
 
 " 
 
 
 
 " 
 
 1144 
 
 M 
 
 
 
 
 
 
 •' 
 
 42 = 142 
 
 SM 
 
 104 
 
 108 
 
 125 
 
 167 
 
 183 
 
 117 
 
 41=141 
 
 SM 
 
 98 
 
 98 
 
 108 
 
 135 
 
 190 
 
 116 
 
 1 
 
 S 
 
 96 
 
 98 
 
 124 
 
 176 
 
 187 
 
 115 
 
 51=151 
 
 s 
 
 96 
 
 96 
 
 108 
 
 147 
 
 173 
 
 114 
 
COMPARING ALL THE INDEX NUMBERS 245 
 
 TABLE 28 (Continued) 
 
 Identifi- 
 cation 
 Number 
 
 Class 
 
 OP 
 
 Formula 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 (Inverse) 
 
 Order op 
 
 Merit 
 
 Poor Index Numbers 
 
 11 
 
 S 
 
 95 
 
 96 
 
 119 
 
 158 
 
 172 
 
 113 
 
 21=121 
 
 s 
 
 96 
 
 97 
 
 121 
 
 167 
 
 180 
 
 112 
 
 101 
 
 s 
 
 96 
 
 97 
 
 121 
 
 167 
 
 179 
 
 111 
 
 251 =351 
 
 s 
 
 97 
 
 97 
 
 111 
 
 153 
 
 169 
 
 110 
 
 102 
 
 s 
 
 102 
 
 99 
 
 113 
 
 162 
 
 208 
 
 109 
 
 243 
 
 M 
 
 102 
 
 103 
 
 119 
 
 179 
 
 174 
 
 108 
 
 245 
 
 M 
 
 " 
 
 " 
 
 " 
 
 
 " 
 
 '« 
 
 247 
 
 M 
 
 " 
 
 " 
 
 " 
 
 
 " 
 
 " 
 
 249 
 
 M 
 
 " 
 
 " 
 
 " 
 
 
 " 
 
 " 
 
 343 
 
 M 
 
 " 
 
 " 
 
 •• 
 
 
 " 
 
 " 
 
 345 
 
 M 
 
 " 
 
 " 
 
 '• 
 
 
 " 
 
 " 
 
 1343 
 
 M 
 
 " 
 
 " 
 
 " 
 
 
 " 
 
 " 
 
 5343 
 
 M 
 
 " 
 
 " 
 
 •• 
 
 
 " 
 
 " 
 
 211 
 
 S 
 
 99 
 
 98 
 
 117 
 
 165 
 
 205 
 
 107 
 
 9 
 
 2 
 
 101 
 
 102 
 
 118 
 
 181 
 
 187 
 
 106 
 
 52=152 
 
 S 
 
 97 
 
 97 
 
 115 
 
 159 
 
 165 
 
 105 
 
 7 
 
 2 
 
 101 
 
 102 
 
 118 
 
 181 
 
 187 
 
 104 
 
 14 
 
 2 
 
 102 
 
 102 
 
 117 
 
 168 
 
 190 
 
 103 
 
 15 
 
 2 
 
 100 
 
 98 
 
 111 
 
 145 
 
 167 
 
 102 
 
 13 
 
 2 
 
 99 
 
 98 
 
 111 
 
 147 
 
 169 
 
 101 
 
 301 
 
 S 
 
 99 
 
 98 
 
 117 
 
 164 
 
 193 
 
 100 
 
 8 
 
 2 
 
 99 
 
 97 
 
 111 
 
 152 
 
 167 
 
 99 
 
 10 
 
 2 
 
 99 
 
 97 
 
 111 
 
 155 
 
 169 
 
 98 
 
 16 
 
 2 
 
 101 
 
 102 
 
 117 
 
 169 
 
 189 
 
 97 
 
 241=341 
 
 SM 
 
 101 
 
 103 
 
 116 
 
 150 
 
 186 
 
 96 
 
 22=122 
 
 S 
 
 102 
 
 99 
 
 113 
 
 162 
 
 194 
 
 95 
 
 31=131 
 
 SM 
 
 99 
 
 99 
 
 119 
 
 164 
 
 191 
 
 94 
 
 34 
 
 M 
 
 101 
 
 105 
 
 118 
 
 166 
 
 182 
 
 93 
 
 221 =321 
 
 S 
 
 99 
 
 98 
 
 117 
 
 164 
 
 187 
 
 92 
 
 33 
 
 M 
 
 100 
 
 99 
 
 107 
 
 156 
 
 169 
 
 91 
 
 43 
 
 M 
 
 101 
 
 100 
 
 108 
 
 164 
 
 168 
 
 90 
 
 45 
 
 M 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 47 
 
 M 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 49 
 
 M 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 143 
 
 M 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 145 
 
 M 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 1143 
 
 M 
 
 '• 
 
 " 
 
 " 
 
 " 
 
 " 
 
 " 
 
 36 
 
 M 
 
 101 
 
 104 
 
 118 
 
 165 
 
 182 
 
 89 
 
 201 
 
 S 
 
 98 
 
 97 
 
 116 
 
 164 
 
 182 
 
 88 
 
 37 
 
 M 
 
 101 
 
 100 
 
 109 
 
 164 
 
 188 
 
 87 
 
 35 
 
 M 
 
 100 
 
 99 
 
 107 
 
 160 
 
 169 
 
 86 
 
 2 
 
 S 
 
 100 
 
 96 
 
 110 
 
 153 
 
 177 
 
 85 
 
246 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 28 {Continued) 
 
 Identifi- 
 
 Class 
 
 cation 
 
 OF 
 
 Number 
 
 Formula 
 
 1914 I 1916 
 
 1916 1917 
 
 1918 
 
 Fair Index Numbers 
 
 1134 
 
 M 
 
 101 
 
 103 
 
 118 
 
 163 
 
 182 
 
 84 
 
 1133 
 
 M 
 
 101 
 
 100 
 
 108 
 
 163 
 
 171 
 
 83 
 
 9051 
 
 
 102 
 
 103 
 
 114 
 
 160 
 
 182 
 
 82 
 
 134 
 
 M 
 
 101 
 
 103 
 
 117 
 
 163 
 
 181 
 
 81 
 
 29 
 
 1 
 
 101 
 
 101 
 
 116 
 
 170 
 
 182 
 
 80 
 
 23 
 
 1 
 
 100 
 
 99 
 
 111 
 
 154 
 
 173 
 
 79 
 
 133 
 
 M 
 
 101 
 
 100 
 
 108 
 
 160 
 
 174 
 
 78 
 
 136 
 
 M 
 
 101 
 
 103 
 
 117 
 
 162 
 
 181 
 
 77 
 
 231 = 331 
 
 SM 
 
 100 
 
 100 
 
 117 
 
 163 
 
 187 
 
 76 
 
 1003 
 
 1 
 
 100 
 
 101 
 
 116 
 
 171 
 
 183 
 
 75 
 
 24 
 
 1 
 
 101 
 
 101 
 
 116 
 
 165 
 
 183 
 
 74 
 
 25 
 
 1 
 
 100 
 
 99 
 
 113 
 
 152 
 
 172 
 
 73 
 
 1013 
 
 1 
 
 100 
 
 99 
 
 113 
 
 154 
 
 173 
 
 72 
 
 27 
 
 1 
 
 100 
 
 101 
 
 116 
 
 171 
 
 182 
 
 71 
 
 38 
 
 M 
 
 101 
 
 102 
 
 117 
 
 158 
 
 180 
 
 70 
 
 1014 
 
 1 
 
 101 
 
 101 
 
 116 
 
 165 
 
 183 
 
 69 
 
 30 
 
 1 
 
 99 
 
 98 
 
 113 
 
 159 
 
 174 
 
 68 
 
 135 
 
 M 
 
 101 
 
 100 
 
 108 
 
 162 
 
 178 
 
 67 
 
 1004 
 
 1 
 
 99 
 
 99 
 
 113 
 
 158 
 
 173 
 
 66 
 
 39 
 
 M 
 
 101 
 
 100 
 
 109 
 
 164 
 
 178 
 
 65 
 
 28 
 
 1 
 
 100 
 
 99 
 
 113 
 
 157 
 
 172 
 
 64 
 
 6023 ('13-' 14) 
 
 100 
 
 100 
 
 112 
 
 154 
 
 173 
 
 63 
 
 32 = 132 
 
 SM 
 
 100 
 
 102 
 
 116 
 
 162 
 
 184 
 
 62 
 
 26 
 
 1 
 
 101 
 
 101 
 
 115 
 
 165 
 
 183 
 
 61 
 
 233 
 
 M 
 
 101 
 
 102 
 
 112 
 
 161 
 
 175 
 
 60 
 
 237 
 
 M 
 
 101 
 
 101 
 
 113 
 
 161 
 
 184 
 
 59 
 
 235 
 
 M 
 
 101 
 
 102 
 
 112 
 
 163 
 
 176 
 
 58 
 
 40 
 
 M 
 
 101 
 
 102 
 
 117 
 
 160 
 
 180 
 
 57 
 
 Good Index Numbers 
 
 335 
 
 M 
 
 101 
 
 101 
 
 113 
 
 162 
 
 180 
 
 56 
 
 1333 
 
 M 
 
 101 
 
 101 
 
 113 
 
 163 
 
 176 
 
 55 
 
 5333 
 
 M 
 
 101 
 
 101 
 
 113 
 
 162 
 
 179 
 
 54 
 
 333 
 
 M 
 
 101 
 
 101 
 
 113 
 
 161 
 
 177 
 
 53 
 
 239 
 
 M 
 
 101 
 
 101 
 
 113 
 
 162 
 
 179 
 
 52 
 
 6023 ('13 & "18) 
 
 99 
 
 99 
 
 114 
 
 160 
 
 180 
 
 51 
 
 6023 ('13-'16) 
 
 100 
 
 100 
 
 114 
 
 157 
 
 175 
 
 50 
 
 209 
 
 
 
 100 
 
 100 
 
 115 
 
 167 
 
 178 
 
 49 
 
 213 
 
 
 
 101 
 
 100 
 
 114 
 
 157 
 
 179 
 
 48 
 
 207 
 
 
 
 100 
 
 100 
 
 115 
 
 166 
 
 177 
 
 47 
 
 215 
 
 
 
 100 
 
 100 
 
 114 
 
 156 
 
 178 
 
 46 
 
 223 
 
 
 
 100 
 
 100 
 
 114 
 
 159 
 
 178 
 
 45 
 
 225 
 
 
 
 100 
 
 100 
 
 114 
 
 159 
 
 177 
 
 44 
 
 229 
 
 
 
 100 
 
 100 
 
 114 
 
 164 
 
 178 
 
 43 
 
 227 
 
 
 
 100 
 
 100 
 
 114 
 
 164 
 
 177 
 
 42 
 
 110 
 
 
 
 100 
 
 100 
 
 114 
 
 162 
 
 179 
 
 41 
 
 109 
 
 
 
 100 
 
 100 
 
 115 
 
 163 
 
 178 
 
 40 
 
COMPARING ALL THE INDEX NUMBERS 247 
 TABLE 28 (Continued) 
 
 Identifi- 
 cation 
 Number 
 
 Class 
 
 OP 
 
 Formula 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 8NVERSE) 
 RDER OF 
 
 Merit 
 
 Very Good Index Numbers 
 
 1 
 
 6053 ('13-'18) 
 
 99.8 
 
 99.9 
 
 114.0 
 
 161.6 
 
 177.9 
 
 39 
 
 54* 
 
 
 
 100.3 
 
 100.1 
 
 114.4 
 
 161.1 
 
 177.4 
 
 38 
 
 108 
 
 
 
 100.2 
 
 99.6 
 
 114.0 
 
 160.3 
 
 177.9 
 
 37 
 
 53t 
 
 
 
 99.9 
 
 99.7 
 
 114.1 
 
 162.1 
 
 177.9 
 
 36 
 
 6053 (' 
 
 13-'16) 
 
 100.0 
 
 100.0 
 
 114.0 
 
 161.9 
 
 178.2 
 
 35 
 
 4153 
 
 
 
 100.1 
 
 100.0 
 
 114.4 
 
 162.4 
 
 178.3 
 
 34 
 
 309 
 
 
 
 100.2 
 
 99.9 
 
 114.3 
 
 162.3 
 
 178.4 
 
 33 
 
 107 
 
 
 
 100.1 
 
 99.9 
 
 114.4 
 
 161.8 
 
 176.6 
 
 32 
 
 4154 
 
 
 
 100.1 
 
 99.9 
 
 114.1 
 
 161.2 
 
 176.8 
 
 31 
 
 6053 (' 
 
 13-'14) 
 
 100.1 
 
 100.1 
 
 113.9 
 
 161.3 
 
 177.7 
 
 30 
 
 123 
 
 
 
 100.1 
 
 99.9 
 
 113.8 
 
 162.1 
 
 177.8 
 
 29 
 
 3153 
 
 
 
 100.2 
 
 99.9 
 
 114.2 
 
 162.1 
 
 176.9 
 
 28 
 
 307 
 
 
 
 100.1 
 
 99.8 
 
 114.2 
 
 161.0 
 
 177.3 
 
 27 
 
 *54=4, 5, 18, 19, 59. t53=3, 6, 17, 20, 60. 
 
 Excellent Index Numbers 
 
 323 
 
 
 
 100.13 
 
 99.89 
 
 113.99 
 
 161.90 
 
 177.98 
 
 26 
 
 124 
 
 
 
 100.16 
 
 99.85 
 
 114.25 
 
 161.74 
 
 178.16 
 
 25 
 
 3353 
 
 
 
 100.14 
 
 99.90 
 
 114.35 
 
 161.94 
 
 177.36 
 
 24 
 
 7053 
 
 
 
 100.09 
 
 99.96 
 
 114.03 
 
 161.53 
 
 177.90 
 
 23 
 
 126 
 
 
 
 100.12 
 
 99.85 
 
 114.20 
 
 161.18 
 
 177.36 
 
 22 
 
 325 
 
 
 
 100.12 
 
 99.85 
 
 114.19 
 
 161.28 
 
 177.35 
 
 21 
 
 1104 
 
 
 
 100.15 
 
 99.84 
 
 114.18 
 
 161.58 
 
 177.92 
 
 20 
 
 5307 
 
 
 
 100.15 
 
 99.82 
 
 114.21 
 
 161.67 
 
 177.84 
 
 19 
 
 1103 
 
 
 
 100.13 
 
 99.91 
 
 114.26 
 
 161.93 
 
 177.72 
 
 18 
 
 125 
 
 
 
 100.12 
 
 99.87 
 
 114.19 
 
 161.37 
 
 177.34 
 
 17 ' 
 
 4353 
 
 
 
 100.13 
 
 99.92 
 
 114.26 
 
 161.78 
 
 177.52 
 
 16 
 
 3154 
 
 
 
 100.12 
 
 99.92 
 
 114.28 
 
 161-77 
 
 177.78 
 
 15 
 
 1303 
 
 
 
 100.14 
 
 99.88 
 
 114.22 
 
 161.75 
 
 177.82 
 
 14 
 
 1123 
 
 
 
 100.14 
 
 99.89 
 
 114.17 
 
 161.62 
 
 177.87 
 
 13 
 
 1124 
 
 
 
 100.12 
 
 99.91 
 
 114.28 
 
 161.78 
 
 177.73 
 
 12 
 
 Superlative Index Numbers 
 
 5323 
 
 
 
 100.13 
 
 99.87 
 
 114.09 
 
 161.59 
 
 177.67 
 
 11 
 
 1323 
 
 
 
 100.13 
 
 99.90 
 
 114.23 
 
 161.70 
 
 177.80 
 
 10 
 
 1153 
 
 
 
 100.13 
 
 99.89 
 
 114.20 
 
 161.70 
 
 177.83 
 
 9 
 
 1353 
 
 
 
 100.13 
 
 99.89 
 
 114.22 
 
 161.71 
 
 177.79 
 
 8 
 
 1154 
 
 
 
 100.12 
 
 99.90 
 
 114.24 
 
 161.73 
 
 177.76 
 
 7 
 
 2154 
 
 
 
 100.14 
 
 99.90 
 
 114.21 
 
 161.69 
 
 177.72 
 
 6 
 
 2353 
 
 
 
 100.13 
 
 99.89 
 
 114.22 
 
 161.60 
 
 177.67 
 
 5 
 
 2153 
 
 
 
 100.12 
 
 99.89 
 
 114.23 
 
 161.52 
 
 177.63 
 
 4 
 
 8054 
 
 
 
 100.12 
 
 99.89 
 
 114.21 
 
 161.56 
 
 177.65 
 
 3 
 
 8053 
 
 
 
 100.12 
 
 99.89 
 
 114.21 
 
 161.56 
 
 177.65 
 
 2 
 
 353* 
 
 
 
 100.12 
 
 99.89 
 
 114.21 
 
 161.56 
 
 177.65 
 
 1 
 
 *353=103, 104, 105, 106, 153, 154, 203, 205, 217, 219, 253, 259, 303, 305. 
 
248 THE MAKING OF INDEX NUMBERS 
 
 It will be seen that of the 12 formulse which we found 
 in the last chapter, each on its own independent merits, 
 to be the closest mates to the ideal, Formula 353, two 
 (307 and 309) are classed as " very good" ; six (323, 3353, 
 325, 5307, 4353, 1303) are classed as " excellent," and 
 four (5323, 1323, 1353, 2353) are classed as "superlative." 
 That is, all of the formulse selected as best on independent 
 grounds also prove to be among the very best when ranked 
 on the basis of agreement with 353. 
 
 And yet, interspersed with these 12 are others just as 
 close to Formula 353, though not exactly fulfilUng both 
 tests. Most of these are the various combinations of 
 53 and 54. These two formulse are so extremely close 
 to each other that any method of spUtting their hair's 
 difference will necessarily agree almost absolutely with 
 353. Thus the formula closest to Formula 353 is 8053, 
 the arithmetic average of 53 and 54. Although 8053 
 does not fulfill either test, it comes very close to fulfilling 
 both and to coinciding with 353 which fulfills both ex- 
 actly. All of the other " superlative " index numbers 
 are combinations of 53 and 54. 
 
 § 2. Chart Giving Index Numbers in Order 
 
 Graphically, we can get a much quicker and clearer 
 view than is possible by mere numerical figures. Chart 49 
 gives the same 119 ranks as were represented in Table 28. 
 But the chart includes, in addition, the chain figures, 
 represented, in the usual way, by small balls. ^ These 
 
 * The distance between each ball and the curve exhibits the disparity 
 between the fixed base and the chain figures. This distance for, say, the 
 year 1918 represents the net cumulative effect of the disparities of all 
 the preceding years. In order to show how much disparity there has 
 been in the last year elapsed, a dark vertical line is inserted (i.e. extending 
 from the 1918 ball to the point where that ball would have been had it 
 remained the same distance from the ciuve as the ball of the last year, 
 1917) ; and likewise for each other year. 
 
COMPARING ALL THE INDEX NUMBERS 249 
 
 RANKING AS TO ACCURACY ^" 
 
 OF 
 
 ALL INDEX NUMBERS 
 
 (I). Worth/ess Index Numbers 
 (Prices) 
 
 151(51) 
 
 73 't^ '15 '16 77 78 
 
 Chart 49 (1). These index numbers, ranked as the least accurate, include 
 one, the simple arithmetic (1), in very common use, and another, the sim- 
 ple aggregative (51), in occasional use. The six index numbers, not only 
 disagree widely with the ideal (353) used as a standard, but also with each 
 other, and also as between the fixed base and chain figures of each (as 
 shown by the balls and the dark verticals — the displacement of each ball 
 from the curve indicating the cumulative divergence of the chain figures, 
 and the dark vertical indicating the year-to-year divergence). 
 
250 
 
 THE MAKING OF INDEX NUMBERS 
 
 (2). Poor Index Numbers 
 
 (Prices) 
 
 '13 U '15 '16 '17 'Id 
 
 Chart 49 (2). The same divergencies in less degree are here in evidence, 
 except that 101 and 21 agree. The list includes two which have been 
 actually suggested, the simple harmonic (11) suggested by Coggeshall and 
 the doubly biased arithmetic (9) suggested by Palgrave. 
 
COMPARING ALL THE INDEX NUMBERS 251 
 
 C2)cont Poor Index Numbers 
 (Prices) 
 
 Chart 49 (2, continued). This set includes the simple median (31). 
 
252 
 
 THE MAKING OF INDEX NUMBERS 
 
 (5). Fair Index Numbers 
 (Prices) 
 
 '13 '/^ 75 'le '17 '16 
 
 Chart 49 (3). The same divergencies, still less marked, are noted. Of 
 these the most usable is 9051, a quickly calculated rough-and-ready index 
 number. 
 
 balls and the dark vertical lines attached to them will be 
 more especially discussed later. At present they may be 
 ignored by the reader so that his attention may be con- 
 centrated on the ranking. 
 
COMPARING ALL THE INDEX NUMBERS 253 
 
 (5)cont Fair Index Numbers 
 (Prices) 
 
 '13 '14 '15 '16 '17 76 
 
 Chart 49 (3, continued) . This list includes one fcrm of Professor Day's 
 index number (6023). 
 
 § 3. The Index Numbers Converge toward 
 Formula 353 
 The most striking fact in Table 28 and Chart 49 is the 
 steady natural convergence of the index numbers toward 
 
254 
 
 THE MAKING OF INDEX NUMBERS 
 
 W. Good Index Numbers 
 (Prices) 
 
 239 
 
 6023 CiSand'ia) 
 
 6023 ('13-16) 
 
 V3 7-? 75 7^ '17 '/a 
 
 Chart 49 (4). The disagreements have here largely disappeared, whether 
 as between each curve and Formula 353, or as among themselves, and also 
 as between the fixed base and chain series. This list includes two forms of 
 Professor Day's index number (6023). 
 
 Formula 353. This would not be true if we had arbi- 
 trarily chosen some widely different curve as the standard 
 of reference, such as, say, 2 or 44. It is noteworthy that 
 Formula 353 and those practically coincident with it 
 constitute the only type of index number out of all the 
 numerous varieties which can boast of having many like 
 
(5) Very Good. ($) Excellent and 
 (7) Superlative Index Numbers 
 (Prices) 
 
 6053 CI3-I8) , 
 
 108 
 
 53(3.6,17.20.60) 
 
 60S3a3-l6) 
 
 4/55 
 
 309 
 
 107 
 
 (5). Very Cood Ind Nos. 
 
 (6).Excellent Ind Nos. 
 
 (7). Superlative Ind. Nos. 
 
 5325 
 1323 
 1153 
 1353 
 1154- 
 215't 
 2353 
 2153 
 BOSf 
 8053 
 
 35 3(103, W4J05. 
 106,153.154.203205 
 2/7,219.253253,303^05) 
 
 '/J 7* '/5 '16 17 76 
 
 Chart 49 (5, 6, 7). All the divergencies continue to disappear until they 
 become imperceptible. The " very goods " include Laspeyres' (53), Paasche's 
 (54), and Lehr's (4153 and 4154). The "excellents" include one of Walsh's 
 (1123), and Lehr's rectified by Test 2 (4353). The "superlatives" include 
 the above Walsh's rectified by Test 2 (1323), two of Walsh's (1153 and 
 1154), the same rectified by Test 2 (1353), Edgeworth's and Marshall's 
 (2153), another of Walsh's (2154), the rectification by Test 2 of the two lat- 
 ter (2353), Drobisch's (8053), and the " ideal" 353, used as standard for 
 all of the Charts 49. 
 
256 THE MAKING OF INDEX NUMBERS 
 
 it. Thus, if any one should contend that Formula 2 
 was the best index number and should try to arrange the 
 formulae in the order of closeness to 2, he would not find 
 the picture altogether unlike that now before us. No. 2 
 would stand very much alone, its closest neighbors all 
 being distant from it. Furthermore, the index numbers 
 which we have chosen as the best would, in such an 
 arrangement, though no longer placed at the culminating 
 end of the list, still keep close together. As the list now 
 stands, almost no index numbers, far away from 353 but 
 neighbors to each other, are close enough neighbors to have 
 any strong family resemblance. There is one exception ; 
 namely, the pair of 101 and 21, which have already been 
 noted as the best in the hierarchy of simple index numbers. 
 
 Again, the 119 varieties in our chart vary about equally 
 on the opposite sides of Formula 353, even though the 
 modes and medians are included, as is shown by the follow- 
 ing averages of Table 29.^ 
 
 Except for the " worthless " class each class averages 
 very close to 353, showing that the variations above and 
 below are about equal, as was to be expected. What has 
 been said would still be true even if we should leave out 
 of consideration so many types of averaging 53 and 54. 
 In short. Formula 353 (or any equivalent) is the evident 
 goal of the complete set toward which, as toward no 
 other, they tend to converge. 
 
 § 4. Many Besides Formula 353 Pass Muster 
 
 It will be seen that by pronouncing Formula 353 to be 
 the best index number, it is not implied that it is separated 
 by a wide gulf from all others. On the contrary, one of 
 
 ^ Strictly the geometric average should be used ; but, except for the first 
 few index numbers (where it was used), this would not differ appreciably 
 from the arithmetic, which, for ease of calculation, was used in all other 
 cases. 
 
COMPARING ALL THE INDEX NUMBERS 257 
 
 TABLE 29. AVERAGES OF EACH OF THE VARIOUS CLASSES 
 OF INDEX NUMBERS 
 
 Classes 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Worthless 
 
 100. 
 
 101. 
 
 118. 
 
 164. 
 
 193. 
 
 Poor 
 
 96. 
 
 99. 
 
 115. 
 
 162. 
 
 181. 
 
 Fair 
 
 101. 
 
 101. 
 
 114. 
 
 161. 
 
 179. 
 
 Good 
 
 100. 
 
 100. 
 
 114. 
 
 161. 
 
 178. 
 
 Very good 
 
 100.1 
 
 99.9 
 
 114.1 
 
 161.6 
 
 177.6 
 
 Excellent 
 
 100.13 
 
 99.88 
 
 114.20 
 
 161.65 
 
 177.71 
 
 Superlative 
 
 100.13 
 
 99.89 
 
 114.21 
 
 161.64 
 
 177.72 
 
 Average of all 
 classes 
 
 99.35 
 
 100.06 
 
 114.37 
 
 161.83 
 
 179.46 
 
 353 
 
 100.12 
 
 99.89 
 
 114.21 
 
 161.56 
 
 177.65 
 
 our main conclusions is that there are others which are 
 really just as accurate. It is only by literally splitting 
 hairs that we can claim any superiority in accuracy of 353 
 over its fellow " superlatives," and then only with doubt. 
 There is less room for doubt as to the superiority of 353 
 over the ''excellent" index numbers, but the degree of its 
 superiority is neghgible. In fact, judged by ordinary 
 practical standards, we can extend the equality to the 
 " very good " or even to the " good." 
 
 To put these comparisons in figures, let us take 1917 
 in which the variations are almost always the greatest. 
 Among the " superlative " the smallest index number is 
 161.52, and the largest 161.73, while the " ideal," Formula 
 353, is 161.56. Among the " excellent " the smallest is 
 161.18 and the largest is 161.94. Among the " very 
 good " the smallest and largest are 160.3 and 162.4. 
 Among the " good," they are 156 and 167 ; among the 
 
258 THE MAKING OF INDEX NUMBERS 
 
 " fair," 152 and 171 ; among the " poor," 145 and 181 ; 
 and among the " worthless," 135 and 196. 
 
 In percentages these figures show the maximum de- 
 viation from the ideal (161.56) to be as follows : among the 
 " superlative," .1 per cent ; among the " excellent," .2 per 
 cent ; among the " very good," .8 per cent ; among the 
 " good," 3.7 per cent ; among the " fair," 6.2 per cent ; 
 among the "poor," 11.7 per cent; among the "worth- 
 less," 21 per cent. 
 
 How far can we go in letting the less accurate index 
 numbers pass muster as good enough ? The answer will 
 vary, of course, according to the standards we set in any 
 particular case. In practice, it is seldom that our stand- 
 ards require a closer approximation than two per cent. 
 On this basis we may admit as usable index numbers all 
 of the 11 " superlative," the 15 " excellent," the 11 " very 
 good," and most of the 16 " good," or nearly 40 per cent 
 of the 134 index numbers in all. These are all in the " " 
 or middle tine class, i.e. they include all except the biased 
 and the freakish index numbers, which is in accordance 
 with the findings of the previous chapter. 
 
 § 5. Comments on Modes, Medians, and Simples 
 
 A glance at the class symbols in the second column of 
 Table 28 shows that " S " (or the simples and their 
 derivatives) and " M " (the modes and medians) are mostly 
 far away from Formula 353 ; that the " 2's " are next 
 farthest from 353, the " I's " next, and the "O's " last. 
 
 The rankings of the simples are of interest. When 
 we were comparing the simples among themselves, we con- 
 denmed the arithmetic and harmonic and their antitheses 
 (Formulae 1, 2, 11, 12) on the sole ground of bias; we 
 did not condemn them on the ground of freakishness of 
 weighting, for then the simple, or even, weighting was 
 
COMPARING ALL THE INDEX NUMBERS 259 
 
 assumed to be correct. But now that we are applying 
 higher standards and comparing the simples themselves 
 with the best weighted index number (Formula 353), we 
 condemn every simple formula, even Formula 21, on the 
 ground of freakish weighting, while still condemning 
 Formulae 1, 2, 11, 12 on the further ground of bias. These 
 latter formulse are thus doubly bad, combining both 
 freakishness and bias, despite the fact that, in some cases, 
 the two happen to neutrahze each other. Thus Formula 
 2 for prices happens to agree closely with 353 for 1914 and 
 1918, but not for other years. Consequently these four 
 formulse stand at, or near, the extreme top of Table 28. 
 
 It should also be noted that the modes in particular 
 occur in clusters in almost random order, and not in the 
 order of their conformity to tests. Normally, as shown 
 in the cases of the other varieties, the " rectification " 
 process actually rectifies. For instance, in Table 28, 
 we find that the primary and biased weighted geometries, 
 Formulse 23, 24, 25, 26, 27, 28, 29, 30, precede the (singly) 
 rectified Formulse 123, 1123, 124, 1124, 125, 126, 223, 
 225, 227, 229, and these, in turn, precede the doubly 
 rectified 323, 325, 1323, 5323 (except that 323 and 325 
 are slightly out of the prescribed order). Moreover, all 
 these several geometries have each a separate rank in the 
 list, whereas all of the 25 modes (including even the 
 simples) occur in a few clusters and with almost no regard 
 to any systematic order. For instance, we find the un- 
 rectified modes, Formulse 44, 46, 48, 50, not preceding the 
 rectified Formulae 144, 146, 1144, but clustered in exactly 
 the same rank with them and with one another. Again, 
 we find 243, 245, 247, 249, 343, 345, 1343, 5343 Ukewise 
 clustered at identically the same rank. And the last- 
 named cluster (consisting of rectified index numbers) 
 precedes, instead of follows as it should, the cluster, 43, 
 
260 THE MAKING OF INDEX NUMBERS 
 
 45, 47, 49, 143, 145, 1143 (comprising mostly unrectified 
 index numbers) . The medians behave considerably better 
 but they also are immobile as contrasted with others. 
 
 The table completes the evidence that what makes a 
 bad index number is either freakishness or bias, and that 
 the bias can be thoroughly eliminated by the rectifica- 
 tion process, while freakishness cannot. Barring index 
 numbers subject to these defects, all index numbers are 
 good. In other words, all in group "0," which lie in 
 the middle tine of the five-tined fork, are good. With 
 few exceptions, every good index number obeys at least 
 one of the two tests. The exceptions are Formulae 53, 54, 
 6023, 6053, 7053, 8053, 8054, all of which, while they fail 
 to obey either test, come very near to obeying both tests. 
 
 § 6. The Simple Median Nearer the Ideal than 
 the Simple Geometric 
 
 We are now ready to return to the unfinished discussion 
 of the median. It is one of the interesting and surprising 
 results of the comparisons in the complete list of formulae 
 that the simple median has a better rating than any other 
 simple index number. The order of increasing merit of 
 the simples as here shown is: Formulae 41 (worthless), 
 1 (worthless), 51 (worthless), 11 (poor), 21 (poor), 31 
 (poor). The median thus not only outranks the mode, 
 which was to be expected, and the simple arithmetic, so 
 much in vogue, but even the geometric. After what has 
 been said as to the freakishness of the median and the 
 virtues of the geometric, it might have been expected that 
 the median would rank among the worst of the simples, 
 and that Formula 21 would rank as the best. And, as 
 we have seen, when we assume that simple or equal weighting 
 is the right weighting, the order of merit would make 
 Formula 21 best and 31 far inferior. But, of course, 
 
COMPARING ALL THE INDEX NUMBERS 261 
 
 simple weighting never really is the right weighting, and 
 our table of merit is based not on simple but on true 
 weighting. On such a scale, Formula 31 seems to outrank 
 21. 
 
 The comparison of Formulse 21 and 31 as to nearness 
 to 353 may be presented numerically as follows : 
 
 TABLE 30. ACCURACY OF SIMPLE GEOMETRIC AND SIMPLE 
 MEDIAN, JUDGED BY THE STANDARD OF FORMULA 353 
 FOR 36 COMMODITIES 
 
 (Prices) 
 
 Formula No. 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1913 
 
 21 
 
 100 
 
 96 
 
 97 
 
 121 
 
 167 
 
 180 
 
 31 
 
 100 
 
 99 
 
 99 
 
 119 
 
 164 
 
 191 
 
 353 
 
 100 
 
 100 
 
 100 
 
 114 
 
 162 
 
 178 
 
 Evidently, the median (31) is somewhat nearer the ideal (353) than is 
 the geometric (21) in 1914, 1915, 1916, and 1917, — by three per cent, 
 two per cent, two per cent, and two per cent respectively. It is farther 
 away only in 1918, — by six per cent. 
 
 We may further test the conclusion reached by comparing Formulae 
 21 and 31 as applied to quantities. The figures follow : 
 
 TABLE 31. ACCURACY OF SIMPLE GEOMETRIC AND SIMPLE 
 MEDIAN, JUDGED BY THE STANDARD OF FORMULA 353 
 FOR 36 COMMODITIES 
 
 (Quantities) 
 
 FORMUIiA No. 
 
 1913 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 21 
 
 100 
 
 98 
 
 111 
 
 121 
 
 119 
 
 115 
 
 31 
 
 100 
 
 99 
 
 107 
 
 117 
 
 119 
 
 121 
 
 353 
 
 100 
 
 99 
 
 109 
 
 119 
 
 119 
 
 125 
 
 Here again Formula 31 shows to better advantage than 21, being one 
 per cent superior in 1914 and five per cent superior in 1918, and scoring 
 a tie in 1916, 1917, and 1915. In 1918 one unimportant commodity, 
 skins, the quantity of which fell enormously while most others rose, 
 spoils the sensitive geometric, — even making the average movement 
 seem to be downward when it is really upward, — but it has no influence 
 on the insensitive median. 
 
262 
 
 THE MAKING OF INDEX NUMBERS 
 
 Further confirmation of our conclusion is found by a study of the 1437 
 commodities used by the War Industries Board. The weighted aggregative 
 (Formula 53) was the index number employed there, and may here be 
 used as our standard in lieu of 353. I have computed the simple geometric 
 and median. The results, which are per prices, are as follows : 
 
 TABLE 32. ACCURACY OF SIMPLE GEOMETRIC AND SIMPLE 
 MEDIAN, JUDGED BY THE STANDARD OF FORMULA 53 
 FOR 1437 COMMODITIES 
 
 (Pre-war year July, 1913 -July, 1914 = 100 percent) 
 
 Formula No. 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 21 
 31 
 53 
 
 101 
 101 
 101 
 
 101 
 
 100 
 
 99 
 
 108 
 101 
 
 102 
 
 138 
 122 
 126 
 
 174 
 162 
 175 
 
 198 
 196 
 194 
 
 We note that Formula 31 is as close to 53 as is 21 for 1913 and closer 
 in 1914 by one per cent, in 1915 by five per cent, 1916 by six per cent, 
 and 1918 by one per cent. The only case to the contrary is 1917, where 
 Formula 21 is the closer by seven per cent. Thus Formula 31 is superior 
 to 21 for the prices of 1437 commodities, just as it was for the prices and 
 quantities of the 36. Unfortunately we lack the data for calculating the 
 quantity ratios of the 1437 commodities. 
 
 Chart 50 shows the relations of the simple median and simple geometric 
 for 1917 for commodities taken from our list of 36 by lot, beginning with 
 three and including every odd number. It will be seen that, as compared 
 with 353 (the dotted line) it is nip and tuck between 21 and 31 as to their 
 closeness to 353, except where the number of commodities falls below 11, 
 when 21 is decidedly the better. In spite of its insensitiveness, as shown 
 by its few changes. No. 31 averages as close to 353 as does 21. 
 
 Is this apparent superiority of Formula 31 an accident? It is hard 
 to say, but I am inclined to think that, at any rate, 21 is not on the average 
 superior to 31. 
 
 Professor Edgeworth has advocated the simple median on the ground 
 that it cannot be so easily influenced by extreme aberrations of one or two 
 individual commodities of "mall importance. In simple (or equal) weight- 
 ing, commodities of small importance are in some cases endowed with 
 undue influence. This is so in the case of the geometric (or, for that matter 
 also the arithmetic and harmonic), whereas in the case of the median such 
 extreme variations produce no disturbance whatever. This argument 
 of Edgeworth's is sound, although at first sight it seems to conflict with 
 some of the lines of reasoning heretofore used in this book. While Pro- 
 fessor Edgeworth praises the median because it is not exaggeratedly sen- 
 sitive, it would seem that I have condemned it just because it is not a 
 sufficiently sensitive barometer. This conflict of opinion, however, is 
 more apparent than real. Insensitiveness is an unmitigated evil in a 
 
COMPARING ALL THE INDEX NUMBERS 263 
 
 carefully weighted index nximber, for it prevents some of the commodities 
 from having their proper influence. This is likewise true also of a simple 
 index number provided its (even) weighting happens not to be too far from 
 the true weighting. But when this even weighting happens to differ enor- 
 mously from the true weighting, as is frequently — probably usually 
 — the case, the matter is not so easily disposed of. In that case it may well 
 be that the insensitiveness of the median by preventing an undue influence 
 of extreme aberrations of unimportant commodities may more than make 
 up for any delinquency in preventing the due influence of important com- 
 modities. Such a net benefit is pretty certain to accrue when the unim- 
 portant commodities are the most extreme in their aberrations, and the 
 important ones, the least. And, possibly, this is what we usually find. It 
 follows that when we are forced to use a simple index number as a make- 
 
 Simp/e Geometric and Simple Median, Compared wifii Idea! 
 for Different l^umbers of Commodities 
 
 \5% 
 
 3 5 7 9 II 13 15 17 19 21 25 25 27 29 31 33 35 
 
 NUMBER OF COMMODITIES 
 Chakt 50. Showing that the simple median (31) and simple geometric 
 (21) are, on the whole, about equally near the ideal (353) for the seventeen 
 different numbers of conamodities except in the cases of 3, 5, 7, and 9 com- 
 modities, when the geometric is distinctly nearer. 
 
 shift for a carefully weighted one because we lack the data for weighting, 
 the simple median may as well masquerade as a weighted index number 
 as the simple geometric. The median cannot go far wrong when the 
 really important commodities do not disperse very widely, whereas the 
 geometric is apt to be thrown out of this middle course by giving a vastly 
 exaggerated influence to a few unimportant but widely aberrant com- 
 modities. 
 
 Whether the important commodities really do usually keep near the 
 middle of the road as compared to the unimportant is doubtful, however. 
 Of the 36 commodities whose prices changed from 1913 to 1914, the middle 
 18 price relatives were much less important than the other 18, i.e. than the 
 9 whose price relatives were the highest and the other 9 whose price relatives 
 were the lowest. The relative unimportance of the middle 18 is best meas- 
 ured by their total weights (taken, say, as the mean between the 1913 
 and 1914 values). This total in 1914 was equal to only 3692 out of a 
 total for all 36 commodities of 13024, or considerably less than half. In 
 
264 THE MAKING OF INDEX NUMBERS 
 
 other words, tlie average commodity near the middle of the price movement 
 was less important than the average commodity near the extremes of 
 movement. This is true in all the years. In 1915, the middle 18 commod- 
 ities had weights of 4062 out of a total of 13588 ; in 1916, 4746 out of 15157 ; 
 in 1917, 5776 out of 17857; in 1918, 6086 out of 19307, — in all cases less 
 than half. Nevertheless, in spite of these facts for the 36 commodities, 
 the simple median, as we have seen, is slightly nearer the ideal than is 
 the simple geometric. 
 
 Our conclusion is that the simple median, except when 
 there are very few commodities, is probably at least as 
 good on the average as a substitute for a weighted index 
 number as is the simple geometric. 
 
 Precisely the same argimients for and against the simple median (com- 
 pared with the simple geometric) apply also to the simple mode. But 
 in this case the balance is certainly against the mode, the mode being far 
 more freakish than the median. The mode, Formula 41, is further from 353 
 than is 2 1 in Chart 49. This is for prices. The same is true as to quantities. 
 The same is also true for the 1437 commodities, the simple mode (41) 
 being 99, 99, 99, 108, 145, 173, as against the geometric (21) which runs 
 101, 101, 108, 138, 174, 198, while the aggregative (53) used as the standard 
 by which to judge between 21 and 31, runs 101, 99, 102, 126, 175, 194. 
 
 § 7. Slight Revision of the Order of the Best Formulae 
 
 The order of merit, which we have found, was deter- 
 mined quite mechanically and doubtless this order, 
 toward the end where the competition for first place is so 
 close, is somewhat accidental and would vary considerably 
 if calculations were made with other data. The last score 
 or two of formulae are practically all alike in accuracy. 
 If we are to discriminate at all among these it is better 
 not to be guided wholly by mechanical methods. We 
 may revise slightly the order of precedence. It is doubtless 
 an accident that places 2153 one place nearer the ideal 
 than 2353 which, on independent grounds, should be the 
 better formula. Doubtless ordinarily it is the closer to 
 353 and is actually found to be so in other cases. 
 
 Without arguing all the fine distinctions which might be 
 drawn, I shall, somewhat dogmatically, pronounce my 
 
COMPARING ALL THE INDEX NUMBERS 265 
 
 own final judgment as to the true order of precedence in 
 accuracy, taking into account all the considerations in 
 this and the preceding chapter. This (increasing) order 
 of merit is : 309, 307, 5307, 1303, 4154, 4153, 3154, 3153, 
 4353, 3353, 1124, 1123, 124, 126, 123, 125, 1154, 1153, 
 2154, 2153, 323, 325, 8054, 8053, 1323, 1353, 5323, 
 2353, 353. These I should call the 29 best formulae 
 with only infinitesimal preferences among them. The 
 list has been intended to include all formulae satisfying 
 both tests (barring medians, modes, simples, and their 
 derivatives). It will be noticed, however, that the list 
 includes a number of formulae obeying only one test and 
 two (8053 and 8054), very excellent ones, obeying neither. 
 
 This list contains none of the formulae in common use, 
 most of which are objectionable because of bias or freakish- 
 ness. This sheaf of 29 accurate formulae represents the 
 best of the large crop reaped from the seed of the 46 
 primary formulae. The 29 are all within less than one-half 
 of one per cent of the *' ideal," 353. So far as accuracy 
 is concerned any one of them is good enough to serve 
 for all practical purposes. Moreover, none outside of 
 this list need ever be used for any purpose where great 
 accuracy is demanded, although about as many other 
 formulae are accurate enough for most purposes. As 
 to other considerations than accuracy, more will be said 
 later. 
 
 Few writers besides Walsh have tried to go outside 
 what are here called the primary formulae. The usual 
 attitude is to observe regretfully that " different ways of 
 computing index numbers lead to different results," 
 and then either to shrug the shoulders in despair of any- 
 thing better, as much as to say " you pays your money and 
 you takes your choice," or vaguely to contend that "some 
 kinds of index numbers are good for some purposes and 
 
266 THE MAKING OF INDEX NUMBERS 
 
 some for others." In view of what we have found as to 
 bias, rectification, and the close agreements in the results, 
 I do not see how any reasonable man can henceforth 
 continue to take either of these views. 
 
 § 8. Conclusions 
 
 What, then, are the results of the comparisons among 
 the 134 varieties of index numbers? The chief results 
 seem to be : 
 
 1. The only really unreliable class of formulae are those 
 which are distinctly freakish, whether because of a freak- 
 ish type, as in the case of the modes and, in less degree 
 the medians, or on account of a freakish weighting, as in 
 the case of the simples. 
 
 2. Formulae which are merely biased can always be 
 thoroughly rectified by mating with formulae of equal, 
 but opposite, bias. 
 
 3. Consequently, in Table 28, all the biased formulae 
 (unless of freakish origin) take their places with great 
 regularity of order ; first, the doubly biased (the " 2 + " 
 and " 2 — " classes occurring side by side), and then the 
 singly biased in the same way. 
 
 4. Any type of formula, with the single exception of 
 the incorrigible mode (which in our Table 28 never scores 
 better than " poor "), can, by passing through our two 
 rolling mills of rectification (Test 1 and Test 2), be straight- 
 ened out into a good index number. All the roads lead 
 to Rome, — whether the roads be the arithmetic, the 
 harmonic, the geometric, or the aggregative. 
 
 5. Even the median, which is fairly freakish by nature, 
 turns out in the end, when doubly rectified, to be at least 
 ''good" (viz. Formulae 335, 1333, 5333, 333, — also 
 239, although only once rectified). Probably, if a large 
 number (instead of only 36) of commodities were taken, 
 
COMPARING ALL THE INDEX NUMBERS 267 
 
 the median would come considerably closer to the 
 " ideal." 
 
 6. As to the mode also, some improvement may be 
 expected by increasing the number of commodities. Un- 
 fortunately, we lack the data for testing weighted modes, 
 and their rectifications, for a large number of commodities. 
 Judging from such indications as are at hand, I venture 
 the guess that, for 100 or 200 commodities, the rectified 
 weighted mode would agree with the ideal within, say, 
 two or three per cent. 
 
 7. Just as any type of index number (with the possible 
 exception of the freakish mode) can be rectified to agree 
 approximately with the ideal, so any system of weighting,^ 
 excepting such freakish weighting as the "simple," can 
 be rectified. It matters not whether an index number, 
 to start with, be weighted according to systems 7, 77, 777, 
 IV, or any crosses between them. After rectification by 
 both tests the resulting index number will invariably 
 emerge (except for modes) as competent. In fact, in 
 one case, even simple weighting turns out fairly well. 
 The simple median, after twofold rectification, becomes 
 a " fair " index number. 
 
 8. Every doubly rectified index number (excepting the 
 modes and simples) is at least " good." Four (medians) are 
 classed as '' good " ; two (arithmetic-harmonic) are classed 
 as '' very good " ; six (arithmetic-harmonic, geometric, 
 and aggregative) are classed as " excellent " ; and five (ge- 
 ometric and aggregative) are classed as " superlative." 
 
 9. Some 53 index numbers will pass muster as at least 
 " good," of which the five worst are medians and the 11 
 best are aggregatives and geometries (the '* superlative "). 
 
 ^ Because so much importance has hitherto been attached to the prob- 
 lem of weighting, I have included an Appendix (II) on "The Influence of 
 Weighting." But it is not essential to the course of the argument of this 
 book. 
 
268 THE MAKING OF INDEX NUMBERS 
 
 All the intervening 37 index numbers are aggregatives, 
 geometries, and arithmetic-harmonics (unless we call 
 Formulae 207 and 209 arithmetics alone, and 213 and 
 215 harmonics alone). 
 
 10. Consequently, the nature of the index number 
 formula (whether arithmetic, harmonic, geometric, median, 
 aggregative, and whether weighted by one system or 
 another) sinks into insignificance as compared with its 
 conformity to the two tests. The only things which are 
 really necessary for a first class index number are : 
 
 a. Absence of freakishness ; 
 
 h. Conformity to Tests 1 and 2. 
 
 The conformity to Test 1 implies, as has been seen, 
 absence of bias. If our standards of a good index number 
 are not high, we need not insist on conformity to tests, 
 but instead on " absence of bias." 
 
 11. Table 28 also shows that Test 1 is a better cor- 
 rective of bias than Test 2, while Test 2 is a better cor- 
 rective of freakishness. Thus, as a rectification of the 
 biased arithmetic Formula 7, Formula 107 obeying Test 1 
 outranks 207 obeying Test 2, and likewise 109 outranks 
 209, 123 outranks 223, 125 outranks 225. But as a 
 rectification of the freakish median 33, Formula 233 out- 
 ranks 133, and 235 outranks 135. Again, as a rectifi- 
 cation of the freakish simple 21, Formula 221 outranks 
 121 ; while, likewise, 231 outranks 131 ; 241 outranks 141 ; 
 251 outranks 151. 
 
 12. The most accurate formulae are those toward the 
 end of the list, including especially : Formulae 353, 8053, 
 2153, 1353, 1323,5323. 
 
 13. If the data for quantities are available only for 
 the base year or a series of years, the best available index 
 numbers of prices are : Formulae 53, 6053, 6023. 
 
COMPARING ALL THE INDEX NUMBERS 269 
 
 14. If only roughly estimated or guessed weights 
 can be used, the best formula is Formula 9051. 
 
 15. If we cannot, or will not, estimate or guess at the 
 weights, the best index numbers are: Formulae 21, 101, 
 31, of which 31 is probably slightly more accurate unless 
 there is good reason to believe that the true weights of 
 the various commodities really are approximately equal, 
 or unless the number of commodities is very small. 
 
 We may restate and summarize our main conclusions 
 as follows : 
 
 Always barring the mode (as a freak type) and the 
 simple (as a freak weighting), type and weighting have no 
 material influence on our final results, after the rectifi- 
 cation processes. After those processes are completed, 
 all the results are substantially the same. This will 
 seem a startling conclusion and quite contrary to common 
 opinion ; for current views do not recognize the existence 
 of bias in the index numbers used nor reaUze that it can 
 be rectified. 
 
CHAPTER XIII 
 THE SO-CALLED CIRCULAR TEST 
 
 § 1. Introduction 
 
 It will be remembered that the fault we first found in 
 certain index numbers, e.g. the simple arithmetic, was 
 that it would not work consistently as between two times, 
 or between two places, like New York and Philadelphia. 
 Test 1 required such consistency and our ideal formula, 353, 
 and many others meet that test. Can we and ought we 
 to extend this requirement for consistency as between 
 the two times, or the two places, which the index num- 
 ber compares (and, of course, it only compares two) to a 
 general consistency between all the times or places to 
 which we apply a. set oi index numbers? 
 
 Hitherto this has been taken for granted by all stu- 
 dents of index numbers. The small balls ought, it has 
 been assumed, always to he on the curve. If they, or any 
 of them, are separated by a gap from the curve, then it 
 would seem there must be, to that extent, something 
 wrong in the index number which permits such an 
 inconsistency. 
 
 By the so-called " circular test," taking New York as 
 base (= 100) and finding Philadelphia 110, then taking 
 Philadelphia as base (= 110) and finding Chicago (115) 
 we ought, when we complete the circuit and take Chicago 
 as base (= 115), to find, by direct comparison. New York 
 100 again. Or again, if Chicago is found to be 115 via 
 Philadelphia, it ought consistently to be 115 when cal- 
 culated directly. 
 
 Still again, instead of taking percentages, let us take 
 
 270 
 
THE SO-CALLED CIRCULAR TEST 271 
 
 easy fractions. Let New York be unity, Philadelphia 
 double this or 2, Chicago 50 per cent more, or 3. Then 
 New York should be (according to the circular test) one 
 third of Chicago, or 1 again. The three links around 
 the circle are here f , f , i, and these, multiplied together, 
 give unity or one hundred per cent.^ For a single com- 
 modity, of course, this holds good. If the price of sugar 
 is twice as high in Philadelphia as New York, 50 per cent 
 higher in Chicago than Philadelphia, then self-evidently, 
 in New York the price of sugar must be a third as high as 
 Chicago. If this is true of one commodity, why not of 
 an average for many ? 
 
 But the analogy of the circular test with the time re- 
 versal test, while plausible, is misleading. I aim to show 
 that the circular test is theoretically a mistaken one, that 
 a necessary irreducible minimum of divergence from 
 such fulfillment is entirely right and proper, and, therefore, 
 that a 'perfect fulfillment of this so-called circular test 
 should really be taken as proof that the formula which 
 fulfills it is erroneous. 
 
 § 2. Illustration of Non-fulfiUment by Case of 
 Three Very Unlike Countries 
 
 We can see best by a concrete example. Let us take 
 three places which, to fix our ideas, we shall call Georgia, 
 Norway, and Egypt. Take a list of 15 commodities of 
 which 5, led by lumber, are important in both Georgia 
 and Norway ; 5, led by cotton, are important in both 
 Georgia and Egypt ; and 5, led by paper, are important 
 in both Egypt and Norway. Let us further suppose 
 that the lumber group, important in both Georgia and 
 Norway, have about the same prices in Georgia and Nor- 
 
 1 For the algebraic expression of the circular test, see Appendix I (Note 
 to Chapter XIII, §1). 
 
272 THE MAKING OF INDEX NUMBERS 
 
 way, and that they so dominate the price comparison 
 between these two countries that the index number is 
 about the same in both countries, the other two groups 
 of commodities in these two countries not greatly inter- 
 fering with this equahty, because one is unimportant in 
 Georgia and the other is unimportant in Norway. Like- 
 wise, in comparing Georgia and Egypt, the cotton group 
 so dominates the Georgia-Egypt index number as to 
 make Georgia and Egypt about the same price level. 
 
 We might conclude, since " two things equal to the 
 same thing are equal to each other," that, therefore, the 
 price levels of Egypt and Norway must be equal, and this 
 would be the case if we thus compare Egypt and Norway 
 via Georgia. But evidently, if we are intent on getting 
 the very best comparison between Norway and Egypt, we 
 shall not go to Georgia for our weights. In the direct 
 comparison between Norway and Egypt the weighting is, 
 so to speak, none of Georgia's business. It is the concern 
 only of Egypt and Norway. In such a direct comparison 
 between Norway and Egypt, the paper group, which 
 played little part in the other two comparisons now tends 
 to dominate the situation ; and if these 5 commodities are 
 higher in price in Norway than in Egypt, that fact may 
 suffice to make the whole Norwegian price level some- 
 what higher than the Egyptian. 
 
 § 3. Comparisons by Index Numbers Differ 
 in Kind 
 
 The paradox of finding the price levels of Norway and 
 Egypt different, although by separate comparisons the 
 price level of each is the same as that of Georgia, is no 
 more strange than that we may find two people each 
 resembling in their features a third person without re- 
 sembling each other. Since an index number is a com- 
 
THE SO-CALLED CIRCULAR TEST 273 
 
 posite dependent on heterogeneous elements, a variation 
 in the composition will change the comparison qualita- 
 tively. There is really, therefore, no contradiction or 
 absurdity in the apparent inconsistencies ; for the three 
 comparisons are all different in kind. If the three groups 
 (lumber, cotton, paper) prominent in the Georgia, 
 Norway, Egypt comparisons, instead of merely dominat- 
 ing the respective comparisons, were completely to monop- 
 olize them, any mystery about their inconsistencies 
 would disappear. We would have three index numbers of 
 only one commodity each : lumber for comparing Georgia 
 and Norway (there being no other common commodity), 
 cotton for comparing Georgia and Egypt (this being 
 the only commodity in common), and paper, the only 
 common commodity, for comparing Norway and Egypt. 
 Our supposedly inconsistent comparisons reduce to the 
 initial facts, viz. that lumber is the same price in Georgia 
 as in Norway, and cotton in Egypt as in Georgia, while 
 paper is higher in Norway than in Egypt, in which three 
 statements are surely no mutual inconsistencies. The 
 fact that lumber and cotton show certain comparisons 
 for Norway and Egypt relatively to a third country 
 is no reason why a commodity quite different from either 
 lumber or cotton should show any particular comparison 
 between Norway and Egypt compared directly. Similarly, 
 even if not so self-evidently, the fact that index num- 
 bers in which lumber and cotton are important show 
 certain comparisons, is no reason why an entirely dif- 
 ferent index number in which they are unimportant should 
 show any particular comparison. 
 
 In short, each dual comparison is a separate problem 
 differing in kind from every other and, therefore, requiring 
 no exact correspondence such as would be required if they 
 were not different. If they were really the same, e.g. if we 
 
274 THE MAKING OF INDEX NUMBERS 
 
 had one and the same commodity to deal with, it would 
 be absurd and impossible to find, say, the price of coffee the 
 same in Norway as in Georgia, the same in Egypt as in 
 Georgia, but yet higher in Norway than in Egypt. 
 
 The truth is, if we were to find any other result than 
 what we have found, we would know that that result was 
 wrong. Such a formula would prove too much, for it would 
 leave no room for qualitative differences. Index numbers 
 are to some extent empirical, and the supposed inconsist- 
 ency in the failure of (variably weighted) index numbers to 
 conform to the circular test, is really a bridge to reality. 
 That is, the so-called " inconsistency " is just what is 
 needed to reconcile our theory with common sense, which 
 tells us at once that we cannot consistently compare far- 
 distant times and climes by means of averages of widely 
 varying elements. Either we must give up the attempt, 
 or we must content ourselves with an artificially rigid 
 system of weights which contradicts the facts. 
 
 § 4. The So-called Circular Test can be Fulfilled 
 Only if Weights are Constant 
 
 The only formulae which conform perfectly to the cir- 
 cular test are index numbers which have constant weights, 
 i.e. weights which are the same for all sides of the "tri- 
 angle " or segments of the " circle," i.e. for every pair 
 of times or places compared. Thus, if all the 15 com- 
 modities, lumber, paper, cotton, etc., are arbitrarily 
 assigned weights which remain the same in all three 
 comparisons, in defiance of the actual differences, then 
 the index number ought to show that if Norway and 
 Egypt have the same price level relatively to Georgia, 
 they will have the same price level relatively to each 
 other. And this is precisely what we do find of the simple 
 
THE SO-CALLED CIRCULAR TEST 275 
 
 or 'jonstant weighted geometric, for instance, and the sim- 
 ple or constant weighted aggregative.^ 
 
 But, clearly, constant weighting is not theoretically 
 correct. If we compare 1913 with 1914, we need one set 
 of weights; if we compare 1913 with 1915 we need, 
 theoretically at least, another set of weights. In the 
 former case we need weights involving the quantities of 
 the two years concerned, 1913 and 1914; in the second 
 case we need weights involving the (somewhat different) 
 quantities of the two years, 1913 and 1915. We cannot 
 justify using the same weights for comparing the price 
 level of 1913, not only with 1914 and 1915, but with 1860, 
 1776, 1492, and the times of Diocletian, Rameses II, 
 and the Stone Age ! 
 
 Similarly, turning from time to space, an index number 
 for comparing the United States and England requires one 
 set of weights, and an index number for comparing the 
 United States and France requires, theoretically at least, 
 another. To take extreme cases, it would obviously be 
 improper to use the same weights in comparing the United 
 States, not only with England and France, but with 
 Russia, Siberia, China, Thibet, and Central Africa. 
 In comparing hot with cold cUmates, coal would be 
 weighted heavily in some cases and in others lightly, 
 and ice reversely. Allowances should likewise be made 
 for differences, in different times or climes, in the quanti- 
 ties of wool, silk, rice, quinine, ivory, glass, blubber, 
 breadfruit, sisal, jade, bamboo, steel, cement, automobiles, 
 boomerangs, machine guns, linotype machines, wax 
 tablets, paper, and other things varying in importance 
 geographically or historically. In comparing the prices 
 of our times with those of 1860, it is just as important to 
 have our weights representative of Lincoln's day as to 
 1 See Appendix I (Note A to Chapter XIII, § 4). 
 
276 THE MAKING OF INDEX NUMBERS 
 
 have them representative of ours. So also in comparing 
 our country with China, we must give equal voice to the 
 peculiarities of the two. 
 
 If we start with weights appropriate to the United States 
 of 1922, any comparison between the United States and 
 modern Kamchatka or ancient Babylonia would be one- 
 sided. Even more one-sided would be a comparison, 
 by the use of these same American weights, between the 
 price levels of Kamchatka and Babylonia. Only by 
 employing the weightings of the United States in 1922, 
 once for all, are we enabled to force a fulfillment of the 
 circular test, so that the three comparisons between the 
 United States in 1922, modern Kamchatka, and ancient 
 Babylonia are mutually consistent. For instance, if 
 the price level of the United States equalled that of Kam- 
 chatka and also equalled that of Babylonia, then these 
 two would equal each other. It is clear that constant 
 weighting, though it makes it possible to fulfill the 
 circular test, does so at the expense of forcing the facts, 
 for the true weights are not thus constant.^ 
 
 § 5. How Closely is the So-called Circular 
 Test Fulfilled? 
 
 But the important question is : How near is the circular 
 test to fulfillment in actual cases? If very near, then 
 practically we may make some use of the circular test as 
 an approximation even if it is not strictly vahd. To 
 answer this question, we shall take Formula 353 and the 
 standard set of data for 1913-1918 which we have used 
 hitherto. 
 
 Numerically, by Formula 353, the price level of 1914 
 
 ' In this connection, the mathematical reader may be interested in 
 another way in which, with a limited application, the circular test may be 
 fulfilled. See Appendix I (Note B to Chapter XIII, § 4). 
 
THE SO-CALLED CIRCULAR TEST 
 
 277 
 
 relatively to 1913 is 100.12, showing a rise of .12 per cent. 
 This is the figure obtained by comparing the two years' 
 prices directly, i.e. without the intervention of any other 
 year. But if we compare them via 1915, we get 99.77 
 for 1914, showing a fall of .23 per cent from 1913 instead 
 of the actual rise of . 12 per cent. The following table gives 
 all the comparisons between 1913 and 1914, both directly 
 and also indirectly, via certain other years. 
 
 
 1913 
 
 1914 
 
 True or direct 
 
 100 
 
 100.12 
 
 Indirect via 1915 
 1916 
 1917 
 1918 
 
 100 
 100 
 100 
 100 
 
 99.77 
 100.21 
 100.34 
 
 99.94 
 
 It will be noticed that, although the intervention of an 
 intermediating year does not yield exactly the same result 
 as the direct comparison between the two years concerned, 
 the discrepancies are very slight. This is found to be true 
 of all good index numbers. That is, while there should be 
 some discrepancy and the index numbers which have none 
 at all are therefore in error, a large discrepancy is equally 
 wrong. Formula 141, for instance, exhibits a large 
 discrepancy ; 353, a small one. 
 
 Let us test, by the so-called circular test, Formulae 9 and 353, repre- 
 senting a very bad and a very good index number respectively ; and, for 
 this purpose, let us take the circuit of years 1913-1914-1915-1913 or 0-1- 
 2-0, which triangle of years we shall refer to briefly as "012." 
 
 By Formula 9 the index number for the side of the triangle 0-1, i.e. 
 the index number of prices for 1914 relatively to 1913 as base, is 100.93 
 per cent; the index number for the next side of the triangle, 1-2, is 101.16 
 per cent; and that for the returning side, 2-0, is 102.21 per cent. The 
 product of these three index numbers around the triangular circuit is 
 104.36 per cent, showing that, even in this three-around comparison, the 
 deviation from unity, or 100 per cent, of Formula 9 is very striking. If 
 we should take a foiu'-around, five-around, or six-around case, the gap in 
 
278 THE MAKING OF INDEX NUMBERS 
 
 the circle would be much greater. Evidently the gap, in the case of 9, is 
 partly due to its known upward bias, each of the three factors tending to 
 be larger than it should be. 
 
 Next, then, let us try Formula 353, which has no bias and fulfills both 
 tests. In this case, we find, for the same circuit 0-1-2, the product of the 
 three index numbers for prices,^ 0-1, 1-2, !i-0, is 100.35, or only about one 
 third of one per cent above 100 per cent or unity. The other index num- 
 bers, which like Formula 353 satisfy both Tests 1 and 2, will, in general, 
 deviate from the so-called circular test by about the same gap, aa Table 33 
 shows. 
 
 TABLE 33. THE "CIRCULAR GAP," OR DEVIATION FROM 
 FULFILLING THE SO-CALLED "CIRCULAR TEST" OF 
 VARIOUS FORMULAE 
 
 (In the 3-around comparison of price indexes for years 1913-1914-1915, 
 
 or 0-1-2) 
 
 Formula No. 
 
 Circular Gap 
 (Per Cents) 
 
 323 
 
 +.34 
 
 325 
 
 +.38 
 
 353 
 
 +.35 
 
 1323 
 
 +.34 
 
 1353 
 
 +.34 
 
 2353 
 
 + .34 
 
 5307 
 
 +.40 
 
 5323 
 
 + .36 
 
 This table shows that if we calculate by Formula 323, starting from 
 
 1913 (year 0) and proceeding to 1914 (year 1), then calculate from this 
 
 1914 as a base to 1915 (year 2), and then calculate from this 1915 as a base 
 to 1913 again, instead of finding ourselves exactly where we started, the 
 resulting figure will be slightly above the starting-point, exceeding the 
 original figure by ^ny of 1 per cent. The other seven formulae give almost 
 uniformly the same result, roughly, a third of one per cent. From these 
 examples, and others which will be noted in other connections, it appears 
 that there is a proper and there is an improper deviation from fulfillment 
 of the circular test. The deviation or circular gap of about one-third of 
 one per cent for Formula 353 and other good formulae represents, as it 
 were, an irreducible minimum of legitimate deviation. On the other 
 hand, the big gap for biased formulae, like 9, represents, for the most part, 
 an illegitimate or erroneous gap. At the other extreme, the simple For- 
 mulae 21 and 51 show no gap at all, even the small proper deviation 
 being artificially suppressed by the use of constant weighting. 
 
 * If an index number of quantities be used, the circular gap will be equal 
 but opposite, provided, the index number fulfills Test 2. See Appendix I 
 (Note to Chapter XIII, § 5). 
 
THE SO-CALLED CIRCULAR TEST 
 
 Circular Test 
 
 Gaps for Years 045 
 of Formulae L9,25. 141,151 
 
 279 
 
 T^ 
 
 ,4» 
 
 \5% 
 
 75 7^ 75 7e y/ '/^ 
 
 Chart 51. Showing that, calculating by Formula 1 and starting from 
 1913 (year 0), then proceeding to 1917 (year 4), 1918 (year 5), and back to 
 1913, the year from which we started, we end at a point above that from 
 which we started; by Formula 9 the same circuit ends still higher; by 
 Formula 23 it ends lower ; by Formula 141 (or 41) it ends stilll ower ; by 
 Formula 151 (or 51) it ends at the starting-point. All five end wrongly. 
 
280 
 
 THE MAKING OF INDEX NUMBERS 
 
 Graphically, Chart 51 shows five formulse, all with dififerent behaviours 
 relatively to the "circular test," and none behaving correctly. Each 
 relates to the triangular comparison between the years 1913, 1917, and 1918. 
 Formula 1 is far from conforming to the circular test, returning very far 
 above the starting-point. Formula 9 returns still further above, 23 returns 
 to 1913 below the starting-point, 141 still further below, while 151 returns 
 exactly to the starting-point. 
 
 When the circular test is fulfilled, any indirect comparison between, say, 
 1913 and any other year, say, 1915 via 1914, will agree with the direct com- 
 parison ; consequently, the chain figures will coincide with the fixed base 
 figures, so that there will be no " balls" above or below our curves. The 
 more nearly the circular test is fulfilled, the more nearly will the balls be 
 to the curves. Thus the reader, by studying the balls in relation to the 
 curves in the various diagrams, can readily gain a rough idea of how nearly 
 the circular test is fulfilled. This subject will be referred to again. 
 
 § 6. Complete Tabulation of '* Circular Gap " for 
 Formula 353 
 
 Table 34 gives the gaps (for 353) for every possible triangle. 
 
 TABLE 34. THE "CIRCULAR GAP," OR DEVIATION FROM 
 FULFILLING THE SO-CALLED "CIRCULAR TEST" OF 
 FORMULA 353 (IN ALL POSSIBLE 3-AROUND COMPARI- 
 SONS OF PRICE INDEXES) 
 
 Years op 
 
 Ctpculap Gap 
 
 " Triangle " 
 
 (Peu Centc!) 
 
 0-1-2 
 
 +.35 
 
 0-1-3 
 
 -.09 
 
 0-1^ 
 
 -.21 
 
 0-1-5 
 
 +.17 
 
 0-2-3 
 
 -.25 
 
 0-2-4 
 
 -.16 
 
 0-2-5 
 
 +.30 
 
 0-3^ 
 
 +.32 
 
 0-3-5 
 
 +.30 
 
 0-4-5 
 
 +.06 
 
 1-2-3 
 
 +.19 
 
 1-2-4 
 
 +.40 
 
 1-2-5 
 
 +.48 
 
 1-3^ 
 
 +.45 
 
 1-3-5 
 
 +.05 
 
 1-4-5 
 
 -.33 
 
 2-3-4 
 
 +.23 
 
 2-3-5 
 
 -.24 
 
 2^-5 
 
 -.40 
 
 3-4-5 
 
 +.08 
 
THE SO-CALLED CIRCULAR TEST 
 
 281 
 
 Even the maximum of these circular gaps (that for the triangle of the 
 years 1-2-5, or 1914-1915-1918-1914) is only ^ per cent, or less than 
 one half of one per cent. 
 
 We find the same smallness of the gaps when the " circuit " consists of 
 four or more sides around. 
 
 Table 35 gives all the quadrangular or 4-aroimd comparisons. 
 
 TABLE 35. THE "CIRCULAR GAP," OR DEVIATION FROM 
 FULFILLING THE SO-CALLED "CIRCULAR TEST" OF 
 FORMULA 353 (IN ALL POSSIBLE 4-AROUND COMPARI- 
 SONS OF PRICE INDEXES) 
 
 Years of 
 
 Circular Gap 
 
 Years op 
 
 Circular Gap 
 
 " Quadrangle " 
 
 (Per Cents) 
 
 " Quadrangle " 
 
 (Per Cents) 
 
 0-1-2-3 
 
 +.10 
 
 0-3-2-4 
 
 -.09 
 
 0-1-2^ 
 
 +.19 
 
 0-3-2-5 
 
 -.55 
 
 0-1-2-5 
 
 +.65 
 
 0-3-4-5 
 
 +.38 
 
 0-1-3-2 
 
 +.16 
 
 0-3-5-4 
 
 +.25 
 
 0-1-3-4 
 
 +.24 
 
 0^-1-5 
 
 -.38 
 
 0-1-3-5 
 
 +.22 
 
 0-4-2-5 
 
 -.45 
 
 0-1-4-2 
 
 -.06 
 
 0-4-3-5 
 
 +.02 
 
 0-1-4-3 
 
 -.53 
 
 1-2-3-4 
 
 + .64 
 
 0-1-4-5 
 
 -.15 
 
 1-2-3-5 
 
 +.23 
 
 0-1-5-2 
 
 -.13 
 
 1-2-4-3 
 
 -.04 
 
 0-1-5-3 
 
 -.13 
 
 1-2-4-5 
 
 +.08 
 
 0-1-5-4 
 
 +.11 
 
 1-2-5-3 
 
 +.44 
 
 0-2-1-3 
 
 +.44 
 
 1-2-5^ 
 
 +.81 
 
 0-2-1-4 
 
 +.57 
 
 1-3-2-4 
 
 -.22 
 
 0-2-1-5 
 
 +.18 
 
 1-3-2-5 
 
 -.29 
 
 0-2-3-4 
 
 +.08 
 
 1-3^-5 
 
 +.12 
 
 0-2-3-5 
 
 +.06 
 
 1-3-5-4 
 
 +.37 
 
 0-2-4-3 
 
 -.48 
 
 1^-2-5 
 
 -.07 
 
 0-2-4-5 
 
 -.10 
 
 1^-3-5 
 
 +.41 
 
 0-2-5-3 
 
 -.00 
 
 2-3-4-5 
 
 -.17 
 
 0-2-5-4 
 
 +.24 
 
 2-3-5-4 
 
 +.15 
 
 0-3-1-4 
 
 +.13 
 
 2^-3-5 
 
 +.48 
 
 0-3-1-5 
 
 -.26 
 
 
 
232 
 
 THE MAKING OF INDEX NUMBERS 
 
 Table 36 gives all the 5-around comparisons. 
 
 TABLE 36. THE "CIRCULAR GAP," OR DEVIATION FROM 
 FULFILLING THE SO-CALLED "CIRCULAR TEST" OF 
 FORMULA 353 (IN ALL POSSIBLE 5-AROUND COMPARI- 
 SONS OF PRICE INDEXES) 
 
 Years of S-around 
 
 Circular Gap 
 
 Years of 5-around 
 
 Circular Gap 
 
 Circuit 
 
 (Per Cents) 
 
 Circuit 
 
 (Per Cents) 
 
 0-1-2-3-4 
 
 +.42 
 
 0-2-4-3-5 
 
 -.17 
 
 0-1-2-3-5 
 
 + .41 
 
 0-2^-5-3 
 
 -.40 
 
 0-1-2^-3 
 
 -.13 
 
 0-2-5-1-3 
 
 -.04 
 
 0-1-2^-5 
 
 + .25 
 
 0-2-5-1^ 
 
 +.09 
 
 0-1-2-5-3 
 
 +.35 
 
 0-2-5-3^ 
 
 +.32 
 
 0-1-2-5-4 
 
 +.59 
 
 0-2-5-4-3 
 
 -.08 
 
 0-1-3-2-4 
 
 +.01 
 
 0-3-1-2-4 
 
 +.28 
 
 0-1-3-2-5 
 
 +.46 
 
 0-3-1-2-5 
 
 +.74 
 
 0-1-3^-2 
 
 +.39 
 
 0-3-1-4-5 
 
 +.07 
 
 0-1-3^-5 
 
 +.29 
 
 0-3-1-5-4 
 
 -.20 
 
 0-1-3-5-2 
 
 -.08 
 
 0-3-2-1-4 
 
 +.32 
 
 0-1-3-5-4 
 
 +.16 
 
 0-3-2-1-5 
 
 -.07 
 
 0-1-4-2-3 
 
 -.30 
 
 0-3-2^-5 
 
 -.15 
 
 0-1-4-2-5 
 
 +.24 
 
 0-3-2-5-4 
 
 -.49 
 
 0-1-4-3-2 
 
 -.28 
 
 0-3-4-1-5 
 
 -.70 
 
 0-1-4-3-5 
 
 -.23 
 
 0-3-4-2-5 
 
 -.77 
 
 0-1-4-5-2 
 
 -.45 
 
 0-3-5-1-4 
 
 +.08 
 
 0-1-4-5-3 
 
 -.46 
 
 0-3-5-2-4 
 
 +.15 
 
 0-1-5-2-3 
 
 -.38 
 
 0^-1-2-5 
 
 +.87 
 
 0-1-5-2-4 
 
 -.28 
 
 0-4-1-3-5 
 
 +.43 
 
 0-1-5-3-2 
 
 + .12 
 
 0-4-2-1-5 
 
 +.02 
 
 0-1-5-3-4 
 
 +.19 
 
 0-4-2-3-5 
 
 +.21 
 
 0-1-5-4-2 
 
 +.27 
 
 0-4-3-1-5 
 
 +.07 
 
 0-1-5-4-3 
 
 -.21 
 
 0-4-3-2-5 
 
 -.22 
 
 0-2-1-3-4 
 
 + .12 
 
 1-2-3^-5 
 
 +.31 
 
 0-2-1-3-5 
 
 +.14 
 
 1-2-3-5-4 
 
 +.56 
 
 0-2-1-4-3 
 
 +.89 
 
 1-2-4-3-5 
 
 + .01 
 
 0-2-1-4-5 
 
 +.51 
 
 1-2-4-5-3 
 
 +.04 
 
 0-2-1-5-3 
 
 +.48 
 
 1-2-5-3-4 
 
 +.89 
 
 0-2-1-5-4 
 
 +.24 
 
 1-2-5-4-3 
 
 +.36 
 
 0-2-3-1-4 
 
 + .38 
 
 1-3-2-4-5 
 
 +.11 
 
 0-2-3-1-5 
 
 -.01 
 
 1-3-2-5-4 
 
 -.61 
 
 0-2-3-4-5 
 
 +.13 
 
 1-3-4-2-5 
 
 -.52 
 
 0-2-3-5-4 
 
 -.00 
 
 1-3-5-2-4 
 
 +.03 
 
 0-2-4-1-3 
 
 +.03 
 
 1-4-2-3-5 
 
 -.17 
 
 0-2-4-1-5 
 
 -.23 
 
 1-4-3-2-5 
 
 +.16 
 
THE SO-CALLED CIRCULAR TEST 
 
 283 
 
 Table 37 gives all the 6-around comparisons. 
 
 TABLE 37. THE "CIRCULAR GAP," OR DEVIATION FROM 
 FULFILLING THE SO-CALLED "CIRCULAR TEST" OF 
 FORMULA 353 riN ALL POSSIBLE 6-AROUND COMPAR- 
 ISONS OF PRICE INDEXES) 
 
 Yeahs of 6-around 
 
 Circular Gap 
 
 Years of 6-around 
 
 Circular Gap 
 
 Circuit 
 
 (Per Cents) 
 
 Circuit 
 
 (Per Cents) 
 
 0-1-2-3-4-5 
 
 +.48 
 
 0-2-3-1-4-5 
 
 +.32 
 
 0-1-2-3-5-4 
 
 +.35 
 
 0-2-3-1-5-4 
 
 +.05 
 
 0-1-2-4-3-5 
 
 +.18 
 
 0-2-3-4-1-5 
 
 -.45 
 
 0-1-2-4-5-3 
 
 -.05 
 
 0-2-3-5-1-4 
 
 +.33 
 
 0-1-2-5-3-4 
 
 +.67 
 
 0-2-4-1-3-5 
 
 -.27 
 
 0-1-2-5^-3 
 
 +.27 
 
 0-2-4-1-5-3 
 
 +.08 
 
 0-1-3-2-4-5 
 
 +.06 
 
 0-2^-3-1-5 
 
 +.22 
 
 0-1-3-2-5-4 
 
 +.41 
 
 0-2-4-5-1-3 
 
 +.36 
 
 0-1-3-4-2-5 
 
 +.69 
 
 0-2-5-1-3-4 ! 
 
 -.36 
 
 0-1-3-4-5-2 
 
 -.01 
 
 0-2-5-1-4-3 
 
 +.41 
 
 0-1-3-5-2^ 
 
 -.24 
 
 0-2-5-3-1-4 
 
 +.13 
 
 0-1-3-5-4-2 
 
 +.32 
 
 0-2-5-4-1-3 
 
 -.36 
 
 0-1-4-2-3-5 
 
 -.00 
 
 0-3-1-2-4-5 
 
 +.34 
 
 0-1-4-2-5-3 
 
 -.06 
 
 0-3-1-2-5 -4 
 
 +.69 
 
 0-1^-3-2-5 
 
 +.02 
 
 0-3-1-4-2-5 
 
 + .34 
 
 0-1-4-3-5-2 
 
 -.53 
 
 0-3-1-5-2-4 
 
 -.19 
 
 0-1-4-5-2-3 
 
 -.70 
 
 0-3-2-1-4-5 
 
 +.26 
 
 0-1-4-5-3-2 
 
 -.21 
 
 0-3-2-1-5-4 
 
 -.01 
 
 0-1-5-2-3-4 
 
 -.05 
 
 0-3-2-4-1-5 
 
 -.47 
 
 0-1-5-2-4-3 
 
 -.60 
 
 0-3-2-5-1-4 
 
 -.16 
 
 0-1-5-3-2-4 
 
 -.04 
 
 0-3-4-1-2-5 
 
 + L19 
 
 0-1-5-3-4-2 
 
 +.35 
 
 0-3-4-2-1-5 
 
 -.30 
 
 0-1-5-4-2-3 
 
 +.02 
 
 0-3-5-1-2-4 
 
 +.33 
 
 0-1-5-4-3-2 
 
 +.04 
 
 0-3-5-2-1-4 
 
 +.53 
 
 0-2-1-3-4-5 
 
 +.06 
 
 0-4-1-2-3-5 
 
 +.62 
 
 0-2-1-3-5-4 
 
 +.20 
 
 0-4-1-3-2-5 
 
 +.68 
 
 0-2-1-4-3-5 
 
 +.59 
 
 0-4-2-1-3-5 
 
 -.02 
 
 0-2-1^-5-3 
 
 +.81 
 
 0-4-2-3-1-5 
 
 -.16 
 
 0-2-1-5-3-4 
 
 +.16 
 
 0-4-3-1-2-5 
 
 +.42 
 
 0-2-1-5-4-3 
 
 +.56 
 
 0-4-3-2-1-5 
 
 + .25 
 
 § 7. Discussion of the "Circular Gap" of 
 Formula 353 
 
 Tables 34-37 give all the possible circuits among the 
 years 1913-1918, and the '' gap " found for each circuit 
 according to Formula 353. As we have seen, these devia- 
 
284 THE MAKING OF INDEX NUMBERS 
 
 tions are normal phenomena, not errors, but fortunately 
 they are so small that for practical purposes they are not 
 worth taking into account. The maximum gap among all 
 the 20 possible triangular comparisons is, as already noted, 
 only .48 per cent (for the circuit of the three years 1-2-5). 
 The maximum gap among all the 45 possible quadrangular 
 circuits is .81 per cent (for the years 1-2-5-4), The maxi- 
 mum for the 72 5-around comparisons is .89 per cent (for 
 0-2-1-4-3 or 1-2-5-3-4). Lastly, the maximum for the 
 60 6-arounds is 1.19 per cent (for the years 0-3-4-1-2-5). 
 
 Even these gaps are unusually large. By the ex- 
 pression for the " probable deviation " we estimate 
 that if any one of the 20 3-around figures be selected by 
 lot, it is as likely as not that it will be less than .19 per 
 cent ; while, a like random choice among the 45 4-arounds 
 will, as likely as not, be less than .22 per cent ; of the 5- 
 arounds, .25 per cent ; and of the 6-arounds, .27 per 
 cent. In a word, the circular test is generally fulfilled 
 within one fourth of one per cent ! 
 
 The maximum gap and the probable^ gap for each 
 group are given in Table 38. 
 
 TABLE 38. "CIRCULAR GAPS" FOR FORMULA 353 
 
 Probable (Per Cents) 
 
 3-around .48 .19 
 
 4-around .81 .22 
 
 5-around .89 .25 
 
 6-around 1.19 .27 
 
 Even these infinitesimal results need to be divided in 
 several pieces to give the share of the deviation per- 
 
 * That is, the gap which is exactly as likely as not. This is the usual 
 sense employed in studies of probability, i.e. the "probable error" of 
 the series, i.e. .6745X the standard deviation, or square root of the average 
 square. 
 
THE SO-CALLED CIRCULAR TEST 285 
 
 taining to any individual index number, for it is to be 
 remembered that the 3-around gap is to be distributed 
 among the three sides of the triangle so that to suppress 
 a .19 per cent gap entirely and force a complete fulfillment 
 of the circular test, it would be necessary to " doctor " 
 each of the three index numbers by only .06 of one per 
 cent ! 
 
 Furthermore, the case we are considering of 36 commodi- 
 ties, very widely dispersing in war-disturbed years, is a 
 very extreme and unusual case. In ordinary times the 
 gap would be even less, and this would be true even if a 
 great number of years were taken. Each additional 
 year in the circuit at first increases the probable gap, in 
 the extreme case here considered, by about .03 ; at this 
 rate without allowing for any diminution, it would require 
 a full century probably to bring the circular test gap up 
 to three per cent ! And this is a conservative figure ; for 
 the gap increases with the dispersion and, as has been 
 often noted, the dispersion of our 36 commodities during 
 this war period, 1913-1918, is much greater than usual. 
 
 Sauerbeck's data (for 36 commodities selected as nearly 
 like our 36 as possible) show a dispersion between 1846 
 and 1913, a period of 67 years, of only 42.10 per cent, or 
 less than that (45.09 per cent) of our 36 commodities in five 
 years. It follows, therefore, that, had Sauerbeck been 
 able to use Formula 353, the discrepancy between the 
 fixed base and chain system would have been found to be 
 in 67 years less than the .27 per cent for our 36 commodi- 
 ties in five years, say, less than ^ of one per cent and less 
 than I of one per cent for a century consisting of years 
 no more disturbed than the 67 mentioned ; but apparently 
 the addition to the gap gradually diminishes, so that it 
 would really be even less. It follows that, except for very 
 long periods or for periods of greater dispersion than the 
 
286 THE MAKING OF INDEX NUMBERS 
 
 Circular Test 
 
 Largest Gaps 
 
 3-Arouncl 
 4 -Around 
 6 -Around 
 Q- Around 
 
 y3 74 75 7S 77 78 
 
 Chart 52. The circular test gap (at the left of each of the four circuits), 
 even at its greatest, as here charted for Formula 353, is remarkably small 
 in all cases. It slightly increases as the circuit of year-to-year index num- 
 bers becomes more circuitous, reaching over one per cent in the 6-around 
 circuit, 1913-'16-'17-'14-'15-'18-'13. 
 
 years of the World War, if such be possible, or both, the 
 circular test is always satisfied by the ideal Formula 353 
 for all intents and purposes. 
 
THE SO-CALLED CIRCULAR TEST 287 
 
 Graphically, the four maximum gaps for Formula 353 
 are given in Chart 52. The hnes return so nearly to the 
 starting point in each case that the observer has to look 
 closely to see the gap. The " probable " gap is not 
 pictured but would be in all cases about half the .48 per 
 cent gap in the chart, the maximum for the 3-around 
 comparison. 
 
 § 8. Comparing the Circular Gaps of the 134 
 Different Formixlae 
 
 Since the circular gap is the proper and necessary result 
 of the ceaseless changing of the weights in our year-to- 
 year comparisons, it is interesting to note that, among the 
 best types of index numbers, the various gaps roughly 
 correspond. 
 
 Since no other index number has been worked out for all possible com- 
 parisons as was Formula 353, we cannot study other formulae by exactly 
 the same methods as we have just studied 353. The only comparisons 
 available are those furnished by the contrasts between the ordinary fixed 
 base and chain index numbers. 
 
 Graphically, in Chart 49, the little vertical black lines (as explained in 
 detail in the fine print below the chart) measure the deviations of each point 
 from the position it would occupy had it fulfilled the circular test. Near the 
 end of the list in Chart 49, the balls have substantially the same relative 
 positions for all the curves, as do also the tiny vertical dark lines indicating 
 the year-to-year deviations under the circular test. We have to count 
 off nearly 40 curves (from the "ideal" at the bottom) before we reach 
 one which shows an appreciable difference in the position of its balls. 
 Beyond this point, as we encounter the less exact formulae, we find an 
 increasing variability of the position of the last ball which never again 
 sits close on the curve as in Formula 353 and the neighboring curves. 
 
 There are three ways or methods by which the eye can sense the degree 
 of deviation of the four balls from any curve. The first and easiest is 
 merely to note the position of the last ball, i.e. that for 1918, which expresses 
 the net cumulative result of all foiu* deviations. But this method gives 
 merely the final result and ignores the intervening history. The four 
 successive deviations, Hke four successive tosses of a coin, will occasionally 
 (once in 16 rounds), all accumulate in one direction; on the other hand, 
 though all four deviations may be great, they may happen largely to 
 offset each other. 
 
 The second way of reading the deviations, therefore, is to run the eye 
 over all four balls and note, in a general way, how far they vary from the 
 
288 THE MAKING OF INDEX NUMBERS 
 
 curve. For curves near the bottom of Chart 49, the two methods show 
 the same results, but for curves near the top they show some different 
 results. The second method also may sometimes give an incomplete 
 picture. For instance, as between the two curves — that for the fixed 
 base drawn in black and that which we imagine as connecting the balls — 
 the only disagreement may be all in the second link, 1914- 1915. After that 
 point the curves may run exactly parallel ; in which case, the second, third, 
 and fourth balls inherit the exact deviation of the first and the eye will 
 be apt to count this one deviation four times, — in Charts 48, eight times. 
 It is clear that the proper way to measure the four deviations is the 
 third way, namely, to examine each separately as a year-to-year matter. 
 This is indicated, in Chart 49, by the vertical dark broad line. This 
 line shows, not how far the ball is from the curve, but how much farther 
 or nearer it is than the 'preceding ball. If a ball is in exactly the same position 
 relatively to the curve as the preceding baU, — if, for instance, they are both 
 just a quarter of an inch below the curve, — there will be no dark line. It 
 is the displacement from this position which the dark line measures ; that 
 is, the extent to which the chain figure has gotten out of line in either direc- 
 tion since the last year.^ 
 
 The eye can readily sense the totaUty of these black 
 lines for any curve and compare that totality with that for 
 any other curve. It requires only a glance at Chart 49 
 to see that the " worthless " and " poor " index numbers 
 have the dark lines very much in evidence except in a 
 few cases (where they are made to disappear entirely 
 by artificially assuming the weights constant). The 
 " fair " index numbers show less blackness ; the " good " 
 still less ; the " very good " very much less. The " excel- 
 lent " still less and the " superlative" the least of all — 
 so little, in fact, as scarcely to be perceptible to the eye. 
 And this seems reasonable. For while, as we have seen, 
 there must be some deviation to express truly the effect of 
 varied weighting, we have found the effect really negligible. 
 
 § 9. Status of all Formulae Relatively to the 
 So-called Circular Test 
 
 So negligible is this normal gap as compared with the 
 ordinary effects of bias or freakishness, that when these 
 
 1 See Appendix I (Note to Chapter XIII, § 8). 
 
THE SO-CALLED CIRCULAR TEST 
 
 289 
 
 effects are present they dominate. Thus we have three 
 chief cases to distinguish : (1) where bias or freakishness 
 is responsible for the gap ; (2) where the gap is forcibly 
 suppressed by constant weighting, and (3) the remaining 
 cases where the gap is normal. 
 
 TABLE 39. LIST OF FORMULA IN (INVERSE) ORDER OF 
 CONFORMITY TO SO-CALLED CIRCULAR TEST 
 
 Formula 
 No. 
 
 Rank 
 
 FORMUIiA 
 
 No. 
 
 Rank 
 
 Formula 
 
 No. 
 
 Rank 
 
 43 
 
 35 
 
 27 
 
 15 
 
 125 
 
 5 
 
 201 
 
 34 
 
 37 
 
 
 126 
 
 
 243 
 
 33 
 
 237 
 
 
 227 
 
 
 245 
 
 " 
 
 1 
 
 14 
 
 325 
 
 
 247 
 
 " 
 
 209 
 
 
 1104 
 
 
 249 
 
 " 
 
 333 
 
 
 1153 
 
 
 44 
 
 32 
 
 2 
 
 13 
 
 1154 
 
 
 46 
 
 " 
 
 10 
 
 
 1303 
 
 
 48 
 
 " 
 
 207 
 
 
 2154 
 
 
 SO 
 
 " 
 
 1333 
 
 
 3154 
 
 
 41=141 
 
 31 
 
 11 
 
 12 
 
 3353 
 
 
 9 
 
 30 
 
 211 
 
 
 4153 
 
 
 35 
 
 29 
 
 233 
 
 
 5307 
 
 
 1133 
 
 " 
 
 335 
 
 
 54** 
 
 
 13 
 
 28 
 
 14 
 
 11 
 
 301 
 
 
 15 
 
 27 
 
 30 
 
 
 309 
 
 
 7 
 
 26 
 
 235 
 
 
 353 t 
 
 
 12 
 
 25 
 
 5333 
 
 
 1014 
 
 
 32 = 132 
 
 24 
 
 16 
 
 10 
 
 1124 
 
 
 40 
 
 23 
 
 225 
 
 
 1353 
 
 
 38 
 
 22 
 
 223 
 
 
 2353 
 
 
 39 
 
 " 
 
 229 
 
 
 5323 
 
 
 31=131 
 
 21 
 
 231 =331 
 
 
 8053 
 
 
 241 =341 
 
 " 
 
 8 
 
 
 8054 
 
 
 33 
 
 20 
 
 24 
 
 
 101 
 
 
 34 
 
 " 
 
 53* 
 
 
 1123 
 
 
 135 
 
 " 
 
 102 
 
 
 1323 
 
 
 215 
 
 " 
 
 108 
 
 
 2153 
 
 
 1013 
 
 " 
 
 1004 
 
 
 123 
 
 
 239 
 
 19 
 
 26 
 
 
 323 
 
 
 25 
 
 18 
 
 28 
 
 
 21=121 
 
 
 133 
 
 " 
 
 307 
 
 
 22 = 122 
 
 
 134 
 
 " 
 
 109 
 
 
 51=151 
 
 
 136 
 
 " 
 
 124 
 
 
 62=152 
 
 
 213 
 
 " 
 
 1103 
 
 
 221=321 
 
 
 23 
 
 17 
 
 3153 
 
 
 261 =361 
 
 
 1134 
 
 " 
 
 4154 
 
 
 6023 
 
 
 29 
 
 16 
 
 4353 
 
 
 6053 
 
 
 36 
 
 " 
 
 107 
 
 5 
 
 9021 
 
 
 42=142 
 
 " 
 
 110 
 
 " 
 
 9051 
 
 
 1003 
 
 " 
 
 
 
 
 
 *53 = 3 = 6= 17 = 20 = 60. 
 ** 54 =4=5=18 = 19 =59. 
 j- 353 =103 = 104 =105 = 106 =153 =154 =203 =205 =217 =219 =253=259 =303 =305. 
 
290 
 
 THE MAKING OF INDEX NUMBERS 
 
 The formulae in class 2, — those conforming to the test by force, so to 
 speak, are 121 (=21), 122 (=22), 151 ( = 51), 152 (=52), 321 (=221), 
 351 ( = 251), 6023, 6053, 9021, 9051, only ten i formula} in all, all geometries 
 and aggregatives. Those in class 3 can be set off less definitely as the 
 gradations are so gradual. Practically, however, they are identical with 
 the "superlative" group which v/e set apart — also somewhat arbitrarily 
 — on the score of nearness to the ideal, Formula 353. 
 
 In Table 39 the formulae are roughly ranked solely according to the 
 degree of conformity to the so-called circular test.'^ 
 
 From this table it is clear that (excepting those at the bottom of the 
 list which hold their rank unfairly, by stereotyped weights) Formula 353 
 
 Dispersion 
 (Measured by Standard Deviations) 
 ( Prices t Fixed dose) 
 
 353 
 
 \5% 
 
 'i3 */^ 75 7e 77 78 
 
 Chaet 53P. Showing the average dispersion of the 36 price relatives 
 taken relatively to the fixed base, 1913, on either side of the ideal (353). 
 
 ^ Not counting Formula 7053 (discussed in the next chapter) which 
 might be added to the list, although on a slightly different basis. 
 
 ^ The rank of each is reckoned roughly by adding together the dark 
 lines in Chart 49 (after first applying to the several lines for the several 
 years rough equalizing coefficients based on the standard deviations of 
 the 36 commodities somewhat on the analogy of the method used for 
 reckoning the order of merit or accuracy in Table 28). 
 
THE SO-CALLED CIRCULAR TEST 
 
 291 
 
 and its former rivals hold close to first place here also ; and that, with few 
 exceptions, the ranking here corresponds roughly to the former ranking in 
 respect of nearness to 353. This confirms Walsh's conclusion on the same 
 subject on the basis of which he accorded the first prize to 353.^ 
 
 Thus, we find that theoretically and practically the best formulae should 
 not and do not yield index numbers which will check 'perfectly when the 
 circular test is applied. It is true that the best forms of index numbers, 
 as determined by other standards, usually check more closely under this 
 test than do the poorest. This is not, however, because the circular test 
 is a valid test of good index numbers for it is not, but merely because any 
 large defects of a formula which would classify it as a poor one under 
 Tests 1 and 2 are likely to classify it as a poor one under the circular test. 
 
 In fact, the effects of the change in the relative weights of different 
 
 Dispersion 
 
 (Measured by Stcmdard Deviations) 
 (Quantities^ Fixed Base) 
 
 353 
 
 *13 'M 75 7S *t7 78 
 
 Chakt 53Q. Analogous to Chart 53P. 
 
 commodities make themselves felt so slowly that the best formulae yield 
 results which check under the circular test to a degree of accuracy far be- 
 yond that required for any practical use to which index numbers are now 
 put. In other words, this means that a single series of index numbers 
 {i.e. one index number for each year) which is calculated by any one of 
 the best formulae will permit the comparison of price levels of any two 
 years to a degree of accuracy beyond anything which is likely to be re- 
 quired for practical purposes. 
 
 Practically, then, the test may be said to be a real test. 
 Theoretically it is not ; for the ranking of formulae ought, 
 
 ^ The Problem, of Estimation, p. 102. 
 
292 
 
 THE MAKING OF INDEX NUMBERS 
 
 in strictness, to be relative not to a perfect fulfillment of 
 the test but to the irreducible minimum exhibited by 
 Formula 353 (or its peers). That is, we should condemn 
 the ten formulae which close the gaps entirelj'- just as truly 
 as those where the gap is larger. Thus the test is not 
 an essential one in the theory of index numbers.^ 
 
 Dispersion 
 (Measured by Standard De¥iations) 
 
 (Prices, Chain) > ^ -^ ^^^ 
 
 'a '1^ 75 le 77 73 
 
 Chart 54P. Showing the average dispersion of the price relatives taken 
 each year relatively to the preceding year, chain fashion. 
 
 § 10. Macaulay's and Ogbum's Theorem 
 
 Professor Frederick R. Macaulay, referring to arithmetic index numbers, 
 says : ^ "the chain numbers draw away (upwards) from the fixed base num- 
 bers" because of a "greater tendency to rise and a less tendency to fall 
 (in percentages) with the smaller relatives than with the larger relatives." 
 
 ^ There are other and still less essential tests which might be considered 
 and were discussed by me in my Purchasing Power of Money (Appendix 
 to Chapter X). See Appendix I (Note to Chapter XIII, § 9). 
 
 ^ American Economic Review, March, 1916, p. 208. 
 
THE SO-CALLED CIRCULAR TEST 293 
 
 Macaulay verifies this conclusion by actual instances. It is also confirmed 
 by the present study, for we find that the typically arithmetic index numbers, 
 Formula 1 (the simple) and Formula 1003 (the cross weighted) as well 
 as 7 and 9 show a cumulative upward tendency of the balls.^ Ma- 
 caulay's and Ogburn's same reasoning could be applied reversely to the 
 harmonic to show accumulation downward. This is illustrated by For- 
 mula 11, 13, 15, 1013. 
 
 The principle involved may be stated in this form : the chain arithmetic 
 has a greater upward bias than the fixed base arithmetic, while, likewise, 
 the chain harmonic has a greater downward bias than the fixed base har- 
 monic. 
 
 Dispersion 
 (Measured by Sfandard Deviations) 
 (Quantities^ Chain) 
 
 353 
 
 '13 74 75 7S 77 78 
 
 Chart 54Q. Analogous to Chart 54P. 
 
 Gra'phically, there is a simple way of picturing this principle. We have 
 seen that where there is bias in a price index, this bias increases rapidly with 
 the dispersion of the price relatives. The reason the bias of the chain 
 system increases faster than that of the fixed base system is that the dis- 
 persion in the chain system increases faster than in the fixed base system. 
 This fact is evident from Charts 53 P and 53Q which show that the stand- 
 ard deviation on the fixed base system, while it increases with the years, 
 increases more and more slowly. The dispersion starts off with a spurt, 
 the first two lines diverging from the curve at a big angle. But year by 
 year (in general) the angle (relatively to the central curve) diminishes. With 
 the chain system, however, a new start is made every year so that we 
 have a succession of spurts with no subsequent tendency to slow up as in 
 the fixed base system. Each line in Charts 54P and 54Q for the standard 
 
 ^ Professor William F. Ogburn has shown this algebraically, on the basis 
 of probability theory. See Appendix I (Note to Chapter XIII, § 10). 
 
294 
 
 THE MAKING OF INDEX NUMBERS 
 
 deviation has a slope diverging from the curve at an angle greater than the 
 corresponding line for that same year in the fixed base system of Chart 53. 
 The same slowing up is seen in Chart 55 which shows the dispersion for 
 Sauerbeck's index number of prices, the dispersion being reckoned rela- 
 tively to the earliest year, 1846, as fixed base.^ 
 
 Dispersion 
 (Measured by Sfandard Deyiations) 
 
 ^Prices, Fixed Base) 
 (Sauerbeck's Figures) 
 
 / ^X 
 
 / 
 
 \ 
 
 \ 
 
 \5% 
 
 / 
 
 \ 
 
 '46 '5S ise ye 'ss ve vs 13 
 
 Chart 55. Showing the average dispersion of 36 of Sauerbeck's price 
 relatives, analogous to the 3P of this book, taken relatively to a fixed base, 
 1846. The dispersion in the five years, 1913-1918 (shown in Chart 53P) 
 exceeds the dispersion shown in this Chart for 67 years. 
 
 In short, the acceleration of the chain bias is due to the retardation of 
 the fLxed base dispersion. The same tendency for the dispersion on the 
 fixed base system to slow up as time goes on may, of course, be shown by 
 the method of "quartiles" or "deciles" relatively to the median. The 
 
 1 Sauerbeck's index number itself is on the base 1867-1877. These 
 charts may also be used in connection with the discussions on bias, in 
 relation to dispersion, of Chapter V. 
 
THE SO-CALLED CIRCULAR TEST 295 
 
 many curves of this sort worked out by Wesley C. Mitchell show this slow- 
 ing up tendency clearly.^ 
 
 § 11. The " Circular Test " Reduced to a 
 " Triangular Test " 
 
 Before leaving the so-called circular test, it may be 
 worth while to note that it may be considered, at bottom, 
 to be simply a triangular test. If any formula (besides 
 satisfying the time reversal test) will satisfy the circular 
 test for any 3-around circuit it will necessarily satisfy it 
 for a 4-around, 5-around, or any other larger number of 
 steps. This extension beyond the original three is easily 
 proved.^ 
 
 § 12. Historical 
 
 The basic idea of the circular test was first explicitly 
 propounded by Westergaard, who maintained that a 
 change in the base ought not to affect the relative sizes 
 of the index numbers of the different years. Walsh, in 
 his Measurement of General Exchange Value, greatly 
 emphasized this idea. He expresses it in the shghtly 
 modified form which, afterward, in his Problem of 
 Estimation, he called the " circular test." He took the 
 ground that, hke other tests taken individually, it is of 
 itself only negative, capable of disproving, but not of 
 proving, an index number. He noted that several old 
 and famihar formulae, obviously faulty for their failure 
 to fulfill other and simpler tests, completely conform to 
 this one. The only formulae which he found to conform 
 perfectly had constant weights.^ He sought for such con- 
 
 1 Wesley C. Mitchell, Business Cycles, pp. Ill, 137, University of 
 California Press, 1913. 
 
 2 See Appendix I (Note to Chapter XIII, § 11). 
 
 ^ See Walsh, Measurement of General Exchange Value, do. 334, 335, 393, 
 397, 398, 399, 431. 
 
296 THE MAKING OF INDEX NUMBERS 
 
 formity among the formulae recommendable for reasons 
 derived from the study of the nature of exchange values 
 and of averages, but he was unable to find any formulae 
 that acciu'ately satisfy this test. 
 
 Among the formulae which, for such reasons, he could 
 recommend, he counted as best those which came nearest 
 to satisfying this test. His latest conclusion is that the 
 formula which I have called " ideal " comes nearest to 
 satisfjdng this test, and he, therefore, agrees with, me in 
 my conclusion that this formula is the best, but for very 
 different reasons. Its failure perfectly to satisfy this test 
 is regarded by him as a blemish or shortcoming. 
 
 Much intellectual labor has thus been expended in a 
 vain effort to find a formula which will yield the absolutely 
 consistent results required by the circular test and still 
 be satisfactory in other respects. 
 
 The simple or the constant weighted geometric index 
 number was favored by Jevons and Walras and several 
 later writers, including Flux and March, chiefly, it would 
 seem, because it satisfies this test, always giving self- 
 consistent results whatever year-to-year calculations are 
 made. 
 
CHAPTER XIV 
 
 BLENDING THE APPARENTLY INCONSISTENT RESULTS 
 
 § 1. Introduction 
 
 I THINK most students of index numbers would be 
 inclined to say of the circular test that theoretically it 
 ought to be fulfilled, but that practically it is not ; and 
 evidence would be cited from index numbers, like Formula 
 1, which have large circular gaps. We have found in 
 Chapter XIII that the exact opposite is true ; that 
 theoretically the circular test ought not to be fulfilled, 
 but that practically it is fulfilled by the best index num- 
 bers, and our evidence is the infinitesimal gap worked out 
 for Formula 353 and the other curves in the " superlative " 
 group. 
 
 Theoretically, every pair of years has its own particular 
 index number dependent on the prices and quantities 
 pertaining to those particular years, regardless of any other 
 year or years. As a consequence of this individualism 
 of index numbers there is, theoretically, a lack of team 
 play, as it were, between the index numbers connecting 
 different years and there is, in consequence, an appearance 
 of mutual inconsistency. It follows that, to secure the 
 theoretically most perfect result, for the sake of finding 
 the very best for each pair of years, we should, for a given 
 series of years and with a given formula, work out every 
 possible index number connecting every possible pair 
 of years among all the years considered. Thus, for the 
 six years taken for the calculations of this book, we should, 
 theoretically, work out the index number 
 
 297 
 
298 THE MAKING OF INDEX NUMBERS 
 
 between 1913 as base and each of the other five years 
 
 a IQI4 " " " " " " " " " 
 
 1915 " " " " 
 
 1916 '' " " " 
 
 1917 " " " " 
 
 1918 " *' " " 
 
 That is, we should use every year as base for all the rest. 
 This would give us a complete set of index numbers be- 
 tween every possible pair of years, each separate figure 
 having its own special meaning, and to be used only for 
 the one comparison, i.e. between the two years for which 
 it is calculated. 
 
 This would make 30 separate index numbers. In this 
 list of 30, every pair of years enters twice, in opposite 
 directions ; once when one of the two years is the base 
 and again when the other is the base. Thus there are 
 only 15 pairs of years, each compared through two index 
 numbers, which are reciprocals when Test 1 is met. Of 
 these 15, we have, as the reader will remember, actually 
 worked out index numbers for nine by each of our 134 
 formulsB, namely, the five on 1913 as base, which consti- 
 tute the " fixed base" series ; and the five which constitute 
 the " chain " system,^ less one duphcation, inasmuch as 
 the first figure (that for 1914) is common to both the 
 fixed base and chain systems. The other six, not worked 
 out, are those connecting years 1914 and 1916, 1914 
 and 1917, 1914 and 1918, 1915 and 1917, 1915 and 1918, 
 1916 and 1918. 
 
 For a series of ten years, there would be, instead of 15 
 
 such " permutations," — - — , or 45 separate index num- 
 bers, of which nine (connecting 1913 with each of the nine 
 
 ^ The complete fixed base series and some of the chain series for all the 
 134 formulae are given, as previously noted, in Appendix VII. 
 
BLENDING THE INCONSISTENT RESULTS 299 
 
 other years) would be the ordinary fixed base series and 
 eight others would be added in the " chain/' For 20 
 
 years there would be , or 190 separate index num- 
 
 bers. For 100 years there would be ^^^ ^ ^^ ^ or 4950 
 
 separate index numbers. 
 
 To calculate such an enormous quantity of separate 
 index numbers, for the sake of finding the very best for 
 each pair of years, and to do so every time we are con- 
 fronted with the problem of tracing price movements 
 through a series of years, would clearly entail very great 
 labor and expense. Would it be worth while? If not, 
 that is, if, in practice, we must forego a theoretically 
 perfect set of index numbers for every possible pair of 
 years, what will be the best course to pursue from a 
 practical point of view ? Shall we content ourselves with 
 the fixed base set and use that series, not only for its proper 
 purpose of comparing the fixed base year with each other 
 year, but also for the theoretically improper purpose of 
 comparing any other two years ? If so, shall we use the 
 first year as the base from which to make our once-for-all 
 set of computations, or shall we, for base, adopt an average 
 covering several years? Or shall we employ the chain 
 system which is theoretically proper only for comparing 
 any two successive years but improper for comparing any 
 other two years? Or shall we use both the fixed base 
 and chain systems? We are now ready to work out 
 answers to these questions. 
 
 § 2. Formula 353 Calculated on Each Separate Year 
 
 as Base 
 
 To illustrate these problems, if we take 353 as our formula and 1913 
 as base, we get the following results : for 1916, 114.21, and for 1918, 177.65. 
 But, theoretically, this does not justify us in assuming that the price levels 
 
300 THE MAKING OF INDEX NUMBERS 
 
 of 1916 and 1918, compared directly and properly with each other, stand 
 as 114.21 to 177.65. Again, the chain system gives correctly the com- 
 parison only between two consecutive years. Thus, it tells us that the 
 price levels of 1916 and 1917 stood in the ratio of 114.32 and 162.23 and 
 that the levels of 1917 and 1918 stood in the ratio of 162.23 and 178.49. 
 But theoretically these do not justify us in assuming that the price levels 
 of 1916 and 1918 stand in the ratio of 114.32 and 178.49. The 
 theoretically correct comparison between 1916 and 1918 must be made, 
 neither by reference to the first year, 1913, nor by reference to the inter- 
 mediate year, 1917, but directly. That is, either 1916 must be the base and 
 1918 calculated from it, or vice versa. 
 
 By such direct comparison, taking 1916 as the base and calling it, not 
 100 but 114.32 (to facilitate comparison with the above figures), we find 
 that prices actually rose between 1916 and 1918 in the ratio of 114.32 to 
 178.36 instead of, as per the chain series, from 114.32 to 178.49 or, as 
 per the fixed base (1913) series, from 114.21 to 177.65. 
 
 Table 40 gives the complete set of index numbers for the years 
 1913-1918 with each year as base. The first line gives the index numbers 
 with 1913 as the fixed base, taken as 100 per cent, as usual. In this series, 
 the index number for 1914 is, for instance, 100.12. The next line gives 
 the index numbers with 1914 as base, taken not as 100 but, to facilitate 
 comparisons, as 100.12 (as in the Une above). Thus, with 1914 as such 
 a base, 1915 is 100.23. The third Une gives the index numbers with 1915 
 as base taken (from the Une above) as 100.23; for instance, with 1915 
 as such a base 1916 is 114.32, and so on, each successive year being thus 
 taken as base but not as 100 (excepting 1913). 
 
 The figures mentioned as base figures are italicized in a diagonal 
 and they themselves constitute the chain figures. That is, the diagonal 
 series is the chain series. By this device, for example, the right and bottom 
 corner figure, 178.49, serves the double purpose of being at once in the 
 chain and in the 1918 fixed base series just as the diagonally opposite 
 (left upper) corner figure (100.00) serves the corresponding double purpose 
 of being at once the beginning of the chain and of the 1913 fixed base 
 series. In the same way, the second row of figures is the fixed base series 
 where 1914 is the base, and is taken not as 100, but as 100.12, the chain 
 figure. Thus aU figm-es in the diagonal serve as the base for all the years 
 on the same line as well as a link in the chain (the diagonal). 
 
 If such a table were to be used in practice it would be used as follows. 
 The first line, or ordinary fixed base figures (1913 being the base), would 
 be used only for comparing any given year such as, say, 1917 ivith this base, 
 1913, and not for comparing it (1917) with any other year such as, say, 
 1915. If we wished to compare 1917 with 1915 we should find in the 
 table the line in which one of these two years is the base (an italicized 
 figure), for instance, the third line. There 1915 is the base, and is taken as 
 100.23. On this base, 1917 is found to be 161.86. Consequently, the 
 best measure for the rise of prices between 1915 and 1917 is this rise from 
 100.23 to 161.86. It is, strictly, not the rise given in the first line in the 
 table, by the ordinary fixed base system. It is there represented as a rise 
 from 99.89 to 161.56 although in this case the two comparisons differ 
 almost inappreciably. 
 
BLENDING THE INCONSISTENT RESULTS 301 
 
 Sp 
 
 HH ^ 
 
 m 
 
 OS 
 
 1-1 
 
 t^ lo CO i^ t^ ^^ 
 
 CO rH »0 00 l>; t~; 
 
 »C »0 •<* rj5 T)^ ^ 
 IM (M (M (M (M ®Q 
 
 l-H 1-H I—l 1— 1 1— 1 T-l 
 
 ^ 
 
 05 (Tl 
 
 (M oq 
 
 rH i-t 
 
 1-1 
 at 
 
 00 -* CO a> Oi oi 
 
 05 (M t^ TfH -^ rt< 
 
 00 Oi 00 00 q6 00 
 rH 1— 1 1— 1 1— 1 1-s T— 1 
 
 •^ 
 t^ 
 
 06 oi 
 
 rH rH 
 
 '^ 
 
 '"' ""* 
 
 
 lO CO T}< ^ Tt< IC 
 
 00 o; i> ^-. !>. CO 
 00 00 00 od 00 00 
 
 rH 1— 1 rH >H r-H 1— 1 
 
 00 tH 
 t>; 
 
 06 OJ 
 
 
 i-< rH 
 
 t4 
 
 O (N ©» cq 00 1-H 
 
 ""! "^ ^ '^ ^ '^ 
 
 oj 00 oo 00 00 00 
 o o o o o o 
 
 "-H rH >-H rH ,— 1 I— 1 
 
 t^ 
 
 06 oi 
 
 
 1— 1 r-i 
 
 <* 
 
 1-1 
 
 fO eo eo lo i-H CO 
 CO eo CO T-^ t>. o 
 
 05 Cj 05 05 00 oi 
 05 O5 05 05 05 05 
 
 rH CO 
 
 oi oi 
 
 05 05 
 
 s 
 
 © CO rH 00 CO 
 CO 05 10 10 
 
 00 
 
 t^ 
 
 oi d 
 Oi 
 
 tH 
 
 u 
 
 00 
 
 rH 
 
 10 10 CO 05 Ci 
 
 CD 05 00 CO ■* -^ 
 i>^ i> 06 06 00 oi 
 l> t> t^ t^ t^ f^ 
 
 1— T 1^ rH tH tH >-s 
 
 05 
 <N 0> 
 
 06 t>^ 
 
 5 
 
 CO >-< CO CO eo CO 
 »o c^_ 00 cq e^ (M_ 
 rH ,-; ^ c<i ^ c^ 
 
 CO CO CO CO Co CO 
 
 1-H 1— 1 i-H T— 1 -"^ rH 
 
 05 CO 
 GO 
 
 CO CO 
 
 rH T— 1 
 
 <0 
 lH 
 
 s 
 
 rH rt (M <^ (M ,-H 
 (N 1-H CO 50 CO -* 
 
 rf! Tj! T^^ -^ Tl^ '* 
 
 1-H rH 1-H >-H T— 1 1-H 
 
 00 CO 
 
 (M 
 
 1-H T— 1 
 
 
 
 lO 
 
 s 
 
 OJ CO s:i CO CO CO 
 
 00 iM <^ <N ■* 
 
 oi d cS d d CO 
 02 c> 
 
 00 CO 
 
 ^ 05 
 
 d C3i 
 
 05 
 1-H 
 
 
 Cq ^ (M T-H CO CO 
 
 rH rs i-H CO t^ ■^ 
 
 di '^ d> d Q d 
 000000 
 
 rH 01 
 
 CO 
 
 ^^ rH 
 
 CO 
 
 a 
 
 10 (M 1> 
 
 CO rH '* "* 
 
 <:S d cS d d 
 000000 
 
 (M 
 (M 
 
 d d 
 
 
 1-H rH 
 
 » « 
 
 CO ^io CO f^^ 
 
 rH 1-H 1-H rH i-H rH 
 
 ^ II r 
 
 
 
 
 CO CO CO CO CO CO 
 10 >0 lO 10 10 UO 
 CO CO CO CO CO CO 
 
 CO CO 
 
 
 ■2S 
 
 
 bO 
 
 tl 
 
 .U 
 
 
 ■^ 
 
 td 
 
 > 
 
 „s 
 
 ^ 
 
 
 >. 
 
 «M 
 
 XI 
 
 
 
 
 
 j^ 
 
 .g 
 
 
 
 
 
 l>^ 
 
 -aJO 
 
302 THE MAKING OF INDEX NUMBERS 
 
 In the above comparison 1915 was taken as the base year and 1917 as 
 the given year. We could, of course, reverse the bases, taking the fifth 
 line where 1917 is 162.23, for base, in which case the given year 1915 is 
 100.46, thus giving the rise of prices between 1915 and 1917 as 100.46 to 
 162.23; this comparison is, of course, exactly the same as the first (i.e. 
 100.23 : 161.86 : : 100.46 : 162.23) because, as we know, our formula (353) 
 satisfies the time reversal test. 
 
 § 3. The Differences Due to Differences of Base are 
 
 Trifling 
 
 By Table 40 we may very readily see the trifling effects of shifting the 
 base from one year to another. For 1913 the figures (for prices) in the left- 
 most vertical column vary only from 100 to 100.47; for 1914, from 100.12 
 to 100.76; for 1915, from 99.89 to 100.46; for 1916, from 114.11 to 114.41 ; 
 for 1917, from 161.21 to 162.23; for 1918, from 177.65 to 178.80. These, 
 which are the extreme discrepancies brought about for each year by shifting 
 the base each year, range only from one third of one per cent to two thirds 
 of one per cent ! 
 
 Let us take the last and largest of these and state the meaning of the 
 discrepancy. It is the discrepancy between, on the one hand, 177.65 as 
 the index number for 1918 on the base 1913 taken as 100 per cent, and, 
 on the other hand, 178.80 for 1918 on the base 1915 taken as 100.23. 
 And, to proceed back to 1913, this last named figure on the diagonal, 
 100.23, was found as the index number for 1915 on 1914 as base taken as 
 100.12 (preceding line), which, in turn (next preceding line), was found as 
 the index number for 1914 on 1913 as base taken as 100.00. In other 
 words, by the true direct comparison, taking 1913 as 100 per cent, we 
 find that the index number of 1918 is 177.65 per cent; but by the indi- 
 rect comparison, starting with the same base and proceeding one link to 
 1914 (diagonal), thence another link (diagonal) to 1915, and then jump- 
 ing (level) to 1918, we get, not 177.65, but 178.80, or two thirds of one 
 per cent more. 
 
 Thus the difference between the various barometers of price-and-quan- 
 tity-changes given in the table are trifling. Nevertheless, it is interesting 
 to note that, as between 1914 and 1915 where the two index numbers are 
 virtually equal, there is enough difference to tip the scales from one direc- 
 tion to the other. According to the first line, or ordinary fixed base system, 
 1913 being the base, the price level seems to fall between 1914 and 1915 
 (from 100.12 to 99.89, or a quarter of one per cent) and a slight fall between 
 the same years (1914 and 1915) is likewise indicated in the last three lines, 
 i.e. with 1916, 1917, or 1918 as base; whereas by the direct, or true, com- 
 parison between 1914 and 1915, i.e. with 1914 as base or 1915 as base (see 
 second line and third line), we note that the price level is found to rise 
 from 100.12 to 100.23, or one ninth of one per cent. 
 
 The reader will notice that each italicized chain figure (say for 
 1915) is duplicated immediately above and also immediately below: — 
 above, because the italicized 1915 figure was purposely taken from 
 
BLENDING THE INCONSISTENT RESULTS 303 
 
 the line above to start off the calculations on the new 1915 base; and 
 below (the 1916 line) because the 1915 year is there calculated backward 
 from the 1916 base by a formula which complies with Test 1. In a word, 
 in the 1915 line, 1916 is calculated from 1915 ; and in the 1916 line 1915 is 
 calculated from 1916, with a formula which works both ways, i.e. com- 
 plies with Test 1. 
 
 Graphically, Chart 56, plotting Table 40, shows the 
 results of applying Formulae 53 and 54 and their cross, 353, 
 on each of the six bases. The upper three sets give these 
 18 curves (six for each formula) individually, separated by 
 spaces, while the lower three give a composite of each 
 set. 
 
 It is clear that the differences are extremely trifling, 
 and, for 353, scarcely perceptible. The preceding table 
 and chart thus show in another way what we saw in the 
 last chapter specifically by means of the circular test, 
 namely, how remarkably little difference it makes what 
 the base or bases may be from which we calculate Formula 
 353. 
 
 In \dew of this virtual agreement between the curves, 
 whatever year is taken as the base, it is perfectly clear 
 that for Formula 353 (and the same would be true of any 
 other good formula) it would be a waste of time, in the 
 practical calculation of index numbers, always to cal- 
 culate all possible inter-year indexes. Any one series 
 will suffice. 
 
 In short, while theoretically the circular test ought not 
 to be fulfilled, and shifting the base ought to yield 
 inconsistencies, the inconsistencies yielded are so slight 
 as practically to be negligible. To use for each formula 
 all the six curves (for six years — more, for more years) 
 would only multiply the time, labor, and expense by a 
 large factor, without serving any useful purpose. In 
 fact, it would be a positive nuisance. A single curve 
 will suffice for all practical purposes. 
 
304 
 
 THE MAKING OF INDEX NUMBERS 
 
 § 4. Index Numbers on Different Bases may 
 well be Blended 
 
 Every one of the six curves is strictly correct only 
 for the limited comparison for which it is constructed. 
 
 Comparison For Six Bases 
 of Formulae 53, 5^, 353 
 (Prices) 
 
 13 
 
 '14 
 
 '15 
 
 '16 
 
 '17 
 
 16 
 
 Chart 56P. These curves, especially the three lower, which are mere 
 composites of those above {i.e. found by plotting all on the same scale, in- 
 stead of separating them as above), indicate that the differences resulting 
 from a shift of base are least for 353, but comparatively slight for 53 and 
 54 also. 
 
 There remains the practical question : if we are not going 
 to use all six, what single curve is the best one to use 
 in their place, for the general purpose of all com- 
 
BLENDING THE INCONSISTENT RESULTS 305 
 
 parisons over a series of years ? Doubtless the very best 
 as to accuracy, were it practicable, is the blend or average 
 of all six. This blend constitutes Formula 7053, if it can 
 be dignified by the name of formula. It is, of course, 
 merely an average of the six sets of particular figures de- 
 rived by Formula 353. This is a compromise single series 
 
 Comparison For Six Bases 
 
 of Formulae 55,54,353 
 (Quantities) 
 
 Bales 
 
 353 
 
 53 Combined 
 5^ Combined 
 353 Combined 
 
 /J '14 15 16 17 
 
 Chart 56Q. Analogous to Chart 56P. 
 
 IQ 
 
 of six figures that can be substituted for the whole table 
 of figures, for the purpose of blending all separate exact 
 comparisons into one general nearly exact comparison. 
 With reference to these averages, no figure in the table 
 deviates by as much as one half of one per cent. The 
 " probable error " of any figure (for price indexes, for 
 1917) is two tenths of one per cent, and, for the other 
 years, less. In other words, it is ju^t as likely as not that 
 
306 THE MAKING OF INDEX NUMBERS 
 
 any figures of Table 40 for 1917 taken at random will 
 differ from the mean (or Formula 7053) figure for 1917 
 {viz., 161.53) by less than two tenths of one per cent.^ 
 
 This blend may be compared to the " chromatic " 
 scale on the piano. This chromatic scale is found by 
 " tempering " the " natural " scale. By the " natural " 
 scale a piano would have but one key ; to obtain other 
 keys would require a separate piano for each, all out of 
 tune with one another. These are blended into one by 
 the chromatic scale by slight readjustments of the various 
 notes. These adjustments change the number of vibra- 
 tions in the natural scale in one case by as much as 1 in 122, 
 or some ten times as great an adjustment as we are called 
 upon to make in our present problem of adjusting index 
 numbers. In other words, the " tempering " of the piano 
 or " chromatic " scale relatively to the violin or " natural " 
 scale, though imperceptible to almost any human ear, 
 is ten times as great as the " tempering " which is neces- 
 sary to secure Formula 7053. 
 
 § 5. The Three Practical Substitutes for Blending 
 
 But to calculate Formula 7053 every time we have an 
 index number to compute would require, first, calculating 
 each of the constituent curves and this, as has been said, 
 could be done only at prohibitive costs. From a practical 
 point of view, there are only three single curves worth 
 considering: (1) that obtained by using the first year 
 1913 as base (the ordinary fixed base Formula 353 or its 
 rivals) ; (2) that by using the chain of successive bases 
 (also by 353 or its rivals) ; and (3) that by using 6053 
 (or its rival 6023), which are like 53 (or 23), except that 
 
 1 Formula 7053, as here used, begins in 1913 with 100.22. For conven- 
 ience, we may reduce this to 100 and reduce all the figures for all the other 
 years accordingly. Both forms are given in the preceding table, but 
 only the last named in Appendix VII. 
 
BLENDING THE INCONSISTENT RESULTS 307 
 
 the base is not a single year but an average formed from 
 several or all the years concerned. Such a formula may 
 be called aggregative {or geometric) formula weighted I with 
 broadened base. One of its chief claims to consideration 
 is that it requires fewer statistical data to be furnished 
 than does 353. 
 
 To determine which of these three (353 fixed base, 
 353 chain, or 6053 broadened base) is the most accurate, 
 
 TABLE 41. FOUR SINGLE SERIES OF SIX INDEX NUMBERS 
 AS MAICESHIFTS FOR THE COMPLETE SET OF TABLE 40 
 
 (Prices) 
 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Formula 6053 (broadened 
 
 base, 1913-1918) 
 Formula 353 (fixed base, 
 
 1913) 
 Formula 353 (chain) 
 
 100. 
 100. 
 100. 
 
 99.79 
 100.12 
 100.12 
 
 99.85 
 
 99.89 
 
 100.23 
 
 114.04 
 114.21 
 114.32 
 
 161.59 
 161.56 
 162.23 
 
 177.88 
 177.65 
 178.49 
 
 Formula 7053 (blend) 
 
 100. 
 
 100.09 
 
 99.96 
 
 114.03 
 
 161.53 
 
 177.90 
 
 This table shows that the chain system is the most erratic of the three 
 as compared with Formula 7053 and that there is practically no choice 
 between the other two. 
 
 The figures for quantities show the same result. 
 
 TABLE 42. FOUR SINGLE SERIES OF SIX INDEX NUMBERS 
 AS MAKESHIFTS FOR THE COMPLETE SET OF TABLE 40 
 
 (Quantities) 
 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1S18 
 
 Formula 6053 (broadened 
 
 base, 1913-1918) 
 Formula 353 (fixed base, 
 
 1913) 
 Formula 353 (chain) 
 
 100. 
 100. 
 100. 
 
 99.00 
 99.33 
 99.33 
 
 108.91 
 109.10 
 108.72 
 
 119.13 
 118.85 
 118.74 
 
 118.99 
 118.98 
 118.49 
 
 125.16 
 125.37 
 124.77 
 
 Formula 7053 (blend) 
 
 100. 
 
 99.37 
 
 109.02 
 
 119.04 
 
 119.00 
 
 125.20 
 
308 
 
 THE MAKING OF INDEX NUMBERS 
 
 it is only necessary to ascertain which of them is nearest 
 to the best blend, namely, 7053. 
 
 Numerically, Tables 41 and 42 on page 307 give these 
 three sets of figures and also the theoretically best blend, 
 Formula 7053, for comparison. 
 
 Optional Varieties of 353 
 , (Prices) 
 
 :i5 
 
 M^ 
 
 15 
 
 16 
 
 17 
 
 16 
 
 Chart 57P. The agreement between the broadened base index num- 
 ber (6053), the blend of the six curves of 353 (7053), and 353 itself (whether 
 with 1913 as a fixed base or with the chain system), is so close that, were 
 precision the only consideration, there would be almost no choice between 
 these four. 
 
 Graphically, Chart 57 gives these three curves and also 
 the theoretically best formula, 7053. They are absolutely 
 indistinguishable to the eye. 
 
 Our conclusion is, then, that either Formula 353, fixed 
 base 1913, or Formula 6053, broadened base 1913-1918, is 
 the best compromise on the score of accuracy. On the 
 score of other and more practical considerations, such as 
 speed of computation, more will be said in a later chapter. 
 
 § 6. Chain vs. Fixed Base System 
 
 The chain system is of little or no real use. The chief arguments in 
 favor of the chain system are three : (1) that it affords more exact com- 
 
BLENDING THE INCONSISTENT RESULTS 309 
 
 parisons than the fixed base system between the current year and the 
 years immediately preceding in which we are presumably more interested 
 than in ancient history; (2) that, graphically, the year-to-year lines of 
 the price curve have the correct current directions, whereas in the fixed 
 base system the year-to-year lines are slightly misleading, merely connect- 
 ing points each of which is really located relatively to the base or origin only, 
 and not to its neighbors ; and (3) that it makes less complicated the neces- 
 sary withdrawal, or entry, or substitution of commodities, as time and 
 change constantly require. 
 
 As to the first argument, though I have myself used it in the past, I 
 have come to a lower estimation of its importance; partly (and chiefly) 
 because the present investigation has shown that, in the case of all good 
 index numbers, there is no really perceptible difference between the chain 
 
 Optional Varieties of 353 
 
 (Quantities) 
 
 13 
 
 74 15 16 17 
 
 Chart 57Q. Analogous to Chart 57P. 
 
 I8> 
 
 and the fixed base figures; partly because, for years to come, we shaU 
 be interested in comparisons with antecedent and pre-war years quite as 
 much as with the immediately preceding years ; and partly because I have 
 come to realize that the ordinary user of index numbers uses chiefly not the 
 diagram but the numerical figure, and he thinks of this figure as relative to 
 the base. Therefore, it is better that it should accurately express the rela- 
 tion to the base. This the fixed base figure does. 
 
 The second argument — the one concerning graphic representation — 
 is sufficiently answered by the fact that the eye is not accurate enough to 
 distinguish between the fixed base and chain base curves given by any of 
 the better formulae. Very minute differences can be perceived only by 
 printed figures. 
 
 Theoretically, it may be said that the graphic curve for the fixed base 
 system is an anomaly. To represent the fixed base and chain curves most 
 appropriately, we ought to draw only the chain curve from year to year, 
 i.e. from ball to ball, whereas, when we use the fixed base points, we ought 
 to connect these, not with each other, but each directly with the base point 
 or origin. 
 
 In Chart 58 (fixed base, using the simple median) the connecting lines 
 between each point and the origin are graphically indicated (dark short 
 lines drawn only part way toward the origin to avoid confusing the eye) ; 
 but these would not give much help to the onlooker were not their ends 
 
310 
 
 THE MAKING OF INDEX NUMBERS 
 
 connected by the dotted curve after the usual fashion of the fixed base 
 curves. 
 
 § 7. Splicing 
 
 The strongest argument for the chain system is the third, i.e. the im- 
 munity it gives from any complications arising out of the withdrawal of 
 any commodity from the index number, or the entry of a new commodity, 
 or both at once, i.e. the substitution of a new for an old. 
 
 Simp/e Median 
 
 and Quartiles dnswn from origin / 
 (Prices) 
 
 73 '14 IS ie 17 76 
 
 Chabt 58. Showing how, strictly speaking, the fixed base index num- 
 bers should be represented — by lines radiating from the fixed base to the 
 given years. The lines are for the median (in the center), those above and 
 below representing the quartiles. The dotted connecting line is needed to 
 help the eye despite the fact that, strictly speaking, its directions do not 
 represent year-to-year index munbers. 
 
 It often happens that we wish to drop some commodity from the list 
 because of its ceasing to be quoted, or of its becoming obsolete or super- 
 seded. And, likewise, it often happens that we wish to include a new 
 commodity because of a new invention or a change in customs. Still 
 oftener must substitutions be made by replacing one grade or style of goods 
 by another. When the chain system is used these operations create no 
 embarrassment, no matter what formula is used ; for, imder this system, 
 
BLENDING THE INCONSISTENT RESULTS 311 
 
 a new start is made each year and the next link can be forged independently 
 of all those preceding. 
 
 But under the fixed base system these changes usually make Gordian 
 knots to cut. In some cases there is no difficulty. Thus, if we drop one 
 brand of, say, condensed milk and substitute another and if the newly 
 marketed brand has, at the time of the change, the same price as the old, 
 it may be substituted without any jar or adjustment, even though it did 
 not exist in the base year. Similarly, if one grade, say, of wheat which did 
 exist in the base year but was not used in the index number, is now sub- 
 stituted for another and, though their prices per bushel do differ, their 
 'price relatives do not differ in terms of their base year prices, we may readily 
 make the transference. Again, if the withdrawal or entry does not change 
 the index nmnber, there is no trouble. This supposition implies, of course, 
 in the case of entry, that the newly entered commodity was also quoted 
 in the base year. But in all other cases under the fixed base system we 
 must make some sort of adjustment. 
 
 Let us assume that the change (whether withdrawal, entry, or substitu- 
 tion) changes the index number, at the time, from 150 under the old way to 
 153 imder the new, or by two per cent. The new figure being two per 
 cent above the old, all futin-e figures calculated by the new way may be 
 presumed to be two per cent too high. Consequently, what is needed is, 
 henceforth, after calculating by the new way, to trim down the result 
 by that much. That is, beginning with the 153, every index number after 
 being duly calculated is to be reduced in the ratio of 153 to 150. 
 
 But in cases where an entirely new commodity enters, so that no base 
 year quotations exist, we cannot enter it at all, in the fixed base system, 
 on all fours with the rest. If it is a case of substitution for a commodity 
 to be withdrawn, we may splice it on to the old series of quotations for 
 the withdrawn conamodity. Thus, if the old commodity, at the time of 
 withdrawal, stood at 120, the new may be arbitrarily entered in its place as 
 120 (despite the fact that there was no 100 for it in the first place) and its 
 future price relatives computed in proportion. If the new commodity is 
 not to be substituted for an old, but added as one more on the list, we may 
 arbitrarily give as its price relative at the time a figure equal to the index 
 number itself. That is, if the index number at the time is 130, the new 
 commodity may start off with 130 as its price relative (despite the fact 
 that there never was any 100 for it). 
 
 In short, the fixed base system is objectionable because it sometimes 
 requires patching. The chain system never does. But this objection to 
 the fixed base system is not very serious. Besides, the patching may be 
 largely or wholly avoided if, as indicated in a later chapter, we take a 
 new start, not every year, but, say, every decade. 
 
 The above explanation is stated in terms of price relatives and applies 
 to all index numbers, except aggregatives. To these an analogous method 
 applies.! 
 
 On the whole, therefore, the fixed base system (at least as applied 
 to Formula 353) is sUghtly to be preferred to the chain, because. 
 
 1 See Appendix I (Note to Chapter XIV, § 7). 
 
312 THE MAKING OF INDEX NUMBERS 
 
 (1) it is simpler to conceive and to calculate, and means something 
 clear and definite to everybody ; 
 
 (2) it has no cumulative error as does the chain system (as is shown by 
 comparison with Formula 7053) ; 
 
 (3) graphically it is indistinguishable from the chain system. 
 
 § 8. Broadening the Fixed Base 
 
 We have considered two of the three series originally 
 contrasted, viz., Formula 353 in the fixed base and chain 
 systems, and between these two we choose the fixed base 
 system. We have also found that in the fixed base system 
 we can always " patch " when commodities are changed 
 in the formula. We have still to consider the broadened 
 base system (which also requires revision from time to 
 time) as compared with the fixed base year system. This 
 is easier to calculate than the blend Formula 7053, and 
 distributes in a simpler way the discrepancies due to differ- 
 ing bases. Moreover it does not require that the cal- 
 culator have at hand all the yearly data needed for 353. 
 He may make his base as broad as the data available, or, 
 as may be necessary to yield a good compromise. 
 
 Broadening the base from one year to several requires : 
 (1) taking as each base price, not one year's price, but an 
 average of several ; and (2) likewise taking as each base 
 weight not one year's but an average of several.^ As 
 stated, the system of weighting is analogous to system I. 
 It is the same throughout the calculation, i.e. constant 
 weights are used for the entire series. For quantity in- 
 dexes, of course, the analogous operations apply. 
 
 We shall consider the advantages of broadening the 
 base as applied to certain types of formulae. First, we 
 shall consider Formula 6053. It is Formula 53, except 
 
 * It may be worth noting, however, that (1) is a superfluous procedure in 
 the cases of Formulae 6023 and 6053, the results being identical (except for 
 a constant) whether one year's price or an average of several is used. 
 
BLENDING THE INCONSISTENT RESULTS 313 
 
 that the base values or quantities are taken as the average 
 of the values or quantities for several years instead of 
 one.^ 
 
 It seems to show no real superiority over 53. The 
 ranking of all index numbers in Table 28 shows For- 
 mula 53 actually closer to 353 than is 6053 (1913-1918), 
 the six years indicated being the broadened base, their 
 average of prices being the base prices in place of the 
 Po's of 53, and their average quantities being the weights 
 in place of the qos of 53. Again, it shows Formula 53 
 nearly as close to 353 as 6053 (1913-1916), and not much 
 less close than is 6053 (1913-1914) .^ 
 
 So far as the aggregative type is concerned, therefore, 
 Formula 53 seems about as good a substitute for 7053 
 as 6053, and, of course, it is easier to compute. If the 
 broadened base Formula 6053 has any advantage over 53, 
 that advantage is too small to show itself in the cases here 
 available, including those for prices and quantities of 
 the 12 crops, and for prices and quantities of stocks on 
 the Stock Exchange given in Chapter XL 
 
 We may, therefore, conclude with reasonable safety 
 that Formula 53 is always a good makeshift for the ideal 
 formula, 353, or for the ideal blend, 7053. Broadening 
 the base to make 6053 seems a superfluous procedure.^ 
 
 1 This derivative of Formula 53 by broadening the base is, of course, the 
 same as that derived from Formula 3 by broadening the base. So derived 
 it might be called 6003. 
 
 ^ The above comparisons were made with Formula 353 fixed base as 
 the standard of comparison, but if Formula 7053 be used instead, we get 
 the same results. 
 
 ' The only case where there might be any really perceptible advantage 
 in Formula 6053 over Formula 53 is in such a case as that of the 12 crops 
 used by Persons and Day, i.e. where there is a large correlation between 
 the price relatives and the quantity relatives so that Formula 53 has a 
 slight bias, second hand, as it were. But even in such a case the advantage 
 is not large, as is clear from the fact that 53 and 54 are so close together 
 (see Charts 47 and 48) and, therefore, so close to 353. 
 
314 THE MAKING OF INDEX NUMBERS 
 
 § 9. The Geometric Formula Weighted / with 
 Broadened Bases 
 
 When we turn from the aggregative type to the geo- 
 metric type, we find a different situation. In this case a 
 broadening of the base (Formula 6023) does help ma- 
 terially. Professors Persons and Day of Harvard have 
 made much use of Formula 6023. Because of their ad- 
 vocacy I have calculated 6023 in order to see whether 
 this process of broadening the base would reduce the 
 
 355 and 6025 Compared /iH 
 
 For 12 Leading Crops (Day& Persons) 
 (Prices) 
 
 •80 "dS '90 '95 DO '05 '10 75 "20 
 
 Chart 59 P. Showing the close agreement between Day's index num- 
 ber (6023) and the ideal (353) for prices of 12 crops with a consistent but 
 faint trace of downward bias in 6023 (1910 is the base). 
 
 downward bias of 23. Evidently it does ; for all the three 
 forms of Formula 6023 which have been calculated lie, 
 in Table 28, nearer 353 than does 23. This is because 
 the price relatives on the broadened base disperse much 
 less widely than do those used in calculating Formula 
 23 and, as we know, bias decreases rapidly with a de- 
 crease of dispersion. The reason why broadening the 
 base makes so much more improvement over Formula 23 
 than over 53 is that there is more room for improvement ; 
 for 23, on 1913 as a base, has a distinct downward bias. 
 
BLENDING THE INCONSISTENT RESULTS 315 
 
 It belongs to group " 1- " in our five-tined f ork. Broad- 
 ening the base to include the two years, 1913 and 1914, re- 
 duces this bias. Broadening it to include four years, 
 1913-1916, reduces it still further. This is shown in the 
 following table : 
 
 533 and 6023 Compared 
 For 12 Leading Crops CDay& Persons) 
 
 (Quantities) 
 
 Chaet 59Q. Analogous to Chart 59 P. The downward bias of 6023 is 
 more evident. (1910 base.) 
 
 TABLE 43. THE INFLUENCE OF BROADENING THE BASE 
 IN REDUCING BIAS 
 
 (Prices) 
 
 FORMUIA No. 
 
 Base 
 
 1913 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 * 23 
 6023 
 6023 
 
 1913 
 (1913-1914) 
 (1913-1916) 
 
 100. 
 100. 
 100. 
 
 99.61 
 
 100.12 
 
 99.93 
 
 98.72 
 99.50 
 99.88 
 
 111.45 
 112.25 
 113.61 
 
 154.08 
 153.53 
 156.61 
 
 173.30 
 173.45 
 175.32 
 
 353 
 7053 
 
 1913 
 (blend) 
 
 100. 
 100. 
 
 100.12 
 100.09 
 
 99.89 
 99.96 
 
 114.21 
 114.03 
 
 161.56 
 161.53 
 
 177.65 
 177.90 
 
 But the figures are still below the standard (either 353, 
 fixed base, or 7053) all along the line. Several other 
 calculations harmonize with this conclusion. 
 
316 THE MAKING OF INDEX NUMBERS 
 
 After I had made these calculations for the 36 commodities, Professor 
 Persons published his defense of Day's index number (Formula 6023). ^ 
 His calculations, which are for 12 crops, are reproduced in Charts 59P, 59Q, 
 and 60P, QOQ, and show a remarkably close agreement between Formulae 
 6023 and 353. At the same time they show a slight trace of downward 
 bias remaining in 6023, and completely confirm the above conclusions. 
 The base, in these studies of Day and Persons, is broadened to the five years 
 1909-1913 : that is, the constant weights used, instead of being the values 
 for the one year, 1910, as per Formula 23 (i.e. instead of po3o> etc.), were 
 the average values for the five years named. 
 
 ^^^ and 602^ Compared 
 
 For 12 Leading Crops (Day d Persons) 
 (Prices) 
 
 l« 
 
 W 7/ T? 7J >» 73 78 77 T8 » 
 
 Chart 60P. Analogous to Chart 59P. (1910 base.) 
 
 In Chart 59 P, Formula 6023 is below 353 in four cases —in 1880, 1885, 
 1895, and 1915; and above in three cases — in 1890, 1905, and 1920. 
 In Chart 59Q it is below in seven cases — in 1880, 1885, 1890, 1900, 1905, 
 1915, and 1920; and above in only one case, namely, 1895. In Chart 60P 
 it is below in four cases — in 1914, 1915, 1917, and 1919 ; and above in 
 three cases — in 1913, 1916, and 1918. In Chart 60Q it is below in six 
 cases — in 1911, 1915, 1916, 1917, 1918, and 1919; and above in only 
 one case — 1912. In all the years not mentioned 353 and 6023 coincide. 
 
 All told, Formula 6023 is below in 21 cases and above in eight, thus 
 showing that its innate downward bias has not quite been suppressed by 
 broadening the base. It is also clear from an examination of the charts 
 that, as we proceed in either direction from the base, 1910, the downward 
 bias of 6023 asserts itself increasingly. 
 
 Thus, by including a sufficient number of years — a full assortment 
 of all the chief varieties met with in, say, a complete "business cycle" 
 we can partly ^ eliminate (for a time at least) the bias of Formula 23. The 
 longer and more representative the period, the more nearly will the bias 
 
 1 Warren M. Persons, " Fisher's Formula for Index Numbers," Review 
 of Economic Statistics, May, 1921, pp. 103-13. 
 
 2 See Appendix I (Note to Chapter XIV, § 9). 
 
BLENDING THE INCONSISTENT RESULTS 317 
 
 be eliminated. But in using Formula 6023, the corrective effect of broad- 
 ening the base will wear off and the downward bias gradually reappear 
 after a few years. Thus, by broadening the base from 1913 to 1913-1918, 
 the dispersion of our 36 price relatives in 1918 is reduced from 45.09 per 
 cent to 20.23 per cent. This results, as Table 48 shows,' in an even greater 
 reduction of the bias — from 7.01 per cent to 1.67 per cent, and, as has 
 just been stated, accounts for the improvement in the index number from 
 broadening the base. But, as we have seen in Chapter V, the dispersion 
 always tends to increase with the lapse of time. Sauerbeck's index number 
 has a broad base (1867-77). Yet the dispersion of the price relatives 
 used by him amounted, in 1920, to 129 per cent. This, as noted later, 
 has given the index number an upward bias of 36 per cent. If Sauerbeck's 
 index number had been calculated by Formula 6023 instead of by For- 
 mula 1 (or 6001) its bias today would have been approximately as great in 
 the opposite direction since, as is shown in Table 7, Formulae 1 and 23 
 
 353 and 6023 Compared 
 For 12 Leading Crops (Day d Persons) 
 (Quantities) 
 
 tW V '12 73 74 IS 10 77 VS If 
 
 Chakt 60Q. Analogous to Chart 60P. (1910 base.) 
 
 have about the same joint errors (except in opposite directions, of course). 
 The Day index, if continued long enough, will inevitably deteriorate in the 
 same way. 
 
 The general conclusion is that broadening the base of the weighted 
 geometric, by which process Formula 23 is converted into 6023, partially 
 eliminates the bias in the weighting of 23, but not entirely. Consequently, 
 the aggregatives, Formulae 6053 and 53, which are virtually free from bias, 
 are probably slightly better makeshifts for 353 than is the geometric 6023, 
 which has a very distinct bias. 
 
 § 10. Averaging Various Individual Quotations for One 
 and the Same Commodity 
 
 Broadening the base implies an average of the data for a series of years 
 and so raises the question of how that average is to be constructed. As a 
 matter of fact, I have used the simple arithmetic average. We need not 
 discuss this at any great length, inasmuch as we have found broadening the 
 base of little or no importance. 
 
 1 See Appendix I (Note to Chapter V, § 11). 
 
318 THE MAKING OF INDEX NUMBERS 
 
 Essentially the same problem enters, however, whenever, as is usually 
 the case, the data for prices and quantities with which we start are aver- 
 ages instead of being the original market quotations. Throughout this 
 book "the price" of any commodity or "the quantity" of it for any one 
 year was assumed given. But what is such a price or such a quantity? 
 Sometimes it is a single quotation for January 1 or July 1, but usually it 
 is an average of several quotations scattered through the year. The 
 question arises : On what principle should this average be constructed ? 
 The -practical answer is any kind of average since, ordinarily, the variations 
 during a year, so far, at least, as prices are concerned, are too little to make 
 any perceptible difference in the result, whatever kind of average is used. 
 Otherwise, there would be ground for subdividing the year into quarters 
 or months until we reach a small enough period to be considered practically 
 a point. The quantities sold will, of course, vary widely. What is needed 
 is their sum for the year (which, of course, is the same thing as the simple 
 arithmetic average of the per annum rates for the separate months or other 
 subdivisions). In short, the simple arithmetic average, both of prices and 
 of quantities, may be used. Or, if it is worth while to put any finer point 
 on it, we may take the weighted arithmetic average for the prices, the 
 weights being the quantities sold. 
 
 This problem of averaging the individual price quotations of one in- 
 dividual commodity in order to obtain "the price" for it for the year is, 
 of course, quite different from, and much simpler than the main problem 
 of this book, which is the problem of constructing index numbers from 
 such yearly figures for many commodities after they are individually 
 obtained to start with. 
 
 § 11. Conclusions 
 
 It appears that broadening the base to secure a blend 
 is always disappointing. In the case of the aggregative 
 it seems superfluous ; for we cannot find that, in practice, 
 it is any improvement over Formula 53. Moreover a 
 blend is a blur and disappoints our natural desire for 
 definiteness. It is neither flesh, fish, nor fowl. In the 
 case of the geometric it fails to suppress completely all 
 traces of weight bias. 
 
 The chief conclusions of this chapter and the last are : 
 
 1. Theoretically, a complete set of index numbers 
 among a number of years consists of all the possible index 
 numbers between every pair of years, using Formula 353 
 or any of its peers. 
 
 2. Practically, the apparent inconsistencies between 
 
BLENDING THE INCONSISTENT RESULTS 319 
 
 these index numbers coupling every pair of years is negli- 
 gible so that the calculation of so many would be a waste 
 of time, effort, and money. 
 
 3. Even were such multiple calculations practicable, — 
 connecting every possible pair of years — they would 
 not be helpful but confusing, like the conflicting natural 
 scales in music. We would be inclined to " temper " 
 or " blend " them into a single series. The ideally best 
 blend would probably be an average (Formula 7053) of the 
 index numbers formed by calculating 353 on all possible 
 bases. 
 
 4. Practically (and so barring blends, like Formula 7053, 
 of the different index numbers themselves), there remain 
 three courses to pursue : 
 
 (a) to employ one fixed base system, using Formula 353 
 or one of its peers ; 
 
 (6) to employ the chain base system, using Formula 353 
 or one of its peers ; 
 
 (c) to employ the broadened base system (such as 
 Formula 6053). 
 
 All three are in exceedingly close agi'eement. 
 
 5. Of these three systems the chain is subject to cu- 
 mulative error and ought not to be used (unless, possibly, 
 as supplementary to the fixed base system). 
 
 6. Of the two remaining systems, the fixed base sys- 
 tem (Formula 353) is somewhat preferable to the broad- 
 ened base system, partly because it is slightly closer to 
 the best blend (7053) and partly because it itself is not 
 a blend at all and, therefore, not blurred. 
 
 7. In those frequent cases, however, where the data 
 are lacking for some years and so do not permit of using 
 Formula 353, or its rivals, a broadened base is to be 
 used. 
 
 8. Two broadened base formulae are practicable for 
 
320 THE MAKING OF INDEX NUMBERS 
 
 this purpose; the aggregative 6053 and the geometric 
 6023. As between these two, while both are good, 
 Formula 6053 seems clearly the better because there is 
 no bias even if only two years are included in the base, 
 or even only one, the formula then reducing to 53. It 
 often happens that only one year's quantities are known, 
 in which case Formula 23 or 53 must be used. Formula 
 23, however, is not usable because of its downward bias, 
 whereas 53 is good, practically as good as 6053. 
 
 § 12. Historical 
 
 The fixed base system has always been the principal 
 method of presenting index numbers, sometimes the first 
 year being used as the base and sometimes a series of 
 years. The broadened base system has been in common 
 use beginning, apparently, with Soetbeer and Laspeyres. 
 Professor Alfred Marshall suggested the chain system 
 in the Contemporary Review, March, 1887, and in the 
 same year Professor Edgeworth and the Committee on this 
 subject, of which he was secretary, recommended the chain 
 system to the British Association for the Advancement of 
 Science. Walsh advocated and adopted it in his book. 
 The Measurement of General Exchange Value. Professor 
 A. W. Flux discussed the effect of changing bases in a 
 paper in the Manchester Literary and Philosophical 
 Society, 1897, and ten years later, in the Quarterly Journal 
 of Economics, discussed the chain method, but without 
 using that term. The term " chain " seems first to have 
 been used by me in the Purchasing Power of Money, in 
 1911, where I commended it, unduly, as I now believe.^ 
 
 ^ Besides the historical sections scattered through the book, of which the 
 above is the last, the reader will find in Appendix IV a brief sketch of 
 " Landmarks in the History of Index Numbers." 
 
CHAPTER XV 
 
 SPEED OF CALCULATION 
 
 § 1. Time Studies 
 
 Hitherto we have ignored the very practical question 
 of speed and ease of calculation. Table 44 gives the 
 results of time studies for calculating the index numbers 
 of prices by the various formulae. The table is constructed 
 on the assumption of 36 prices and quantities^ supplied 
 to the computer. He is furnished with a computing 
 machine and logarithmic tables. The time required to 
 construct index numbers for either prices or quantities for 
 the years 1914-1918 by Formula 51 (fixed base) is taken 
 as unity. In the case of the particular computer who 
 gave himself to these time studies, Formula 51 required 
 56 minutes. As he was probably slightly more rapid 
 than the average computer, we may think of the time for 
 51 as one hour, and of all the other figures in the table as, 
 therefore, representing hours. In every case the time of 
 calculation was that required to calculate the five index 
 numbers, to two decimal places.^ The absolute times 
 would be different, of course, if there were a different 
 number of commodities, a different number of years, or a 
 different decimal figure to be calculated. But the figures 
 given in the table are all relative to the time of calculating 
 Formula 51 (or 151) and this relative time would not be 
 
 ^ Except in the ease of the simples, for which no quantities are needed, 
 and in the case of Formula 9051, for which it is assumed that guessed round 
 weights (1, 10, 100, and 1000) are supplied. 
 
 ^ Except for the modes which were calculated only to the decimal 
 point. They could not be calculated beyond the decimal point by the 
 rough method here used. 
 
 321 
 
322 
 
 THE MAKING OF INDEX NUMBERS 
 
 greatly affected by any changes in the number of commod- 
 ities, or of years, or of decimal points to be computed. 
 
 TABLE 44. 
 
 RANK IN SPEED OF COMPUTATION OF 
 FORMULAE 
 
 
 Time of Computation as Multiple of 
 Time Required by Formula 51 (Fixed 
 
 Rank in Speed of 
 Computation 
 
 Formula No. 
 
 Base) Taken as Unity 
 
 
 Fixed Base 
 
 Chain 
 
 (Fixed Base) 
 
 5343 
 
 64.3 
 
 64.5 
 
 109 
 
 5307 
 
 62.1 
 
 62.2 
 
 108 
 
 5333 
 
 51.5 
 
 51.6 
 
 107 
 
 1303 
 
 45.3 
 
 45.5 
 
 106 
 
 345 
 
 44.6 
 
 44.6 
 
 105 
 
 5323 
 
 44.2 
 
 44.3 
 
 104 
 
 1343 
 
 42.7 
 
 42.8 
 
 103 
 
 4353 
 
 39.4 
 
 39.5 
 
 102 
 
 335 
 
 38.1 
 
 38.3 
 
 101 
 
 3353 
 
 37.8 
 
 37.9 
 
 100 
 
 7053 
 
 37.5 
 
 
 99 
 
 343 
 
 37.3 
 
 37.5 
 
 98 
 
 245 
 
 37.1 
 
 37.3 
 
 97 
 
 247 
 
 11 
 
 << 
 
 (( 
 
 1333 
 
 36.3 
 
 36.4 
 
 96 
 
 307 
 
 35.5 
 
 35.6 
 
 95 
 
 1323 
 
 35.1 
 
 35.2 
 
 94 
 
 309 
 
 34.8 
 
 35.0 
 
 93 
 
 1353 
 
 34.5 
 
 34.7 
 
 92 
 
 235 
 
 33.9 
 
 34.0 
 
 91 
 
 237 
 
 It 
 
 (( 
 
 11 
 
 225 
 
 31.9 
 
 32.0 
 
 90 
 
 227 
 
 (< 
 
 a 
 
 (( 
 
 207 
 
 31.7 
 
 31.8 
 
 89 
 
 215 
 
 (( 
 
 << 
 
 <( 
 
 126 
 
 31.6 
 
 31.7 
 
 88 
 
 325 
 
 31.5 
 
 31.6 
 
 87 
 
 323 
 
 31.3 
 
 31.4 
 
 86 
 
 333 
 
 30.y 
 
 31.0 
 
 85 
 
 146 
 
 29.7 
 
 29.8 
 
 84 
 
 108 
 
 29.3 
 
 29.4 
 
 83 
 
 243 
 
 29.2 
 
 29.0 
 
 82 
 
 1124 
 
 29.1 
 
 29.3 
 
 81 
 
 249 
 
 28.4 
 
 28.6 
 
 80 
 
 1123 
 
 28.0 
 
 28.1 
 
 79 
 
 1144 
 
 27.9 
 
 28.0 
 
 78 
 
 241 = 341 
 
 27.1 
 
 27.2 
 
 77 
 
 1143 
 
 26.6 
 
 26.8 
 
 76 
 
SPEED OF CALCULATION 
 TABLE 44 {Continued) 
 
 323 
 
 
 Time of Computation as Multiple of 
 
 
 Formula No. 
 
 Time Required by Formula 51 (Fixed 
 Base) Taken as Unity 
 
 Rank in Speed of 
 Computation 
 
 
 Fixed Base 
 
 Chain 
 
 (Fixed Base) 
 
 136 
 
 26.5 
 
 26.6 
 
 75 
 
 125 
 
 26.1 
 
 26.3 
 
 74 
 
 233 
 
 26.0 
 
 25.8 
 
 73 
 
 1104 
 
 25.3 
 
 25.5 
 
 72 
 
 239 
 
 25.2 
 
 25.4 
 
 71 
 
 1004 
 
 24.9 
 
 25.0 
 
 70 
 
 1014 
 
 a 
 
 << 
 
 11 
 
 1134 
 
 24.7 
 
 24.8 
 
 69 
 
 145 
 
 24.3 
 
 24.4 
 
 68 
 
 1103 
 
 24.2 
 
 24.3 
 
 67 
 
 1154 
 
 24.1 
 
 24.3 
 
 66 
 
 107 
 
 23.8 
 
 24.0 
 
 65 
 
 1003 
 
 23.7 
 
 23.9 
 
 64 
 
 1013 
 
 li 
 
 « 
 
 u 
 
 229 
 
 23.6 
 
 23.8 
 
 63 
 
 1133 
 
 23.4 
 
 23.5 
 
 62 
 
 144 
 
 23.0 
 
 24.8 
 
 61 
 
 124 
 
 23.0 
 
 23.1 
 
 60 
 
 143 
 
 22.8 
 
 24.4 
 
 59 
 
 209 
 
 22.8 
 
 23.0 
 
 58 
 
 213 
 
 << 
 
 a 
 
 <( 
 
 123 
 
 22.7 
 
 22.9 
 
 57 
 
 3154 
 
 <( 
 
 (< 
 
 a 
 
 26 
 
 22.5 
 
 22.6 
 
 56 
 
 28 
 
 (( 
 
 ti 
 
 << 
 
 223 
 
 21.4 
 
 23.5 
 
 55 
 
 46 
 
 21.1 
 
 21.3 
 
 54 
 
 48 
 
 (< 
 
 it 
 
 <« 
 
 135 
 
 21.1 
 
 21.2 
 
 53 
 
 301 
 
 21.0 
 
 21.2 
 
 52 
 
 4154 
 
 20.8 
 
 20.9 
 
 51 
 
 231=331 
 
 20.6 
 
 20.8 
 
 50 
 
 110 
 
 20.5 
 
 20.7 
 
 49 
 
 109 
 
 20.4 
 
 20.5 
 
 48 
 
 134 
 
 19.8 
 
 21.6 
 
 47 
 
 4153 
 
 19.6 
 
 19.8 
 
 46 
 
 133 
 
 19.5 
 
 21.2 
 
 45 
 
 36 
 
 19.5 
 
 19.7 
 
 44 
 
 38 
 
 << 
 
 11 
 
 << 
 
 1153 
 
 18.7 
 
 18.9 
 
 43 
 
324 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 44 (Continued) 
 
 
 Time op Computation as Multiple of 
 Time Required by Formula 51 (Fixed 
 Base) Taken as Unity 
 
 Rank in Speed of 
 Computation 
 
 Formula No. 
 
 
 
 
 Fixed Base 
 
 Chain 
 
 (Fixed Base) 
 
 8 
 
 18.4 
 
 18.6 
 
 42 
 
 16 
 
 11 
 
 ti 
 
 u 
 
 30 
 
 18.2 
 
 18.3 
 
 41 
 
 221 =321 
 
 17.6 
 
 17.6 
 
 40 
 
 3153 
 
 17.3 
 
 17.4 
 
 39 
 
 25 
 
 17.0 
 
 17.9 
 
 38 
 
 27 
 
 It 
 
 <( 
 
 tt 
 
 29 
 
 17.0 
 
 17.2 
 
 37 
 
 44 
 
 16.9 
 
 17.0 
 
 36 
 
 50 
 
 (1 
 
 It 
 
 ti 
 
 6023 C13-'16) 
 
 16.5 
 
 16.5 
 
 35 
 
 24 
 
 16.1 
 
 18.3 
 
 34 
 
 45 
 
 15.7 
 
 15.9 
 
 33 
 
 47 
 
 (( 
 
 (( 
 
 (( 
 
 49 
 
 « 
 
 It 
 
 tt 
 
 201 
 
 <( 
 
 it 
 
 tt 
 
 211 
 
 It 
 
 It 
 
 tt 
 
 34 
 
 15.3 
 
 15.4 
 
 32 
 
 40 
 
 << 
 
 (( 
 
 (( 
 
 2353 
 
 14.9 
 
 15.1 
 
 31 
 
 6023 ('13-'14) 
 
 14.6 
 
 14.6 
 
 30 
 
 6023 ('13&'18) 
 
 <( 
 
 tt 
 
 (( 
 
 10 
 
 14.3 
 
 HA 
 
 29 
 
 353* 
 
 <( 
 
 tt 
 
 tt 
 
 8054 
 
 tt 
 
 tt 
 
 tt 
 
 35 
 
 14.1 
 
 14.3 
 
 28 
 
 37 
 
 (( 
 
 tt 
 
 <( 
 
 39 
 
 (1 
 
 tt 
 
 tt 
 
 8053 
 
 tt 
 
 tt 
 
 tt 
 
 2154 
 
 14.0 
 
 14.1 
 
 27 
 
 42 = 142 
 
 13.9 
 
 14.1 
 
 26 
 
 7 
 
 13.0 
 
 13.1 
 
 25 
 
 9 
 
 « 
 
 tt 
 
 tt 
 
 15 
 
 K 
 
 tt 
 
 tt 
 
 32 = 132 
 
 12.9 
 
 13.1 
 
 24 
 
 43 
 
 12.6 
 
 15.9 
 
 23 
 
 102 
 
 12.6 
 
 12.7 
 
 22 
 
 14 
 
 12.0 
 
 13.4 
 
 21 
 
 22 = 122 
 
 11.9 
 
 11.9 
 
 20 
 
 23 
 
 11.6 
 
 17.2 
 
 19 
 
 ♦Identical with 103, 104, 105, 106. 153, 154, 203, 205, 217, 219, 253, 259, 303, 305. 
 
SPEED OF CALCULATION 
 
 325 
 
 TABLE 44 {Continued) 
 
 FOBMULA No. 
 
 Time of Computation as Multiple of 
 Time Required by Formula 51 (Fixed 
 Base) Taken as Unity 
 
 Rank in Speed of 
 Computation 
 
 
 Fixed Base 
 
 Chain 
 
 (Fixed Base) 
 
 33 
 
 11.0 
 
 14.3 
 
 18 
 
 2 
 
 10.5 
 
 10.6 
 
 17 
 
 12 
 
 (( 
 
 u 
 
 ti 
 
 2153 
 
 9.6 
 
 9.8 
 
 16 
 
 54 = 4 = 5 = 
 
 8.7 
 
 8.9 
 
 15 
 
 18 = 19=59 
 
 
 
 
 41 = 141 
 
 8.5 
 
 8.6 
 
 14 
 
 251=351 
 
 7.8 
 
 7.8 
 
 13 
 
 31 = 131 
 
 7.5 
 
 7.6 
 
 12 
 
 101 
 
 7.4 
 
 7.6 
 
 11 
 
 13 
 
 6.6 
 
 13.1 
 
 10 
 
 6053 ('13-' 18) 
 
 6.5 
 
 6.5 
 
 9 
 
 21 = 121 
 
 6.4 
 
 6.4 
 
 8 
 
 6053 ('13-' 16) 
 
 6.1 
 
 6.1 
 
 7 
 
 6053 ('13-' 14) 
 
 6.6 
 
 5.6 
 
 6 
 
 52 = 152 
 
 5.5 
 
 5.5 
 
 5 
 
 53=3=6 = 
 
 5.3 
 
 8.9 
 
 4 
 
 17=20 = 60 
 
 
 
 
 1 
 
 5.1 
 
 5.3 
 
 3 
 
 11 
 
 11 
 
 << 
 
 <( 
 
 9051 
 
 2.0 
 
 2.0 
 
 2 
 
 51 = 151 
 
 1.0 
 
 1.0 
 
 1 
 
 § 2. Comments on the Table of Speed of Computation 
 of Formulae 
 
 It will be seen that the first prize for speed goes to 
 Formula 51 ; to calculate this requires only one hour. 
 The booby prize is captured by a mode, 5343 ; this re- 
 quires 64.3 hours. All the other formulae occupy the 107 
 intermediate ranks. 
 
 Our ideal, Formula 353, requiring 14.3 hours, ranks 
 twenty-ninth. In speed it surpasses all the twelve other 
 formulae mentioned in Chapter XI as rivaling 353 in 
 accuracy. One of the 13, next to the slowest in the whole 
 
326 THE MAKING OF INDEX NUMBERS 
 
 table, is Formula 5307, requiring 62.1 hours. Another, 
 the closest competitor with 353 for the place of honor 
 for accuracy. Formula 5323 — the best product of the . 
 geometric type — requires 44.2 hours, or over three times 
 as long as 353. 
 
 Among other ranks in the table we note, beginning near 
 the top, or slow, end. Formula 7053, requiring 37.5 hours ; 
 1123, one of Walsh's favorites for accuracy, 28 hours; 
 Lehr's4154, 20.8 hours; 1153, another favorite of Walsh, 
 18.7 hours; 6023, the favorite of Professors Day and 
 Persons, 16.5 hours (when four years are combined in 
 the broadened base) or 14.6 hours (when two years are 
 combined). All these take longer than 353 (14.3 hours). 
 
 Among those requiring less time, the one I would 
 especially note is Formula 2153, which our table of rank 
 in accuracy shows to be practically identical with 353.^ 
 The time for Formula 2153 is only 9.6 hours.^ Formula 
 6053 (with a four years' base) requires only 6.1 hours (as 
 against 16.5 for its rival, 6023). Formula 53 requires 
 only 5.3 hours, and 9051, when only round weights, multi- 
 ples of 10, are used, needs but 2 hours. 
 
 The chain system usually requires five or ten minutes' 
 
 ^ For proof see Appendix I (Note to Chapter XV, § 2) . 
 
 " Professor Persons ("Fisher's Formula for Index Numbers," Review 
 of Economic. Statistics, May, 1921, p. 104) gives some time tests for his 
 Formulae 6023 and 353, which give very different results from those of the 
 tables here given. There are two reasons for this difference. In the first 
 place, Persons's comparison between Formulae 6023 and 353 apparently 
 omits the preliminary work of calculating the weights for 6023 and so does 
 not give a complete comparison. Our figures show that Formula 6023 
 (four year base) requires 16.5 hours and 353, 14.3 hours — a small differ- 
 ence, but in favor of 353. 
 
 The second point is that Formula 2153 can be used as a short cut for 353, 
 reducing the time to 9.6 hours, or nearly half of that for 6023, for which no 
 corresponding short cut is available. 
 
 Why Persons's time estimate for 353 chain should be double that of 353 
 fixed base, I do not understand. In any time study I have made, the dif- 
 ference between these two is much smaller. 
 
SPEED OF CALCULATION 327 
 
 more time than the j&xed base system although in a few 
 cases it actually requires less (because of certain items 
 being duplicated in that system and so needing to be 
 calculated but once). 
 
 It will be noted that, in many cases, the most accurate 
 index numbers require very little time for calculation 
 while the least accurate require a great deal of time. 
 Thus the modes are very time-consuming, and this de- 
 spite the fact that they are worked out only up to the 
 decimal. If they were accurately worked out by formulae 
 instead of roughly calculated by ocular inspection, and 
 if they were carried to the same two decimal places as 
 are used for the other formulae, the times consumed would 
 be several-fold more than the figures entered in the table. 
 As it is, the slowest formula is a mode, 5343 ; the other 
 modes in order are : Formulae 345, 1343, 343, 245, 247, 
 146, 243, 249, 1144, 341, 1143, 145, 144, 143, 46, 48, all in 
 the slower half of the list, and of the remaining modes, 44, 
 45, 47, 49, 142, 43, 141, none can boast of speed, — even 
 the fastest of them (141 or 41) ranking fourteenth. Nor 
 are the medians as fast as tradition has led us to beUeve. 
 The modern use of calculating machines has put the 
 median to shame. The fastest median, the simple For- 
 mula 31 (or 131), stands twelfth, requiring 7^ hours. 
 
 For practical use, even when the highest accuracy is 
 demanded, we never need to go beyond the fastest 16 
 formulae. The sixteenth formula is 2153, which, we have 
 seen, is, to all intents and purposes, always identical with 
 the ideal 353. And of these first sixteen the only ones 
 which have any valid claim to be used in actual practice 
 are Formula 2153 (sixteenth, requiring 9.6 hours), 31 
 (twelfth, requiring 7.5 hours), 21 (eighth, requiring 6.4 
 hours), 6053 (seventh, requiring 6.1 hours), 53 (fourth, 
 requiring 5.3 hours), and 9051 (second, requiring 2 hours). 
 
328 THE MAKING OF INDEX NUMBERS 
 
 It will be noted that several of the index numbers 
 used or recommended by others are not included in the 
 above list. The simple arithmetic index number, Formula 
 1, stands well as to speed of calculation, ranking third 
 and requiring only 5.1 hours. But, as we have seen, it 
 ranks among the " worthless " formulae in accuracy. If 
 a simple index number is really necessary, because of lack 
 of data for weighting, Formula 21 and 31 are far more 
 accurate than Formula 1 and do not take very much 
 longer to calculate (6.4 and 7.5). Usually, however, the 
 round weight Formula 9051, which is shorter to calculate 
 and at the same time more accurate than Formula 1, 
 can be used. Formula 53, which is still more accurate and 
 requires but a trifle more time, can be used if quantities 
 are known. Formula 54 need almost^ never be used. 
 It has often been recommended, but, in accuracy, it is 
 exactly as far from the ideal 353 on one side as 53 is on the 
 other, while 53 can be calculated nearly twice as quickly 
 as 54. 
 
 Formula 6023, recommended by Professors Day and 
 Persons of Harvard, is inferior both in accuracy and 
 speed to 6053 and 2153. Formulae 1123, 1153, and 
 1154, formerly recommended as the theoretically best by 
 Walsh, are probably not quite as accurate as 2153 (as 
 
 ^ The only case where Formula 53 cannot be used in place of 54 is when 
 the base weights (qos) are lacking while the current year weights (^I's) are 
 available. The only instance of such a case which has come to my atten- 
 tion is that of foreign exchange. The Federal Reserve Bulletin now pub- 
 lishes an index number of the Foreign Exchanges relatively to their " pars." 
 These pars {e.g. $4.86f for sterhng) are the base pnces (po's). But there 
 are no corresponding base quantities (go's) since the " base," in this case, 
 is of no historical year; in fact for some countries the "par" was 
 never historically realized. But the current quantities iqi's) are avail- 
 able. Here Formula 54 is indicated (or one of its equals, 4, 5, 18, 19). 
 There is scarcely any other unbiased formula available. At present the 
 Federal Reserve Bulletin uses Formula 29, which has an upward bias, — and 
 a large one when, as at present, the exchanges have a wide dispersion. 
 
SPEED OF CALCULATION 329 
 
 shown in our table of ranks and in our discussion in 
 Chapter XII), and require from twice to three times as 
 long to calculate. In his last book Walsh has also rec- 
 ommended^ Formula 2153 for adoption in practice, as 
 well as 353 as probably the most perfect theoretically. 
 
 The most important result of this chapter is that 
 Formula 2153 may be used as a short-cut method of com- 
 puting Formula 353, it being so close an approximation 
 to 353 as practically to be identical. It gives almost 
 the same result (within less than one part in 2500) and 
 in 9.6 hours instead of 14.3 hours. It should, therefore, 
 in practice be used when yearly data permit. 
 
 When yearly data are incomplete, we should use one 
 of the following formulae : 6053, 53, 9051, 21, 31, according 
 to the completeness of available data, as set forth in 
 Chapter XVII, § 8. 
 
 ^ For further discussion, see Chapter XVII, § 8. 
 
CHAPTER XVI 
 
 OTHER PRACTICAL CONSIDERATIONS 
 
 § 1. Introduction 
 
 We have studied the accuracy of the various possible 
 formulae for index numbers and their comparative speeds 
 of computation. These are the two chief considerations 
 in constructing an index number. But the problem of 
 accuracy was not fully covered ; for our study was confined 
 to the question of the formula and did not cover the data 
 that went into the formula. Hitherto, in this book, 
 by the *' accuracy " of an index number has been meant 
 its accuracy as a measure of the average movement of 
 the given set of prices (or quantities, as the case may be) 
 We have found, for instance, that Formula 353 enables us 
 to measure the average changes of the prices of the 86 
 specified commodities within less than one part in a thou- 
 sand. Yet the index numbers which have been thus com- 
 puted and found to possess a marvelously high degree of 
 accuracy, as a measure of the movements of those com- 
 modities, do not, of course, pretend to any such degree of 
 accuracy, as a measure of the movement of the prices of 
 all the commodities, perhaps many hundreds, which we 
 would wish to be represented. To obtain such precision 
 in measuring the general movement of all the prices we 
 would need to have and to use them all. Practically, such 
 completeness of data is never possible. We must content 
 ourselves with samples. We want to find, therefore, an 
 index number constructed from a relatively small num- 
 ber of commodities which shall measure, as accurately 
 
 330 
 
OTHER PRACTICAL CONSIDERATIONS 331 
 
 as possible, the movement not only of this small number 
 included, but also of those excluded. ' 
 
 Thus are opened up two new lines of investigation 
 with regard to the accuracy of index numbers, namely, 
 the influence of (1) the assortment of samples and (2) the 
 number of samples. Each of these subjects offers a field 
 of study which has scarcely yet been touched. I shall 
 try here merely to utilize what has already been accom- 
 plished by Mitchell, Kelley, Persons, and others, and to 
 urge that their important work be followed up either by 
 them or by other investigators. 
 
 These two subjects are probably quite as important 
 as the choice of the formula. Certain it is that, when the 
 number of samples used is small, an unwise choice can 
 spoil the result. It is also doubtless true that even the 
 best available assortment and number of commodities 
 cannot yield the same degree of accuracy as the merely 
 mathematical accuracy of the formulae. I venture to 
 express the guess that, when thoroughgoing studies are 
 made in these two fields, it will be concluded that we can 
 seldom reduce the errors, or fringe of uncertainty, of our 
 index numbers to less than one or two per cent. As com- 
 pared with such errors, small though they be, the errors 
 which we have found present in the formulae are quite 
 negligible. In short, in view of the rather rough work 
 required of it, the formula (whether it be 353 or any other 
 among over a score of the best formulae) may be regarded 
 as a perfectly accurate instrument of measurement. 
 
 § 2. The Assortment of Samples 
 
 What is a wise assortment depends greatly on the 
 purpose of the index number. If, for instance, the pur- 
 pose is to represent the general movement of wholesale 
 prices of foods in the United States, there should be more 
 
332 THE MAKING OF INDEX NUMBERS 
 
 samples of meats than of fish and more of cereals than of 
 garden vegetables. The assortment should also include 
 representatives of the various stages of production. 
 Again, if all stages are included in one line of goods, e.g. 
 wheat, flour, and bread, the corresponding stages should 
 be included in other lines such as corn, hogs, and pork. 
 
 The price movements of any raw material and its 
 finished products, such as cotton and cotton goods, pig iron 
 and wire nails, or wheat and flour will tend to resemble 
 each other. On the other hand, there will be a cross- 
 wise correspondence between all raw materials as con- 
 trasted with all finished products — cotton, pig iron, and 
 wheat, on the one hand, moving somewhat alike, while 
 cotton goods, wire nails, and flour will move somewhat 
 alike. As shown by Mitchell,^ the raw materials fluctuate 
 more widely than do the finished products. Again, goods 
 finished for consumers for family use have a resemblance 
 to each other as compared with goods finished for indus- 
 trial use, the latter fluctuating more than the former. 
 Every group having any distinctive character should be 
 represented in due proportion to the others. The price 
 quotations should also be fairly assorted geographically. 
 
 This process of fair sampling is intimately related to 
 the process of fair weighting, for which, in fact, it may 
 roughly be used as a substitute. The Canadian Depart- 
 ment of Labor and the British Board of Trade endeavor 
 to obviate the need of any specific weighting by represent- 
 ing the important groups of goods by a large number 
 of commodities, or series of quotations, while representing 
 the unimportant commodities by a small number and 
 then taking a simple average. By means of such precau- 
 tions, a simple index number virtually loses its freakish 
 weighting, and becomes roughly equivalent to a weighted 
 
 1 Bulletin 284, United States Bureau of Labor Statistics, pp. 44, 45. 
 
OTHER PRACTICAL CONSIDERATIONS 333 
 
 index number. The simple geometric formula (21) is 
 thereby made nearly as good as the well-weighted formula 
 (1123), a vast improvement, and Formula 1 brought nearer 
 to 1003, an improvement, but not so vast, for the upward 
 bias remains, though the freakishness has gone. Thus, the 
 Canadian index number has the bias of Formula 1003 while 
 the British Board of Trade index number has very nearly the 
 excellence of 1123. Bradstreet's index number (Formula 
 51), thanks to a good selection of data, has also been 
 converted from what would otherwise be a worthless index 
 number into a fairly good index number, being virtually 
 9051, or a close approach to 53. Without such precautions 
 great distortion occurs. 
 
 During the Civil War the Economist index number 
 became erratic because, out of 22 commodities, no less than 
 four were cotton and cotton products. As the Civil War 
 raised cotton prices enormously, the Economist index 
 number showed in 1864 a rise of 45 per cent over the 
 price level of 1860, whereas Sauerbeck's index number of 
 45 commodities showed for the same period only a 12 
 per cent rise (both series being recomputed from 1860 
 as base). Again, in the Aldrich Report of 1893, the simple 
 average included 25 kinds of pocket knives, making pocket 
 knives 25 times as important as wheat, or com, or coal. 
 
 At best, however, such multiplication of commodities 
 is only a rough substitute for actual weighting. On the 
 other hand, even when weights are used they need to be 
 adjusted to fit in with the numbers of commodities included 
 under the various groups. Thus the War Industries Board, 
 having included seven groups (foods, clothing, rubber- 
 paper-fiber, metals, fuels, building materials, chemicals) 
 subdivided into 50 classes (nearly 1500 separate commodi- 
 ties or series of quotations), proceeded to weight them in 
 two stages. In the first place, each commodity was 
 
334 THE MAKING OF INDEX NUMBERS 
 
 weighted according to statistics or estimates of the pro- 
 duction or volume of business done in that commodity. 
 Then, in the second place, inasmuch as some of the 50 
 classes were more fully represented than others, i.e. 
 were represented by a larger number of commodities, 
 the classes which were meagerly represented in number 
 of commodities were the more liberally weighted to com- 
 pensate. The weights first assigned to them as individual 
 commodities were magnified or multipHed by factors 
 called " class weights " to make them represent more 
 adequately the large class to which they belong. This, 
 or some equivalent procedure, should always be employed 
 where the highest accuracy is desired. 
 
 In short, either insufficient weights should be compen- 
 sated for by duplicating samples (as in the Canadian and 
 Board of Trade index numbers), or insufficient samples 
 should be compensated for by additional weighting (as 
 in the War Industries Board index numbers). Except 
 as a substitute for weighting, samples need not be multi- 
 plied greatly. In fact, where it is desired to save labor by 
 restricting the number of commodities, those selected 
 should be so assorted as to differ from each other in charac- 
 ter as much as possible rather than to resemble each other 
 as much as possible. As Professor Kelley says, the prices 
 included should be correlated not so much with each 
 other as with those excluded.^ Where the samples are 
 thus well selected, the index number will not only rep- 
 resent well the price movements of the commodities in- 
 cluded, but also those excluded, usually the larger group. 
 
 § 3. The Basis of Classification 
 
 As Professor Mitchell points out, there is no consistent 
 basis of classification in the grouping employed by the 
 
 ' "Certain Properties of Index Numbers," Quarterly Publication of 
 the American Statistical Association, pp. 826-41, September, 1921. 
 
OTHER PRACTICAL CONSIDERATIONS 335 
 
 United States Bureau of Labor Statistics and others. 
 Sometimes the basis is physical appearance {e.g. as in the 
 case of "metals"), use served {e.g. "house furnishing 
 goods "), place of production (" farm products "), the 
 industry concerned (" automobile supplies "), etc. Mitch- 
 ell thinks, on the whole, that the most useful classifica- 
 tions are raw versus manufactured; the raw being sub- 
 divided into farm crops and animal, forest, and mineral 
 products, and the manufactured being subdivided into 
 goods for personal consumption, such as sugar, and goods 
 for business consumption, such as tin plates. 
 
 I venture to express the opinion that we shall ulti- 
 mately find two chief bases or groups for classifying goods. 
 
 (1) We need a basis for setting off the particular field 
 which the index number is to represent. Since this may 
 be any field whatever in which we are interested, the basis 
 for including or excluding commodities may be physical 
 appearance, use served, or anything else, according to the 
 field to be studied. For instance, a leading paper manu- 
 facturer has constructed for use in his business an index 
 number of the costs involved in the manufacture of 
 paper. These comprise wood pulp, labor, and all other 
 items entering into that cost. 
 
 (2) On the other hand, the basis on which, within 
 the particular field thus marked out, the samples should 
 be assorted is none of those bases above mentioned but 
 rather the behavior of the prices. All behaviors should be 
 fairly represented. In the paper manufacturer's index 
 number of costs, both labor and wood pulp should be 
 represented, not because they are so widely different in 
 physical nature, but because the price of wood pulp and 
 the price of labor behave differently. If it were true 
 that they always rose and fell together a sample of either 
 would serve perfectly for both. 
 
336 THE MAKING OF INDEX NUMBERS 
 
 One of the most interesting kinds of index numbers is 
 Professor Persons' s new index number for use as a barom- 
 eter of trade. In this case the selection of the ten com- 
 modities included is based, not on any of the usual 
 criteria, but on their previous behavior in relation to the 
 business cycle. 
 
 § 4. The Number of Quotations Used 
 
 Ideally, the quotations should be as inclusive as possible 
 of the quotations properly belonging to the class being 
 studied. In reality, however, we are restricted by expense 
 or other practical obstacles. If the assortment is good, 
 the number is not very important. The War Industries 
 Board used 1474 commodities, or series of quotations. 
 But the resulting index number differs only sUghtly 
 (seldom by one per cent) from that of the United States 
 Bureau of Labor Statistics for about 300 commodities. 
 
 Wesley C. Mitchell, in Bulletin 284 of the United States 
 Bureau of Labor Statistics, has compared the index 
 number of the Bureau for about 250 commodities with the 
 index number for 145, 50, 40, and 25 commodities, taking 
 care to retain a similar representation of the various con- 
 stituent groups of conunodities in the cases of the 145 
 and 50 commodities, but making the 40 representative 
 on another principle, and choosing the 25 at random. 
 He found that the 145 index differed on the average from 
 the index of the Bureau of Labor Statistics, by less than 
 one per cent, the 50 index by less than two and one half per 
 cent, the 40 index by less than 5.4 per cent, and the 25 index 
 (taken in two ways) by less than four and three per cent. 
 
 I have made a similar comparison of various numbers 
 of commodities from the list published weekly in Dun's 
 Review. Beginning with 200 commodities and succes- 
 sively halving, I have taken the sub-lists of 100 commodi- 
 
OTHER PRACTICAL CONSIDERATIONS 
 
 337 
 
 ties, 50, 25, 12, 6, and 3, so selected as to be, so far as 
 possible, equally and fairly representative of the various 
 classes of commodities in exchange.^ These were calcu- 
 lated (relatively to 1913 as a base) by Formula 53 (or 3). 
 The results are plotted in Chart 61. They show a rather 
 surprising resemblance. Taking 200 as a standard of 
 comparison, and gauging the closeness of the others to 
 this by the average ^ of their deviations from it, we find the 
 following figures : 
 
 TABLE 45. DEVIATIONS FROM 200 
 COMMODITIES INDEX 
 
 Number op Commodities 
 
 Deviations 
 
 Included in Index 
 
 (Pek Cents)] 
 
 100 
 
 1.78 
 
 50 
 
 2.05 
 
 25 
 
 1.61 
 
 12 
 
 2.64 
 
 6 
 
 4.31 
 
 3 
 
 3.65 
 
 ^ The similarity in assortment is, of course, necessarily rough. It is 
 impossible, for instance, to assort three or six commodities so as to 
 include a sample in every one of the eight classes used for the 200 
 commodities. The actual assortments are shown in the following table : 
 
 PERCENTAGES OF AGGREGATE VALUE OF THE 200, 100, 50, 25, 12, 6, 
 AND 3 COMMODITY INDEXES RESPECTIVELY, IN EACH GROUP 
 
 s 
 
 O a 
 
 a 
 
 
 g» 
 
 
 
 ^ g 
 
 < 
 
 < 
 
 Z 
 
 § 
 
 
 
 
 
 o 
 
 1" 
 
 6 ^ 
 la >- 
 
 0. 
 
 
 
 2 
 
 1^ 
 
 5h 
 
 200 
 
 27.48 
 
 20.80 
 
 11.39 
 
 8.61 
 
 18.47 
 
 6.54 
 
 3.85 
 
 2.86 
 
 100. 
 
 100 
 
 27.18 
 
 22.73 
 
 9.29 
 
 10.94 
 
 18.22 
 
 5.13 
 
 3.35 
 
 3.16 
 
 100. 
 
 50 
 
 30.57 
 
 29.19 
 
 12.75 
 
 3.45 
 
 13.04 
 
 6.85 
 
 2.35 
 
 1.80 
 
 100. 
 
 25 
 
 23.69 
 
 30.06 
 
 13.72 
 
 5.24 
 
 16.64 
 
 8.11 
 
 1.09 
 
 1.45 
 
 100. 
 
 12 
 
 29.73 
 
 34.76 
 
 9.40 
 
 6.90 
 
 13.73 
 
 3.35 
 
 .76 
 
 1.37 
 
 100. 
 
 6 
 
 40.40 
 
 35.97 
 
 5.80 
 
 0.00 
 
 13.28 
 
 4.55 
 
 0.00 
 
 0.00 
 
 100, 
 
 3 
 
 55.75 
 
 25.92 
 
 0.00 
 
 0.00 
 
 18.33 
 
 0.00 
 
 0.00 
 
 0.00 
 
 100. 
 
 2 Calculated as the square root of the average of the squares of the deviations. 
 
338 THE MAKING OF INDEX NUMBERS 
 
 From this table and Chart 61, it is clear that the mere 
 number of commodities is of only moderate importance. 
 A small number may be nearly as good as a large number 
 provided they be equally well selected or assorted. 
 
 According to the theory of probabilities, the probable 
 error of the mean of any number of observations is in- 
 
 ^/Tect of 
 Number of Commodities 
 
 on Index Nos. 
 
 25 COMMODITIES 
 
 ^° £2l:iM0Diries 
 
 100 COMMODITIES 
 200 COMMODITIES 
 
 1921 
 APRS 
 
 I 3% 
 
 NOV.II 
 
 1922. 
 JAN.I3 
 
 J'JN.3 AU6.5 
 
 Chart 61. Comparing the index numbers of 200, 100, 50, 25, 12, 6, 
 and 3 commodities, each group having roughly similar proportions of farm 
 products, foods, clothing, fuel and lighting, metals, building materials, 
 drugs, and miscellaneous. 
 
 versely proportional to the square root of the number. 
 This rule would apply here if all commodities were inde- 
 pendent and equally important. We could then say, 
 for instance, that 50 commodities would show twice the 
 error which four times that number, or 200 commodities, 
 would show, and the latter, in turn, twice that of 800. 
 By this law of the square root, accuracy increases very 
 slowly with an increase in number. 
 
OTHER PRACTICAL CONSIDERATIONS 
 
 339 
 
 In actual fact the improvement in accuracy with an 
 increase in the number of commodities is even slower than 
 this rule would lead us to expect. From the preceding 
 table of deviations I think it may be inferred by rough 
 averages^ that, in order to reduce the error by half we 
 must multiply the number of commodities not by four but 
 by thirty-five. If this be true, the index number of the 
 War Industries Board with its 1366 commodities is only 
 twice as accurate as an index number formed from 40 
 commodities, other things equal. 
 
 This slowness of improvement in index numbers with an 
 increase in the number of commodities is largely because 
 the number of commodities does not represent their 
 importance or weights. These weights for the 100, 50, 
 25, 12, 6, and 3 commodity groups (in dollars and in per 
 cents of the weights of the 200) are as follows : 
 
 TABLE 46. COMPARISON OF THE AGGREGATE VALUE OF 
 THE 100, 50, 25, 12, 6, AND 3 COMMODITIES WITH THE 
 AGGREGATE VALUE OF THE 200 COMMODITIES 
 
 No. OF Commodities 
 
 Aggregate Value 
 (in Millions or Dollars) 
 
 Aggregate Value 
 (in Per Cents) 
 
 200 
 
 18266 
 
 100 
 
 100 
 
 12079 
 
 66 
 
 50 
 
 6572 
 
 36 
 
 25 
 
 4331 
 
 24 
 
 12 
 
 3284 
 
 18 
 
 6 
 
 2416 
 
 13 
 
 3 
 
 1751 
 
 10 
 
 If we use these weights instead of the number of the 
 commodities the resulting law of increasing accuracy 
 with increase in weights of commodities included is more 
 nearly in accord with that required by the theory of 
 probabihty. As Table 46 shows, when a small number 
 
 ^ Obtained by plotting the table of standard deviations in relation to 
 the number of commodities on doubly logarithmic, or ratio chart, paper. 
 
340 THE MAKING OF INDEX NUMBERS 
 
 of commodities is used, we naturally choose the most 
 important, which means those having the greatest weights. 
 If, now, we calculate the relationship between the errors 
 of the index numbers of 100, 50, 25, 12, 6, and 3 commodi- 
 ties, on the one hand, and, on the other, not the total 
 number of commodities but their total weights, we find 
 that, on the average, in order to reduce the error by half, 
 we must multiply the total weight of the commodities 
 by ten, whereas probability theory requires four. 
 
 Incidentally, by extending graphically these rough 
 laws connecting error and the number or weight of com- 
 modities, it may be estimated that the probable error of 
 the index number of the 200 commodities as samples as 
 compared with an index number calculated from an 
 absolutely complete set of commodities is about 1^ per 
 cent.^ But in order to obtain a trustworthy empirical 
 formula we would need very much fuller data than 
 those here given. I hope someone will make a thorough 
 enough study of this subject to obtain such a for- 
 mula. 
 
 An index number, really valuable, has been computed 
 for as few as 10 commodities, — that recently constructed 
 by Professor Persons to be used for forecasting. Seldom, ^ 
 however, are index numbers of much value unless they 
 consist of more than 20 commodities ; and 50 (the number 
 of classes used by the War Industries Board) is a much 
 better number. After 50, the improvement obtained from 
 increasing the number of commodities is gradual and it is 
 doubtful if the gain from increasing the number beyond 
 200 is ordinarily worth the extra trouble and expense. 
 
 ^ This same result obtains whether the numbers or the weights are 
 used. For another (Kelley's) method of reckoning this probable error, 
 see Appendix I (Note to Chapter XVI, § 4). As there shown, Kelley's 
 method yields, as its result, a little less than 1 per cent in one case and 1.3 
 per cent in another. 
 
OTHER PRACTICAL CONSIDERATIONS 341 
 
 § 5. Errors in the Data 
 
 It is, of course, vital that the original data shall be as 
 accurate as possible. That is, the markets used, the sources 
 of quotations, and the collecting agency should be the most 
 reliable and authoritative. Nevertheless, the net effect 
 on the index number of inaccuracies in the original data 
 is smaller than would naturally be supposed, especially 
 if a large number of commodities are used. If there be 
 100 commodities and an average or typical group of ten 
 among them are each ten per cent too high, the net effect on 
 the index number is to make it only one per cent too high! 
 And the chances against all ten thus erring in the same 
 direction is negligible. The errors would probably largely 
 offset each other, so that the probable error in the index 
 number which would result from ten typical commodities, 
 out of 100, being each ten per cent wrong, but some too 
 high and others too low at random, would not be one per 
 cent, but only about one fourth of one per cent. If 
 every one of the 100 commodities is subject to an error 
 of ten per cent in either direction at random, the net 
 resultant error in the index number would probably not 
 be over two and one half per cent. 
 
 From such surprising examples we see : (1) that even 
 rough data are valuable if we have enough of them, and 
 (2) that, under conditions of ordinary and reasonable 
 accuracy of the data, the inaccuracies which actually 
 enter have a negligible influence on the result, probably 
 less than one tenth of one per cent in the case of such an 
 index number as that of the United States Bureau of 
 Labor Statistics. 
 
 What has been said applies to the price data (for an 
 index number of prices). The quantity data, which are 
 needed only for the weights, require even less accuracy. 
 
342 THE MAKING OF INDEX NUMBERS 
 
 As is shown in Appendix II, § 7, the effect of a change 
 in a weight is only a small fraction of that of a change in the 
 price relative. If the data for any or all of the weights 
 were wrong by 50 or 100 per cent, the effect on the index 
 number would seldom amount to one per cent. 
 
 § 6. The Errors of Four Standard Index Numbers 
 
 We have now seen that the accuracy of an index number 
 depends upon four circumstances : 
 
 (1) the choice of the formula, 
 
 (2) the assortment of items included, 
 
 (3) the number of items included, 
 
 (4) the procuring of the original data. 
 
 At present, the chief source of error in standard or current 
 index numbers is in the formula. This book shows that 
 this source of error can (if full data are available) be elimi- 
 nated entirely — or, to be exact, can be reduced to much 
 less than a tenth of one per cent. 
 
 We may now summarize the whole subject of the 
 degree of accuracy of index numbers by citing four actual 
 examples : the index numbers of wholesale prices of the 
 War Industries Board, the United States Bureau of 
 Labor Statistics, the Statist's or Sauerbeck's, and Pro- 
 fessor Day's index numbers of prices and quantities of 
 12 crops. In each case I shall estimate or guess at the 
 errors due to each of the four sources of possible error and 
 the extent to which such errors were avoidable. 
 
 The War Industries Board index number, which is for 
 the years 1913-1918, is probably the most accurate index 
 number ever constructed owing to the huge number of 
 commodities included and the fact that the data for 
 quantities are available. 
 
 1. The error in this index number due to errors of the 
 formula (53) is usually less than one fourth of one per 
 
OTHER PRACTICAL CONSIDERATIONS 343 
 
 cent, but reaches about one half of one per cent for 1918^ 
 (the figure used being below the ideal, 353). 
 
 2. The error due to errors in the assortment of items 
 included (corrected by class weighting) is, I imagine, 
 always less than one per cent. 
 
 3. The error due to the number of commodities (over 
 1300) not being complete is, I imagine, less than one half 
 of one per cent. 
 
 4. The error due to errors in the original data is pre- 
 sumably less than one tenth of one per cent. 
 
 The net error due to all four sources is, I imagine, 
 usually, if not always, less than one per cent. All of the 
 errors were doubtless unavoidable excepting that due to 
 the choice of the formula, and this probably accounts 
 for perhaps a third, or a half, of the net error. That is, 
 this most precise of index numbers might have been twice 
 as precise as it is had Formula 353 (or any of its peers) 
 been used as the formula instead of 53. Had this been 
 done it would have been worth while to use a figure beyond 
 the decimal point. It is a pity that the highest available 
 degree of precision was not reached, as such a good oppor- 
 tunity for calculating 353 seldom occurs, owing to the non- 
 availabihty in most cases of statistics of yearly quantities. 
 
 We may next consider Day's index numbers of prices 
 and quantities of 12 crops. 
 
 1. As to ^he formula, or instrumental error, the calcula- 
 tions of Professor Persons comparing Professor Day's 
 (6023) with the " ideal " (353) shows an error in the price 
 index of usually less than one fourth of one per cent, 
 exceeding one per cent only once, when it was 1.6 per cent. 
 
 * As judged from the 90 raw materials for which the War Industries 
 Board publishes the full data needed for calculating Formula 353. 
 Charts for Formulae 53 and 54 (the latter calculated by me) for these 90 
 commodities are given in Chapter XI. Formula 353, of course, exactly 
 splits the difference between 53 and 54. 
 
344 THE MAKING OF INDEX NUMBERS 
 
 For quantities, the error is usually less than one per cent, 
 the maximum being 1.5 per cent. 
 
 2. As to assortment, I can only guess roughly that the 
 error from this source would be inside of one or two per 
 cent. 
 
 3. As to the number of commodities, I would guess, 
 say, two per cent. 
 
 4. As to accuracy of data, I would guess that the index 
 number would not be affected more than one per cent. 
 
 The total net error is probably usually within three or 
 four per cent, although if all the errors happened to be in 
 the same direction and all large they might make a total 
 of five or six per cent. I assume that all of these errors 
 are unavoidable, except that due to choosing a formula 
 with a slight downward bias. Had Formula 353 been 
 chosen instead of 6023, the error would have been reduced, 
 but seldom by as much as one per cent. While the gains 
 in accuracy by using a better formula would be small 
 as compared with the errors from other and less avoidable 
 sources, they would have been worth while, to say nothing 
 of the gain in speed of computation. As indicated in Chap- 
 ter XIV, § 9, the formula error is bound in the future to 
 grow indefinitely. 
 
 Our next example is that of the United States Bureau 
 of Labor Statistics. The errors are probably about the 
 same as for the War Industries Board : 
 
 1. Formula. 53 is used, erring, say, usually less than 
 one fourth of one per cent, and, at most, say, one half 
 of one per cent. 
 
 2. Assortment. Say, less than one per cent. 
 
 3. Numbers of commodities. Say, less than one per 
 cent. 
 
 4. Data. Say, less than one tenth of one per cent. 
 The total or net error is presumably usually within one 
 
OTHER PRACTICAL CONSIDERATIONS 345 
 
 or two per cent, almost all being presumably unavoidable 
 — that from formula included, owing to the non-avail- 
 ability of j^early quantities. 
 
 Finally, let us look at Sauerbeck's, or what is now the 
 Statist's, index number. 
 
 1. Formula. This error is of two parts, that due to 
 the bias of the arithmetic type and that due to the freak- 
 ishness of the simple weighting. The first can be esti- 
 mated with considerable certainty, if we calculate the 
 standard deviation and use our formula connecting the 
 standard deviation and the bias. I have worked out the 
 standard deviation for 1920, relatively to the base 1867- 
 1877, for the 45 commodities. This is 129 per cent, from 
 which we know that the upward bias is 36 per cent. For 
 1913 the standard deviation was 33 per cent and the bias 
 4.1 per cent so that, not only is the Sauerbeck-Statist in- 
 dex number for 1920 distorted upward by this cause by 36 
 per cent relatively to the original base, but it is distorted 
 relatively to 1913 by 31 per cent. As to the error of 
 f reakishness of weighting this may be said to be practically 
 the same thing as the error of assortment. 
 
 2. Assortment. Say, one per cent. 
 
 3. Number of commodities. Say, one or two per cent. 
 
 4. Data. Say, one tenth of one per cent. 
 
 The net error is probably, say, 35 to 40 per cent. In 
 this case the source of almost all the error is the bias in 
 the formula which reaches so high a figure partly because 
 of the long lapse of time since the base period and partly 
 because of the great dispersion due to the confusion pro- 
 duced by the World War. This source of error is, of 
 course, avoidable. This Sauerbeck-Statist index number 
 has done pioneer work and deserves that respect always 
 due to long and faithful service. But it is now both too 
 old and too old-fashioned to be of great service in the future. 
 
346 THE MAKING OF INDEX NUMBERS 
 
 § 7. A New Index Number 
 
 I have worked out a new index number of wholesale 
 prices of 200 commodities by a method which combines 
 speed of computation with as much accuracy as the data 
 afford. This I hope later to publish weekly. The data 
 include only base year quantities and the formula used 
 is a combination of 53, employing base year quantities, 
 and 9051. For the 28 most important commodities the 
 method of 53 is used, i.e. each price quotation is multiplied 
 by the best obtainable statistical figure for the quantity 
 marketed of that commodity, while for the other 172 
 commodities the round figures 1, 10, 100, or 1000 are used, 
 whichever in any given case is nearest the statistical 
 figure. No sacrifice of accuracy is made by using such 
 round figures for so many unimportant conunodities, 
 as I have proved by certain tests.^ 
 
 In this way we avoid the necessity of having labo- 
 riously to calculate to any greater degree of precision than 
 that which is attainable. This saving of useless labor is 
 enormous. To calculate this index number of 200 com- 
 modities, once the data are given, requires (for the 
 calculation of a single index number) only two and a half 
 hours as contrasted with the eight hours which would be 
 required if all of the 200 statistical weights were used. As 
 to precision reached, I believe the error is nearly as 
 small as that of the United States Bureau of Labor 
 Statistics, the total error being, say, usually less than 
 two per cent. 
 
 § 8. An Index Number should be Easily Understood 
 
 It is practically important that an index number, 
 besides being accurate and quickly calculated, should be 
 
 » See Appendix I (Note to Chapter XVI, § 7). 
 
OTHER PRACTICAL CONSIDERATIONS 347 
 
 easily understood. In this respect the aggregatives 
 have an obvious advantage over all other types. Any- 
 one can understand Formula 53, especially if the base 
 number be taken not as 100, but as the sum of values ( SpoQ'o) 
 in the base year. In this case the index number is simply 
 the number of dollars which a given bill of goods costs 
 from time to time. Formula 54 is almost as simple, 
 being merely calculated the other way around. Formula 
 2153 is next in simplicity, the bill of goods being the 
 average of the above two. Formula 353 is a little harder 
 for the man in the street to understand, but is intelligible 
 as the mean of 53 and 54. These are easier to under- 
 stand than any arithmetic average, still easier than any 
 harmonic, and far easier than the geometries. 
 
 Another advantage of the aggregatives is that of sim- 
 plicity and ease of manipulation. When we wish index 
 numbers of foods, clothing, etc., as subheads under a gen- 
 eral group from which we also want an index number, the 
 aggregative is the most easily and most intelligibly added, 
 combined, averaged, and otherwise manipulated — far 
 more easily than the medians, in particular. 
 
 § 9. Ranking of Formulae by Four Criteria 
 
 We may here summarize the whole subject of ranking 
 the formulae in Table 47, in the third column of which I 
 have arbitrarily ranked the 20 simplest in the order of their 
 simplicity of formula, — in other words, their intelligibility. 
 The three other columns give, likewise, the 20 best rank- 
 ing formulae in respect of accuracy, speed, and conform- 
 ity to the so-called circular test.^ 
 
 * The order of "accuracy " is the revised order given in Chapter XII, § 7. 
 The order of conformity to the circular test here given, above the line divid- 
 ing those which fully conform from the rest, is also revised arbitrarily. The 
 order given in Chapter XIII, § 9, is obviously largely accidental, being based 
 on only four data for each formula. 
 
348 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 47. (INVERSE) ORDER OF RANK OF FORMULA 
 
 Of the 20 First in Acciiracy 
 u « u u "Speed 
 " " " " "Simplicity 
 " " " " •' Circular Test Conformity 
 
 Accuracy 
 
 Speed 
 
 Simplicity 
 
 Circular Test 
 Conformity 
 
 3353 
 
 22 
 
 52 
 
 2153 
 
 1124 
 
 23 
 
 6023 
 
 323 
 
 1123 
 
 33 
 
 23 
 
 325 
 
 124 
 
 2\ 
 
 353 
 
 8054 
 
 126 
 
 12] 
 
 8054 
 
 8053 
 
 123 
 
 2153 
 
 8053 
 
 1323 
 
 125 
 
 54 
 
 1153 
 
 1353 
 
 1154 
 
 41 
 
 9021* 
 
 5323 
 
 1153 
 
 351 
 
 9011* 
 
 2353 
 
 2154 
 
 31 
 
 101 
 
 9001* 
 21 
 
 353 
 
 2153 
 
 21 
 
 
 323 
 
 13 
 
 11 
 
 22 
 
 
 325 
 
 6053 
 
 2153 
 
 51 
 
 
 8054 
 
 21 
 
 6053 
 
 52 
 
 
 8053 
 
 52 
 
 54 
 
 321 
 
 
 1323 
 
 53 
 
 53 
 
 351 
 
 
 1353 
 
 1] 
 
 31 
 
 6023 
 
 
 5323 
 
 11] 
 
 1 
 
 6053 
 
 
 2353 
 
 9051 
 
 9051 
 
 9021* 
 
 
 353 
 
 51 
 
 51 
 
 9051 . 
 
 
 * Index numbers by these "rough weight" formulse were not computed for this book. 
 Consequently they enter into the competition only in column 3 (and one of them, 9021, in 
 column 4) and not elsewhere, where computation is involved. The reason for omitting their 
 computation is the impossibility of selecting the rough weights. Or rather there are an 
 infinite number of sets of rough or guessed weights which might be used. The rough weights 
 in the case of 9051, on the other hand, are definitely ascertainable, being 1, 10, 100, etc. ; 
 for the unique idea of Formula 9051 is a minimum of calculation, — the rough and ready 
 summation of the original data after "imply shifting decimal points. 
 
 Approximate ranks can be assigned to these formula, however. With any reasonable 
 selection of rough weights. Formulae 9001 and 9011, if computed and ranked for columns 
 1, 2, 4, would be too inaccurate to find a place among the 20 most accurate (column 1), 
 and would find no place among the 20 in column 4, but would take rank, in speed, below the 
 middle of column 2. As to Formula 9021, this is already ranked in the last two columns; 
 it would find no place in the accuracy list (column 1) but would find a place in the upper 
 half of the speed list (column 2). 
 
 We see that, for accuracy, 353 takes first rank, for speed 
 in computation, 51, for the convenience of conformity to 
 the so-called circular test, any of the ten below the divid- 
 
OTHER PRACTICAL CONSIDERATIONS 349 
 
 ing line in the fourth column, and, for simplicity, 51. In 
 this table, the formulae which occur but once are itali- 
 cized. As none of these are near the goal in any list 
 they certainly need never be used. Eight occur three 
 tunes (21, 51, 353, 2153, 6053, 8053, 8054, 9051) and only 
 one (2153) occurs in all four columns. Taking into 
 account accuracy, speed, ease of manipulation, and 
 intelligibility, Formula 2153 seems, on the whole, to take 
 the highest rank for ordinary practical use. 
 
 § 10. Conclusions 
 
 The " instrumental error," or error in the index number 
 as an instrument of measure, can be reduced by the right 
 choice of formula so low as to be negligible as compared 
 with the errors from other sources — particularly the 
 assortment of the commodities included and their number. 
 The greater the number of commodities, other things 
 equal (including assortment), the more accurate it is ; but 
 the increase in accuracy is very slow, requiring perhaps a 
 thirty-five-fold increase in numbers to cut the error in 
 two. 
 
 Of the four chief sources of error, formula, assortment, 
 number of commodities, and original data, the two first 
 are usually most at fault. The error in the Sauerbeck- 
 Statist index number today reaches over 35 per 
 cent from the first source alone. If an index number 
 be constructed in the best possible way, not only from 
 accurate data and with an adequate number of commodi- 
 ties, say, several hundred, but from data carefully 
 assorted for the purpose in view, and with a first-class 
 formula, such as 353 or 2153, it can probably be made 
 accurate within close to one per cent. 
 
 General conclusions as to ranking are stated at the top 
 of this page. 
 
CHAPTER XVII 
 SUMMARY AND OUTLOOK 
 
 § 1. Introduction 
 
 An index number of prices is intended to measure such 
 magnitudes as the " price level " of one date or place 
 relatively to that of another. It is an average of " price 
 relatives." These price relatives (or movements of the 
 prices of individual commodities) usually disperse or scat- 
 ter widely. The dispersion or scattering of the price 
 relatives used in this book (for the years 1913 to 1918), 
 was especially great. Thus the price of wool in 1918 
 (relative to 1913) was 282 per cent and that of rubber, 68 
 per cent. Evidently their average or index number 
 (reckoned arithmetically) was 175 per cent. 
 
 Since an index number for any date is always relative 
 to some other date it necessarily implies two dates or peri- 
 ods and only two. When we calculate a series of index 
 numbers for a series of years each individual index number 
 connects one of the years with some other year. The 
 usual way is to take some one year, such as the first year, 
 as the "base" and calculate the index number of each 
 other year relatively to that common base. This is 
 called the " fixed base system." Another way is that of 
 the " chain " system by which the index number of each 
 year is first calculated as a " link " relatively to the pre- 
 ceding year and then multiplied by all the preceding links 
 back to the base year. 
 
 350 
 
SUMMARY AND OUTLOOK 351 
 
 § 2. Varieties of Types, Weightings, and Tests 
 
 There are only six types of index number formulae 
 which need to be considered : the arithmetic, harmonic, 
 geometric, median, mode, and aggregative — all defined in 
 Chapter II. Of these, the mode and, in general, the me- 
 dian, may be ignored as too sluggish or unresponsive to 
 email influences to make them sensitive and accurate 
 barometers of price movements. 
 
 As shown in Chapters III and VIII, there are six chief 
 ways of "weighting " the price relatives entering into any 
 index number (except the aggregative) viz., (1) simple 
 or even weighting, each price relative (like the 282 and 
 C8) being counted once ; (2) by base year values (desig- 
 nated in the book as weighting /), wool being counted 
 twice to rubber once if the value of the wool sold in the 
 base year is twice that of the rubber; (3) by given 
 year values (weighting IV) ; (4) and (5) by '^ hybrid" 
 values, each weight being formed by multiplying the price 
 of either year by the quantity of the other year (weight- 
 ings II and ///) ; and, finally (6) by crosses or means 
 between the weightings I and IV, or II and III. 
 
 Of these six systems of weighting the middle four are 
 fundamental enough to tabulate : 
 
 I by base year values (po?0) etc.) 
 
 // '' hybrid " (pogi, " ) 
 
 III " hybrid' " (pi^o, " ) 
 
 IV " given year " (piQ'ij " ) 
 
 In the case of the aggregative index numbers, since the 
 weights (of an index number of prices) are, in this case, 
 merely quantities (not, as in other cases, values), we have 
 only four methods of weighting, viz., (1) simple ; (2) by 
 base year quantities (weighting I) ; (3) by given year 
 
352 THE MAKING OF INDEX NUMBERS 
 
 quantities (weighting I V) ; and (4) by a cross or mean 
 between the last two (7 and IV). 
 
 There are two chief tests of reversibiUty for an index 
 number formula (F) : First, it should give consistent 
 results if applied forward (Poi) and backward (Pio) 
 between the two dates " " and " 1 " (i.e. Poi X Pio 
 should = 1). This has been called Test 1, or the time re- 
 versal test. To illustrate, if the index number shows 
 1918's prices average twice those of 1913, the same for- 
 mula should, when applied the other way round, show 
 1913's prices to average half those of 1918. Secondly, 
 the formula should give consistent results if applied to 
 prices and to quantities {i.e. Poi X Qoi should = Toi) 
 
 ^.e. -^^). This has been called Test 2, or the factor 
 
 reversal test. To illustrate, if we know that the total value 
 of the commodities has doubled and our index number 
 of prices shows that prices have, on the average, doubled, 
 the same index number formula should, when applied to 
 quantities, show that quantities, on the average, have re- 
 mained the same. 
 
 In short, we can check up the forward and backward 
 index numbers by the principle that their product should 
 be unity, and we can check up the price and quantity 
 index numbers by the principle that their product should 
 be the value ratio. 
 
 § 3. Bias 
 
 But many kinds of index numbers do not thus check 
 up. For instance, the arithmetic index number does not. 
 The product of any arithmetic index number, taken for- 
 ward, multiplied by the arithmetic with the same weights 
 but taken backward, fails to meet Test 1 by always and 
 necessarily exceeding unity {i.e. Poi X Pio > 1)- 
 
SUMMARY AND OUTLOOK 353 
 
 Thus, if we designate by 100 per cent the 1913 price of 
 each of the 36 commodities used in this book, the prices 
 of bacon, barley, beef, etc., in 1917 are respectively 193 
 per cent, 211 per cent, 129 per cent, etc., the simple arith- 
 metic average of which figures — i.e. the simple arith- 
 metic index number for 1917 — is 176 per cent ; while if, 
 reversely, we call every price in 1917 100 per cent, the 
 prices of bacon, barley, beef, etc., in 1913 are respectively 
 52 per cent, 47 per cent, 77 per cent, etc., the simple arith- 
 metic average of which figures is 63 per cent. But these 
 two arithmetic index numbers, 176 per cent and 63 per 
 cent, are mutually inconsistent since the ratio of 176 to 
 100 is not the same as the ratio of 100 to 63, i.e. 1.76 and 
 .63 are not reciprocals. In other words, 1.76 X .63 is not 
 1.00, but is 1.11, so that the 1.76 and the .63 are too big 
 by a '' joint error " of 11 per cent, or about 5.5 per cent 
 apiece. The 11 per cent is their " joint error " and the 
 5.5 per cent imputed to each index number is its "up- 
 ward bias," a tendency to exaggerate inherent in the 
 arithmetic process of averaging. 
 
 Similarly, the harmonic process of calculating index 
 numbers has a downward bias {i.e. Poi X Fio< 1). 
 
 It is of interest to observe that the 11 per cent or other 
 figure calculated for the " joint error " of any two forward 
 and backward index numbers, or of any two price and 
 quantity index numbers, is always an absolutely true 
 figure. We can always know to a certainty how greatly 
 the product of the two index numbers errs. But — and 
 this is of still greater interest — the ascription of half of 
 the joint error to each of the two is merely a guess, based 
 on considerations of probability. We can never say with 
 certainty how far wrong any one index number may be. 
 The " absolutely correct " figure always eludes us. We 
 have no absolute criterion of correctness but only of in- 
 
354 THE MAKING OF INDEX NUMBERS 
 
 correctness. Nevertheless — and this is of the greatest 
 interest — we can, on grounds of probability, narrow 
 down the fringe of doubt until it is practically negligible. 
 
 Besides the above mentioned cases of bias lurking in 
 two types of index numbers — the arithmetic and the 
 harmonic — there is another sort of bias pertaining to 
 certain systems of weighting. It might seem, at first sight, 
 that any of the six systems of weighting would be as 
 likely to afford errors in one direction as the other — that, 
 for instance, the use of base year values as weights would 
 be no more likely to yield a small index number than 
 would the use of given year values, nor the use of given 
 year values to yield a large index number any more than 
 that of base year values. But such equal liability to err 
 in either direction is not found. Of the six systems of 
 weighting (applicable to all the types of index numbers, 
 except the aggregative), only the simple weighting and 
 the cross weighting are not definitely biased in some one 
 direction. 
 
 As to the other four weightings (7, 77, 777, IV), it 
 was shown in Chapter V that the formulae with weight- 
 ings 7 and 777 have necessarily a positive joint error, as 
 likewise do the formulae with weightings 77 and IV. It 
 was also shown that weightings 7 and 77 give almost iden- 
 tical results, as also do 777 and IV. Practically, there- 
 fore, the four systems yield only two results : 7, 77 and 
 777, 7F with a positive joint error between these two. If 
 we apportion this joint error equally, we may say that 7 
 and 77 have a definite downward bias, and 777 and IV 
 a definite upward bias. It was shown that the reason 
 these weight biases exist is because 7 and 77 give too 
 much weight to the smaller price relatives while 777 and 
 IV give too much weight to the larger price relatives. 
 
 Bias must be eliminated in order to obtain a good index 
 
SUMMARY AND OUTLOOK 355 
 
 number. To be completely free of bias a formula of un- 
 biased type, such as the geometric, needs also to have un- 
 biased weighting, such as cross weighting. A biased type, 
 however, can be remedied by the use of an oppositely 
 biased weighting, or vice versa. Thus, in the case of an 
 arithmetic formula weighted by base year values, the 
 upward type bias is offset by the downward weight bias. 
 Reversely, in the case of a harmonic weighted by given 
 year values, the downward type bias is offset by the up- 
 ward weight bias. 
 
 Some formulae, however, have both type bias and 
 weight bias. Thus the arithmetic formula, if weighted 
 by given year values, has a double dose of upward bias 
 {i.e. both the upward type bias inherent in the arithmetic 
 process of averaging and the upward weight bias inherent 
 in the given year weighting). Reversely, the harmonic 
 formula, if weighted by base year values, has a double 
 dose of downward bias (i.e. both the downward type bias 
 inherent in the harmonic process of averaging, and the 
 downward weight bias inherent in base year weighting). 
 
 Other formulae, of course, have just a single dose of bias 
 due either to the type or the weighting. Thus the geo- 
 metric formula weighted by given year values has simply 
 the upward weight bias from the given year weighting 
 without any type bias, while, reversely, the geometric 
 weighted by base year values has only the downward 
 weight bias from the base year weighting without any 
 type bias. Again the cross weight arithmetic has simply 
 upward type bias pertaining to the arithmetic process 
 without any weight bias, while, reversely, the cross weight 
 harmonic has only the downward type bias of the harmonic 
 without any weight bias. 
 
 The bias of any index number (whether type bias or 
 weight bias, or both) increases with the dispersion of the 
 
356 THE MAKING OF INDEX NUMBERS 
 
 price relatives and in a rapidly increasing ratio. Con- 
 sequently a biased formula, while it has only a slight error 
 when there is little dispersion, has an enormous error when 
 (as happens with the lapse of time) there is a great dis- 
 persion. 
 
 As to the aggregative, the two types of weighting, I 
 and IV, are not biased. The aggregative / (Formula' 
 53^ in our series of numbers) is known as Laspeyres' 
 formula and the aggregative IV (Formula 54) is known as 
 Paasche's formula. These two formulae are identical re- 
 spectively with arithmetic I (Formula 3) and harmonic 
 IV (Formula 19), as well as with certain others. 
 
 Although only two (arithmetic and harmonic) of the 
 six types of formulae and only four (7, II, III, IV) of 
 the six kinds of weighting are " biased," i.e. liable to err 
 in a given direction, they are all subject to some error and 
 so may be called more or less " erratic." When a formula 
 is especially erratic it is called " freakish." The mode 
 and, less markedly, the median are freakish types and 
 simple weighting is freakish weighting. The weighted 
 aggregatives are only slightly erratic ; the joint error of 
 the forward and backward aggregative index numbers is 
 very small. 
 
 § 4. Derivation of Antithetical Formulae 
 
 By means of the two reversibility tests we find that 
 each formula has its special '' time antithesis " and its 
 special " factor antithesis." As shown in Chapter IV, 
 the time antithesis is derived by reversing the times, i.e. 
 taking the index number backward, and, then inverting 
 the result (dividing it into unity), while the factor antithe- 
 sis is derived by reversing the factors, i.e. taking the index 
 
 ^ For the mnemonic system of numbering the various formulae see 
 Appendix V, § 2. 
 
SUMMARY AND OUTLOOK 357 
 
 number for quantities instead of for prices and then di- 
 viding the result into the value ratio. That is, the time 
 
 antithesis of any index number formula, Pqi is ^- while 
 
 the factor antithesis of Poi is ■—. 
 
 By these processes, applied to the various types and sys- 
 tems of weighting already described, we are provided 
 with 46 primary formulae. As shown in Chapter VII, 
 these are arrangeable in sets of four each, or " quartets " 
 (some of which may be reduced to " duets ")• In each 
 quartet, each horizontal pair of formulae are antitheses 
 by Test 1, and each vertical pair are antitheses by Test 
 2, thus forming two pairs of time antitheses and two 
 pairs of factor antitheses. 
 
 These 46 primary formulae comprise : the simples, the 
 weighted I, II, III, IV, just cited, and the factor antith- 
 eses of all these. Of these 46 not a single one conforms 
 to the factor reversal test, and only four (the simple 
 geometric, median, mode, and aggregative) to the time 
 reversal test. 
 
 § 5. Rectification 
 
 By crossing (i.e. taking the geometric mean of) any 
 pair of time antitheses, we derive a formula which conforms 
 to the time reversal test; and by crossing any pair of 
 factor antitheses, we derive a formula which conforms to 
 the factor reversal test ; while by doing both, we derive 
 a formula which conforms to both tests. 
 
 Instead of crossing formulae, we may, as already stated, 
 cross their weights. By this alternative process we may 
 also derive formulae that conform to Test 1. The two 
 alternative processes do, however, present certain contrasts. 
 For instance, in order to secure conformity to Test 1, 
 formula crossing must be accompHshed through the geo- 
 
358 THE MAKING OF INDEX NUMBERS 
 
 metric mean (except that, in two cases, those of the geomet- 
 ric and aggregative index numbers, the aggregative mean 
 is also an available method). Weight crossing, on the 
 other hand, may be accomplished through the arithmetic, 
 harmonic, or geometric means and, of these three methods, 
 the arithmetic probably yields the most accurate result. 
 
 Any given cross weight formula and the corresponding 
 cross formula always agree very nearly. By means of 
 crossing formulae we mcrease our list of 46 " primary " 
 formulae to the " main series " of 96 and, by crossing the 
 weights, we enlarge the series to 134. 
 
 By rectification any bad formula can be reformed. 
 Bias can be eUminated, while freakishness can be reduced 
 but not entirely ehminated. 
 
 § 6. Base Shifting 
 
 The so-called circular test requires that, in a given series 
 of years no matter which year is taken for base, the re- 
 sulting index numbers shall stand in the same ratios each 
 to each ; consequently, if we calculate index numbers 
 from year to year, or from place to place around a speci- 
 fied circuit of years or places, we shall end at the same 
 figure from which we started. But this circular test is, 
 strictly, not a fair test ; for shifting the base ought to change 
 these relations. A direct comparison between two par- 
 ticular years is the only true comparison for those two 
 years. Comparisons between those two particular years, 
 via other years, ought not necessarily to give the same re- 
 sult ; on the contrary, there ought, in general, to be a 
 discrepancy or gap. Nevertheless, in the cases of our 
 most exact formulae, this gap is, in actual fact, found to 
 be negligible, being only a small fraction of one per cent. 
 That is, the circular test, although theoretically wrong, 
 is practically fulfilled by the best formulae. 
 
SUMMARY AND OUTLOOK 359 
 
 Consequently, it is not necessary, in practice, to calcu- 
 late an index number between every possible pair of years. 
 A single series will be sufficiently accurate for all these 
 inter-year comparisons. For this piu"pose, we may use the 
 chain system, the fixed base system, the base being one 
 year only, or a broadened fixed base system, i.e. one in 
 which the base is an average over several years. Of 
 these three, the chain system is strictly correct only for 
 consecutive years ; for longer comparisons {i.e. when 
 reckoned back relatively to the original base), it is subject 
 to cumulative error. Of the remaining two, the broad- 
 ened fixed base system seems, on the whole, better than 
 the fixed base system with its single year as base, although 
 we may often be forced to use the latter for lack of data 
 necessary for calculating any broader base. Moreover, 
 in the case of the aggregative, the preference is inappre- 
 ciable. 
 
 Index numbers are more frequently used to compare 
 each year with the base than to compare successive years. 
 The fixed base system, when used for comparing two years 
 neither of which is the base, is always subject to some 
 error. But this error is usually sHght and is not cumula- 
 tive. Only for long or for very dispersive periods, if at 
 all, is any other index number needed in addition to those 
 of the fixed base system. 
 
 §7. Formulae Compared 
 
 To find the best formula we first eliminate as " freakish " 
 the simples and their derivatives, and the modes and me- 
 dians and their derivatives. All the remaining formulse 
 fall into five groups, which may be plotted as a five-tined 
 fork, the middle tine portraying the formulae without 
 bias, the two tines nearest portraying the formulae having 
 a single dose of bias, and the two outer tines portraying 
 
360 THE MAKING OF INDEX NUMBERS 
 
 the formulae having a double dose of bias. Eliminating 
 all biased formulse, we have remaining only those on the 
 middle tine, 47 in number, all of which agree closely with 
 each other. These consist of rectified formulse and of the 
 Formulse 53 and 54, Laspeyres' and Paasche's. Of these 
 47, the 13 which fulfill both tests agree with one another 
 still better. Of these 13, the " ideal " Formula 353, 
 
 J ^^^^° X —^^, is at least equal in accuracy and is 
 Spogo Spogi 
 probably slightly superior in accuracy to any of the others. 
 
 This Formula 353 is demonstrably correct within less 
 than one eighth of one per cent and probably within a hun- 
 dredth of one per cent, as a measure of the average change 
 of the given data (prices or quantities, etc.) between the 
 two years for which it is calculated. In other words, in 
 the case of Formula 353, we have no perceptible " instru- 
 mental error " to deal with. So far as the mere question 
 of formula is concerned, the index number method is cer- 
 tainly henceforth to be recognized as possessing as high a 
 degree of precision as the majority of physical measures 
 in practical use. 
 
 But there is no thought of maintaining that 353 is the 
 " one and only " formula. On the contrary, a chief con- 
 clusion is that all index numbers which are not freakish or 
 biased practically agree with each other. Even the freakish 
 medians, and probably also the more freakish modes, 
 agree with the good ones fairly well when very large num- 
 bers of commodities are used. In all others, viz., the 
 arithmetic-harmonic, the geometric, and the aggregative, 
 agreement is found to a startling degree. In other words, 
 the idea that index numbers of different types or systems 
 of weighting disagree is, in general, true only before they 
 are " rectified." Those, like Pierson, whose studies have 
 led them to distrust and abandon index numbers as worth- 
 
SUMMARY AND OUTLOOK 361 
 
 less have simply not pushed their studies far enough. 
 Nevertheless, a small grain of truth remains in Pierson's 
 contention. There is no index number which can be spoken 
 of as absolutely " correct." There must, theoretically, 
 always remain a fringe of doubt. All that we can say 
 with certainty is that this fringe of doubt instead of be- 
 ing very large, as Pierson thought, is, for the " ideal " 
 formula, very small — ordinarily less than a tenth or 
 even a hundredth of one per cent. 
 
 § 8. The Eight Most Practical Formulae 
 We have seen that Formula 353 is the best when the 
 
 utmost accuracy is desired. Formula 2153, wo i- gi; Pi 
 
 2(go + gi) Po 
 
 however, which will seldom appreciably differ in its re- 
 sults from 353, is more quickly calculated. 
 
 In case the full data are not available for calculating 
 Formula 353 or 2153, but data are available for calculat- 
 ing 6053, 53, or 54, any of these three will serve excellently 
 as a substitute for 353. If data even for Formula 53 
 are imavailable, round-weights may be guessed at, i.e. 
 9051 may be used as a makeshift for Formula 53. 
 
 If no data at all are available for judging the relative 
 weights so that recourse must be had to simple formulae, 
 the simple median (Formula 31) and the simple geometric 
 (Formula 21) are the best, with possibly a slight prefer- 
 ence for the former in most cases. The simple arithmetic 
 (Formula 1) should not be used under any circumstances, 
 being always biased and usually freakish as well. Nor 
 should the simple aggregative (Formula 51) ever be used ; 
 in fact this is even less reliable. 
 
 The relative accuracy of these eight formulae may 
 roughly be given as follows : 353 is usually correct within 
 one hundredth of one per cent ; 2153 is usually correct 
 
362 THE MAKING OF INDEX NUMBERS 
 
 within one fourth of one per cent ; 6053, 53, and 54 are usu- 
 ally correct within one per cent ; 9051 is usually correct 
 within three per cent; 21 and 31 are usually correct 
 within six per cent. 
 
 These eight important formulse are the only ones 
 which ever need to be used, although not by any means the 
 only ones which may be used. Their computation and 
 that of 8053 are exemplified in figures in Appendix VI, § 2. 
 
 § 9. Suggested Application to the United States 
 
 These eight formulse are to be used according to the 
 adequacy of the data. For the general index number of 
 the United States Bureau of Labor Statistics, full data for 
 quantities, being dependent on the census reports, are 
 available only once in ten years. Consequently, Formula 
 353 can be used only once in ten years. In the intervening 
 period, Formula 53 should be used as, in fact, it is. At 
 the close of each decade the figure reached by Formula 53 
 can be checked up by means of 353 applied to the new 
 data then available. The discrepancy may then be pro- 
 rated over the preceding ten years and these corrected 
 figures be substituted in all future publications for the 
 figures originally obtained by Formula 53, just as is done 
 with population figures. 
 
 The figures for Formula 53 should be calculated by the 
 fixed base method, as at present, and not the chain sys- 
 tem, so that the discrepancy at the end of the decade may 
 be a minimum. The Formula 353 figures, on the other 
 hand, being calculated between successive censuses, 
 would form a chain system, each link being a decade, 
 although, to satisfy scientific curiosity, it would be well, as 
 each new census appears, to calculate from each new census 
 directly to all the preceding censuses. The discrepancies 
 which would be found would inevitably be negUgible. 
 
SUMMARY AND OUTLOOK 363 
 
 § 10. Critique of Formulae Proposed by Others 
 
 It has been necessary to compare many varieties of 
 formulas only to find, in the end, little practical use for 
 most of them. Until complete comparisons were made we 
 could not be sure which agreed or disagreed, which were 
 correct, or which lent themselves to rapid calculation. 
 
 Of the 25 formulae mentioned by previous writers as 
 possibly valuable, we have seen that the following ought 
 never to be used because of bias: 1, 2, 9, 11, 23. The 
 following ought never to be used because of freakishness : 
 41, 51, 52. All the rest may be used under various cir- 
 cumstances (as to availability of data) as may also 
 about 35 other formulae presented in this book for the 
 first time. All these usable formulae will agree under 
 like circumstances with the seven formulae actually rec- 
 ommended as the most practical. 
 
 The only formulae much in use of the 25 formulae men- 
 tioned by previous writers are : 1, 21, 31, 51, 53, 6023, 6053, 
 9021. Of these eight, Formula 21 or Formula 9021, now 
 used by the British Board of Trade, 53 or 6053, used by 
 the United States Bureau of Labor Statistics and the Aus- 
 tralian Bureau of Census and Statistics, and Formula 
 23 or 6023, used by Professors Day and Persons in the 
 Review of Economic Statistics, published by the Harvard 
 Committee on Economic Research, are all good, although 
 the last named will deteriorate as, with the lapse of time, 
 the base period is left very far away. Of the other five, 
 the most thoroughly objectionable are 1 and 51, although 
 1 is the formula most often used. There are two 
 objections to Formula 1, the simple arithmetic, viz. ; (1) 
 that it is " simple," and (2) that it is arithmetic ! — that 
 it is at once freakish and biased. In the case of Sauer- 
 beck's index number, for instance, the bias alone reaches 
 36 per cent ! 
 
, 364 THE MAKING OF INDEX NUMBERS 
 
 The conclusions of the present book depart from pre- 
 vious thought and practice in fundamental method. 
 Hitherto writers have been debating the " best type " 
 (whether arithmetic, geometric, or median) by itself, the 
 *' best weighting " by itself, and the bearing on these of 
 the distribution of price relatives. But from our study 
 it should be clear that it makes little difference what type 
 we start with, or what the weighting is (so long as it is 
 systematic), or what the distribution of price relatives 
 may be so long as we " rectify " the formulae and so elim- 
 inate all these sources of distortion or onesidedness. 
 
 Moreover, even if we do not thus rectify the primary 
 formulae but merely choose from among them, our study 
 helps us do the choosing, so as to avoid bias and minimize 
 error. Thus, as to the long controversy over the relative 
 merits of the arithmetic and the geometric types, our study 
 shows us that the simple geometric, 21, is better than the 
 simple arithmetic, 1, but that, curiously enough, the 
 weighted arithmetic, 3, is better than the weighted geo- 
 metric, 23. 
 
 § 11. Speed of Computation 
 
 The chief practical restriction on the use of the many 
 fairly good formulae is imposed by the time required to 
 calculate them. No formula, for instance, surpasses 
 appreciably in accuracy Formula 5323 and, were it as 
 easily calculated as its equivalent, 353, I would seriously 
 suggest 5323 for practical use. But, on a test problem, 
 it requires 44.2 hours to calculate 5323 while Formula 
 353, which yields precisely as good a result, requires only 
 14.3 hours, and 2153, which yields almost as good a re- 
 sult, requires only 9.6 hours. 
 
 Besides accuracy and speed we need, in practice, to 
 consider two other qualities, viz,, conformity to the 
 
SUMMARY AND OUTLOOK 365 
 
 so-called circular test, and simplicity, or intelligibility to 
 the uninitiated. The best practical all-around formula, 
 taking all four points into account, — accuracy, speed, 
 minimum legitimate circular discrepancy, simplicity — 
 is the Edgeworth-Marshall formula, 2153. 
 
 Formula 353 is " best " only in the sense of accuracy, 
 as the telescopes in the great observatories are best. 
 But smaller, cheaper telescopes, spy glasses, and opera 
 glasses still have their uses. No one would want a Lick 
 telescope on the porch of his summer residence or at the 
 theater. 
 
 § 32. Two Consequences of the Agreement 
 of Index Ntimbers 
 
 Among the consequences of the surprising agreement 
 between the various legitimate methods of calculating 
 index numbers are two which need emphasis here. The 
 first is that all discussion of " different formulae appropriate 
 for different purposes " falls to the ground. The second 
 is that, the supposed differences among formulae once 
 banished, the real problem of accuracy is shifted to the 
 other features of an index number, — the assortment of 
 the commodities included, their number, and data. 
 
 Errors due to mere insufficiency of number are relatively 
 small, while those due to inaccuracies of data are usually 
 negligible, even though these inaccuracies individually 
 be great. Thus the figures for weights in particular may 
 usually be tenfold or one tenth of the true figures with- 
 out appreciably disturbing the accuracy of the resulting 
 index number. Henceforth, the effort to improve the 
 accuracy of index numbers must center chiefly on the 
 assortment of the items to be included. This will differ 
 for the different purposes to which the proposed index 
 number is to be put. 
 
366 THE MAKING OF INDEX NUMBERS 
 
 § 13. Current Ideas 
 
 How do the conclusions reached in this book differ 
 from previous views on index numbers ? Largely, of course , 
 these views are confirmed and supported by new data. 
 The main results of C. M. Walsh's thoroughgoing studies 
 are supported. His three favorite formulae, advocated in 
 his first and larger book, the Measurement of General Ex- 
 change Value, are 1123, 1153, and 1154, all of which are 
 '' superlative " in our hierarchy of index numbers, i.e. 
 practically peers of 353. He also advocated (as Marshall 
 and Edgeworth did before him and as I do) Formula 2153. 
 In his second book. The Problem of Estimation, as already 
 indicated, he reached independently the conclusion that 
 Formula 353 is probably the king of all index number 
 formulae. In like manner, the conclusions of this book 
 support and are supported by most of the work of Jevons, 
 Marshall, Edgeworth, Pigou, Flux, Knibbs, Mitchell, 
 Meeker, Young, Persons, and Macaulay. 
 
 Yet many of the conclusions are new and of these sev- 
 eral run athwart current ideas. The concept of bias, as 
 it applies to the arithmetic and harmonic types, has been 
 implicitly recognized (though not specially named) by 
 Walsh, and, to some extent, by others; but, as applied 
 to systems of weighting, it is new. 
 
 One of the points which, though by implication recog- 
 nized by Walsh, will appear as new to almost everyone 
 else, is that the kind of weighting befitting any index num- 
 ber is different for different types. 
 
 Test 1 has been more or less definitely recognized, but 
 Test 2 is new and no index number hitherto in actual use 
 conforms to Test 2. 
 
 Rectification is a new idea, except as to one special 
 case (namely rectification relatively to Test 1 accompUshed 
 
SUMMARY AND OUTLOOK 367 
 
 by means of weight crossing). Consequently, many of 
 the formulsB derived in the processes of rectification are 
 new and several of these new formulae are, so far as ac- 
 curacy is concerned, practically as good as any formula 
 previously suggested. 
 
 The conclusion that the circular test is theoretically 
 wrong is entirely new ; that it is nevertheless practically 
 right, as applied to all good index numbers, is almost 
 new ; that all index numbers conforming to rational stand- 
 ards of excellence agree to a nicety is new ; that the par- 
 ticular type of formulae and the particular weighting of 
 formulae prior to rectification and the particular sort of 
 dispersion or distribution of the relatives to be averaged, 
 are unimportant, and that only the criteria of goodness 
 are vitally important, is new ; finally, that in selecting an 
 index number formula the purpose to which it is put is 
 immaterial is practically new. 
 
 In view of these divergences from current thought, it 
 is not siu-prising that the conclusions reached often col- 
 lide with current practice. 
 
 § 14. The Future Uses of Index Numbers 
 
 If the conclusions reached are correct, some of the meth- 
 ods of calculating index numbers now most in vogue 
 should be discontinued. It is high time that index num- 
 bers should be so calculated as to enable us to get out of 
 them all there is in them. Their use is rapidly growing 
 and often with little heed paid to the methods of making 
 them. When they are made rightly, as a matter of or- 
 dinary routine, their usefulness will be greatly increased 
 and may be extended to many fields scarcely touched 
 upon as yet. 
 
 Thirty years ago only wholesale price indexes were 
 used and even these were not as numerous, as widely 
 
368 THE MAKING OF INDEX NUMBERS 
 
 known, or as widely used as today, when so many official 
 agencies and so many trade journals publish them. In- 
 dex numbers of retail prices, of wages, and of the prices 
 or sales of stocks were rarities, if not curiosities. Today 
 these are common. In Great Britain alone, three million 
 laborers have their wages regulated annually by an index 
 nmnber of retail prices. We have numerous index num- 
 bers of the stock market, even in daily papers. We now 
 have also index numbers of the cost of living, of the mini- 
 mum of subsistence, and of wages in terms of that mini- 
 mum. Good index numbers of the quantities of goods 
 produced, consumed, or exchanged are also comparatively 
 new. Beginning with the crude efforts of Rawson-Rawson 
 a generation ago, Kemmerer in 1907, and myself in 1911, 
 such index numbers have in the past few years come to 
 have considerable statistical value, and are even becoming 
 differentiated into indexes of production, manufacture, 
 crops, national income, imports, exports, barometers of 
 trade, etc. Another recent application of index num- 
 bers, now current in at least five countries, is that of 
 measuring the trend of the foreign exchanges. 
 
 One of the most interesting recent developments is the 
 application of index numbers to special industries, such as 
 lumber or building {e.g. the Aberthaw Index of the cost of 
 a cement building) ; or even to special individual busi- 
 nesses, such as the American Writing Paper Company 
 {e.g. for paper production costs) ; or even to special de- 
 partments in an individual business {e.g. the price of 
 textbooks of Henry Holt and Company). When the 
 business statistician begins to realize the usefulness of 
 this device in his own business, index numbers will be 
 found sprouting, right and left, to serve the purposes of 
 trade journals, of railways, insurance companies, banks, 
 commercial houses, and large corporations. Their use- 
 
SUMMARY AND OUTLOOK 369 
 
 fulness will be greatly enhanced when the wrong formulae 
 (especially Formula 1) now generally used are replaced 
 by right ones.^ 
 
 But the original purpose of index numbers — to meas- 
 ure the purchasing power of money — will remain a 
 principal, if not the principal, use of index numbers. It 
 is through index numbers that we measure, and thereby 
 realize, changes in the value of money. Whether or not 
 we ever stabilize that value, it is of the greatest impor- 
 tance that we know just how stable or unstable our present 
 money is. This is the chief reason why today we are so 
 much more interested in index numbers than before the 
 war. Index numbers tell us the value of the mark, lire, 
 and franc, at home in terms of goods, as foreign exchange 
 tells us their value abroad in term^s of gold. And if, or 
 when, we do regulate and stabilize the moneys of the world, 
 not simply relatively to each other but relatively to goods, 
 it is the index number which will be requisitioned to 
 measure and guide such regulation. 
 
 Addendum to 1 9 
 
 Since this chapter was put ia paged type, the United States 
 Bureau of Labor Statistics has changed its system of weighting 
 by substituting the newly available data of 1919 for those of 
 1909 hitherto used. Their results enable us to calculate the 
 index number by Formula 353 between the two years, 1909 and 
 1919. This turns out to be 1.4 per cent lower than the 
 Bureau's old figure based on 1909 data and that much higher 
 than its new figure based on 1919 data. The adjustments 
 needed for the intervening nine years barely exceed 1 per cent 
 in any case. 
 
 ^ For a list of current index numbers, see Appendix I (Note to Chapter 
 XVII, § 14). 
 
APPENDIX I 
 
 NOTES TO TEXT 
 
 Note A to Chapter II, § 3. The Word "Aggregative." The word "ag- 
 gregative" is here proposed for general use (after consultation with several 
 experts) in place of "price-aggregate" or any other long phrase. I first 
 favored "aggregatic," a coined word, but Professor Wesley C. Mitchell 
 called my attention to the existence, in the dictionary, of "aggregative." 
 Besides brevity it has several advantages over the "price-aggregate" or 
 "aggregate-expenditure method," or other roundabout inadequate phrases 
 which have been used, including its appUcabUity to quantities, wages, etc., 
 as well as to prices. 
 
 Note B to Chapter II, § 3. The Base Number Need Not be 100. Any 
 other number than 100 may, of course, be arbitrarily taken. As such a 
 common base number, G. H. Knibbs of Australia has used 1000. This 
 would change our index number for 1914 from the above 96.32 to 963.2 and 
 increase tenfold every other index number in the series. The London 
 Economist takes 2200 as the base number, there being originally 22 com- 
 modities in the index number. Analogously, we could here take 3600 as 
 the base number, in which case the index number for 1914, instead of the 
 above 96.32 would be 36 times as much, or the 3467.52 at the foot of the 
 column in the table, saving us the trouble of dividing. Some index num- 
 bers take, as the base number, the number of dollars spent on a given bud- 
 get of commodities in the base year or period. But, in general, the 100 
 per cent figure is found most convenient. 
 
 In Table 2 in Chapter II, § 6, while the base number for each individual 
 link is originally taken as 100 per cent, in the final series the base numbers 
 are 100, 96.32, 97.94, 125.33, etc., the first being used only as base num- 
 ber for the second, the second (96.32) being likewise used only as base 
 number for the third, etc. 
 
 Note to Chapter II, § 11. Proof that for the Simple Geometric, Fixed Base 
 and Chain Methods Agree. To prove algebraically the identity between 
 chain and base averages under the simple geometric formula, 
 
 the 1913-1914 link is V— X ^ X . 
 
 Po Pa 
 
 and the 1914-1915 link is x'^ X ^ X . . . 
 
 Pi Pi 
 
 The chain index number for 1915 relatively to 1913 via 1914 is the product 
 
 371 
 
372 
 
 THE MAKING OF INDEX NUMBERS 
 
 of these two links ; and, in that product, evidently the pi's cancel out as 
 do the p'l's, etc., giving, as the result, 
 
 -^El X — X . . . which is identical with the fixed base formula for 1915 
 
 Po p'o 
 
 relatively to 1913. 
 
 Note to Chapter II, § 13. Method of Finding the Simple Mode. There are 
 
 Finding the Simple Mode 
 
 rnodQ 
 
 NUHBER OF COMMODITIES 
 
 Chart 62. Illustrating the graphical distribution of the price relatives 
 and the method of selecting the mode. (This chart is the only one in the 
 book not a ratio chart ; but, of course, the location of the mode is unaffected 
 thereby.) The top bar represents one commodity (coke), the price relative 
 of which lies in the range 350 to 355 per cent ; the bottom bar represents 
 two commodities (coffee and rubber) in the range 80 to 85 per cent ; while 
 the mode occurs where there are four commodities — the largest niunber 
 — within the range 135 to 140 per cent. 
 
APPENDIX I 373 
 
 many methods of computing the mode, several graphic and several alge- 
 braic. The method here used is the simplest and roughest and is illus- 
 trated in Chart 62, for prices for 1917.^ The largest price relative (351.8) 
 lay between 350 and 355 and is represented by the topmost bar. The 
 smallest (80.3) lay between 80 and 85 and, as there was another (83.5) in 
 that range, these two are represented by the lowermost bar which is, there- 
 fore, twice as long as the uppermost, or first mentioned, bar. Between 
 these extremes are ranged the other price relatives represented by the other 
 bars — usually representing one price relative each but in five cases, 
 including the case of the lowermost, representing two price relatives, 
 and in two cases, representing four. The total of the bars represents 36, 
 the total number of price relatives, or the number of commodities. 
 
 The commonest or most frequent is, therefore, the height of one of the 
 two fourfold bars. The one chosen and marked "mode" has a height of 
 135-140. The chart illustrates the difficulty which often arises of choos- 
 ing between two equal frequencies. Here the lower of the two fourfold 
 bars was chosen because, by taking a range larger than 5 points, the fre- 
 quency within that range is greater for the neighborhood of the lower bar 
 than for that of the upper. 
 
 Note to Chapter II, § 14. Proof that Fixed Base and Chain Methods 
 Agree in Simple Aggregative. The formula for the aggregative index 
 number for 1915 (year "2") relatively to 1913 (year "0") is 
 
 Spo' 
 On a chain basis, the formulae to be multiplied eventually are 
 
 the 1913-1914 link ^^ 
 
 Spo 
 
 and the 1914-1915 Hnk -^^. 
 Spi 
 
 The chain index number for 1915 relatively to 1913, via 1914, is the prod- 
 uct of these two links, i.e. (after canceling) it is — ^ which is identi- 
 
 Spo 
 cal with the above formula for 1915 on 1913 directly as base. 
 
 Note A to Chapter II, § 15. The General Definition of "Average." An 
 average, x, of any series of terms, a, b, c, etc., is any function of these terms 
 such that, if they all happen to be equal to each other, x will be equal to 
 each of them also. 
 
 Thus, taking the simple arithmetic average 
 
 a+b + c-\- . . . 
 
 X = ■ ■ ! 
 
 n 
 
 where n is the number of terms, let us show that this is a true, average ac- 
 cording to the definition. If each of these terms happens to equal every 
 other, having a common value, k, i.e. ii a = b = c = . . . = k, then, evi- 
 dently, 
 
 ' This chart is not a ratio chart, but the results are not affected thereby. 
 
374 THE MAKING OF INDEX NUMBERS 
 
 k -\- k -{- k + . . . nk , 
 
 X = — ■ = — = «, 
 
 n n 
 
 which was to have been proved. 
 
 The simple harmonic is Ukewise a true average ; for, in this case, 
 
 i + i + i + ... 1 + 1 + 1 + .. . „(i) 1 
 
 a b c k k k \kJ k 
 
 which was to have been proved. 
 Likewise, for the simple geometric, 
 
 X = -y/a be. . . = Vfc k k . . . = VA:" = k. 
 
 Likewise, as to the simple median. For the middle term of a, b, c, etc., 
 when they become k, k, k, etc., is k; and, as to the simple mode, the com- 
 monest term among k, k, k, etc., is A;. 
 
 As to the simple aggregative we must start with fractions with specific 
 
 numerators and denominators. Let a be — . 6 be —> c be — > etc. Then 
 
 ABC 
 
 the simple aggregative average of a, b, c, etc., is 
 
 a+ /3 + 7 + . .. 
 
 A+B + C+ ... 
 
 Ifa = b = c = ...=k, then -7 = ^ —7; - ^ ^^^ a = kA; ^ = kB; 
 
 ABC 
 
 7 = kC, etc. Hence, substituting these for a, /3, 7, etc., in the above 
 expression for x, we have 
 
 kA +kB + kC + . . . 
 X = 
 
 A+B + C + . . . 
 k(A+B + C + . . .) 
 
 A+B + C + . 
 
 = k, 
 
 which was to have been proved. 
 
 We have found, then, that all of the six simple averages used in this 
 book are true averages according to the definition. 
 
 Weighting does not affect the matter; because weighting is, by defi- 
 nition, merely counting a term as though it were two terms, or three terms, 
 or any other number of terms. 
 
 The only index numbers used in this book which are not true averages 
 are some, not all, of the even numbered formulae (and derivatives), which 
 are the quotients of a value ratio divided by an average. Our definition, 
 however, may be modified to suit such cases by specifying as the test of 
 an average P of the price ratios, not only that it shall equal the price ratios 
 if they equal each other, but also that at the same time the quantity ratios 
 shall equal each other. 
 
APPENDIX I 375 
 
 Thatis,if p = '^Pl2l^Q 
 
 Spo3o 
 
 9i g 1 
 
 where Q is an average, by the ordinary definition, of — , ^, etc., we are to 
 
 So go 
 prove that P = k when 
 
 Pi _ P'l _ P"i - - /g 
 
 Po p'o p"o 
 and when, at the same time, 
 
 2!'= ^ = . . . = k'. 
 go g'o 
 
 The last equations show that Q, being an average of -?3, etc., must be equal 
 
 go 
 
 to k'. Hence P = ^^i^} ^ &'. 
 
 Spogo 
 
 Since ^ = fc, 
 
 Po 
 
 it follows that Pi = kpo, etc. 
 
 Substituting in the last expression we have 
 
 p _ 2(fcpo)(fc'go) _^ j^, 
 
 Spogo 
 
 _ fcfc'(Spogo) ^ ^, 
 
 spogo 
 
 = k, 
 
 which was to have been proved. 
 
 It should be noted, incidentally, that the definition of an average, as 
 originally stated, is a little broader than that usually employed, which re- 
 quires that an average, to deserve the name, must lie between the highest 
 and the lowest of the terms averaged. This would rule out the geometric 
 average when one of the terms was zero or negative. But as index num- 
 bers are always averages of positive terms this limitation of the geometric 
 does not embarrass us. Even other forms which, under extreme conditions, 
 kick over the traces seldom do so in practice. 
 
 Note B to Chapter II, § 15. Proof that Geometric Lies between Arithmetic 
 Above and Harmonic Below. The rigorous proof of this well known propo- 
 sition (that the geometric average necessarily lies below the arithmetic 
 and above the harmonic) is to be found in standard treatises on algebra.^ 
 But the simple principle involved may be noted here. 
 
 Let us compare first only the arithmetic and the geometric averages of 
 (say) 50 and 200 (the arithmetic being 125 and the geometric, 100). The 
 geometric average is based wholly on the idea of ratios. Relatively to 
 
 ' See, for instance, Chrystal'a Text-Book of Algebra, Part II, p. 46. 
 
376 THE MAKING OF INDEX NUMBERS 
 
 the 100 the 200 is twice as great and the 50 is "twice as small," so that 
 "geometrically," i.e. as to ratios, the two balance each other, one being as 
 much superior in ratio to 100 as the other is inferior in ratio to 100. But 
 these equal ratios on either side of 100 make unequal differences on either 
 side of 100; for the differences are 50 below and 100 above. Hence 100, 
 while midway geometrically between 50 and 200, is lower than midway 
 arithmetically. Hence the arithmetic average lies above the geometric. 
 Similarly, the geometric average of 10 and 1000 is 100, the 1000 being ten 
 times as great as this average and the 10 being "ten times as small" as 
 this average. But this 100, while half-way up from 10 toward 1000 in 
 two equal ratio steps, is not nearly half-way up in two equal difference steps. 
 Similarly, the geometric average of 1 and 10,000 is 100 because two steps 
 of one hundredfold each carries us from 1 to 10,000, the 100 being the half- 
 way step, but arithmetically 100 is far nearer to the 1 than to the 10,000. 
 
 In short, the geometric method gives more influence to the small magni- 
 tudes than does the arithmetic and so results in a smaller average. 
 
 If we take the geometric average of any terms and then take the geo- 
 metric average of their reciprocals, these two geometric averages are re- 
 ciprocals of each other. By ordinary algebra this is almost self-evident, i.e. 
 
 1 1 
 
 ^ 
 
 Ix-x- X 
 
 VaXbXcX ... 
 
 Vo X 6 X c X . 
 
 Just as the arithmetic average is necessarily greater than the geometric 
 average so the harmonic average is necessarily smaller than the geometric 
 average. 
 
 This is due to inverting. Take the same original figures, 50 and 200, 
 whose geometric average is 100. Their reciprocals are rs and ^^ whose 
 geometric average is tot, the reciprocal of the geometric average of the 
 original figures 50 and 200. 
 
 Now this inverting the three numbers 50, 100, 200, has also inverted 
 their order from the ascending order of 50, 100, 200, to the descending order 
 
 of TC, TiTff, Taty- 
 
 But the arithmetic average always lies above the geometric. The arith- 
 metic average, therefore, in the series 50, 100, 200, is on the right of the 
 100 and is on the left of the r^ in the series, tV, riir, ^hs- To be spe- 
 cific, we may insert in both cases the arithmetic average in parenthesis 
 in its proper order, as follow^ : 
 
 50, 100, (125), 200, and i^, (^), tot, ^■ 
 
 Reinverting, we obtain 50, [80], 100, 200 where the 80 in square brackets 
 is the harmonic average of 50 and 200 {i.e. the reciprocal of the arithmetic 
 average of their reciprocals). Evidently this harmonic average is below 
 the geometric. 
 
 It is interesting to note further that, when there are only two numbers to 
 be averaged (a and h), not only, as has just been shown, does the geometric 
 average of a and h (which is Va X h) lie between the arithmetic and har- 
 monic averages of a and b but it is their geometric average ; for the latter, 
 the geometric average of the arithmetic and harmonic averages of a and h, 
 
APPENDIX I 377 
 
 S-^) ( 
 
 a 
 after reduction, comes out V a X b, the geometric average of a and b. 
 
 Note to Chapter III, § 1. An Index Number of Purchasing Power. In 
 the book no use is made of the concept of purchasing power of money. 
 Everything which could be said of purchasing power can be said of prices 
 and it may be confusing to treat of both. An index number of the general 
 purchasing power of a dollar may be defined as the reciprocal of an index 
 number of prices. If either is obtained, the other may be obtained from 
 it by inversion. This index of general purchasing power may also be con- 
 ceived as an average of particular purchasing powers over individual com- 
 modities, each such being defined as the reciprocal of a price, i.e. a "dol- 
 lar's worth" of anything. The ratios of such dollar's worths between any 
 two dates is the reciprocal of the price relative. Any formula for prices 
 in this book may be translated into a formula of purchasing power by 
 substituting for po, etc., the expression l/ro, etc., where r stands for par- 
 ticular purchasing power and by substituting for P, etc., the expression 1/R. 
 
 It wiU be found that a given formula applied to work out an index num- 
 ber of purchasing power will yield the same numerical result as if applied 
 to work out an index of prices reversed in time. From this it follows that 
 the reciprocal of the index number of purchasing power is equal to the time 
 antithesis of the index number of prices. 
 
 Note to Chapter III, § 4. Calculating the Weighted Median and Mode. 
 According to the definition of weights, a term having a weight of "2" is 
 counted as two terms ; and this applies as readily to the median as to any 
 other average when the weight is an integer. 
 
 When the weight is not an integer the same principle applies, though 
 not so simply. In any case it is well first to arrange the price relatives 
 in order of magnitude. Opposite such a column we write, in another 
 column, the weight for each relative. 
 
 This second column has 36 elements. Their total sum, S, is the total of 
 the weights. The median is the position in the first column opposite the 
 
 half-way point in the second. Take, then, half of this sum, -. Then add 
 
 the elements in the second column, from above, tUl a point is reached where 
 
 o 
 adding one element more will make the sum exceed -. Let A be this sum 
 
 or 
 
 slightly smaller than -. Proceed in the same manner from below, obtain- 
 Ji 
 
 c 
 
 ing another sum, B, slightly smaller than -. Then 
 
 A < - and 5 < -. 
 2 2 
 
 This leaves one middle element with a weight which we may call m, the 
 
 CI 
 
 element which makes A ox B exceed — , and such that 
 A + m + B = 5. 
 
378 THE MAKING OF INDEX NUMBERS 
 
 The relative in the first column opposite the weight m in the second may 
 be said to lie opposite the middle of m, so that this particular relative is 
 the required median in case, and only in case, half of the second column 
 falls exactly at the middle of m, i.e. in case 
 
 A + !^ = fi+^ = ^. 
 2 2 2 
 
 In all other cases, the median is not exactly the relative in the first col- 
 unan opposite m, but is an imaginary figure in the first column above or 
 below said relative, which imaginary figure does come opposite the middle 
 of S. This imaginary figure is interp>olated by proportional parts, i.e. by 
 taking the distance in the first column between the two neighboring rela- 
 tives between which the median falls and dividing that distance in the ratio 
 in which in the second column, the middle of S divides the distance between 
 the middle of m and the middle of the neighboring weight. (In practice 
 the operation is simplified by multiplying by two, i.e. by not halving the 
 two weights.) 
 
 The mode is calculated by the same graphic method for the weighted 
 as for the simple index number, i.e. by plotting columns representing the fre- 
 quency (or total of weights) of price relatives which fall between certain 
 equidistant limits, such as 100-120, 120-140, etc., and selecting the rela- 
 tive having the greatest frequency, or highest colmnn. Various devices 
 are resorted to to facilitate the work which need not be particularized, as 
 the result is always somewhat arbitrary in any case. 
 
 Note to Chapter III, § 7. Peculiarities of the Aggregative. It may be 
 worth while here to note that the aggregative is, in every respect, peculiar 
 as compared with the other five types of average. As we have seen, the 
 aggregative average, unUke all the other averages, is not computed from the 
 mere price relatives or ratios of which it is an average, but requires, in 
 addition, the specific numerators and denominators of those ratios (the 
 prices themselves). It follows that, if any particular ratio were "reduced" 
 by division, while that ratio itself would be unaffected, its numerator and 
 denominator would be affected and such a change would, in general, change 
 the resulting index number. For any other type than the aggregative 
 it would make no difference what the numerator and denominator were 
 so long as their ratio did not change. 
 
 Again, as we have already seen, the simple aggregative is not simple in 
 precisely the same sense as the other simple index numbers, because it 
 requires not only the price ratios but the prices. 
 
 Finally, the weights used in the aggregative average are not weights in 
 quite the same sense as are the weights used in the other averages because 
 they are applied not to the terms averaged (i.e. the price ratios) but to 
 their numerators and denominators separately; moreover, these weights 
 are not values, as are the weights of the other averages, but quantities. 
 
 Nevertheless, the aggregative conforms to our general definition of an 
 average given in Appendix I (Note A to Chapter II, § 15) ; the simple 
 aggregative is analogous to the other simples in that, given the initial ma- 
 terials, in this case the prices, they are not reduplicated but are each taken 
 once only ; and, lastly, the weights used conform to the general definition 
 of weights given in Chapter I, § 4. I, therefore, prefer retaining the 
 
APPENDIX I 379 
 
 terms "average," "simple," and "weights" rather than discarding any 
 of them in respect to aggregatives. 
 
 Note to Chapter III, § 11. Formulce S and 17 Reduce to 68, and 5 and 19 
 to 59, by Cancellation. The arithmetic with weighting / (Formula 3) is 
 
 Pi 1 r I P'l , 
 
 Po p 
 
 Poqo + p'oq'o + . . . 
 
 Canceling the two po's in the first term of the numerator, and again can- 
 ceUng the two p'o's in the second, etc., we have 
 
 goPi + q'op'i + . . . 
 
 Poqo + p'oq'o + 
 
 which is identical with the aggregative with weighting / (Formula 53). 
 Similarly, the arithmetic with weighting // (Formula 5) is 
 
 Poqi 2J + p'oq'i ^ + . . . 
 Po p 
 
 Poqi + p'oq'i + . . . 
 
 which reduces, by canceling the po's, the p'o's, etc., to 
 
 giPi + q'lP'i + • • . 
 Poqi + p'oq'i + . . . 
 
 which is identical with aggregative IV (Formula 59). 
 
 The harmonics /// and IV (Formulae 17 and 19) reduce, similarly, to 
 the aggregatives / and IV (53 and 59) respectively. 
 
 Note to Chapter III, § 12. Professor Edgeworth's and Professor Young's 
 " Probability" Systems of Weighting Give Erratic Results. So far as I know, 
 the only systematic methods of weighting not mentioned in the text which 
 have been even hinted at by other writers are those mentioned by Pro- 
 fessor F. Y. Edgeworth and Professor Allyn A. Young, modeled on proba- 
 bility theory. 
 
 Professor Allyn A. Young proposes, when the data are uncertain, the 
 formulae 
 
 V 
 
 ^(qoW) 
 
 and J s(£2W) 
 
 ^ ^(qiW) 
 
 and their geometric mean. His idea in thus using squares of quantities 
 and values is to follow the analogy of the formulae of probability in the 
 method of least squares. 
 
 By these formulae, the index number of prices on 1913 as a fixed base 
 would be : 
 
3S0 
 
 THE MAKING OF INDEX NUMBERS 
 
 Formula 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 
 101.23 
 
 99.99 
 
 108.36 
 
 149.75 
 
 
 ^ ^(qoW) 
 
 164.59 
 
 J MqiW) 
 ^ ^iqiW) 
 
 101.52 
 
 100.52 
 
 108.35 
 
 148.31 
 
 166.68 
 
 Edgeworth, to meet the case of uncertain data, proposes to use as weights 
 the reciprocal of the squares of the deviations from some mean, on the 
 theory that the price relatives which deviate the furthest are the least 
 likely to be true indicators of the general trend of prices and therefore 
 ought to be given the least weight. 
 
 Edgeworth's formula has not been definitely expressed and might be 
 variously interpreted. AppUed to the geometric mean it may be written 
 
 ^^/(#>^Ci) 
 
 iLLY'X 
 
 where d, d', etc., are the percentage deviations of — , — , etc., respectively 
 
 Po p'o 
 from the mean 
 
 »11n T) n 
 
 X 
 
 'Po Po 
 By this formula we would have as our price index number the following : 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 96.16 
 
 96.81 
 
 121.24 
 
 165.53 
 
 179.64 
 
 These results are widely different from our results by ordinary methods. 
 Neither Edgeworth's nor Young's proposed formula seems to fit the case. I 
 agree with Walsh that ordinarily it must be presumed that the price and 
 quantity data are not uncertain but certain, and, if certain, each has a right 
 to be represented, not in proportion to its deviation from some mean, but 
 in proportion to its importance in the usual sense. 
 
 It seems to me that the only proper application of ideas of probability 
 to averaging price relatives is in cases where the data are actually defective 
 or uncertain ; and the only practical way in such cases is, first, to write the 
 formula deemed best and then, if the data are considered as uncertain, cor- 
 rect this formula in the individual cases of uncertainty by multiplying by 
 arbitrary coeflScients of uncertainty. 
 
APPENDIX I 381 
 
 At any rate, there is ordinarily no presumption that the uncertainty of 
 the data varies inversely as their deviation (or as its square) from any nor- 
 mal. Such a use of the deviations might lead to very bizarre results. 
 
 Note to Chapter /F, § 10. The Scope of Our Conclusions. To see clearly 
 the formal framework of our study, let us review it briefly. The problem 
 of finding an index number Pqi for comparing, on the average, the prices (p's) 
 of commodities at two times was mathematically conditioned by certain 
 p's and q's, the q's being the coefficients by which the p's are multiplied to 
 give the values, pq's, of these commodities. So that Spi?i and Zpoqo are 
 the total values of the two groups. 
 
 What we have sought is a formula or formulse for Pqi such that, if ap- 
 plied the other way, Pio, these two appUcations wUl be consistent, i.e. 
 PoiPio will be unity, and such also that if the same formula be applied 
 to the q's as well as to the p's, these two applications will be consistent, 
 i.e. PoiQoi = Spi^i -T- Spogo or PioQio = 2po3o ^ ^piqu 
 
 All our conclusions flow from the above formal background. They are, 
 therefore, of as broad application as is this background. They apply if 
 the p's are wholesale prices and the q's are the amounts imported into the 
 United States. They apply equally well if the p's are retail prices and the 
 q's are the quantities sold by grocers in New York City. They hkewise 
 apply if the p's are rates of wages per hour and the q's the numbers of hours 
 worked, or if the p's are the freight rates and the q's the quantities of mer- 
 chandise transported from New York to Liverpool by all Cunard steamers. 
 They likewise apply if the p's are the prices of industrial stocks and the 
 q's are the number of shares sold by John Smith in January. They hke- 
 wise apply to the right-hand side of the "equation of exchange."^ (Some 
 critics have, because of my interest in the equation of exchange, jumped to 
 the conclusion that my discussion of index numbers is in some way limited 
 to the problem of the equation of exchange !) They hkewise apply if the 
 p's are the lengths of the visiting cards of the "400" and the q's their 
 breadths, pq being their area. 
 
 The results will differ only when the above mathematical conditions 
 differ. Thus, while we could reckon the average change in the length and 
 breadth of visiting cards between two years so as to preserve Tests 1 and 2, 
 we would have to modify our methods if we were to measiu-e the average 
 change in length, breadth, and thickness of dry goods boxes ; for the en- 
 trance of a third factor in addition to the two, p and q, would change 
 the conditions of the problem. Likewise we would need to modify our 
 methods if, for any reason, Tests 1 and 2 are not required. 
 
 What is emphasized is simply that within the formal conditions which 
 apply to the above premises we find an enormous range of problems. 
 
 We may formulate in the most general way the above mentioned con- 
 ditions to which the reasoning of this book applies as follows : 
 
 Given a group of variable magnitudes which, under a set of circum- 
 stances designated by " " are po, p'o, p"o, etc., and which, under a second 
 set of circumstances designated by "1" are pi, p'l, p"i, etc., respectively, 
 and, 
 
 Given another group of variable magnitudes which are in one to one 
 
 1 See Irving Fisher, The Purchasing Power of Money, pp. 26, 53, 388, etc. 
 
382 THE MAKING OF INDEX NUMBERS 
 
 correspondence with the members of the first group, and which, under the 
 set of circumstances "0," are go, q'o, q"o and, imder the second set of cir- 
 cumstances, "1," are 51, q'l, q"\, respectively, and, 
 
 Given an objective relation existing between the corresponding members 
 of the two groups such that the products poqo, p'oq'o, p"oq"o, etc., on the 
 one hand, and piqi, p'lq'i, p"iq"h etc., on the other, possess a real 
 significance in the field of study from which the magnitudes are drawn, 
 such that it will be recognized as suitable for use in checking up with the 
 ratios as described below. 
 
 The problem is to construct an index number 
 
 Poi which shall serve as a fair average of — , ^, ^—^, etc., 
 
 Pa p'o P"q 
 
 and Qoi which shall serve as a fair average of — , ^, ^', etc., 
 
 qo q'o q"o 
 
 and Pio which shall serve as a fair average of _, — , ^— ?, etc., 
 
 Pi P'l P"i 
 
 and Qio which shall serve as a fair average of — , — , ^—^, etc. 
 
 91 q'l q"i 
 
 Under these circumstances it is fair to require the fulfillment of two 
 tests, viz., Test 1 that Poi X Pio = 1 and Qoi X Qio = 1; also Test 2 that 
 
 Poi X Qoi = ^^i2l and Pio X Qio = ^^Ml, The justification of these 
 
 2po5o 2pigi 
 
 relations is that they hold true of the individual magnitudes of which Poi, 
 
 Pio, Qoi, and Qio are averages. For example, we know that ^ x — = 1 
 <J po Pi 
 
 and — X — = 2l21 and we can assign no reason for violating the anal- 
 
 Po go PoQo 
 ogous relationships among the averages, in one direction rather than the 
 other. 
 
 With these preliminaries, all the reasoning which we have followed through 
 this book applies whether the subject matter be wholesale prices and quan- 
 tities marketed, or the length and breadth of visiting cards, or anything 
 whatsoever. It certainly applies not only to "general purpose index num- 
 bers of wholesale prices" but, by merely using a different set of p's and 
 g's, to any special index number of prices, say, of railway freight rates, or 
 to index numbers of retail prices, Raymond Pearl's index mmaber of food 
 prices weighted by their calorific food values (the calories being, in this 
 case, the q's), cost of living, wages, stock or bond prices, or sales, costs of 
 paper manufacture, the rate of interest, crop yields, and many others. 
 
 Only when the problem is one which cannot be covered by the above 
 formal and general statement will our reasoning be inapplicable. I know 
 no problem where index niunbers have yet been employed to which these 
 general conditions do not apply. In practical scientific research, the 
 nearest approach, of which I know, to such a case as that of the dry goods 
 box is to be found in anthropometry. In comparing the shapes and sizes 
 {i.e. " builds ") of two persons, or of the same person in two periods of 
 
APPENDIX I 383 
 
 life, or of two groups of persons, we have such three dimensional problems 
 and their best solutions wUl, of course, differ from those of the two dimen- 
 sional problems of this book. 
 
 Note to Chapter V, § 2. Proof that the Product of the Arithmetic Forward 
 by the Arithmetic Backward Exceeds Unity. The forward formula is ' ■ 
 
 and the backward, — ^LL go that their product is vPo/ ^Vi/ ^ -^g ^^^ 
 
 n r^ 
 
 to prove that this always and necessarily exceeds unity. ^ 
 
 To prove this, we first prove the more elementary theorem that the 
 simple arithmetic average of any number and its reciprocal exceeds unity. 
 Evidently one of the two must exceed imity. Let 1 4- a be that one and 
 
 let be the other. We are to prove that 
 
 1 + a 
 
 1±^>1. 
 
 2 
 On reducing and simplifying, this fraction becomes 
 
 2 + 2a + a2 
 
 2 +2a 
 
 which may be written 
 
 1 + 
 
 2 +2o 
 
 This evidently exceeds unity, which was to have been proved. In other 
 
 words, the two terms, 1 + a H , exceed 2. 
 
 1 + a 
 
 Applying the theorem just proved to the problem in hand, we note that 
 S ( ^Mz ( ^ ), which is to be miiltiphed out, may be written 
 
 (^ + ^ + ^+...(n terms)) 
 \po V a P I 
 
 (PP + PJ + P:!" + ...(„ terms)). 
 ^V\ P\ Pi ' 
 
 On multiplying these two series, we see that the product consists of a series 
 of terms to the nxunber of n^. Some of these terms (namely, those found 
 
 1 It is assumed, of course, that the price ratios — , ~, etc. , are not all equal and that they 
 
 po p'o 
 
 ore all positive magnitudes. 
 
384 THE MAKING OF INDEX NUMBERS 
 
 by the vertical multiplications such as — X — , etc.) are each evidently 
 
 Vo Pi 
 unity. The other terms may be arranged in couplets of reciprocals joined 
 
 by +. Thus the product of the two factors, — X ^— ', may be joined to 
 
 its reciprocal, — X - — , coming from the same two columns. Since these 
 
 Po p"i 
 two terms are reciprocals, one must exceed unity and the other be less than 
 
 unity, i.e. they may be written (1 + a) + ( ) which sum we have 
 
 U +a) 
 
 ■■mf.) 
 
 just shown exceeds 2. It follows that the numerator of — ^^^^ — vtiL will 
 
 have terms to the number of n^, each term being either 1 or else coupled 
 with another term, the two exceeding 2. Hence the numerator is more 
 than n^ while the denominator is exactly n*. Hence the whole fraction 
 exceeds unity. 
 
 Note to Chapter V, § 6. The Two Steps between Weightings I and IV. 
 In the text, systems / and // were summarily limaped together as practi- 
 cally the same, and Ukewise systems /// and IV were lumped together. 
 Let us now cUmb about from one index number to another, all based on 
 the same price list but varied by weighting. Thus, in passing from such 
 an index number with the first weighting (/) to one with the last weight- 
 ing (IV), we shall take two separate steps, the short step from / to // and 
 the long one from // to IV, or, alternatively, the long one, / to III, and 
 the short one, /// to IV. To fix our ideas, let us adopt the last com^se 
 I-III-IV. 
 
 The first step is the passage from / to III, i.e. changing the weight of 
 bacon from poqo to piQo (and likewise changing the weight of barley from 
 p'oq'o to p'lq'o, etc.). Tliis change in the weighting system has the effect, 
 as we have seen, of loading the more heavily those price relatives which 
 are already high, and, therefore, of raising considerably the index number 
 III above / with which we started. This raising always happens whether 
 prices are rising or falling. That is, in this first or "long" step between 
 / and III, there is no uncertainty. Any index number under the system 
 of weighting /// must be larger than under weighting I. 
 
 In the "short" step between III and IV, on the other hand, there is 
 uncertainty. Any index nu.nber under III may be greater or less than 
 under IV and may even possibly happen under very unusual circumstances 
 to be much larger or smaller. It is a fair lottery. The high price relatives 
 may draw either heavy weights or light weights with an even chance each 
 way, as, likewise, may the low price relatives. The net effect wiU prob- 
 ably be an almost complete offsetting so that the final index number (IV) 
 will probably be close to /// and may be either slightly above or below.* 
 
 ' It ia, of course, conceivable that there is a correlation between the prices and quan- 
 tities but this may be in either direction according as the prime mover is supply or demand. 
 In the case in hand there is essentially no correlation and investigation of some New York 
 Stock Exchange prices shows the same absence of correlation. In the case of the 12 crops 
 used by Day and Persons of the Harvard Committee on Economic Research, where supply 
 
APPENDIX I 385 
 
 So that after both steps are taken, and we compare 7 with IV, we cannot, 
 as we could in the case of type bias, be absolutely sure of the result. AU 
 that we can say is that it is exceedingly probable that IV will exceed I. 
 No case to the contrary occurs in the present investigation and it seems 
 very imlikely that such a case will ever be encountered in practice (except 
 for the mode and, in rare cases, the median, or except when there are a 
 very few commodities in the index number). 
 
 But we have not, even yet, thrown our results respecting weight bias 
 into a form quite comparable with that employed for type bias. In taking 
 each of the two steps, the "long" and the "short," we have used only 
 forward index numbers. But now, after putting the two steps together, 
 we are ready to revert to the original method, that of multiplying together 
 forward and backward index numbers of the same kind. 
 
 Thus, we have shown that (in all probability) geometric IV forward is 
 always a larger index number than its time antithesis, geometric I forward. 
 But geometric I forward is the reciprocal of geometric IV backward. In 
 proof : Geometric / (23) forward is 
 
 ^Po 
 
 Geometric IV (29) forward is 
 Spigi 
 
 Spogo//pi\po3o ^, {p'i\p'oq'o ^^ 
 
 \\po> ^Vo> ^'■' 
 
 Geometric IV backward is found by interchanging the "O's" and "I's" 
 in geometric IV forward and is 
 
 spogo 
 
 )go//Po\po3o ^ (p'o\p'(^'o 
 
 Evidently the first of these three formulae (geometric / forward) is the 
 reciprocal of the last (geometric IV backward), which was to have been 
 proved. It follows that the product of geometric IV forward X geometric 
 IV backward is the same as 
 
 geometric IV forward X ( - 
 
 1 
 
 ^geometric / forward/ 
 
 which is 
 
 geometric IV 
 geometric / 
 
 and this (since IV always exceeds 7) is greater than unity. Consequently 
 the original product, geometric IV forward X geometric IV backward, 
 exceeds unity. In other words, they have a positive joint error. In still 
 
 was the dominant variable in changing the quantities marketed and where there is an in- 
 verse correlation between quantity and price, weighting system I makes for a higher index 
 number than II, and /// than IV. Yet it is noteworthy that the effect on the curves given 
 in Chapter XI is almost negligible. 
 
386 THE MAKING OF INDEX NUMBERS 
 
 other words, geometric IV has an upward bias, and this bias is acquired 
 through weighting, in exactly the same sense as type bias previously found 
 for the arithmetic and harmonic. 
 
 Likewise, we could trace the transformation of an index number by chang- 
 ing its system of weighting from II to /// via I or IV, i.e. first by the short 
 step- from // to I and then the long one from I to III, or first by the long 
 step from II to IV and then the short step from IV to III. In all cases, 
 changing a. quantity element in the weights has only a smaU effect, which 
 must, in general, be assumed equally likely to be in either direction ; but 
 a change of the price element in the weight has a larger effect and in a 
 definite direction. 
 
 By such reasoning we may impute upward bias to geometric IV, geo- 
 metric ///, median IV, median ///, mode IV, mode ///, while similarly, 
 / and // have a downward bias for the same three types (geometric, me- 
 dian, mode). 
 
 The arithmetic and the harmonic remain. We are to show that, for 
 instance, arithmetic / F forward X arithmetic 7 F backward exceeds unity. 
 In algebraic terms, in the first place, the arithmetic IV backward is the 
 same as the reciprocal of harmonic / forward. 
 For 
 
 arithmetic IV forward is 
 
 £o. 
 
 2pi 
 arithmetic IV backward is 
 
 Sp.g.Pl 
 
 2pigi 
 
 Its reciprocal is 
 
 spogo^"' 
 . Pi. 
 
 Spoffo 
 
 spogo^" 
 
 Pi 
 
 This is harmonic I, which was to have been proved. 
 Hence (to retain for comparison the "spelled out" method just em- 
 ployed), 
 
 arithmetic IV forward X arithmetic IV backward 
 
 is arithmetic IV forward X 
 
 harmonic I forward^ 
 arithmetic IV forward 
 
 harmonic / forward 
 
 That this exceeds unity, or that the numerator exceeds the denominator, 
 remains to be proved. We shall see that, not only does the numerator 
 exceed the denominator, but that the numerator exceeds arithmetic I, 
 
APPENDIX I 387 
 
 or arithmetic 77, or harmonic 777, or harmonic IV, and that these exceed 
 the denominator. 
 
 In the first place, arithmetic IV exceeds arithmetic 77 as we have al- 
 ready proved with certainty and, since arithmetic 7 in all probability 
 agrees closely with arithmetic 77, it follows that, in all probability, arith- 
 metic IV exceeds arithmetic 7. But we have seen that arithmetic 7 is 
 identical with harmonic 777 (both being Laspeyres') while arithmetic 77 
 and harmonic IV are both Paasche's. And we know that harmonic 777 
 exceeds with absolute certainty harmonic 7. Thus, the numerator is 
 greater than and the denominator is less than (indifferently) arithmetic 
 7 and 77, or harmonic 777 and IV, which was to have been proved. 
 
 Note to Chapter V, § 9. Formula 9 after Reversing Subscripts and In- 
 verting Becomes 13. Formula 9 (or arithmetic IV) forward being 
 
 ♦_ go 
 
 Spiji 
 
 its backward application, found and represented by the dotted line in 
 Charts 18P and 18Q by reversing the subscripts, is, as shown in the last 
 note, 
 
 Pi^ 
 
 SpoSo ' 
 
 the reciprocal of which (represented graphically by prolonging the dotted 
 line in Charts 18P and 18Q) is 
 
 Spo?o 
 
 Spogo^° 
 
 Pi 
 
 But this is Formula 13 (or harmonic 7), which was to have been proved. 
 
 Note to Chapter V, § 11. Bias and Dispersion in Formuloe. Any bias, 
 as has been seen, is defined in terms of some joint error. Thus the joint 
 error (in this case joint bias {B)) of the arithmetic forward and backward 
 is given by the formula 1 -1- B = arithmetic forward X arithmetic back- 
 ward, or, calling arithmetic forward A and remembering that arithmetic 
 backward is the reciprocal of the harmonic forward (which we may call 
 
 1 A 
 H), we have 1 + B — A X — = —. But the bias of the arithmetic for- 
 
 II H 
 ward is not the whole of B since 1 -1- S expresses the full ratio of the up- 
 ward biased A to the downward biased H and so involves a double appli- 
 cation of bias. Thus, we may define the bias, b, as half of B, or rather 
 half geometrically (as in compound interest) according to the formula 
 
 (1+ &)2 = 1 -H B, 
 
388 'THE MAKING OF INDEX NUMBERS 
 
 whence * 
 
 Our main fonnulae, then, are 
 
 A 
 1 + 6 = -=» 
 VAH 
 
 1 H •' 
 
 1 + & VAH 
 
 either of which may be derived from the other, the former expressing the 
 upward bias of A relatively to VAH , and the latter expressing the down- 
 ward bias of H relatively to VAH. 
 
 We next need a "dispersion" index, d, to represent the degree of the 
 divergences of the price ratios from each other. Let us begin with the 
 case of only two commodities, considered of equal importance, their price 
 relatives (or quantity relatives, or whatever the subject matter may be) 
 being r and r', where r is the larger. The total divergence from each other, 
 D, of r and r' may be defined by 1 + D = r/r'. But a preferable magni- 
 tude to use is not this total divergence between the two but the average 
 dispersion, d, from a common mean, best taken as their geometric mean so 
 that d is half of D geometrically, i.e. 
 
 (l+d)^ = l+D=^,. 
 r 
 
 Hence 
 
 -. 1 J fr r y/rr' 
 \r' VrP , r 
 
 1 +d * r Vrr' ^ .: ' 
 From these equations we may derive 
 
 r = (1 + d) V^^ ' 
 
 r' =l^^)lV'^'. 
 VI +d) 
 
 Since there are but two relatives to be averaged, r and r', their simple 
 
 ' In the same way beginning with the harmonic, instead of the arithmetic, and using 6' 
 
 If 
 
 to express a downward bias we could derive 1—6' =—==. But since by multiplying 
 together the equations for 1 + 6 and 1 — 6' we get (1 + 6)(1 — 60 " 1 we can better 
 use, instead of 1 —h\ the equal expression -;; — — r and dispense with the use of &' altogether. 
 
 1+0 
 
APPENDIX I 
 
 389 
 
 arithmetic average (il) is A = —^ — and their simple harmonic average 
 
 (H) is H = 
 
 r r' 
 
 In these, we substitute the above expressions for 
 
 r and r' giving 
 
 H = 
 
 r + r' L 
 
 l+d + 
 
 L_1 
 
 (l+d) 
 
 2 
 
 2Vrr^ 
 
 1+i 
 
 r 'r' 
 
 + (1 + d) 
 
 r (1 + d) 
 whence, dividing the equations and canceling VrP, 
 
 'a +d) + 
 
 (l+d) 
 
 A^ 
 H 
 
 In other words, the result is independent of the actual magnitude of the 
 price relatives and dependent only on the ratio (1 +d) of their divergence 
 from their mean. Anticipating this result, we might have substituted 
 for the above proof the following simplification : 
 
 Let the (geometric) average of the two price relatives be considered as 
 
 100 per cent, or unity, the upper one, l+d, and the lower, -. Then 
 
 l+d 
 
 and 
 
 whence evidently 
 
 a+d) + 
 
 A = 
 H = 
 
 a+d) 
 
 (l+d) 
 
 + (l+d) 
 
 A 
 H 
 
 l+d + 
 
 1 +d 
 
 as before. 
 
 A 
 But we already know that (1 + 6)* is also equal to — . Hence we have 
 
 H 
 
 (after extracting the square roots) 
 
 l+d + 
 
 1+6 = 
 
 l+d 
 
390 THE MAKING OF INDEX NUMBERS 
 
 as the equation expressing the relation between the bias 6 and the disper- 
 sion index d. 
 
 That bias and dispersion are both relative to the same axis or mean pro- 
 portional can readily be shown in several ways from the above equations. 
 The mean proportional with reference to which h was reckoned was '^AH 
 and that with reference to which d was measured was Vrr', and these two 
 expressions are readily shown to be equal. 
 
 From this formula it will be seen that the bias increases very rapidly 
 with an increase in the dispersion, that when the dispersion is zero the 
 bias is zero, when the dispersion is 5 per cent the bias is negligible, when 
 the dispersion is 50 per cent the bias is 8.34 per cent as the following table 
 shows : 
 
 TABLE 48. FOR FINDING THE BIAS CORRESPONI>- 
 ING TO ANY GIVEN DISPERSION 
 
 (Both in per cents) 
 
 Dispersion (d) 
 
 BlA8 (b) 
 
 5 
 
 .12 
 
 10 
 
 .45 
 
 20 
 
 1.67 
 
 30 
 
 3.46 
 
 40 
 
 5.72 
 
 50 
 
 8.34 
 
 100 
 
 25.00 
 
 So much for the simple case of two price relatives and where the disper- 
 sion is self-evident. Where there are more than two the dispersion must 
 be some sort of average. To obtain such an average, we substitute, in 
 thought, two imaginary price relatives for all the actual ones, the disper- 
 sion of each of these two from their mean being an average of all the actual 
 deviations of the 36 from their mean. Various such averages have been 
 used to measure dispersion. That usually employed is the "standard 
 deviation" obtained by taking the average of the squares of the deviations 
 of the individual price relatives (each deviation being measured from the 
 arithmetic average) and extracting the square root. Another is analogous 
 to the above but is geometric in nature instead of arithmetic. It is found 
 by taking the standard deviation of the logarithms of the price relatives 
 and then taking the anti-logarithm of that. Another is the average 
 "spread" between the median and the two "quartUes." 
 
 Of these the middle one seems the best adapted to the present purpose. 
 It is certainly better adapted theoretically than the first (the ordinary 
 arithmetically defined standard deviation), because the price relatives and 
 quantity relatives with which we have to deal are widely varying and have 
 "skew" distribution varying more upward than downward which the 
 geometric or logarithmic standard deviation tends to eliminate. 
 
APPENDIX I 
 
 391 
 
 Practically, however, the arithmetic and geometric standard deviations 
 agree surprisingly well in spite of the skewness and greatness of the dis- 
 persion. This will be seen from the following table : 
 
 TABLE 49. STANDARD DEVIATIONS (FOR PRICES) 
 (In per cents) 
 
 Fixed Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Arithmetic S. D 
 
 Geometric S. D 
 
 10 
 11 
 
 16 
 17 
 
 24 
 21 
 
 58 
 39 
 
 45 
 33 
 
 Chain of Bases 
 
 
 
 
 
 
 Arithmetic S. D. 
 
 Geometric S. D 
 
 10 
 
 11 
 
 12 
 12 
 
 27 
 
 22 
 
 29 
 
 22 
 
 20 
 22 
 
 We may then picture the dispersing terms (price relatives, or quan- 
 tity relatives, or whatever the terms under consideration may be) as all 
 reduced to two imaginary terms, say price relatives, one Ijdng above the 
 (geometric) average and representing all the actual price relatives above 
 the average, and the other lying below that average and representing all 
 the actual price relatives below the average, and each diverging from that 
 average in the ratio 1 + d (and from each other in the ratio (1 + d)^). In 
 this empirical way we reduce the complex case of many price relatives 
 to the original and simpler case of only two price relatives. 
 
 The question now arises : Will the dispersion index d as thus defined 
 {i.e. as the geometrically, or logarithmically, determined standard devia- 
 tion) be actually related to the bias h according to the formula 1 + & = 
 
 i— which we found to be true in the simple two term case? 
 
 The answer is, yes, very closely. 
 
 First we shall show that the above empirical relation between the bias h 
 and the dispersion index d can be made absolutely exact for the case of any 
 number of commodities if we suitably change the definition of rf to a fourth 
 form, in terms of A and H, as follows : 
 
 To maintain absolutely the equation 
 
 l+d + 
 
 l-h6 = 
 
 1 +d 
 
392 
 
 THE MAKING OF INDEX NUMBERS 
 
 we simply use it and the equation 
 from which to derive 
 
 {l+d) + 
 
 1 
 
 a+d) 
 
 -\h = . 
 
 2 \^ VaH 
 
 Solving this quadratic equation for 1 + d and reducing we have 
 
 1 +(i = 
 
 Va^ - ah + a 
 Vah 
 
 This new determination of d is relative to ^ AH as before. 
 
 The above formulae will seWe also for the harmonic except that whereas 
 
 1 + d is the magnitude pertaining to the arithmetic, is the magni- 
 
 \ ■\- d 
 tude pertaining to the harmonic, the d being the same. 
 
 It only remains to show that this special form of dispersion index (in 
 terms of A and H and therefore also, of coiuse, in terms of the original 
 data themselves) is, in actual fact, very close to the geometric (logarith- 
 mically calculated) standard deviation, as the following figures show : 
 
 TABLE 50. SPECIAL DISPERSION INDEX COMPARED 
 WITH STANDARD DEVIATION (LOGARITHMICALLY 
 CALCULATED) FOR THE 36 COMMODITIES (SIMPLE) 
 
 (In per cents) 
 
 
 Special 
 
 Stand ABD 
 
 1914 
 
 11.5 
 17.3 
 21.5 
 39.2 
 33.7 
 
 11.5 
 
 1916 
 
 17.2 
 
 1916 
 
 21 4 
 
 1917 
 
 38.7 
 
 1918 
 
 33.1 
 
 
 
 For the weighted arithmetic and harmonic the case is only slightly dif- 
 ferent. We then have, for instance. 
 
 l+d + 
 
 1+5=- 
 
 1 +d 
 
 VA'H' 
 
 where A' and ff' are weighted arithmetic and harmonic index numbers 
 whence 
 
APPENDIX I 
 
 393 
 
 1+d = 
 
 VA'^ - A'H' +A' 
 VaW' 
 
 This also is close to the (logarithmically calculated) standard deviation as 
 the following table (in which the weighted averages have the mean weights 
 Vpo2oPi2i> etc., as per Formulas 1003 and 1013) shows : 
 
 TABLE 51. SPECIAL DISPERSION INDEX COMPARED WITH 
 STANDARD DEVIATION (LOGARITHMICALLY CALCU- 
 LATED) FOR THE 36 COMMODITIES (WEIGHTED) 
 
 (In per cents) 
 
 
 Special 
 
 Standard 
 
 1914 
 
 8.3 
 15.3 
 19.2 
 39.1 
 26.2 
 
 7.7 
 
 1915 
 
 15 1 
 
 1916 
 
 19.2 
 
 1917 
 
 38.9 
 
 1918 
 
 26 5 
 
 
 
 So much for the type bias as applied to the simple arithmetic or har- 
 monic, and as applied to their mean weighted forms. We have still to 
 consider the weight bias of the various systems of weighting. 
 
 Summarizing the proof in its simplest form, let us assume only two com- 
 modities as before, their price relatives ( — and ^ ) being 1+d and 
 
 VPo p'oj 
 
 . As to the weights poqo, p'oq'o and piji, p'lq'i, we may call po and 
 
 1+d 
 
 p'o 100 per cent, or 1, so that pi and p'l are 1+d and 
 
 while as 
 
 1 +d' 
 
 to the quantities we assume they do not change, i.e. qo = q\ and q'o = q'\ 
 (which may be called q and q' simply) and that they are such as to make 
 equal the average weights of the two price relatives over the two years, 
 
 i.e. VpogoPi^i = '^p'oq'op'iq'i. 
 
 Substituting in this equation the above values for the p's, viz., po = 1, 
 
 Pi = 1 + d, and p'o = 1, p'l =7-—:. 
 
 1+0 
 
 V(l + d)go2i = yj q'oq'i 
 
 i.e. (remembering the above q equalities, go = qi and q'o = q'l), 
 
 vcr+«?-Vi^^ 
 
394 THE MAKING OF INDEX NUMBERS 
 
 whence, (1 + d)q^ = Sl. or (1 + d)V = 9^ or (1 + d^ = -^%r 2^ = 
 
 1 + d q^ q 
 
 1 + d or, letting q = 1, simply, q' = 1 + d. 
 
 Summarizing, we may now substitute, in any formula to be investigated, 
 the following magnitudes : po = 1, p'o = 1, go = 1, q'o = I + d, pi = 
 
 1 + '^' P'l = T-T-7' 91 = 1, 9'i = 1 + d. 
 
 1 +d 
 
 Applying these, we find that Formulae 53, 54, 353, 123, 125, 323, 325 
 (some of which have not yet been explained) reduce to unity so that we 
 may consider the bias of the formulae to be investigated as measured rela- 
 tively to any one of these as a basis. The bias of any formula becomes 
 simply the value of that formula after substituting the above eight values 
 for po, p'o, go, q'o, Vh P'h Qh q'l- 
 
 The following are the results for index numbers by Formulae 1003, 7 
 or 9, 27 or 29. 
 
 ^+'^+rTl d- 
 
 1003 1 + & = o whence h = 2 (\ j^ d) ^^^ 
 
 7 or 9 l+& = l+dH ^ 1 whence h = -^— (2) 
 
 1+d 1+d 
 
 -^ d- 
 
 27 or 29 l+& = (l+d)« + d whence & = -^ + .. . (3) 
 
 2 + 
 
 the terms omitted in the last being negligible. 
 
 Equation (1) gives the bias of the singly biased arithmetic 
 
 and of the singly biased harmonic. 
 
 Equation (2) gives the bias of the doubly biased arithmetic 
 
 and of the doubly biased harmonic. 
 
 Equation (3) gives the bias of the singly biased geometric. 
 
 The equations are given in terms of upward bias but the corresponding 
 
 downward biases also (i.e. of Formulae 1013; 13 and 15; 23 and 25) are im- 
 
 pUcitly given merely by inverting, i.e. taking . 
 
 1+0 
 
 Evidently (as equation (2) shows) Formula 9, or Palgrave's formula, has 
 a double dose of upward biao as compared with the bias (shown by equa- 
 tion (1)) for 1003, the mean weight arithmetic. That is, besides the type 
 bias, which Formula 1003 has, there is the weight bias of 9 and the one is 
 equal to the other. The weight bias (given by equation 3) of the 
 geometric, Formula 29, is evidently larger than either of the (single) 
 biases as given in the first two equations. It is larger than the first, both 
 because its denominator is less by d and because there are other terms to 
 be added, although d is so small compared with 2 and with 2 + d and the 
 additive terms are also so small, each involving a power of d, that the 
 entire difference between the last equation and the first is negligible. 
 
 The above equations are not only absolutely true under the special con- 
 
APPENDIX I 
 
 395 
 
 ditions assumed but are approximately true in actual cases such as that 
 of the 36 commodities. The dissimilarity between the equations for the bias 
 of the arithmetic and harmonic index numbers (1003 and 1013) and that 
 for the weighted geometries (23, 25, 27, 29) might lead one to suppose that 
 they would give widely different results. But when we calculate them we 
 find they agree almost exactly, as the following table shows, giving the 
 bias (5) of both corresponding to various standard deviations (d). 
 
 d 
 
 h 
 
 h 
 
 
 Arithmetic 
 
 Geometric 
 
 
 Harmonic 
 
 (23, 25, 27, 
 
 
 (1003, 1013) 
 
 29) 
 
 5 
 
 .12 
 
 .12 
 
 10 
 
 .45 
 
 .45 
 
 20 
 
 1.67 
 
 1.67 
 
 30 
 
 3.46 
 
 3.48 
 
 40 
 
 5.72 
 
 5.77 
 
 50 
 
 8.34 
 
 8.45 
 
 200 
 
 25.00 
 
 25.99 
 
 We could, of course, make the equations absolutely exact by suitably 
 adapting the definition of dispersion to each particular case. But the object 
 of this Appendix note has been to show how the size of the bias is related 
 to the size of the dispersion of the original data. Where there is only 
 slight dispersion the error caused by using a biased formula is small but as 
 the dispersion increases the error thus introduced increases, and in a much 
 faster ratio. Consequently, in cases of wide dispersion, such as those of 
 the 36 commodities (for 1917 relatively to 1913), the upward bias of For- 
 mula 1, for instance, or the downward bias of 23, is very great. 
 
 For any particular set of statistics we can, by calculating the standard 
 deviation or dispersion index, and from it the bias of any biased formula, 
 tell in advance whether the use of that formula wiU introduce too large an 
 error to make its use permissible. ^ 
 
 Note to Chapter VI, § 1. If One Formula is the Time Antithesis of 
 Another, the "Other" is of the "One." This is very simply shown. Let 
 Poi stand for any index number, taken forward, i.e. for time "1" relatively 
 to time "0. " Our twofold procedure gives : 
 
 Starting with Poi 
 
 (1) By reversing the times, Pio 
 
 (2) By inverting the last, 
 
 Pio 
 which, therefore, is the time antithesis of the original Pqi- We are to show 
 
 1 See, for instance. Chapter XVI, § 6, for discussion as to the large bias in Sauerbeck's 
 index numbers. 
 
396 THE MAKING OF INDEX NUMBERS 
 
 that starting with the last formula and applying the same twofold pro- 
 cedure we shall reach, as its time antithesis, the original formula. 
 
 Starting, then, with — — 
 Pio 
 
 (1) By reversing the times, — — 
 
 (2) By inverting the last, Poi 
 which was to have been found. 
 
 Note to Chapter VII, § 6. The Cross between Two Factor Antitheses 
 Fulfills Test 2. Disciission. Let Poi be any given formula. Its factor an- 
 tithesis is _Ei^ -4- Qoi where Qoi is, of course, the formula corresponding 
 
 Spogo 
 to Poi appUed to quantities. Their cross or geometric average is 
 
 4 
 
 2po3o 
 
 This last formula fulfills Test 2 because its factor antithesis is, inter- 
 changing p's and q's, 
 
 V 
 
 QoiX^^i^-^Poi 
 
 250P0 
 
 and this, multiplied by the preceding, gives ^'^' , as the test requires. 
 
 Sgopo 
 We have considered the rectified formula for prices a cross between the 
 
 original formula Pqi and its time antithesis, -J- Qoi. 
 
 2pogo 
 But, evidently, the same expression may be written more symmetrically : 
 
 while, likewise, the rectification of Qoi is 
 
 2po?o A/ Poi' 
 
 In these forms for the rectified formulae the two factors are not index 
 numbers. The first factor, iu both cases, is the mean between the value 
 ratio and unity, or 100 per cent. Thus, if the value ratio is 121 per cent, 
 its square root, or the mean between it and 100 per cent, is 1 10 per cent. This 
 is what each index number, that for prices and that for quantities, would 
 be if they were equal ; that is, it is their geometric mean or average. 
 
 The other factor, in each case, is the multiplier or corrector of that aver- 
 age, which is necessary, in the one case, to produce the rectified price index, 
 and, in the other, to produce the rectified quantity index. These two 
 factors are reciprocals of each other, one magnifying and the other 
 reducing the average in a certain proportion. Thus if Poi is two per cent 
 greater than Qoi, this two per cent is apportioned equally on both sides of 
 
APPENDIX I 397 
 
 the mean, 110, — the rectified P being 110 X Vi^ (or about one per cent 
 above 110) and the rectified Q being 110 X VxM (or about one per cent 
 below 110). 
 
 The first factor \\ ^'^^ might be called the half-way ratio, being at 
 ^ 2po9o 
 once the mean between 100 per cent and the value ratio and also between 
 th e rec tified P and Q (or unrectified, for that matter) while the second factor 
 
 [— ^ or ■%/— might be called the jmce multiplier or quantity multiplier. 
 
 Qoi ' ^01 
 
 In these terms we may say that the rectified index numbers of prices 
 and quantities are each obtained from the half-way ratio by means of price 
 and quantity multipliers. 
 
 The reader may be interested in following through the application of 
 the preceding remarks to the rectification of Formula 3 (which is the same 
 as of 4, 5, 6, 17, 18, 19, 20, 53, 54, 69, or 60), the results of which are very 
 simple. 
 
 Thus, for prices, the result is 
 
 
 
 The four magnitudes entering into this expression are, of course, the 
 same as those entering into that already given for 103P and 103Q. By 
 merely a change in the order four different formulae are formed, two for 
 103P and two for 103Q. 
 
 Note A to Chapter VII, § 8. Given Two Time Antitheses, Their Respective 
 
 Factor Antitheses are Time Antitheses of Each Other. Let Poi and _j— be 
 
 any time antitheses and let Qoi and — — (that is, the same formulae applied 
 
 to quantities) likewise be time antitheses of each other. Then the factor 
 antitheses of the first two are 
 
 ^Ml4-0„xand?Mi^ 1 
 2po3o Spogo Qio 
 
 These are evidently time antitheses of each other because by interchanging 
 the "O's" and "I's" of either formula and then inverting, we turn each 
 into the other. 
 
 Note B to Chapter VII, § 8. Given Two Factor Antitheses, Their Re' 
 spective Time Antitheses are Factor Antitheses of Each Other. Let Poi and 
 
 — ^^ T- Qoi, be any two factor antitheses. Evidently their respective 
 2pogo 
 
 time antitheses, viz. — and Qia -i- ■ '^° , are also factor antitheses of 
 
 Pio 2pigi 
 
 each other. 
 
398 THE MAKING OF INDEX NUMBERS 
 
 Note to Chapter VII, § 9. Rectification May he First of Time Antitheses 
 and then of Factor Antitheses, or Vice Versa, or Simultaneously. In general 
 terms any quartet of formulae is 
 
 Pox 
 
 1 
 
 Pio 
 Zpigi 
 
 Spogo 
 1 
 
 The two crosses of time antitheses are 
 
 V 
 
 Poi X- 
 
 f/2pi£l\ 
 
 X 
 
 Spo^o 
 
 \QoJ \ ^ 
 
 (1) 
 
 (2) 
 
 the latter, (2), of which reduces to 
 
 2pogo 
 
 4 
 
 (2) 
 
 QoiX-^ 
 
 which is the factor antithesis of the former, (1), being obtainable from 
 it by interchanging the p's and q's and dividing into —El3l, 
 
 2po2o 
 
 The two crosses of factor antitheses are 
 
 Qro / 
 
 (3) 
 
 (4) 
 
 These are time antitheses of each other ; if in either we reverse and 1 and 
 invert we get the other. 
 
 Inspection will also show that the cross of either of the above pairs of 
 crosses as well as the fourth root of the product of the original quartet will 
 give the same result, viz. 
 
APPENDIX I 399 
 
 4 
 
 4/PoiQio(Spi9i)^ ,g. 
 
 PioQoi(2pogo)2 
 
 This expression (5) is the general formula by which we may rectify any 
 index number formula, Fqi. by both tests at once. 
 
 Note A to Chapter VII, § 19. Crossing the Two Crosses (i.e. the One 
 Obtained Arithmetically and the Other, Harmonically) . While neither arith- 
 metic nor harmonic crossing of two time antitheses will yield an index 
 number fulfilling the time reversal test the geometric cross of these two crosses 
 will do so and will in fact be identical with the geometric cross of the 
 formulae themselves, as the reader can readily prove. 
 
 Moreover, without using any such geometric crossing we can approach 
 the same result as a limit by continued application of the arithmetic and 
 harmonic crossing as follows: (1) cross the original antithetical formulae 
 arithmetically and harmonically ; (2) cross the last two results arithmet- 
 ically and harmonically ; (3) again cross the last two results arithmetically 
 and harmonically; and so on indefinitely. In this series the two terms 
 approach each other so rapidly that two or three steps will suffice, practi- 
 cally, to make them equal. Compare Appendix I, Note to Chapter IX, § 1. 
 
 Note B to Chapter VII, § 19. Two Geometric Time Antitheses May be 
 Crossed Aggregatively as May Two Aggregative Time Antitheses. Any two 
 geometric time antitheses, such as 23 and 29, may be written, in fractional 
 form, as follows : 
 
 ^ 2P»9ypjPogo X p\pWo X . . . 
 
 ^^'^P0V020 X p'oP'<^'o X 
 
 and 
 
 2pi5l; 
 
 29 = VPi P'^' X p'lP'ig'' X 
 
 2pigi 
 
 VpoPi2' X p'oP''8'' X . . . 
 
 If written in the above form they may readily be combined aggregatively 
 by adding the two above numerators for the new numerator and adding 
 the two denominators for the new denominator. 
 
 Likewise the aggregatives (Formulae 53 and 59) may be crossed aggre- 
 gatively, the result being 
 
 Spo3o + Spogi 
 
 Each of these aggregative crosses (the aggregative cross of the geometries 
 and the aggregative cross of the aggregatives) conforms to the time test, 
 as may readily be proved by the twofold procedure. The last named ag- 
 gregative cross (between the two aggregative time antitheses) is interest- 
 ing mathematically because its factor antithesis turns out to be a new 
 and curious average of Formulae 53 and 59 very different from any of the 
 
 other averages used in this book, viz. 1 + (53) h- 1 + - — -. 
 
 (59) 
 
400 
 
 THE MAKING OF INDEX NUMBERS 
 
 These aggregative means agree closely with the geometric means. 
 
 Thus the geometric is the only one of our six types of averages which 
 can be used universally ^ot crossing formulce themselves (any two time an- 
 titheses or any two factor antitheses) so as to satisfy the time reversal 
 and factor reversal tests. Of the other types of average only the aggre- 
 gative will satisfy the time reversal test and its application is limited to 
 crossing two geometric time antitheses or two aggregative time antitheses, 
 as just shown. 
 
 Note to Chapter VIII, § 6. Formulce IOO4, IOI4, II24, 1134, 1144 are 
 Factor Antitheses of 1003, WIS, 1123, 1133, 1143, Respectively, Although 
 
 Different Cross-Weiglttings of S^and^^ 
 (Prices') 
 
 II$3&II54 
 2154.2,5, 
 3154.31^3 
 
 41^^,^,54 
 
 13 74 15 16 17 16 
 
 Chart 63 P. There is close agreement between the four methods of 
 crossing weights. The antithesis of each also agrees closely with its 
 original, being sensibly identical therewith except in the last two cases and 
 absolutely so in the first. 
 
 Derived Otherwise. We are to show that if (1) P'oi and P"oi, differing 
 only in weights, be combined so as to form another formula, Pqi, by crossing 
 
 their weights, and if (2) their factor antitheses ( -^^ -4- Q'oi and 
 
 ^2pogo 
 
 -5- Q "01 1 be likewise combined to form another ( namely, ^'^^ -5- Qoi ) 1 
 2po9o / V 2po3o / 
 
 the latter will be the factor antithesis of Pq\. 
 
 When this is stated algebraically it becomes almost self-evident. 
 
 If P'oi and P"oi be combined into Poi, andif their factor antitheses, namely. 
 
 Spo3o 
 
 Q'o, and ?Mi + 
 Spojo 
 
 Q"o 
 
 be combined into 
 
APPENDIX I 
 
 401 
 
 Spogo 
 
 -i-Qo 
 
 this is evidently the factor antithesis of Poi (Qoi being of the same model 
 as Foi since by hypothesis the former is of the same model as P'oi and P"oi, 
 and the latter as Q'oi and Q"oi, while all these four are of the same model 
 as each other). 
 
 Note to Chapter VIII, § 10. Unlike Formula Crossing, Weight Crossing 
 May be Not Only Geometrically but Arithmetically and Harmonically Done. 
 
 Different Cross-We/ghtings of 5^ and 5^ 
 
 (Quantities) ^^^^^^^ 
 
 /l53SHI5t 
 
 2153.2154 
 5154.3,53 
 
 13 
 
 M '15 16 17 
 
 Chart 63Q. Analogous to Chart 63 P. 
 
 18 
 
 It will be remembered that the geometric method of crossing weights gives 
 the same result from crossing weights / and 7F as from crossing weights 
 // and ///. But this is not true of the arithmetic or harmonic methods of 
 crossing weights. Just as the cross formulce, 123 and 125, slightly differ 
 from each other (as do 133 and 135, 143 and 145), so do their cross weight 
 analogues slightly differ from each other if the crossing is performed 
 arithmetically, and also if it is performed harmonically. 
 
 Since crossing the weights by means of the arithmetic method or by 
 means of the harmonic method has never been suggested by other writers, 
 except as applied to the aggregative index number, they have been cal- 
 culated here only for that type of index number. The results do not, of 
 course, differ very appreciably from those of the geometric method and 
 the same agreement between the results of crossing by the various possible 
 methods would be found, though not quite to the same degree, if the other 
 types were calculated. 
 
 The identification numbers of the arithmetic cross weighted index numbers 
 begin with 2000 ; and the identification numbers of the harmonic with 3000. 
 
 As to those beginning with 4000, Formula 4153 is a cross weight (of 53 and 
 64) by means of a weighted arithmetic mean of the weights. Formula 4154 
 is its factor antithesis and 4353 the cross (geometrical) of 4153 and 4154. 
 
 Graphically, Charts 63P and 63Q show the closeness of the four methods 
 of crossing the weights of Formulae 53 and 54. They could scarcely fail 
 
402 
 
 THE MAKING OF INDEX NUMBERS 
 
 to agree closely because Fonnulse 53 and 54 are themselves so close together. 
 It is noteworthy that Formula 4153 differs more from its factor antithesis 
 than any other combination of 53 and 54 differs from its factor antithesis. 
 
 Charts 64P and 64Q show the final result after double rectification of 
 all the cross weight formulse as compared with the cross Formula 353. 
 They are quite indistinguishable from each other and from Formula 353. 
 That is, all of the foregoing new cross weight formulae lie in practical coin- 
 cidence with the middle tine of the five tine fork. So close are the new 
 middle tine curves to Formulae 1153, 1154, etc., that the differences are 
 of no practical significance. 
 
 It is worth noting, however, that of the four methods of weight crossing, 
 namely, those used in Formulae 1153, 2153, 3153, 4153, we can show reason 
 for decided preferences. These will soon be discussed. The only point to be 
 emphasized here is that Formula 2153 formed by arithmetically averaging 
 the weights of Nos. 53 and 54 is the only one of the four which necessarily 
 falls between 53 and 54, or necessarily agrees with these if they agree with 
 each other. 
 
 We are not justified in taking for granted, as has been done hitherto, 
 that any cross weight formula lies between the two original formulae (as 
 is the case with cross formulae). Examination shows that it is not true of 
 the geometric, harmonic, or Formula 4153. 
 
 Let us take up these three in order. First, consider the geometric 
 method of crossing the weights. Suppose that of the price relatives to be 
 averaged, half are 100 per cent and the remaining half are 300 per cent. 
 Next let us suppose the numerical values of the weights for the base year 
 to be (for the first 18 relatives of 100 each) respectively 2, 0, 2, 0, 2, 0, etc., 
 in alternation, and the numerical values of the given year weights (for the 
 same 18 relatives) to be 0, 2, 0, 2, 0, 2, etc., in alternation ; while for 
 the second 18 price relatives, of 300 each, the weights are all unity. 
 
 For convenience we may tabulate : 
 
 Price Relativeb 
 
 Weighting 
 
 Base Year 
 
 Given Year 
 
 First half • 
 
 100 per cent 
 100 per cent 
 100 per cent 
 100 per cent 
 etc. 
 
 2 
 
 2 
 
 
 
 2 
 
 2 
 
 Second half • 
 
 300 per cent 
 300 per cent 
 300 per cent 
 300 per cent 
 etc. 
 
 1 
 
 1 
 1 
 1 
 
 1 
 1 
 1 
 
 1 
 
APPENDIX I 403 
 
 It is clear that, under the base year system of weighting, in the first half 
 every even item has a zero weight and disappears leaving only the odd terms 
 to be averaged. But these are all alike (100 per cent) and have each the 
 same weights (2). In the second half the price relatives are all 300 and 
 have weights 1. It foUows that the average of all reduces to an average of 
 nine terms each weighted as though it were two and 18 terms each weighted 
 once ; in other words, an average of two sets of 18 terms each, or a simple 
 average of 100 per cent and 300 per cent. 
 
 Turning to the given year weights we find the same result ; for in that 
 case every odd term disappears in the first half, again leaving nine doubly 
 weighted lOO's to be averaged in with 18 singly weighted 300's. 
 
 It follows that the resulting index numbers are the same, whether base 
 year weights or given year weights are used. In either case, we have the 
 same figures 300 and 100 to be averaged equally weighted, so that the aver- 
 age of 300 and 100 must be the same in both cases. (This must be true 
 whether this average be arithmetic, geometric, or harmonic. If the average 
 is arithmetic, the index number is 200 ; if geometric, 173 ; if harmonic, 150.) 
 
 So much for crossing the formulae. 
 
 When we cross the weights the result is surprisingly different. For the 
 w eights in the first half are all zero (V2 X 0, Vo X 2, V2 X 0, 
 Vo X 2, etc.) ! The weights in the second half are all unity. Hence, 
 the entire first half disappears and the average becomes the average of 
 18 terms of 300 per cent each, which is 300 per cent. 
 
 We have here, therefore, a case where the results of base year weighting 
 and of given year weighting agree (being each, say, 200) whereas when we 
 take the geometric mean of the weights we get 300 ! 
 
 It stands to reason, I think, that if base year weighting and given year 
 weighting both give identical index numbers (as 200), any mean weighting 
 which is worth while ought to give the same result (200), and not be ca- 
 pable of giving a result (300) larger than either. 
 
 Again, if the base year and given year weighting give different results, 
 such as 149 and 151, we may reasonably demand that the result of using 
 mean weights shall Ue between these figures instead of lying far outside, 
 Uke 300. 
 
 Of course, what has been proved by using zero weights would be true, 
 though in less degree, if weights not zero, but very snmll, were used. 
 
 This possibility of miscarriage is even greater in the case of the harmonic 
 average. For each harmonic average lies on the opposite side of the geo- 
 metric from the arithmetic. 
 
 We find some examples of such miscarriages of the cross weighted 
 formulae. The median shows such a miscarriage. Thus the base year 
 weighting (Formula 33) gives (for quantities, for 1918) 122.39 and the 
 given year weighting (Formula 39), 123.50, but the geometrically cross 
 weighted median (1133), instead of lying between 122.39 and 123.50, is 
 122.27. A few of the chain figures (for quantities 1917 and 1918) are 
 still further out of line. 
 
 For the aggregative Formula 1153 (with geometric cross weights) and 
 3153 (with harmonic cross weights) the figures in a few cases do not remain 
 between those for 53 and 54 but likewise jump over the traces. 
 
404 
 
 THE MAKING OF INDEX NUMBERS 
 
 The only case where this happens with the geometric cross weights is 
 for prices for 1918 (chain) where Formiolae 53 and 54 give 178.56 and 178.43 
 while 1153 gives 178.37. 
 
 The harmonic likewise escapes the confines of Formulae 53 and 54 in 
 several instances for the fixed base index numbers. Thus for prices : 
 
 For 1917, Formulae 53 and 54 give 162.07 and 
 
 161.05 
 whereas Formula 3153 gives 162.11 
 
 For 1918, Formulae 53 and 54 give 177.87 and 
 
 177.43 
 whereas Formula 3153 gives 176.94 
 
 353 Compared with 
 its Cross -Weight Rivals 
 (Prices) 
 
 ■13 M 15 16 17 75 
 
 Chart 64P. On the score of accuracy there is almost no preference 
 between the doubly rectified cross weight formulae and 353. 
 
 As to Formula 4153, it presents the allurement of using a weighted aver- 
 age of weights. But this overdoes the effort to use weights somewhat as 
 a double negative overdoes negation. 
 
 A simple illustration will suffice to show that Formula 4153 fails to 
 split the difference between 53 and 54 and that its results are unfair. Sup- 
 pose the price of wheat in 1913 was po = $1 a bushel and in 1914, pi = $20 
 a bushel, while rubber was p'o = $20 a pound in 1913 and p'l = $1 a 
 pound in 1914; and that their quantities were 90 = 3 million bushels and 
 g'o = 3 million pounds respectively in 1913, and qi = 300 million bushels 
 and q'\ = 300 million pounds respectively in 1914. Then, by Formula 
 63, we find the average price change of these two commodities to be 
 
 P'^° + P^g'" = 20X3+ 1X3 ^ 20+ 1 ^ 100 ,,^t. 
 Pogo + p'03'0 1X3+20X3 1+20 
 
APPENDIX I 
 
 405 
 
 By Formula 54, we have 
 
 Pigi + p\q\ ^ 20 X 300 + 1 X 300 ^ 20 + 1 
 Poqi + p'oq'i 1 X 300 + 20 X 300 1+20 
 
 = 100 per cent. 
 
 Thus FormulsB 53 and 54 agree. But Formula 4153 does not lie be- 
 tween, i.e. does not agree with both. 
 
 Formula 4153 is 
 
 oq/ I X3 +20 X 
 \ 1+20 
 
 pi / Pogo + PigA ^ p'j ( P'oq'o + P\q'i \ 
 \ Po + Pi / \ p'o + p'l ) 
 
 \ Po + Pi J \ p'o + p'l ) 
 
 \a.i /' 20 X 3 + 1 X 300 \ 
 / V 20 + 1 / 
 
 I.e. 
 
 '1 X 3 + 20 X 300 
 
 ) + 20( 
 
 20 X 3 + 1 X 300 
 
 1 + 20 ; \ 20 + 1 
 
 353 Compared with 
 
 its CrosS'Vfeight Rivals, 
 (Quantities) 
 
 ) 
 
 = 912 per cent. 
 
 '13 
 
 7^ 75 7S 77 
 
 Chart 64Q. Analogous to Chart 64 P. 
 
 73 
 
 Each bracket is an average. Inside the brackets the use of the prices 
 1 and 20 as weights for averaging the quantities 3 and 300 gives the greater 
 weight to the 300 in the left brackets and to the 3 in the right brackets. 
 Hence the resulting average, i.e. the value of the bracket, is nearer 300 
 in the case of the left brackets and nearer 3 in the other two. In other 
 words, that quantity always dominates which pertains to the year in which 
 the commodity happens to have the higher price. 
 
 Now it stands to reason that this is unfair, not only because the result 
 (912 per cent) lies outside the two coincident results (100 per cent) of For- 
 mulae 53 and 54, but also because their equality itself stands to reason. 
 
406 THE MAKING OF INDEX NUMBERS 
 
 Formula 53 gives the index number when the quantities are 3 and 3 ; and 
 Formula 54 gives the index number when the quantities are 300 and 300. 
 This is clearly as it should be since the weighting is purely relative. 
 If then the base year weighting and given year weighting are thus relatively 
 the same for the two commodities we surely have no right to spoil this same- 
 ness by any combination of these two methods of weighting. 
 
 The numerical example given shows that weighting the quantities by 
 prices (before averaging them for use as weights for prices) introduces a 
 wrong principle. While it does not bias the result it produces a haphazard 
 favoritism, favoring pi in the numerator or po in the denominator. This 
 is unfair, for favoring pi in the numerator relatively to p'l in the numerator 
 influences the resulting ratio in the same direction as favoring po in the 
 denominator relatively to p'o in the denominator. 
 
 Formula 4153 represents distinctly the most erratic of the methods of 
 crossing weights. The geometric will follow closely the arithmetic, both 
 being simple ; and the harmonic will be close to the geometric. But For- 
 mula 4153 introduces in the weighting a new disturbing element. Accord- 
 ingly, we find that Formula 4153 does not remain between 53 and 54 as 
 often even as do 1153 or 3153. 
 
 We find for prices (fixed base) the following cases where Formula 4153 
 falls outside the range between 53 and 54. 
 
 For 1916 Formulae 53 and 54 give 114.08 
 and 114.35 
 
 For 1916 Formula 4153 gives 114.44 
 
 For 1917 Formulae 53 and 54 give 162.07 
 
 and 161.05 
 
 For 1917 Formula 4153 gives 162.40 
 
 For 1918 Formulae 53 and 54 give 177.87 
 
 and 177.43 
 
 For 1918 Formula 4153 gives 178.26 
 
 For quantities we find similar discrepancies for 1918. There are like dis- 
 crepancies in the chain numbers. 
 
 After rectification by Test 2 the results (for Formula 4353) are appre- 
 ciably improved. 
 
 Formula 2153 remains as the only cross weight formula which always 
 and necessarily falls between 53 and 54. 
 
 Formula 2153 is obtained by crossing the weights of 53 and 54 arith- 
 meiically (by taking the simple arithmetic average of their weights). We 
 shall show first that this cross weight formula is identical with the cross 
 formula obtained by crossing 53 and 54 aggregatively. In its r61e as a cross 
 weight formula (arithmetically crossed) it is 
 
APPENDIX I 407 
 
 In its r61e as a cross formula (aggregatively crossed) it is 
 
 Spogi + spogo 
 
 That the two are identical is evident by canceling the "2's" in the first 
 and multiplying out. 
 
 The last formula, being a mean or average of 53 and 54, must necessarily 
 lie between 53 and 54, which was to have been proved. 
 
 In this connection it is interesting to note that, besides Formula 2153, 
 there could be constructed other formulae which are both cross formulcB 
 and cross weight formulce. Formula 2 153 is such as between 53 and 54, aggre- 
 gative index numbers. But similar results can be had with arithmetic 
 index numbers and also with harmonic index numbers. In each of these 
 cases we get precisely the same result by taking two formulae (say, 3 and 9, 
 or 5 and 7, or 13 and 19, or 15 and 17) of the same model and crossing their 
 weights arithmetically as by crossing the formulae themselves aggregatively. 
 
 Note to Chapter IX, § 1. The {Geometric) Cross of Formulce 8053 and 
 8054 is Identical with 353. Using a for Formula 53, and 6 for 54, 8053 is 
 
 , 8054 is . Their cross or geometric mean is 
 
 a 
 
 V 
 
 2-±i X ^— = VSS = VSS X 54 - 353. 
 
 a 
 
 Note to Chapter XI, § 4. // the Mode is Above the Geometric Forward It 
 is Below Backward. This is most easily made evident by considering 
 Charts IIP and IIQ. We saw that the arithmetic forward and backward 
 are not prolongations of each other because the arithmetic fails to satisfy 
 Test 1 ; and the same is true of the harmonic forward and backward. But 
 for any formula which does satisfy Test 1, the forward and backward forms 
 will be prolongations of each other. This is true of all the simple index 
 numbers (except the arithmetic and harmonic) including the geometric 
 and mode. Consequently, we have the picture simply of two straight 
 lines intersecting at the origin, one for the geometric forward and back- 
 ward, and the other for the mode forward and backward. It is, therefore, 
 clear that if on one side of the origin the mode lies above the geometric, 
 it must lie below it on the other. 
 
 Note to Chapter XI, § 10. Derivation of Probable Error of Any of the 13 
 Formulce Considered as Equally Good Observations. Assuming that the 13 
 index numbers are equally good, the formula for their probable error, i.e. 
 the as-likely-as-not deviation (from their mean) of any of the 13 observa- 
 
 tions taken at random is .6745-1 / where d denotes the deviation from 
 
 their mean of any of the observations, and n denotes the number (in this 
 case 13) of the observations. 
 
408 THE MAKING OF INDEX NUMBERS 
 
 The expression for the "probable error" of the mean itself is the preced- 
 ing expression divided by Vn. 
 
 Note to Chapter XI, § 11. Does "Skewness" of Dispersion Matter f 
 Hitherto one of the chief questions investigated by students of index num- 
 bers is the question of the distribution of the data averaged, the sort of 
 dispersion, whether in particular it is, or is not, "skew." Thus we know, 
 from the work of Wesley C. Mitchell and others, that price relatives dis- 
 perse far more widely upward than downward, the reason obviously being 
 that there is more room for dispersion upward. In the downward direc- 
 tion they are limited by zero while upward there is no limit. 
 
 It has been assumed that the character of this distribution will have a 
 determining influence in the choice of the best index number. Much is 
 made of this consideration by Walsh, Edgeworth, and others. Elaborate 
 arguments have been constructed to show that the geometric mean or some 
 other is the appropriate mean to use in constructing index numbers based 
 on the idea that the dispersion is supposed to be more symmetrical "geo- 
 metrically" than it is "arithmetically." 
 
 It will be noted that in this book we have had no occasion whatever to 
 invoke this consideration. In choosing the formula for an index number 
 the skewness or asymmetry of distribution of the terms averaged is 
 of absolutely no consequence. This may seem a most revolutionary 
 idea. There has been a growing tendency to take account of the 
 distribution of the data in any social problem before deciding on 
 whether the geometric or the arithmetic process of averaging should be 
 used. I am offering no objection to this in general. On the contrary it 
 is of great importance for many purposes in social problems. Even aver- 
 aging human heights and weights should take the character of the distri- 
 bution into account. 
 
 But in the realm of index numbers the case is different and for a very 
 simple reason. Unlike heights or weights, price relatives or quantity rela- 
 tives are ratios of two terms either of which two may be taken as the nu- 
 merator. Any ratio is necessarily a double ended affair. If used in one di- 
 rection the ratios disperse in one way while if used in the other direction 
 they disperse in precisely the opposite way. The large ratios for one of 
 the two ways become the small ratios the other way and in the same rela- 
 tive degree. Thus, if sugar rises from 10 cents to 20 cents and wheat from 
 $1 to $3 between two times or places the price relatives are 200 per cent 
 and 300 per cent, the wheat relative being a half greater than the other. 
 But, reversing the direction of the comparison, the price relatives are 50 per 
 cent and 33^ per cent, the sugar relative being now a half greater than the other. 
 
 Charts IIP and IIQ illustrate the reversal of the dispersion through the 
 reversal of the times. 
 
 When, therefore, we rectify by Test 1 thus taking account, in equal terms, 
 of these two opposite dispersions, any skewness of distribution enters in 
 both ways and cancels itself out. Consequently, in our final results, such 
 as 309, 323, and 353, there is no trace of any effect of skewness. These 
 three, so far as they differ at all, differ sometimes in one direction and 
 sometimes in the other, although 309, for instance, is made up from index 
 numbers affected greatly by skewness of distribution. 
 
APPENDIX I 
 
 409 
 
 D'istribution of 14^7 Price Relatives 
 
 (Forward and Backward) 
 
 \000- 
 
 FORWARD 
 
 
 10- 
 
 J" 
 
 BACKWARO 
 
 itUMBER OF COMMODITIES 
 
 Chart 65. Showing how, in the ratio chart, 
 the distribution of the price relatives, taken for- 
 ward and backward, is exactly reversed in skew- 
 ness and order of averages except as to the arith- 
 metic and harmonic (which exchange places). 
 
410 THE MAKING OF INDEX NUMBERS 
 
 If we plot the two distributions on an ordinary frequency curve such as 
 Chart 62 it is true that the dispersion in both cases will be wider at the 
 top than at the bottom (or, as it is usually plotted, at the right than at the 
 left). But, and this is significant, the commodities which are at the top 
 in one case are at the bottom in the other and vice versa. 
 
 The real reason for the greater dispersion upward than downward lies 
 in the arithmetical scale by which we measure. If we use the ratio chart 
 we cannot even say that the distribution is skew, and if skew, in any par- 
 ticular direction. Chart 65 shows the distribution of the 1437 price rela- 
 tives of the War Industries Board for 1917 relatively to the year July, 1913- 
 July, 1914 and (in fainter and dotted lines) the distribution reversed. It 
 will be observed that the skewness is reversed, the mode being the least 
 of the five averages in the original distribution and the greatest in the 
 dotted figure. The order of the five averages is reversed in the two dis- 
 tributions except that the arithmetic and harmonic exchange places as 
 usual. When ratio charting is used we may say that a "normal distri- 
 bution" is one which is symmetrical about the mode, or geometric, or me- 
 dian, which three will normally agree, while the arithmetic will always be 
 above and the harmonic below these three ; there is exactly as much chance 
 of skewness in one direction as the other. 
 
 Note to Chapter XI, § 13. Formulce 53 and 5 If. are Sometimes Slightly 
 Biased. Whether 54 is greater or less than 53 depends on whether the 
 price relatives are positively or negatively correlated with the quantity 
 relatives. 
 
 The price relatives and the quantity relatives (1913 being the base) for 
 the 36 commodities used here are correlated as follows : 
 
 1914 + .265 
 
 1915 -f- .023 
 
 1916 + .035 
 
 1917 - .133 
 
 1918 - .250 
 
 These correlations are mostly too small to have much significance and 
 are about equally positive and negative. A clearer and more consistent 
 case of correlation between price and quantity movements is given by Pro- 
 fessor Persons, who finds that for 12 leading crops the price and quantity 
 movements are negatively correlated with the high coefficient of — .88. 
 When the correlation is positive it means that the weights {i.e. the q's) 
 in Formula 53, which has tne system called weighting /, are analogous in 
 this respect to the system weighted / for all the other types of index num- 
 bers. It will be remembered that for the arithmetic, harmonic, geometric, 
 and median (and, theoretically, the mode), weighting / imparted a down- 
 ward bias and weighting IV imparted an upward bias. 
 
 This was due to the price element in the weights which in weighting IV 
 tended to associate a large weight with a large price relative and a small 
 weight with a small price relative, thus overweighting the high and pro- 
 ducing the upward bias; with weighting / the opposite situation holds 
 true. 
 
 But in the case of the aggregative type, the weights contain no price 
 
APPENDIX I 411 
 
 element, as the weights are mere quantities. Yet the same effect is pro- 
 duced if these quantities are positively correlated with the price move- 
 ments ; for we then have the same tendency to an association of large 
 weights with the large price relatives and small with the small ; only that 
 tendency is much weaker — unless the correlation is 100 per cent so that 
 the quantities behave exactly as though they were prices. 
 
 We would expect, then, that wherever correlation is positive we would 
 find the aggregative IV, or 59 (or 54), above the aggregative /, or 53, just 
 as we found, for the other types, 3 below 9, 13 below 19, 23 below 29, and 
 33 below 39. And this is just what we do find except that the differences 
 for the aggregative are much less than for the other types. On the other 
 hand, where the correlation is negative we would expect to find the oppo- 
 site and so we do. That is, in our calculations for the 36 commodities, 53 
 lies below 54 (or 59) in 1914, 1915, 1916, when the correlation is positive, 
 but above in 1917 and 1918 when the correlation is negative. This is 
 shown in the upper tier of Charts 39P and 39Q. 
 
 But for Persons' figures for prices and quantities of 12 crops (relatively 
 to the base 1910) 53 is always (except once) above 54 (or 59) showing a 
 definite upward bias of 53 due to the definite and high negative correlation, 
 i.e. to the fact that big crops make for low prices and vice versa. This is 
 shown in Charts 47P, 47Q, 48P, 48Q. 
 
 Some readers will be asking whether there is not always some upward 
 bias in 53 and downward bias in 54 aside from mere error in either direction. 
 The answer is that, while in the case of crops a negative correlation is found 
 because crops here represent supply, prices are affected also by demand, 
 and the quantities in our formulae are about as likely to represent changes 
 in demand as changes in supply. As prices go up with increased demand 
 and down with increased supply the chances seem about even whether the 
 actual quantities marketed will be positively or negatively correlated with 
 prices, and all the figures we have, except these crop figures, sustain this 
 conclusion. 
 
 Moreover, the same logic applies, not only to this comparison of aggre- 
 gatives, but to comparisons of two arithmetics, or two harmonics, etc., 
 where the weighting systems differ only as to the quantity element. In 
 all these cases weighting I and weighting II differ from each other only as 
 to the quantities as do III and IV from each other. Thus the same rea- 
 soning by which aggregatives / and IV differ appHes to arithmetic I and 77, 
 or to arithmetic 777 and 7F as well as to the corresponding harmonics and 
 the corresponding geometries. An inspection of the charts shows just what 
 we are thus led to expect. In all these cases, both for the price indexes 
 and the quantity indexes, with trifling exceptions, the 7 is below the 77 
 (and the 777 below the IV) for 1914, 1915, and 1916 and above for 1917 
 and 1918. 
 
 The only exceptions are for the quantity indexes of 1918 for the har- 
 monics and geometries where 77 very sUghtly exceeds 7, presumably owing 
 to some disturbing influence of the greatly aberrant quantity, skins. 
 
 This faithful correspondence between correlation coefficients and the 
 influence of quantities in the weighting of the price index is certainly re- 
 markable when we consider how infinitesimal are the influences thus traced. 
 
412 THE MAKING OF INDEX NUMBERS 
 
 Even the sluggish median reflects the same influences with few exceptions. 
 We can also say that almost always the larger the correlation coefficient 
 the larger the divergence found between 53 and 54. Thus the behavior 
 of all of our weighting systems has been pretty fully analyzed. The 
 large differences made (to price indexes) are those made by price elements 
 in the weights and the small by the quantity elements in proportion to 
 their correlation with the price relatives. 
 
 We see, then, that Laspeyres' and Paasche's formulre (53 and 54) are 
 usually close to each other even when slightly biased. In order to study 
 the consequences of a really wide difference between them we pick out 
 from among our 36 commodities "rubber" and "skins" and calculate the 
 index number for these two only, and then do the same for "lumber" and 
 "wool." The first pair are chosen to make 53 most exceed 54, and the 
 second to make 54 most exceed 53. The reason is that the first pair, 
 rubber and skins, happen, during the period covered, to have had their 
 prices most affected by supply so that their quantities and prices tended to 
 move in opposite directions. The quantity of rubber marketed rose and 
 its price fell; the quantity of skins fell and the price rose enormously. 
 Lumber and wool, on the other hand, were affected chiefly by demand. An 
 increase of demand drove up the price of wool much beyond the average 
 rise of prices, while the quantity marketed also increased ; contrariwise, a 
 decrease of demand kept the price of lumber far behind the average while 
 the quantity marketed decreased. 
 
 That is, the p's and q's of rubber are correlated negatively, as are those of 
 skins, while the p's and q's of lumber are correlated positively as are those of wool. 
 
 As we have seen, when negative correlation prevails, 53 exceeds 54, and 
 when positive, 54 exceeds 53. In the present case the figures are as given 
 in Table 52 and Table 53. Here, occasionally, are considerable differences 
 between the results obtained by using Formula 53 or Formula 54. In 
 the less extreme case of lumber and wool, the maximum excess of 54 over 
 53 is only about eight per cent (for 1918), while, in the much more extreme 
 case of rubber and skins 53 exceeds 54 by 32 per cent in 1918. 
 
 One reason why the figures were worked out for such non-representative 
 cases was to discover whether Formula 2153 would still be able to serve as a 
 good short cut for 353. Table 54 and Table 55 show that it would be a good 
 substitute for the less extreme case of lumber and wool, but not always 
 very good for the other. 
 
 It wiU be seen that, in the less extreme case of lumber and wool 2153 devi- 
 ates from 353 more than a third of one per cent in only one instance, that of 
 quantities in 1918, when the deviation amounts to nine tenths of one per 
 cent. In the more extreme case of rubber and skins, 2153 deviates by 
 over one per cent four times out of ten, the deviation reaching five per 
 cent for prices in 1918 (when 53 exceeds 54 by 32 per cent and 353Q is over 
 200 per cent).^ Such deviations are, of course, quite impossible when, in- 
 stead of two culled commodities, a larger number of commodities, unculled, 
 are included. 
 
 Note to Chapter XII, § 1. Method Used for Ranking Formulae in Close- 
 ness to S5S. The method of ranking the 134 index numbers relatively to 
 > See Appendix I, Note to Chapter XV, S 2. 
 
APPENDIX I 413 
 
 Formula 353 as ideal consists in : (1) finding the difference between any 
 given index number and the ideal for each year (1914-1918) ; (2) reducing 
 these differences to percentages of the ideal index number; (3) further 
 adjusting them in inverse proportion to the dispersion index referred to in 
 Appendix I, Note to Chapter V, § 11; and (4) taking the simple arith- 
 metic average of these deviations disregarding plus and minus signs. 
 
 This method of grading our formulae is not the most accurate possible 
 but is accurate enough for our purpose and much more easily computed 
 than the most accurate. The resulting order of formulae is probably al- 
 most exactly the same as if a greater refinement of method were employed. 
 The third step is inserted on the theory that a year of very wide dispersion, 
 like 1917, would naturally show wider differences among formulae than 
 would a year of small dispersion, like 1914, and that, therefore, in reckoning 
 the distance of any index number from the ideal a small percentage distance 
 in 1914 should count as much as a large one in 1917. 
 
 Note to Chapter XIII, § 1. The Algebraic Expression of the Circular 
 Test. Let the three cities, or years, be designated as 1, 2, and 3, and let 
 the index numbers representing the ratios between their price levels be 
 Pn, P23, P31 (and also, of course, their reverse, P21, P32, P13). The pro- 
 posed test is that any particular index formula should yield results which 
 will make Pu X P23 = P13 or will make P12 X P23 X P31 = 1. These 
 
 two conditions are equivalent if Pis = — — (i.e. if our "time reversal" test 
 
 P31 
 
 is satisfied) as is evident by substituting — for Pu in the first formula 
 
 P31 
 
 (P12P23 = P13), and clearing fractions. The result is evidently the second 
 (P12P23P31 = 1). In other words, the product of the three index numbers 
 taken in the same direction around the triangle is required, by the supposed 
 test, to be unity. 
 
 Note A to Chapter XIII, § 4. The Simple or Constant Weighted Geo- 
 metric (9021) Conforms to the Circular Test. That the simple geometric 
 (21) or constant weighted geometric (9021) conforms to the circular test 
 is easily shown. Formula 9021 is 
 
 W^W 
 
 where w, w', etc., are constant weights, i.e. the same for all the years, 0, 1, 
 etc. The above formula is written for the index number of year " 1 " rela- 
 tively to year "0," i.e., as we pass from "0" to "1." Passing from "1" 
 to "2" we have the following : 
 
 \1MSP^ 
 
 '1 , V . 
 
 ^Pl/ \p 
 
 and to complete the circuit, passing from "2" to "0,' 
 
 2ti 
 
 "^^^XAW---- 
 
414 
 
 THE MAKING OF INDEX NUMBERS 
 
 < 
 
 zn 
 
 '^ 
 I— I 
 
 < 
 
 So 
 
 
 o 
 
 I— I 
 
 
 
 ■* in 
 
 lO CO 
 
 
 
 lO ■* 
 
 t^ IM 
 
 
 ^ 
 
 CO O 
 
 t^ lO 
 
 
 y 
 
 O 'H 
 
 lO o 
 
 
 CO 
 
 (N ^ 
 
 
 <-H O 
 
 00 --• 
 
 
 t- 
 
 CO CO 
 
 T-i t^ 
 
 
 
 ■* o 
 
 o o 
 
 
 iH 
 
 (M Tl< 
 
 00 CO 
 
 
 
 CO 
 
 (N IN 
 
 ID 
 
 
 
 
 
 
 
 > 
 
 
 05 00 
 
 CO o 
 
 to 
 
 >* lO 
 
 CO -* 
 
 1 
 
 •H 
 
 CO CO 
 
 ^ CD 
 
 H 
 
 Ti 
 
 <N 00 
 
 O 00 
 
 
 
 <N 
 
 (M I-H 
 
 
 CO CO 
 
 CO lo 
 
 
 lO 
 
 CO ^ 
 
 CO Oi 
 
 
 05 -* 
 
 00 05 
 
 P 
 
 ^ 
 
 Oi <£> 
 
 t^ CD 
 
 a 
 
 
 1— I 
 
 1-t 1—1 
 
 
 <n CO 
 
 Oi rfi 
 
 
 ^ 
 
 05 o 
 
 IM O 
 
 
 
 t^ 00 
 
 CO <M 
 
 
 1H 
 
 ■-H 00 
 
 
 
 
 
 I-H f—t 
 
 1-1 
 
 Q O 
 
 o o 
 
 §g 
 
 
 oH 1—1 
 
 I-H 1-H 
 
 
 
 IM CO 
 
 >* CO 
 
 
 2 
 
 O CO 
 
 O Oi 
 
 
 o> 
 
 00 lO 
 
 tH 00 
 
 
 r* 
 
 
 Oi CO 
 
 
 lO -H 
 
 O lO 
 
 
 S 
 
 (M t^ 
 
 .-( (N 
 
 
 e> 
 
 d CO 
 
 rH CO 
 
 
 1-1 
 
 00 —1 
 
 O 00 
 
 
 
 (N 
 
 1-H 
 
 
 •* CO 
 
 <M CO 
 
 
 <0 
 
 05 lO 
 
 IM O 
 
 
 
 (M i-H 
 
 lO 00 
 
 
 «■( 
 
 00 O 
 
 Oi 00 
 
 OS 
 
 
 i-H 
 
 
 (>■ 
 
 
 
 
 
 
 
 
 
 lO Tt< 
 
 ^ 00 
 
 ■< 
 
 IS 
 
 O <N 
 
 t^ o 
 
 >J 
 
 t4 
 
 05 lO 
 
 Tj< ^ 
 
 tf 
 
 1-1 
 
 i-H 
 
 l> l> 
 
 o 
 
 
 
 
 
 
 
 K 
 
 
 O '-H 
 
 CD t^ 
 
 P^ 
 
 1-1 
 
 at 
 
 CO CO 
 
 (M CO 
 
 
 CO ^ 
 
 O Oi 
 
 
 iH 
 
 I> o 
 
 00 l> 
 
 « 
 
 2 <=> 
 
 oo 
 
 
 
 o o 
 
 o o 
 
 
 »H 
 
 
 »-H 1-H 
 
 
 
 u 
 
 
 
 
 « 
 
 
 
 
 ^ CO 
 
 
 
 
 -Q S 
 
 
 
 
 
 CO **< 
 
 «5 ITS 
 
 t— I 
 
 w 
 
 H 
 Q 
 
 h.5 
 
 o 
 o 
 
 3s 
 
 ZC t> 
 
 SI 
 
 Wo 
 
 >>* 
 
 OM 
 H W 
 
 Q 
 o 
 
 HH 
 Pi 
 
 
 
 
 t^ CD 
 
 Oi IM 
 
 
 
 O 00 
 
 "* CO 
 
 
 »H 
 
 00 t- 
 
 1^ to 
 
 
 t4 
 
 00 CO 
 
 Oi o 
 
 
 I-H 
 
 1-H 
 
 
 'Cj* 1-1 
 
 Oi ^ 
 
 
 t- 
 
 (M 00 
 
 CO ■* 
 
 
 e> 
 
 r^ 1^ 
 
 >* Oi 
 
 
 1-1 
 
 05 lO 
 
 o o 
 
 
 
 I-H 
 
 
 » 
 
 
 Oi ^ 
 
 CO CO 
 
 
 «e 
 
 (N lO 
 
 CD CD 
 
 
 
 
 
 -< 
 
 
 IM Tt< 
 
 Oi 1-H 
 
 ^ 
 
 iH 
 
 O CD 
 
 O I-H 
 
 tf 
 
 ^ 
 
 
 1—1 I-H 
 
 1-H 1-H 
 
 
 CO CO 
 
 lO CO 
 
 
 lO 
 
 iH 
 
 o o 
 
 CO Oi 
 
 a 
 
 -* CO 
 
 1-1 C<1 
 
 ■< 
 
 •H 
 
 Oi >o 
 
 o o 
 
 
 
 
 
 c? 
 
 
 
 
 
 lO t^ 
 
 ■* CD 
 
 
 1-1 
 
 05 t» 
 
 IM IM 
 
 
 Tt< (N 
 
 00 00 
 
 
 
 Oi <M 
 
 Oi Oi 
 
 
 m 
 
 o o 
 
 o o 
 
 
 at 
 
 1H 
 
 o o 
 
 o o 
 
 
 
 1-H 1-H 
 
 
 
 O 1> 
 
 Oi IM 
 
 
 CO 
 
 1-1 
 
 Oi 1-H 
 
 CO o 
 
 
 CO (M 
 
 1-1 Tt< 
 
 
 iH 
 
 CO 00 
 
 lO CD 
 
 
 
 I-H IM 
 
 1-H 1-H 
 
 
 O t^ 
 
 Tf< O 
 
 
 iH 
 
 (M IM 
 
 <M ■* 
 
 
 CO 00 
 
 00 Tj* 
 
 
 1H 
 
 I-H ^H 
 
 IM CO 
 
 
 
 1-H IM 
 
 
 
 CO Oi 
 
 1-( Tj< 
 
 
 0> 
 
 CD (M 
 
 lO ■* 
 
 
 T-l Tt< 
 
 to t^ 
 
 
 T< 
 
 O CO 
 
 o o 
 
 S 
 > 
 
 
 1-H 1-H 
 
 1-H 1-H 
 
 
 
 
 p 
 
 
 i-i CD 
 
 Oi o 
 
 ■<! 
 
 la 
 
 1-H CO 
 
 o t- 
 
 H 
 
 at 
 
 O lO 
 
 CO »* 
 
 « 
 
 
 O (M 
 
 o o 
 
 
 
 1-H ^H 
 
 8 
 
 
 
 
 
 
 
 OS 
 
 
 CD CO 
 
 CO Oi 
 
 d 
 
 •* 
 
 CO to 
 
 t- t^ 
 
 
 o> 
 
 O ii 
 
 O Q 
 
 
 *H 
 
 o o 
 
 O ^ 
 
 
 
 1-H 1-H 
 
 1—1 I-H 
 
 
 « 
 
 oo 
 
 OO 
 
 
 
 o o 
 
 O O 
 
 
 a 
 
 T-H I-H 
 
 I-H rH 
 
 
 
 tH 
 
 
 
 
 <u 
 
 
 
 
 -o ^ 
 
 
 
 
 
 CO ^ 
 
APPENDIX I 
 
 415 
 
 
 
 
 (N T)< 
 
 
 
 00 
 
 CO O) 
 
 
 
 T-l 
 
 •* t^ 
 
 
 
 ^ 
 
 
 
 
 »0 1-H 
 
 
 
 t- 
 
 <N CO 
 
 
 
 
 
 
 
 e> 
 
 r)H lO 
 
 
 
 IH 
 
 HO lO 
 IN (N 
 
 
 
 t^ Oi 
 
 
 
 «o 
 
 00 rH 
 
 
 CO 
 
 
 CO -^ 
 
 
 s 
 
 <H 
 
 05 05 
 
 
 
 
 T-l rH 
 
 
 H 
 
 
 
 
 15 
 
 
 
 
 a 
 
 
 -* CO 
 
 
 lO 
 
 (M a> 
 
 
 
 ^ Tt* 
 
 
 
 1-1 
 
 
 
 
 CD CO 
 
 
 
 ^ 
 
 <X3 l> 
 
 
 
 
 
 
 
 o 
 
 (N (M 
 
 
 
 
 r-l i-H 
 1—1 tH 
 
 
 n 
 
 oo 
 
 
 
 
 ^^ o 
 
 
 
 ^ 
 
 1-1 fH 
 
 
 
 
 CO TjH 
 
 
 
 iH 
 
 <M --• 
 
 
 
 0> 
 1-1 
 
 Oi lO 
 
 
 
 I> t^ 
 
 
 
 ■* CO 
 
 
 
 iH 
 
 l> 05 
 
 
 
 0> 
 
 1-H 1> 
 
 
 
 
 Oi 00 
 
 
 
 lO 1-1 
 
 
 
 «o 
 
 lO Tjl 
 
 
 
 
 
 
 
 e> 
 
 1-H O 
 
 
 
 Ti 
 
 05 0» 
 
 
 
 t^ 00 
 
 
 g 
 
 lO 
 
 00 CO 
 
 
 K 
 
 1-1 
 
 l> 1> 
 
 
 Ph 
 
 
 
 
 
 1-1 o> 
 
 
 
 ^ 
 
 00 t^ 
 
 
 
 
 Oi o 
 
 
 
 iH 
 
 t^ t- 
 
 
 
 n 
 
 o o 
 
 
 
 r4 
 
 o o 
 
 
 
 1-H tH 
 
 
 
 
 CO CO 
 
 
 
 
 lO lO 
 
 
 
 
 CO I-H 
 
 
 
 
 (M 
 
 1-3 
 O 
 O 
 
 O 
 
 
 o 
 
 >H 
 
 P5 
 
 CQ 
 W 
 
 pq 
 
 :? 
 
 w 
 p 
 
 
 
 
 I> 00 
 
 
 
 00 
 
 rH CO 
 
 
 
 iH 
 
 at 
 
 1-1 (M 
 
 
 
 lH 
 
 O O 
 
 
 
 l> rH 
 
 
 
 
 00 c^ 
 
 
 
 CD t^ 
 
 
 
 1-1 
 
 o o 
 
 1—1 1— 1 
 
 
 
 CO CD 
 
 
 
 <e 
 
 CO CO 
 
 
 
 
 
 
 fa 
 
 ei 
 
 o o 
 
 
 y 
 
 
 1—1 I— 1 
 
 
 
 1—1 1—1 
 
 
 H 
 
 
 
 
 
 
 
 
 
 
 ■<*i »o 
 
 
 IS 
 
 '"i ""! 
 
 
 
 o> 
 
 <N C<J 
 
 
 
 tH 
 
 o o 
 
 
 
 lO lO 
 
 
 
 iH 
 
 (N (M 
 
 
 
 00 00 
 
 
 
 1H 
 
 o as 
 
 
 
 M 
 
 o o 
 
 
 
 s 
 
 o o 
 
 
 
 1—1 1— 1 
 
 
 
 
 00 CO 
 
 
 
 00 
 
 lo CO 
 
 
 
 
 
 
 
 e> 
 
 t^ r^ 
 
 
 
 1-1 
 
 1— ( 1-H 
 
 
 
 00 05 
 
 
 
 t" 
 
 IM CO 
 
 
 
 
 
 
 
 
 
 
 
 v4 
 
 CO CO 
 
 1-H I— 1 
 
 
 
 t^ (N 
 
 
 
 to 
 
 '^ to 
 
 
 
 
 
 
 
 0> 
 
 CD CD 
 
 
 
 1H 
 
 O O 
 
 
 
 a o 
 
 
 
 IS 
 
 00 Oi 
 
 
 
 
 
 
 
 at 
 
 CO CO 
 
 
 K 
 
 1-1 
 
 o o 
 
 
 (S 
 
 
 I-H 1-H 
 
 
 
 00 00 
 
 
 
 «* 
 
 1> 1-- 
 
 
 
 1H 
 0> 
 
 o o 
 
 
 
 
 o o 
 
 1—1 1-H 
 
 
 CO 
 
 o o 
 
 
 
 
 o o 
 
 
 
 iH 
 
 rH 1-H 
 
 
 
 
 CO CO 
 
 
 
 
 lO o 
 
 
 
 
 CO 1-H 
 
 
 
 
 (N 
 
 
416 THE MAKING OF INDEX NUMBERS 
 Multiplying all three together we have 
 
 ^ Vpo Pi Vil \P Pi P 2/ 
 
 which by cancellation reduces to unity, thus satisfying the circular test. 
 The above proof includes the simple geometric as a special case simply by 
 putting w = w' = w" = . . . = 1, 
 
 That the simple aggregative (51) or constant weighted aggregative (9051) 
 conforms to the circular test is likewise easily shown. Formula 9051 is 
 
 — — for the step from to 1 
 
 Ziypo 
 
 — ?^ for the step from 1 to 2 
 /Zwpi 
 
 ?^^ for the step from 2 to 
 
 the product of which is unity. This includes the simple as a special case 
 where w = w' = w" = . . . = 1. 
 
 Note B to Chapter XIII, § 4. A Formula Fulfilling Tests 1 and 2 May he 
 Modified to Fidfill the Circular Test as Applied to Three Specific Dates. It 
 may interest the mathematically inclined reader to observe that, not only 
 can conformity to the circular test be gained by making the weights arti- 
 ficially constant in the face of the facts, but that such conformity limited 
 to a specific triangle of dates can be attained by a mutual adjustment of 
 the true formulae for the three inconsistent comparisons. 
 
 Let the original index number be Poi which may be of any variety. Let 
 its rectification by Test 2, in the usual way, be P'oi. Let the time antith- 
 esis of P'oi be 1/P'io. With these P"s as our starting point we are to de- 
 rive P"'s which fulfill Test 1 and from these P"'s we are to derive P""s 
 which fulfill the circular test (so far as the three particular dates are con- 
 cerned). 
 
 The rectification by Test 1 of P"oi, is evidently, by the usual method, 
 
 P"oi = -»/_-£I, which, by multiplying numerator and denominator by 
 
 » P'lO 
 
 VP'oi reduces to 
 
 VP'oiP'io 
 
 P'lO 
 
 Likewise P"io = 
 
 VP'ioP'oi 
 
 the last two expressions having the same denominators. 
 
 It is easy to show (by multiplying the last two equations together), and 
 in fact it has previously been shown, that these rectified formulae fulfill 
 the time reversal test, i.e. that 
 
 P"oi X P"io = 1. 
 
APPENDIX I 417 
 
 This may, for present convenience, be called the circular test applied to 
 two dates. (Unlike what follows for three dates, this two date test applies 
 to any two dates.) 
 
 Next, for the three specific dates or places, 1, 2, 3, such as, say, Georgia, 
 Norway, and Egypt, we are to secure further "rectified" P""s such that 
 
 P 12 X i 23 X i 31 = 1. 
 
 These required formulae are 
 
 ^ 12—3 
 
 Pf/r 
 
 23 — 
 
 Pirr 
 
 31 — 
 
 -VP"l2P"2zP' 
 
 Pn 
 23 
 
 a/P"23P"3iP' 
 
 P"31 
 
 '\ P" Z\P"\2P" a 
 
 For proof, multiplying the above three equations together, we have 
 
 p" p" p/' 
 
 ■pft, Titn -puf ^ "^ "-^^ 31 
 
 C^ M" 23^ 31 == ■» , = 1 
 
 ■^J{P'\2Y{P'\Z)\P"ZXY 
 
 which was to have been proved. 
 
 Moreover, in obtaining the P" formula3 which satisfy the circular test, 
 we have not lost the fulfillment of the time reversal test for two dates, nor 
 lost the fulfillment of Test 2. This applies to the P""s as well as the P"'s. 
 For instance, as to Test 1, 
 
 P"21 
 
 V P 2lP 13P 32 
 
 and, multiplying this by the first formula above, we have 
 
 pin w -pill __ P 12 X X 21 
 
 ^(P"l2 X P"2.) (P"23 X P"32) (P"31 X P"l3) 
 
 1 
 
 = 1 
 
 \/l X 1 X 1 
 which was to have been proved. 
 
 If desired, by retracing our steps, by successive substitutions, we can, of 
 course, obtain P"'i2 in terms of P12, etc. 
 
 Thus the P'", formulae satisfy the circular test both as applied to the 
 three particular dates, and as applied to any two dates (time reversal test). 
 
 But this fulfillment of the circular test applies only to three specific 
 dates. K we change date 3 this will change P"'i2. The index number be- 
 tween dates 1 and 2 has thus no fixed value but has a different value for 
 every different date 3. Moreover, if we attempt to go further, and find 
 a formula, P"" which satisfies the circular test for four dates, such that it 
 will still hold for every three and for every two, we encounter difficulties ; 
 for the P""i2 which fulfills the circular test for 1, 2, 3, will differ from that 
 which fulfills the circular test for 1, 2, 4. We shall not even have a single 
 value of P""i2 which can serve in all comparisons, dual, triple, quadruple. 
 
418 THE MAKING OF INDEX NUMBERS 
 
 Note to Chapter XIII, § 5. The Meaning of "Equal and Opposite " Cir- 
 cular Gaps. 
 
 Let P12P23 . . . Pni = 1 + a 
 Let Q12Q23 . . . Qai = I — h, 
 
 where a and h represent the circular "gaps." Since we are assuming 
 that Test 2 is fulfilled, P12 X Q12 = 1 ; P23 X Q23 = 1 ; . . . ; PuiQm = 1 ; 
 and, therefore, 
 
 (P12Q12) (P23Q23) . . . (PnlOnl) = 1, 
 
 i.e. 
 
 (1 + a) (1 - 6) = 1, 
 
 which is the theorem which was meant when it was stated, for brevity, that 
 the deviations a and b were "equal and opposite." 
 
 That is, 1 + a and 1 — 6 are reciprocals. Moreover, if, as in the case 
 of Formula 353, a and b are very small they will also be numerically 
 equal, to several decimal places. 
 
 Note to Chapter XIII, § 8. For Formulm Failing in Test 1 It Makes a 
 Difference Whether or Not We Pass All Around the Triangle in One Direction. 
 In case the index number under consideration does not obey the time re- 
 versal test, the dark vertical line does not, strictly speaking, measure the 
 deviations from the circular test, if by that phrase is meant the discrepancy 
 found after going all around the triangle in one direction. In such a case 
 the dark vertical line for 1915 is the discrepancy found by going from 1913 
 around tivo sides of the triangle in one direction (e.g. from 1913 to 1914 and 
 then to 1915) and comparing the position thus reached with that reached 
 by another start from 1913 in the opposite direction along the third side, 
 i.e. from 1913 to 1915. In such cases, where the time reversal test is not 
 fulfilled, there are thus several discrepancies pertaining to any triangle of 
 comparisons (instead of only one as for Formula 353 and the other formulae 
 which do fulfill this test). Taking the years 1, 2, 3, we have the circular 
 gap, 1-2-3-1 or 3-2-1-3 ; also the following others : 1-2-3 compared with 
 1-3 ; 2-3-1 compared with 2-1 ; 3-1-2 compared with 3-2 ; 3-2-1 com- 
 pared with 3-1. But, in the case where the time reversal test is fulfilled, all 
 these deviations reduce to the same, except that the reversing of the direc- 
 tion around the triangle has the effect of changing the sign of the figure, 
 so to speak. Thus the triangular ratio for 0-1-2 is 100.35 per cent in 
 Table 34 while, for 2-1-0, It is 1/100.35 per cent, or 99.65 per cent, so 
 that, in the first case, the triangular deviation from unity is +.35 per cent 
 and, in the second, —.35 per cent. 
 
 Note to Chapter XIII, § 9. The Relation of This Book to the Appendix on 
 Index Numbers in the Author's " Purchasing Power of Money." This book 
 has centered on the idea of reversibility as the supreme sort of test for an 
 index number. In my earlier book. The Purchasing Power of Money, in 
 the Appendix to Chapter X, I have employed other tests. The difference 
 between the two studies is one of emphasis. Nothing in the earlier study 
 needs to be abandoned (with the exception of the circular test), and the 
 conclusions of that study are, in general, consistent with those of the present 
 
APPENDIX I 
 
 419 
 
 study. The fundamental difference in method between the two is that, 
 in the earlier study, attention was concentrated on the algebraic properties 
 of the formulae whereas, in the present, attention is concentrated on their 
 numerical results. 
 
 The present study had its origin in the attempt to compare the numerical 
 results of formulae differing in algebraic properties. But as soon as these 
 numerical results were calculated, they revealed new directions in which 
 to study the reasons for the differences and similarities, directions of study 
 of far greater practical importance than the algebraic properties of the 
 formulae. 
 
 But now that our new study is completed, we may compare it with the 
 old. In the old, 44 formulae were studied, the original numbering of which, 
 translated into our new numbering, is as follows : 
 
 TABLE 56. CROSS REFERENCES BETWEEN THE NUMBERS 
 FOR FORMULA TABULATED IN "THE PURCHASING 
 POWER OF MONEY" AND THE NUMBERING USED 
 IN THIS BOOK 
 
 Number in " Pur- 
 
 New 
 Number 
 
 Number in " Pur- 
 
 New 
 Number 
 
 chasing Power 
 OF Money" 
 
 chasing Power 
 OF Money" 
 
 1 
 
 51 
 
 23 
 
 4153 
 
 2 
 
 52 
 
 24 
 
 4154 
 
 3 
 
 1 
 
 25 
 
 9 
 
 4 
 
 2 
 
 26 
 
 10 
 
 5 
 
 11 
 
 27 
 
 7 
 
 6 
 
 12 
 
 28 
 
 8 
 
 7 
 
 21 
 
 29 
 
 9001 
 
 8 
 
 22 
 
 30 
 
 omitted 
 
 9 
 
 31 
 
 31 
 
 15 
 
 10 
 
 32 
 
 32 
 
 16 
 
 11 
 
 54 
 
 33 
 
 13 
 
 12 
 
 53 
 
 34 
 
 14 
 
 13 
 
 8053 
 
 35 
 
 29 
 
 14 
 
 8054 
 
 36 
 
 30 
 
 15 
 
 353 
 
 37 
 
 23 
 
 16 
 
 353 
 
 38 
 
 24 
 
 17 
 
 2153 
 
 39 
 
 27 
 
 18 
 
 2154 
 
 40 
 
 28 
 
 19 
 
 3153 
 
 41 
 
 25 
 
 20 
 
 3154 
 
 42 
 
 26 
 
 21 
 
 1153 
 
 43 
 
 omitted * 
 
 22 
 
 1154 
 
 44 
 
 omitted 
 
 * But calculated in Appendix III. 
 
420 THE MAKING OF INDEX NUMBERS 
 
 There were, in the earlier study, eight tests, each of which was applied 
 in two ways, first, for dual comparisons (between two years only) and, 
 secondly, for comparing a series of years. In the folding table opposite 
 p. 418 of the Purchasing Power of Money each index number was credited 
 with a score of "^" for every test which it fulfilled in a dual comparison 
 only and "1" for every test which it fulfilled in a series of years. Since, 
 as we have seen, in Chapter XIII of this book, only dual comparisons have 
 theoretical validity, we here ignore the distinction between the "i " and " 1." 
 In the earlier study each test was stated with reference to the applica- 
 tion of the formula to the equation of exchange to fulfill which any 
 formula for prices must be accompanied by its factor antithesis (there 
 called simply its "antithesis") for the quantities. Each test was stated 
 both in reference to prices and quantities, and the fulfillment of either was 
 credited as a good mark for the other, its factor antithesis, because the 
 two were running mates in the equation of exchange. Inasmuch as we 
 here seek to rectify the formulae so that the running mates may be of the 
 same kind, there is no real need of such mutual crediting. We need con- 
 sider, therefore, only the tests for one of the two factors, say, for prices 
 (p's) and omit those separately stated (Tests 2, 4, 6) for quantities (g's). 
 We may also ignore Test 7, "changing of the base," as this has been 
 fully considered in the present book. 
 
 There are left four tests included in the old book and not hitherto made 
 use of in the new, namely : (1) Proportionality. An index number of prices 
 should agree with the price relatives if those agree with each other. (2) De- 
 terminateness. An index number of prices should not be rendered zero, 
 infinity, or indeterminate by an individual price becoming zero. (3) With- 
 drawal or Entry. An index number of prices should be unaffected by the 
 withdrawal or entry of a price relative agreeing with the index number. 
 (4) Commensurability. An index number of prices should be unaffected by 
 changing any unit of measurement of prices or quantities. 
 
 The last test eliminates all of the "ratios of averages" as shown in Ap- 
 pendix III and also Formula 51 in our numbered series, together with 
 those derived from, or dependent on 51, viz. 52 and 251. All the other 
 formulae obey this test, which may be considered of fundamental impor- 
 tance in the theory of index numbers. 
 
 The test of proportionality is really a definition of an average.^ It is 
 fulfilled among the primary formulae by all the odd numbered formulae. 
 But none of the even numbered formulae fulfill it (except Laspeyres' and 
 Paasche's, which are also odd numbered). This makes 24 primary for- 
 mulae which fulfill the proportionality test. 
 
 Table 57 gives the fulfillment or non-fulfillment of each formula as to 
 all the above mentioned tests except that of commensurability already fully 
 scored in the paragraph last but one. In the table a " X " signifies ful- 
 fillment and a " — " signifies non-fulfillment. 
 
 From this table it is clear that these tests differ radically from the re- 
 versal tests in the text in that they make very little quantitative discrimi- 
 
 ' Thus the formulse failing to fulfill the proportionality test are not true averages, except 
 under certain conditions. Such a formula for the price index ia an average of the price 
 relatives only when the quantity relatives are all equal. 
 
APPENDIX I 421 
 
 nation. The proportionality test, for instance, tells us that certain other 
 formulae do agree with the relatives when these agree with each other, 
 which agreement is certainly to their credit. Under such simple circum- 
 stances where there is no dispersion all these various index numbers agree 
 with each other. But then, no index number is needed ! When there is 
 dispersion this test disappears and the various index numbers scatter. 
 That is, the test applies when we do not need its help and, when we do, it 
 does not help us. 
 
 On the other hand, the test tells us that certain index numbers do not 
 exactly agree with the relatives even when these agree with each other. 
 This is certainly to their discredit. But, from a practical point of view, 
 we want to know how near to agreement the formula then comes. We 
 find that, in some cases, the disagreement is great and, in others, negligible 
 so that the mere fact of non-agreement is of little practical value. 
 
 It is worth while to note that, in all the formulae such as the "super- 
 lative" which we have selected on other grounds as superior to the rest, 
 the proportionality test is either perfectly fulfilled or almost perfectly ful- 
 filled. That is to say, the proportionality test never conflicts appreciably 
 with our previous conclusions as to what formulae are best, although it 
 does not help us much in sifting them out from the inferior formulae. It 
 is interesting to note that the proportionality test shows some predilection 
 for the aggregative type and very little for the geometric. This is in spite 
 of the fact that the geometric is, par excellence, a proportionality type. The 
 reason is obviously that the factor antitheses of the geometric introduce 
 a discordant element — the value ratio. In the case of the aggregative 
 the value ratio is more congenial. Consequently, of the index numbers 
 which might perhaps be called the two chief rivals for accuracy, 353 and 
 5323, the former conforms to the proportionality test but the latter does 
 not — nor does 5307, the best of the arithmetic-harmonic type. Thus 
 353 has another small feather in its cap. In fact the only other formulae, 
 among those fulfilling both the main tests, which fulfill also the proportion- 
 ality test are 1353, 2353, and 3353, all aggregatives. Thus none of the 
 others, fulfilling both the main tests, are, strictly speaking, true averages. 
 As to the determinateness test the formulae which pass this test perfectly 
 are usually very poor formulae while many of the best ones faU. Formula 
 353 and all the aggregatives pass; but 307, 309, 323, 325, 5307, 5323 faU. 
 Here again 353 scores. 
 
 As to the withdrawal and entry test, it follows the proportionality test 
 among the primary formulae, being fulfilled by all the odd numbered for- 
 mulae but not by the even (except those which are also odd). But when 
 we come to the cross formulae few meet the test. 
 
 All three tests relate to the behavior of the formula under some special 
 circumstances, such as when all the relatives are equal, when one is zero, 
 or when one coincides with the index number, and have little value as a 
 general guide. All the good formulae which fail really pass practically. 
 
 It will be seen, then, that those three tests are of minor importance. This 
 is the reason I have not made use of them in the text. The only parts of 
 my earlier work which have vital importance have been utilized and ampli- 
 fied in the present text. These three minor tests, however, while weak, 
 
422 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 57. SHOWING THE FORMUL/E WHICH FULFILL AND 
 DO NOT FULFILL THREE SUPPLEMENTARY TESTS 
 
 Formula 
 
 Propor- 
 
 Deter- 
 
 MI- 
 
 With- 
 drawal 
 
 Formula 
 
 Propor- 
 
 Deter- 
 mi- 
 
 With- 
 
 DRAWAIj 
 
 No. 
 
 tionality 
 
 AND 
 
 No. 
 
 tionality 
 
 and 
 
 
 
 NATENE88 
 
 Entry 
 
 
 
 nate NESS 
 
 Entry 
 
 1 
 
 X 
 
 _ 
 
 X 
 
 201 
 
 
 
 __ 
 
 _ 
 
 2 
 
 — 
 
 X 
 
 — 
 
 203 
 
 X 
 
 X 
 
 — 
 
 3 
 
 X 
 
 X 
 
 X 
 
 205 
 
 X 
 
 X 
 
 — 
 
 4 
 
 X 
 
 X 
 
 X 
 
 207 
 
 — 
 
 — 
 
 — 
 
 5 
 
 X 
 
 X 
 
 X 
 
 209 
 
 — 
 
 — 
 
 — 
 
 6 
 
 X 
 
 X 
 
 X 
 
 211 
 
 — 
 
 — 
 
 — 
 
 7 
 
 X 
 
 — 
 
 X 
 
 213 
 
 — 
 
 — 
 
 — 
 
 8 
 
 — 
 
 X 
 
 — 
 
 215 
 
 — 
 
 — 
 
 — 
 
 9 
 
 X 
 
 — 
 
 X 
 
 217 
 
 X 
 
 X 
 
 — 
 
 10 
 
 — 
 
 X 
 
 — 
 
 219 
 
 X 
 
 X 
 
 — 
 
 11 
 
 X 
 
 — 
 
 X 
 
 221 
 
 — 
 
 — 
 
 — 
 
 12 
 
 — 
 
 X 
 
 — 
 
 223 
 
 — 
 
 — 
 
 — 
 
 13 
 
 X 
 
 — 
 
 X 
 
 225 
 
 — 
 
 — 
 
 — 
 
 14 
 
 — 
 
 X 
 
 — 
 
 227 
 
 — 
 
 — 
 
 — 
 
 15 
 
 X 
 
 — 
 
 X 
 
 229 
 
 — 
 
 — 
 
 — 
 
 16 
 
 — 
 
 X 
 
 — 
 
 231 
 
 — 
 
 X 
 
 — 
 
 17 
 
 X 
 
 X 
 
 X 
 
 233 
 
 — 
 
 X 
 
 — 
 
 18 
 
 X 
 
 X 
 
 X 
 
 235 
 
 — 
 
 X 
 
 — 
 
 19 
 
 X 
 
 X 
 
 X 
 
 237 
 
 — 
 
 X 
 
 — 
 
 20 
 
 X 
 
 X 
 
 X 
 
 239 
 
 — 
 
 X 
 
 — 
 
 21 
 
 X 
 
 — 
 
 X 
 
 241 
 
 — 
 
 X 
 
 — 
 
 22 
 
 — 
 
 X 
 
 — 
 
 243 
 
 — 
 
 X 
 
 — 
 
 23 
 
 X 
 
 — 
 
 X 
 
 245 
 
 — 
 
 X 
 
 — 
 
 24 
 
 — 
 
 X 
 
 — 
 
 247 
 
 — 
 
 X 
 
 — 
 
 25 
 
 X 
 
 — 
 
 X 
 
 249 
 
 — 
 
 X 
 
 — 
 
 26 
 
 — 
 
 X 
 
 — 
 
 251 
 
 — 
 
 X 
 
 — 
 
 27 
 
 X 
 
 — 
 
 X 
 
 253 
 
 X 
 
 X 
 
 — 
 
 28 
 
 — 
 
 X 
 
 — 
 
 259 
 
 X 
 
 X 
 
 — 
 
 29 
 
 X 
 
 — 
 
 X 
 
 301 
 
 — 
 
 — 
 
 — 
 
 30 
 
 — 
 
 X 
 
 — 
 
 303 
 
 X 
 
 X 
 
 — 
 
 31 
 
 X 
 
 X 
 
 X* 
 
 305 
 
 X 
 
 X 
 
 — 
 
 32 
 
 — 
 
 X 
 
 — 
 
 307 
 
 — 
 
 — 
 
 — 
 
 33 
 
 X 
 
 X 
 
 X 
 
 309 
 
 — 
 
 — 
 
 — 
 
 34 
 
 — 
 
 X 
 
 — 
 
 321 
 
 — 
 
 — 
 
 — 
 
 35 
 
 X 
 
 X 
 
 X 
 
 323 
 
 — 
 
 — 
 
 — 
 
 36 
 
 — 
 
 X 
 
 — 
 
 325 
 
 — 
 
 — 
 
 — 
 
 37 
 
 X 
 
 X 
 
 X 
 
 331 
 
 — 
 
 X 
 
 — 
 
 38 
 
 — 
 
 X 
 
 — 
 
 333 
 
 — 
 
 X 
 
 — 
 
 39 
 
 X 
 
 X 
 
 X 
 
 335 
 
 — 
 
 X 
 
 — 
 
 40 
 
 — 
 
 X 
 
 — 
 
 341 
 
 — 
 
 X 
 
 — 
 
 41 
 
 X 
 
 X 
 
 X 
 
 343 
 
 — 
 
 X 
 
 — 
 
 42 
 
 - 
 
 X 
 
 — 
 
 345 
 
 — 
 
 X 
 
 — 
 
 43 
 
 X 
 
 X 
 
 X 
 
 351 
 
 — 
 
 X 
 
 — 
 
 44 
 
 — 
 
 X 
 
 — 
 
 353 
 
 X 
 
 X 
 
 — 
 
 45 
 
 X 
 
 X 
 
 X 
 
 1003 
 
 X 
 
 — 
 
 X 
 
 46 
 
 — 
 
 X 
 
 — 
 
 1004 
 
 — 
 
 X 
 
 — 
 
 * In case of withdrawal, the test is fulfilled only provided (if the number of terms is first 
 odd) that the median term is also equal to the median of the two neighboring terms, or (if 
 the number of terms is first even) provided the two middle terms are equal. Practically 
 these conditions are fulfilled, at least approximately, in all ordinary circumstances. In 
 case of entry no such reservations are necessary. 
 
APPENDIX I 
 
 423 
 
 TABLE 57 (Continued) 
 
 Formula 
 No. 
 
 Propor- 
 tionality 
 
 Deter- 
 mi- 
 nate NESS 
 
 With- 
 drawal 
 
 AND 
 
 Entrt 
 
 Formula 
 No. 
 
 Propor- 
 tionality 
 
 Deter- 
 
 Ml- 
 
 nateness 
 
 With- 
 drawal 
 
 AND 
 
 Entry 
 
 47 
 
 X 
 
 X 
 
 X 
 
 1013 
 
 X 
 
 
 X 
 
 48 
 
 — 
 
 X 
 
 — 
 
 1014 
 
 — 
 
 X 
 
 
 49 
 
 X 
 
 X 
 
 X 
 
 1103 
 
 X 
 
 
 X 
 
 50 
 
 — 
 
 X 
 
 — 
 
 1104 
 
 
 X 
 
 
 51 
 
 X 
 
 X 
 
 X 
 
 1123 
 
 X 
 
 
 X 
 
 52 
 
 — 
 
 X 
 
 — 
 
 1124 
 
 _ 
 
 X 
 
 
 53 
 
 X 
 
 X 
 
 X 
 
 1133 
 
 X 
 
 X 
 
 X 
 
 54 
 
 X 
 
 X 
 
 X 
 
 1134 
 
 — 
 
 X 
 
 
 59 
 
 X 
 
 X 
 
 X 
 
 1143 
 
 X 
 
 X 
 
 X 
 
 60 
 
 X 
 
 X 
 
 X 
 
 1144 
 
 _ 
 
 X 
 
 
 101 
 
 X 
 
 — 
 
 X 
 
 1153 
 
 X 
 
 X 
 
 X 
 
 102 
 
 — 
 
 X 
 
 — 
 
 1154 
 
 X 
 
 X 
 
 
 103 
 
 X 
 
 X 
 
 — 
 
 1303 
 
 
 
 _ 
 
 104 
 
 X 
 
 X 
 
 — 
 
 1323 
 
 _ 
 
 _ 
 
 
 
 105 
 
 X 
 
 X 
 
 — 
 
 1333 
 
 — 
 
 X 
 
 _ 
 
 106 
 
 X 
 
 X 
 
 — 
 
 1343 
 
 — 
 
 X 
 
 _ 
 
 107 
 
 X 
 
 — 
 
 — 
 
 1353 
 
 X 
 
 X 
 
 _ 
 
 108 
 
 — 
 
 X 
 
 — 
 
 2153 
 
 X 
 
 X 
 
 X 
 
 109 
 
 X 
 
 — 
 
 — 
 
 2154 
 
 X 
 
 X 
 
 
 110 
 
 — 
 
 X 
 
 — 
 
 2353 
 
 X 
 
 X 
 
 _ 
 
 121 
 
 X 
 
 — 
 
 X 
 
 3153 
 
 X 
 
 X 
 
 X 
 
 122 
 
 — 
 
 X 
 
 — 
 
 3154 
 
 X 
 
 X 
 
 
 123 
 
 X 
 
 — 
 
 — 
 
 3353 
 
 X 
 
 X 
 
 — 
 
 124 
 
 — 
 
 X 
 
 — 
 
 4153 
 
 X 
 
 X 
 
 X 
 
 125 
 
 X 
 
 — 
 
 — 
 
 4154 
 
 — 
 
 X 
 
 
 126 
 
 — 
 
 X 
 
 — 
 
 4353 
 
 — 
 
 X 
 
 — 
 
 131 
 
 X 
 
 X 
 
 X 
 
 5307 
 
 — 
 
 _ 
 
 _ 
 
 132 
 
 — 
 
 X 
 
 — 
 
 5323 
 
 — 
 
 _ 
 
 _ 
 
 133 
 
 X 
 
 X 
 
 — 
 
 5333 
 
 — 
 
 X 
 
 _ 
 
 134 
 
 — 
 
 X 
 
 — 
 
 5343 
 
 — 
 
 X 
 
 _ 
 
 135 
 
 X 
 
 X 
 
 — 
 
 6023 
 
 X 
 
 
 X 
 
 136 
 
 — 
 
 X 
 
 — 
 
 6053 
 
 X 
 
 X 
 
 X 
 
 141 
 
 X 
 
 X 
 
 X 
 
 8053 
 
 X 
 
 X 
 
 
 142 
 
 — 
 
 X 
 
 — 
 
 8054 
 
 X 
 
 X 
 
 — 
 
 143 
 
 X 
 
 X 
 
 — 
 
 8353 
 
 X 
 
 X 
 
 — 
 
 144 
 
 — 
 
 X 
 
 — 
 
 9001 
 
 X 
 
 — 
 
 X 
 
 145 
 
 X 
 
 X 
 
 — 
 
 9011 
 
 X 
 
 — 
 
 X 
 
 146 
 
 — 
 
 X 
 
 — 
 
 9021 
 
 X 
 
 — 
 
 X 
 
 151 
 
 X 
 
 X 
 
 X 
 
 9031 
 
 X 
 
 X 
 
 X 
 
 152 
 
 — 
 
 X 
 
 — 
 
 9041 
 
 X 
 
 X 
 
 X 
 
 153 
 
 X 
 
 X 
 
 — 
 
 9051 
 
 X 
 
 X 
 
 X 
 
 154 
 
 X 
 
 X 
 
 
 
 
 
 
 do not contradict but confirm so far as in them lies the conclusions of this 
 book. 
 
 Formula 353, our prize formula by other tests, fulfills perfectly all but 
 one of these three minor tests and fulfills the remaining one — the with- 
 drawal and entry test — so nearly to perfection as to more than satisfy 
 every practical demand. 
 
 This practical fulfillment would be clear a priori even if we were to make 
 no calculation to verify it. For 353 is a cross between 53 and 54 each oj 
 
424 THE MAKING OF INDEX NUMBERS 
 
 which fulfills this test perfectly and which are always close to each other. Fur- 
 thermore, it is clear that a newly entered commodity the price relative 
 for which in 1917 agrees with the value of 353 P (as it was prior to the 
 entry of the new commodity) both being 161.558, could not, if its weight 
 or importance were very small, disturb appreciably the value (161.558) 
 which 353 already had while, on the other hand, if the importance of the 
 new commodity were very great, i.e. if its price relative were heavily 
 weighted, it would so dominate the index number as to make its value 
 practically coincide with its own (also 161.558). Thus, at either extreme, 
 the result would be very close to 161.558 ; and it stands to reason that 
 it could not depart from this very much at intermediate points. 
 
 Formula 353 would fulfill this test at all intermediate points provided 
 
 that — = 353Q. That is, 353P would remain unchanged by entering a 
 
 new commodity such that — = 353P provided also — = 353Q.^ 
 Po go 
 
 If the ratio to be entered, ^, is not equal to 353Q the further away it is 
 
 go 
 
 the more will the new 353P differ from the old. 
 
 To take an example more extreme than any met with even among our 
 extremely erratic 36 quantities, let gi be one tenth of go. Now let us see 
 how far 353P can get from fulfilling the withdrawal and entry test, by (1) 
 talcing the case (that for 1917) where the two constituent elements 53P 
 and 54P are the farthest apart and (2) assuming that while the price ratio 
 
 of the entered commodity agrees with 353P ( i.e. — = 1.61558 ) its quantity 
 
 \ Po / 
 
 ratio is absurdly far from agreeing with 353Q ( i.e. — = — , although 353Q, 
 
 >. \ go 10 
 
 for 1917, is 118.98 per cent, i.e. 1.1898). 
 
 Let Po = 1 and pi = 1.61558. We have now fixed all the conditions 
 except the absolute values of go and gi. If go is very small, say 1 (and so 
 Qi is .1), the effect on the index number is infinitesimal; for, before the 
 entry of po = 1, pi = 1.61558, go = 1 and gi = .1 the 353P was 
 
 / ^Pigo X ^Mi = / 21238.49 J 
 
 A/spogo Spogi \ 13104.818 
 
 25191.136 
 '2pogo ■ ■ Spogi \ 13104.818 ' 15641.85 
 = Vl.62066 X 1.61050 = 1.61558 J 
 
 while after their entry 353P becomes 
 
 4 
 
 Spigo + pigo Spigi + pigi 
 2pogo + pogo Spogi + Pogi 
 
 V 21238.49 + 1.61558 X 1 y 25191.136 + 1.61558 X .1 _ 
 13104.818 + 1X1 15641.85 + 1 X .1 
 
 '"Cf. Truman L. Kelley, Quarterly Publication of the American Statistical Association, 
 September, 1921, p. 835. The apparently different formula given by Professor Kelley 
 reduces to 353Q. 
 
APPENDIX I 425 
 
 Vl.62066 X 1.61050 = 1.61558 
 
 the ratio of which to the original 1.61558 is 1.00000. Evidently, the new 
 figures are too small to influence the result appreciably. 
 
 On the other hand, if qo is very large (gi being always rtr of go and po 
 being 1, and pi being 1.61558), say Qq = 1,000,000,000, the result is: 
 
 4 
 
 21238.49 + 1.61558 X 1,000,000,000 25191.136 + 1.61558 X 100,000,000 
 13104.818 + 1 X 1,000,000,000 15641.85 + 1 X 100,000,000 
 
 = Vl.61558 X 1.61558 = 1.61558 
 
 the ratio of which to 1.61558 is 1.00000, showing that the new figures eclipse 
 the old but yield the same result. 
 
 Between these two extremes qo has a value which makes the maximum 
 discrepancy, i.e. which renders a maximum, or minimum, above or below 
 unity, the ratio 
 
 4 
 
 
 4 
 
 Spigo >^ Spigi 
 
 Spogo Spogi 
 
 This value of go is obtained by diflterentiating and solving for go the 
 
 equation — = 0. 
 dqo 
 
 Before differentiating we may omit the radical sign and omit the denom- 
 inator, for the ratio R is a maximum or minimum according as its square 
 is a maximum or minimum, which in turn is according as its numerator is 
 a maximum or a minimum, the denominator being constant. 
 
 For simplicity we may put Spogo = a, Spogi = b, Spigo = c, Spigi = d. 
 We may also, for convenience, call go = x and gi = kx where A; = x&- 
 
 Thus we are to maximize 
 
 ( c + pix \ /a 
 a + paxj \l 
 
 ' c + pix \ ( d + pifca; \ 
 ^6 -|- Pofcx/ 
 
 or 
 
 to maximize 
 
 (c a d 6 \ 
 
 substitutmg m = — ,n= — , r = — , s = — i 
 Vi Po Pu Pot/ 
 
 ( m -\- x \ / r 4- x \ 
 n + x/\s + xj 
 
 Differentiating this with respect to x, and placing the result equal to 
 zero, we have 
 
 (m+x)(s-r) , (r + x)(n - m) ^ ^ 
 s-{- X n + X 
 
426 THE MAKING OF INDEX NUMBERS 
 
 Solving for x, we have ■ 
 
 2 \4 
 where, for brevity 
 
 _ (m + n) (s — r) + (a + r) (n — m) 
 s — r -\- n — m 
 
 , _ mn (s — r) + rs (n — m) 
 8 — r + n — m 
 
 It remains to evaluate x numerically. 
 
 The result of solving this equation is a; = 50 = 45134.14 so that also 
 Poqo = 45134.14 which makes the new index number, after the entry of 
 the new commodity, 1.61418 and its ratio to the original index niunber, 
 1.61558, .99913, instead of unity as it is if go is very small or very large. 
 
 In other words, the maximum deviation from unity occurs when the 
 new commodity entered has a value in 1913 of 45134.14, or over three times 
 the total value (13104.818) of all the 36 original commodities. Such a 
 gigantic commodity may have a price ratio of 161.558 agreeing with the 
 original index number and yet its entry will change the index number from 
 161.558 to 161.418, because the quantity ratio of the new commodity does 
 not agree with the old quantity index, being .1 instead of 1.1898. Yet even 
 this maximum possible wandering from 161.558 is negligible, being less 
 than one part in a thousand. If the new commodity were not so gigantic 
 this tiny disturbance would be much tinier. Thus this single failure of 
 our ideal formula, 353, to fulfill all tests applied is practically not a failure. 
 
 Note to Chapter XIII, § 10. Ogbum's Formula for Macaulay's Theorem. 
 Professor W. F. Ogburn has derived an interesting and simple formula * 
 for the difference between the chain and fixed base index numbers when 
 both are simple arithmetics. It shows that we may always tell whether 
 the chain or fixed base figures tend to be the greater, by watching a cri- 
 terion. This criterion is found by : 
 
 (1) Subtracting any price relative (say that of bacon) for any given year 
 from the index number of that year ; 
 
 (2) Multiplying the difference thus found by the percentage increase of 
 the price of that commodity (bacon) between said year and the next ; 
 
 (3) Adding the product thus found (which may, of course, be positive or 
 negative) for bacon to the cori-esponding product for barley, etc., through- 
 out the hst. 
 
 If the net sum thus obtained is positive, the chain figures are increasing 
 (between said year and the next) faster than the fixed base figures. If it 
 is negative, the opposite is true. It is usually positive ; because, for in- 
 stance, the lower relatives, affording the largest differences, are the most 
 likely to recover and so have the larger percentage increases to be mul- 
 tiplied by. A low price going still lower is the exception. 
 
 Note to Chapter XIII, § 11. If a Formula Satisfies the Circular Test for 
 
 » See Wesley C. Mitchell, Bulletin No. 284, United States Bureau of Labor Statistics, 
 pp. 88-89, footnote. 
 
APPENDIX I 427 
 
 Every Three Dates It Will Satisfy for Four, or Any Other Number. Thus, 
 let us add Boston to the previous trio of cities (of Chapter XIII, § 1) and 
 let us, in thought, step the price levels up or down from city to city in any 
 desired circuit such as the following : Philadelphia, New York, Boston, 
 Chicago, Philadelphia. What we are about to prove is that, if the test is 
 fulfilled for every triangular comparison among these four cities, it will 
 necessarily be fulfilled for the quadrangular comparison stated. 
 
 By hypothesis (i.e. by the assumed triangular test) we know that passing 
 around the triangle Philadelphia, Boston, Chicago, Philadelphia we re- 
 turn to the same figure, 100 per cent, with which we started. But, by the 
 same hypothesis applied to a different triangle of cities, we know that the 
 price level of Boston, calculated, in the above case, directly from Phila- 
 delphia, is the same as though it were calculated via New York. Conse- 
 quently, we may, without affecting the result, insert New York between 
 Philadelphia and Boston. This converts the original triangular circuit 
 Philadelphia, Boston, Chicago into a quadrangular one, Philadelphia, 
 New York, Boston, Chicago without disturbing the result, namely, that we 
 end in Philadelphia at the same figure with which we started. 
 
 Algebraically, we wish to prove that Pn X P23 X Pu X Pn = 1, and 
 Pn X P23 X P34 X P45 X Pbi = 1, etc., etc. 
 
 Since the triangular test is assumed to be fulfilled, we know that 
 P12 X P23 X P31 = 1. 
 
 But for P31 we may substitute P34 X P41, — since the triangular test 
 
 P34 X P41 = P3 
 
 shows that P34 X P41 X Pu = 1, or (since Pi3 = — ) 
 
 Making this substitution, we have P12 X P23 X P34 X P4i = 1, which 
 is the proposition to have been proved, — for four steps around the circle. 
 
 Again, substituting in the last for Pu the expression Pa X P&i we have 
 P12 X P23 X P34 X P46 X Psi = 1 and, substituting likewise for Psi, we 
 have P12 X P23 X P34 X P46 X Pbi X Pei = 1, etc., etc., which were to 
 have been proved. 
 
 Since all these theorems as to four, five, six, etc.. years follow from that 
 for three only, it is clear that the essential number of years for this supposed 
 test is three. It might, therefore, be called the "triangular" test rather 
 than the "circular" test. 
 
 In other words, the so-called circular test really starts with three years. 
 It cannot start with two and introduce a third, fourth, etc., on the analogy 
 of the above process, as the reader can readily convince himself if he tries 
 it. 
 
 Thus the triangular test is on a different plane from the dual or time re- 
 versal test. The dual test befits an index number because, by its very 
 nature, an index number (such as P12) involves just two times, such as " 1 " 
 and "2," not three. The triangular test introduces an extraneous element 
 not already represented in the index number itself. 
 
 Note to Chapter XIV, §7. Splicing as Applied to Aggregative Index 
 Numbers. The following is quoted from a statement kindly sent me by 
 Mr. Charles A. Bell, of the United States Bureau of Labor Statistics, show- 
 ing the method of spUcing employed by that Bureau : 
 
428 THE MAKING OF INDEX NUMBERS 
 
 In general, the method followed by the Bureau is as follows : When one grade or quality 
 of an article is to be substituted for another, great care is taken that the newcomer shall 
 correspond as closely as possible with its predecessor. In the case of manufactured prod- 
 ucts, as shoes and textiles, the manufacturer furnishing the information is asked to make 
 the selection. In this way the least possible violence is done to the continuity of the price 
 series. In all cases of this kind the best advice available is sought. The two series are 
 then brought together, with overlapping data for at least one full year, in which form the 
 detailed price information is published. The continuous series of price relatives is con- 
 structed through the medium of the overlapping year, which carries with it the assumption 
 that prices of the substituted commodity in previous years, if available, would have shown 
 the same degree of fluctuation as the former commodity. 
 
 In constructing the group and general index numbers, the plan is followed of building 
 two parallel columns of weighted price aggregates for any year in which an addition, a sub- 
 stitution, or a withdrawal takes place. The first column contains items strictly comparable 
 with those for preceding years and the second column contains items strictly comparable 
 with those for succeeding years. The index number for the overlapping year is, of course, 
 based on the items in the first column. The index numbers for subsequent years are found 
 by summing the items for such years and converting them to percentages of the sum in 
 the second column for the overlapping year, then multiplying them by the index number 
 for the overlapping year, thus converting them to the original base. This is, in effect, a 
 chain index system, welded into one with a fixed base. Its elasticity permits the intro- 
 duction or dropping of commodities without serious jar to the structure, although the effort 
 is made to reduce to a minimum consistent with fairness the number of changes in the list 
 of commodities. As you understand, of course, the Bureau is not concerned with price 
 relatives of individual commodities in constructing its index numbers. 
 
 Note to Chapter XIV, § 9. Bias of 602S and 23 Small {in the case of the 
 12 crops) because of Correlation between Price and Quantity Movements. 
 There is another reason why the downward bias of Formula 6023 is so small. 
 This is that the downward bias of 23 itself is small. This is because of 
 the inverse correlation between the price relatives and the quantity rela- 
 tives. It will be recalled that weight bias exists in a price index because 
 of the price element in the weight. In Formula 23, for instance, the index 
 number is an average of price relatives so weighted that a high price 
 relative draws a low price element in its weight and a low, a high. 
 The other, or quantity element, was assumed as likely to lean in one 
 direction as the other. But if this is not true ; if, instead, every high 
 price element has associated with it a low quantity element and vice 
 versa, evidently the weight itself, or product of a low price by a high quan- 
 tity, or a high price by a low quantity, will be devoid of bias. If the price 
 and quantity elements are thus correlated to the extreme limit of 100 per 
 cent, the downward bias of 23 will be completely abolished. In the present 
 case, where the correlation is —88 per cent, the bias is nearly abolished. Were 
 it not for this inverse correlation the downward bias of 6023 (which is 23 
 with broadened base) would be much more in evidence. 
 
 Note to Chapter XV, § 2. Special Proof that 2153 is Extremely Close to 
 S53. Formula 2153 will, under all ordinary circumstances, be sufficiently 
 close to Formula 353 to serve as a short cut substitute. Only where, as in 
 this monograph, the highest accuracy is desired, is it necessary to spend 
 the additional time for calculating Formula 353. Formula 2153 may be 
 either greater or less than 353 according to circumstances. It is desirable 
 to construct a table by which we may know how close 2153 and 353 may be 
 under various circumstances. The two formulse (say, for prices, which we 
 may call Formulae 2153P and 353P) will coincide, of course, if Formulae 
 63P and 54P, of which they are averages, happen to coincide. (In this 
 
APPENDIX I 429 
 
 case, Formulae Nos. 53Q and 54Q will also coincide.) The two (2153P 
 and 353P) will also coincide if 53Q and 54Q happen to be reciprocals of 
 each other, i.e. if one of the latter is above 100 per cent, the base, and the 
 other below it in the same proportion. In all other cases, 2153P and 353P 
 will differ. 
 
 The following formula ^ gives the relative size of 2153P and 353P : 
 
 (2153P) _ 1 + (54Q) _^ jEl 
 (353P) 1 + (53Q) ■ \'53' 
 
 The 54 and 53, under the radical, may be either both "P's " or both "Q's," 
 they being proportional.^ 
 
 The reader can readily verify this formula by substituting in it the ex- 
 pression for Formula 53, etc.^ 
 
 From this formula, it follows that if 
 
 (54P) / ^ (54Q)\ J 
 (53P) V (53Q)/ 
 and if, furthermore, 
 
 (54Q) X (53Q) > 1, 
 
 then 2153P will exceed 353P as also will be the case if both the above in- 
 equalities are reversed. But if only the upper, or only the lower, be re- 
 versed, then 2153P will be less than 353P. 
 The formula may also be written 
 
 1 + (353Q) VE 
 1 + (353Q) VH 
 
 
 From this, knowing |4 and 353Q, we may calculate the different values 
 of the formula for various possible values of || and 353Q. Evidently if 
 either |4 or 353Q is equal to unity, the formula reduces to unity. That 
 is, if either (1) 53 and 54 are close together, or (2) 353Q is close to 100 per 
 cent, then 2153 and 353 are very close together. 
 
 Table 58 tells us how near or far apart are Formulae 2153 and 353, if we 
 know (1) how near or far apart are 53 and 54, and (2) how large or small 
 they (and their average 353) are. 
 
 1 First suggested to me, in substance, by Professor Hudson Hastings of the Pollak 
 Foundation for Economic Research. 
 
 2 By definition 54P = 7 -^ 53Q, likewise 54Q = F -^ 53P ; dividing these and cancel- 
 ing we get the proportion. (F is the value ratio.) 
 
 ' He may also be interested in developing the formulae (corresponding somewhat to the 
 above for 2153), for 2154, 2353, 8053, 8054, 8353, in terms of 353 ; also in terms of 53 and 54. 
 These include the interrelations connecting all available types of averaging the two for- 
 mula, 53 and 54, i.e. the arithmetic (8053), harmonic (8054), geometric (353), and aggre- 
 gative (2153) methods. That 2153 is an aggregative average of 53 and 54, i.e. is 
 
 numerator of Formula 53 + numerator of Formula 54 .^ ^j^^^ .j ^^^.^ ^^ algebraically 
 denominator of Formula 53 + denominator of Formula 54 
 expressed and compared with the ordinary formula for 2153, 
 
430 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 58. FORMULA 2153P AS A PERCENTAGE OF FOR- 
 MULA 353P (According to various values of H and 353Q, both 
 expressed in per cents) 
 
 54 
 
 FoBMULA 3530 
 
 53 
 
 200 
 
 160 
 
 120 
 
 100 
 
 80 
 
 60 
 
 110 
 
 101.6 
 
 101.0 
 
 100.4 
 
 100.0 
 
 99.5 
 
 98.4 
 
 105 
 
 100.8 
 
 100.5 
 
 100.2 
 
 100.0 
 
 99.7 
 
 99.2 
 
 102 
 
 100.3 
 
 100.2 
 
 100.1 
 
 100.0 
 
 99.9 
 
 99.7 
 
 100 
 
 100.0 
 
 100.0 
 
 100.0 
 
 100.0 
 
 100.0 
 
 100.0 
 
 98 
 
 99.7 
 
 99.8 
 
 99.9 
 
 100.0 
 
 100.1 
 
 100.3 
 
 96 
 
 99.1 
 
 99.5 
 
 99.8 
 
 100.0 
 
 100.3 
 
 100.9 
 
 90 
 
 98.3 
 
 99.0 
 
 99.5 
 
 100.0 
 
 100.6 
 
 101.8 
 
 From this table it will be seen that the index number by Formula 2153 
 is always close to that by 353, even under the extreme conditions repre- 
 sented by the four corners of the table — conditions seldom, if ever, reahzed 
 in practice. The upper left corner represents a condition where H is 
 110 per cent, i.e. where Formula 54 exceeds 53 by 10 per cent (a difference 
 probably never reached in practice) combined with the additional fact 
 that the price level is very high (200 per cent). Under these two circum- 
 stances, the ratio of 2153P to 353P is 101.6, i.e. 2153P is 1.6 per cent higher 
 than 353P. 
 
 In the other three corners other extreme circumstances are represented. 
 The table shows that, even if only one of the two conditions is extreme, the 
 two index numbers, 2153 and 353, coincide as perfectly as when neither 
 is extreme. By means of this table, it is easy to tell in any individual case 
 how great an error will be involved by using 2153 instead of 353, and 
 whether the additional accuracy of 353 is worth the additional trouble. 
 In the case of the 36 commodities, there is no instance where 353 would 
 be needed as 2153 is close to 353, always within one tenth of one per cent. 
 The reason is that 53 and 54 are so close together. In the case of Persons' 
 statistics for 12 crops. Formulae 53 and 54 are further apart. But even 
 this fact does not require the u^e of Formula 353 except possibly in the case 
 of the year 1890, where, besides the fact that 14 is low (94.85 per cent), 
 there is the additional fact that 353Q is very low (56.7 per cent). In 
 this case, the ratio of Formula 2153P to 353P is 100.8 per cent. That is. 
 the two differ by three fourths of one per cent. This is the greatest error 
 I can find in any actual case and this, in most cases, would not be con- 
 sidered worth taking into account. 
 
 Note to Chapter XVI, §4. "Probable Error" by Professor Kelley's 
 Method. Professor Truman L. Kelley ^ proposes another method of 
 
 1 "Certain Properties of Index Numbers," Quarterly Publication of the American Statis- 
 tical Association, pp. 826-41, September, 1921, 
 
APPENDIX I 431 
 
 measuring the "probable error" of an index number, meaning the error 
 due to incompleteness of sampling, or smallness of the number of commodi- 
 ties included in the index. His method is to divide the list of commodities 
 into halves, calculate (by the same formula as that used for the entire set) the 
 series of index numbers for each of the halves, take the coefficient of corre- 
 lation, r, between these two series of index numbers, or "sub-indices," take 
 
 their "reUability coefficient," R, which is equal to , take the standard 
 
 1 + r 
 deviation of each of the two series of sub-indices (from the mean of the 
 series), take the average cr of these two and, from this, calculate the stand- 
 ard devi ation o f the original index, for the same period, by the formula 
 
 h JL. J. 
 a' = ff-Y — - — • Having thus obtained R and <r' , he obtains the desired 
 
 "probable error" of the original set, by the formula ^ 
 
 P. E. = .6745(7' Vl -R. 
 
 Applying this formula to our 200 commodities, we find, after dividing 
 them into two groups, A and B, of 100 each, selected by lot, that the stand- 
 ard deviation of A is .0344, and of B. .0351 giving a = -0344 -f- -0351 ^ 
 
 .03475 and r = .790; whence (r' = .0329 and R = .883 and P. E. = .008. 
 
 That is, according to this reckoning, the 200 commodities considered 
 as samples give an index number the probable error of which, in the sense 
 of its deviation from an ideally complete set of commodities, is .008, or a 
 little less than 1 per cent. 
 
 But the two 100 lists, A and B, dififer from the 200 list in not being in- 
 tentionally selected as good samples. In the following example, the 200 
 list is divided into two 100 lists by a mixture of lot and assorting such 
 that, so far as possible, A' and B' are equally well assorted as samples and 
 have equal importance or weights. We find the standard deviation of J.' is 
 .0432, of B' .0316 giving <t = -0432 + .0316 ^ ^g^^ ^^^ ^ ^ ^^^ . whence 
 
 a' = .0324 and R = .668 and P. E. = .013, or 1.3 per cent, a result very 
 close to the 1^ per cent by my own method. 
 
 The fact that the former and more completely random application of 
 Kelley's method gives a smaller result may, I think, properly be called 
 accidental. We would expect the opposite contrast. 
 
 Professor Kelley warns against using his method when the dates for the 
 quotations are too close together. "It is desirable that the time interval 
 between successive indices be sufficient to insure the relative independence 
 of the commodity quotations involved." ^ This seems to me to constitute 
 a serious weakness in the method, a weakness which does not apply to the 
 method in the text. In the present case the time intervals are short, aver- 
 aging less than three months. 
 
 * He also gives (p. 832) a special formula for the probable error in the case of geometric 
 formulse (our Formulae 21 and 9021). 
 « Ibid., p. 830. 
 
432 THE MAKING OF INDEX NUMBERS 
 
 Note to Chapter XVI, § 7. Round Weights for the Majority of Commodi- 
 ties are Sufficiently Accurate. The proof is as follows : First compute the 
 index number as proposed, i.e. with statistical weights for the most impor- 
 tant 28, and the round weights nearest thereto for the 172 others. Thus, 
 for wheat No. 2, red, the statistical quantity 603 was used and multiplied 
 by the price at any time. But for citric acid the quantity is given statisti- 
 cally as 3.36 but the quantity used in my index number is the nearest round 
 number, 10, this being nearer than 1.^ Similarly for turpentine, the quan- 
 tity statistically given is 53 gallons instead of which we use 100, the nearest 
 round number. After doing likewise for each of the 172 commodities we 
 calculate the index number for the 200 commodities. 
 
 Having obtained this index number by using the nearest round weights, 
 we next compare it with what it would be if the exact weights had been 
 used. The two differ by less than one per cent even when the dispersion 
 of prices is as great as for 1916 relative to 1913, a dispersion seldom reached 
 inside of 40 years (as shown in Table 10 in Chapter V). We may therefore 
 rely on this short cut method to give results within one per cent of what the 
 long method would give. This error of less than one per cent is the error 
 of any index number relative to the base. The error from month to month 
 would, of course, be still less. 
 
 Note to Chapter XVII, § 14. List of Calculated Index Numbers 
 
 1. Discontinued Index Numbers 
 
 Ferguson, Roman Empire (301 a.d.) ; Leber, France (900-1847) ; 
 Shuckburgh Evelyn, Great Britain (1050-1800) ; d'Avenel, France 
 (1200-1790); Rogers, Great Britain (1259-1793); Hanauer, France 
 (1351-1875); Vaughan, Great Britain (1352-1650); Wiebe, Great 
 Britain (1451-1600); Dutot, France (1462-1715); Wiebe, France 
 (1493-1600) ; GUliodts, Belgium (1500-1600) ; Carli, Italy (1500-1750) ; 
 Elmes, Great Britain (1600-1800) ; Jevons, Great Britain (1782-1865) ; 
 Roelse, United States (1791-1801); Flux, Great Britain (1798-1869); 
 Hansen, United States (1801-1840); Hurlin, United States 
 (1810-1920); Burchard, United States (1825-1884); Juergens, 
 United States (1825-1863) ; de Foville, France (1827-1880) ; Laspeyres, 
 Germany (1831-1863); Porter, Great Britain (1833-1837); Walker, 
 United States (1834-1859) ; Giffen, Great Britain (1840-1883) ; Falkner 
 (Aldrich Senate Report), United States (1840-1891); Mulhall, Great 
 Britain (1841-1884); Krai, Germany (1845-1884); Bourne, Great 
 Britain (1845-1879) ; Levasseur, France (1847-1856) ; Paasche, Germany 
 (1847-1872) ; Soetbeer, Germany (1847-1891) ; Denis, Belgium (1850- 
 1910) ; Schmitz, Germany (1851-1913) ; Drobisch, Germany (1854-1867) ; 
 EUis, Great Britain (1859-1876) ; Mitchell, Germany, Great Britain, and 
 United States (1860-1880) ; Wasserab, Germany (1861-1885) ; Atkinson, 
 India (1861-1908); Mcllraith, New Zealand (1861-1910); Powers, 
 United States (1862-1895) ; Palgrave, Great Britain and France (1865- 
 1886) ; Jankovich, Austria (1867-1909) ; Daggett, United States (1870- 
 
 1 The half-way point between 1 and 10 is best taken as \/l X 10 = 3.16, rather than a8 
 } (1 + 10) or 5.5, although the difference between the results of using 3.16 or 5.5 is negligible. 
 
APPENDIX I 433 
 
 1894); Walras, Switzerland (1871-1884); van der Borght, Germany 
 (1872-1880) ; Fisher (from Japanese Report of the Commission for In- 
 vestigation of Monetary Systems), China, India, and Japan (1873-1893, 
 except China which commenced in 1874); Flux, France (1873-1897); 
 Hansard, Great Britain (1874-1883); von Inama-Sternegg, Austria 
 (1875-1888) ; Koefoed, Denmark (1876-1919) ; Bureau of Economic Re- 
 search, United States (1878-1900) ; Kemmerer, United States (1879-1908) ; 
 Julin, Belgium (1880-1908); Levasseur, France (1880-1908); Conrad, 
 Germany (1880-1897) ; Einar Rudd, Norway (1880-1910) ; Waxweiler, 
 Belgium (1881-1910) ; Nicolai, Belgium (1881-1909) ; Sauveur, Belgium 
 (1881-1909) ; Zahn, Germany (1881-1910) ; Zimmerman, Germany (1881- 
 1910); Falkenburg, Netherlands (1881-1911); Methorst, Netherlands 
 (1881-1911) ; Alberti, Italy (1885-1911) ; Hartwig, Germany (1886-1910) ; 
 O'Conor, India (1887-1902) ; Eulenberg, Germany (1889-1911) ; Hooker, 
 Germany (1890-1911); Datta and Shirras, India (1890-1912); Imperial 
 Ministry of Commerce and Industry, Petrograd, Russia (1890-1912) ; 
 La R^forme ficonomique, France (1891-1913) ; Flux, Germany (1891- 
 1897) ; Bernis, Spain (1891-1913) ; Barker, United States (1891-1896) ; 
 Calwer, Germany (1895-1909); Fisher, United States (1896-1918); 
 Vossische Zeitung, Germany (1900-1912); Loria, Italy (1900-1909); 
 Ottolenghi, Italy (1910-1918) ; Pearl (U. S. Food Administration), United 
 States (1911-1918) ; Statistical Department of Stuttgart, Germany (Stutt- 
 gart) (1913-1919) ; Mitchell (War Industries Board), United States (1913- 
 1918); Foster, United States (1913-1919); Statistical Department of 
 Nurnberg, Germany (Nurnberg) (1914-1920). 
 
 2. Current Index Numbers 
 
 Argentina: Revista de Economia Argentina, Bunge, wholesale (im- 
 
 ports and exports). 
 
 Ibid., Bunge, retail, 18 commodities. Formula 9001. 
 Ibid., Bunge, cost of Uving, Formula 9001. 
 
 Australia : Quarterly Summary of Australian Statistics, Knibbs, 
 
 (Melbourne) wholesale, 92 commodities. Formula 53. 
 
 Quarterly Statistical Bulletin of New South Wales, whole- 
 sale, 100 commodities. 
 
 Quarterly Summary of Australian Statistics, Knibbs, 
 cost of living, 46 commodities and rent. Formula 53. 
 
 Austria : Mitteilungen des Bundesamtes fur Statistik, Bundesamt 
 
 (Vienna) fur Statistik, retail, 23 commodities. 
 
 Ibid., Paritdtische Kommission, cost of living, 23 com- 
 modities. 
 
 Belgium : Department of Statistics, wholesale, 130 commodities. 
 
 Revue du Travail, wholesale, 209 commodities (more or 
 
 less from time to time). Formula 21. 
 
 Ibid., wholesale, 127 commodities, Formula 21. 
 
 Ibid., retail, 22 commodities, Formula 9001. 
 
 < . Ibid., retail, 30 commodities, Formula 9001. 
 
 Ibid., cost of living, 56 commodities, Formula 1. 
 
434 
 
 THE MAKING OF INDEX NUMBERS 
 
 Bulgaria : 
 Canada: 
 
 China: 
 
 (Shanghai) 
 Czechoslovakia 
 
 Denmark : 
 
 Dutch East 
 
 Indies: 
 Egypt: 
 
 (Cairo) 
 
 Finland: 
 France : 
 
 (Paris) 
 Germany : 
 
 (Halle) 
 
 (Berlin) 
 
 (Berlin) 
 
 (Frankfurt- 
 am-Main) 
 (Hannover) 
 
 Bulletin slalistique mensuel de la Direction GSnerale de la 
 Statistique, wholesale. 
 Ibid., retail, 47 commodities, Formula 3. 
 Labour Gazette, Coats, wholesale, 238 commodities, For- 
 mula 53. 
 
 Ibid., Coats, cost of living, 29 staple foods, 5 fuel and 
 light, clothing, rent, and sundries. 
 
 Federal Reserve BvUetin, wholesale, 101 commodities. 
 Formula 53. 
 
 Monthly Commercial Letter, Canadian Bank of Commerce, 
 Michell, wholesale, 48 commodities. 
 Toronto newspapers, Michell, wholesale, 40 commodities. 
 Formula 1. 
 
 Finance and Commerce (Shanghai), Bureau of Markets, 
 Treasury Department, wholesale, 147 commodities. 
 Monthly Price Bulletin, Statistical Office, Ryba, retail, 
 25 commodities, Formula 1. 
 
 Finanstidende, wholesale, 33 commodities. Formula 9001. 
 Statistiske Efterretninger, cost of living. 
 Statistical Bureau of the Department of Agriculture, whole- 
 sale. 
 
 Monthly Agricultural Statistics, Statistical Department, 
 wholesale, 26 commodities, Formula 21. 
 Ibid., retail, 23 commodities, Formula 9001. 
 Ibid., cost of living. 
 
 Social Tidskrift, cost of Uving, 17 commodities, rent, fuel, 
 a daily newspaper, and taxes. 
 
 Bulletin de la Statistique Generate de France, March, whole- 
 sale, 45 commodities. Formula 1. 
 Ibid., March, retail, 13 commodities. Formula 53. 
 Ibid., March, cost of hving, 13 commodities. Formula 53. 
 Frankfurter Zeitung, wholesale, 98 commodities. Formula 1. 
 Wirtschaft und StcUistik, Statisiisches Reichsamt, whole- 
 sale, 38 commodities. Formula 3. 
 
 Statistische Vierteljahrshefte, Statistisches Amt der Stadt 
 Halle, retail, 41 commodities. 
 
 Wirtschaft und Statistik, Statistisches Reichsamt, cost of 
 living, 17 commodities and rent. 
 
 ■Monatliche Ubersichten ttber Lebensmittelpreise, Calwer, 
 cost of living, 19 commodities. 
 
 Finanzpolitische Korrespondenz, Kuczynski, minimum cost 
 of living, 19 commodities, rent, and miscellaneous, For- 
 mula 53. 
 
 Die Kosten des Emdhrungsbedarfs, Silbergleit, cost of Uving 
 (food). Formula 1. 
 
 Indexziffem (published by Reitz and Kohler, Frankfurt- 
 am-Main), Elsas, cost of living, 40 commodities. Formula 3. 
 Mitteilungen des Statistischen Amts der Stadt Hannover, 
 cost of hving, 37 commodities. Formula 9001. 
 
APPENDIX I 
 
 435 
 
 (Koln) Statistische Monatsherichte, Statistisches Ami, cost of 
 
 living. 
 (Leipzig) Statistisches Amt, cost of living. 
 
 (Ludwigs- Statistische Vierteljahrsberichte der Stadt Ludwigshafen, 
 
 haf en) cost of living. 
 
 (Mannheim) Mannheimer Tageszeitung, Hofmann, cost of living, 79 
 
 commodities, rent, and miscellaneous. 
 Great Britain : Board of Trade Journal, Flux, wholesale, 150 commodi- 
 ties, Formula 21. 
 
 Economist, wholesale, 44 commodities, Formula 1. 
 
 Federal Reserve Bank of New York, Monthly Review, 
 
 Snyder, wholesale, 20 commodities. 
 
 Federal Reserve Bulletin, wholesale, 98 commodities, 
 
 Formula 53. 
 
 Statist, wholesale, 45 commodities, Formula 1. 
 
 Times (London), Crump, wholesale, 70 commodities. 
 
 Formula 21. 
 
 Labour Gazette, cost of living, 41 commodities and rent, 
 
 Formula 9001. 
 Hungary : Szakszervezeti Ertesito, cost of living, 34 commodities. 
 
 India : Labour Gazette, Shirras, wholesale, 43 commodities, For- 
 
 (Bombay) mula 1. 
 
 (Calcutta) Department of Statistics, wholesale, 75 commodities. 
 
 (Bombay) Labour Gazette, Shirras, cost of living, 23 commodities 
 
 and rent. 
 Italy: Annuario Statistico Italiano, wholesale, 13 commodities, 
 
 Formula 1. 
 
 L' Economista, Bachi, wholesale, 100 commodities. 
 
 Formula 1, chain, Formula 21. 
 
 La Riforma Sociale, Necco, wholesale (imports and ex- 
 ports), 19 imports and 12 exports. 
 (Milan) Bollettino municipale mensile, cost of living. 
 
 (Rome) Bollettino del Ufficio del Lavoro, cost of living. 
 
 (Florence) Ufficio di Statistica, cost of living. 
 
 Japan: Bank of Japan, wholesale, 56 commodities. Formula 1. 
 
 (Tokio) Department of Agriculture and Commerce, wholesale, 
 
 39 commodities. 
 
 Oriental Economist, wholesale. 
 Netherlands : Maandschrift van het Centraal Bureau voor de Statistiek, 
 
 wholesale, 53 commodities. Formula 1. 
 (Amsterdam) Maandbericht van het Bureau van Statistiek, retail, 26 
 
 commodities. Formulae 1 and 3. 
 
 Ibid., cost of living. 
 (Hague) Maandcijfers van het Statistish Bureau, cost of living. 
 
 New Zealand : Monthly Abstract of Statistics, Fraser, wholesale, 140 
 
 commodities. Formula 53. 
 
 Ibid., Fraser, cost of hving, 66 commodities and rent. 
 
 Ibid., Fraser, export prices. 
 
 Ibid., Fraser, producers' prices. 
 
436 
 
 THE MAKING OF INDEX NUMBERS 
 
 Norway : 
 
 Peru 
 Poland : 
 (Warsaw) 
 
 Russia : 
 
 (Moscow) 
 South Africa : 
 
 Spain: 
 
 (Barcelona) 
 Sweden : 
 
 Switzerland : 
 
 (Basle) 
 
 (Berne) 
 
 (Zurich) 
 United States : 
 
 Oekonomisk Revue, wholesale, 70 commodities, Formula 1. 
 Farmand, wholesale, 40 commodities, Formula 1. 
 Statisiiske Meddelelser, Del Statistiske Centralbyra, cost of 
 living. Formula 53. 
 
 Direccion de Estadistica, wholesale, 58 commodities, 
 Formula 1. 
 
 Central Statistical Office, wholesale, 68 commodities, For- 
 mula 21. 
 
 Statystyka Pracy of the Central Statistical Office, cost of 
 living, 38 commodities and rent. 
 Ekonomicheskaia Zhizn, retail, 22 commodities. 
 
 Quarterly Abstract of Union Statistics, wholesale, 188 
 commodities, Formula 53. 
 
 Ibid., Cousins, retail, 23 commodities, Formula 53. 
 Ibid., cost of living, 19 commodities and rent, Formula 
 53. 
 
 Instituto Geograjico y Estadistico, wholesale, 74 commodi- 
 ties. Formula 1. 
 Ibid., retail, 28 commodities. 
 
 Bulleti del Museo Social, wholesale, 25 commodities. 
 Goteborgs Handels-och Sjofartstidning, Silverstolpe, whole- 
 sale, 47 commodities. Formula 53. 
 
 Kommersiella Meddelanden, wholesale, 160 commodities, 
 Formula 3. 
 
 Sociala Meddelanden, cost of living, 75 commodities, rent, 
 taxes, and miscellaneous. 
 
 Neue Zurcher Zeitung, Lorenz, wholesale, 71 commodities, 
 Formula 9001. 
 
 Statistische Monatsberichte, retail, 21 commodities. 
 Schxveizerischer Konsumverein, retail, 41 commodities. 
 Halbjahrsberichte des Statistischen Amts der Stadt Bern, 
 retail, 79 commodities. 
 Statistik der Stadt Zurich, cost of living. 
 Annalist, wholesale, 25 commodities. Formula 1. 
 Bradstreet, wholesale, 96 commodities. Formula 51. 
 Babson, wholesale, 10 commodities. Formula 1. 
 Bureau of Labor Statistics, Monthly Labor Review, Stewart, 
 wholesale, 404 commodities (more or less from time to 
 time). Formula 53. 
 
 Dun's Review, Little, wholesale, about 300 commodities. 
 Formula 53. 
 
 Federal Reserve Bulletin, wholesale, 104 commodities. 
 Formula 53. 
 
 Federal Reserve Bank of New York, Monthly Review, 
 Snyder, wholesale, 20 commodities. 
 
 Gibson's Weekly Market Letter, wholesale, 22 commodities. 
 Harvard Review of Economic Statistics, Persons, whole- 
 sale, 10 commodities. Formula 21, 
 
APPENDIX I 437 
 
 San Diego (California) Union, Bissell, wholesale, 60 
 commodities, Formula 21. 
 
 Bulletin, National City Bank of New York, Austin, 
 wholesale (imports and exports), 25 imports and 30 ex- 
 ports, Formula 51. 
 
 Bureau of Labor Statistics, Monthly Labor Review, Stewart, 
 retail, 43 commodities, Formula 53. 
 Ibid., Stewart, cost of living, 184 commodities and rent. 
 Massachusetts Special Commission on the Necessaries of 
 Life, Parkins, cost of living, 78 commodities, Formula 
 9001. 
 
 National Bureau of Economic Research, King, cost of liv- 
 ing for families spending $25,000 per annum, Formula 
 9001. 
 
 National Industrial Conference Board Monthly Service 
 Letters and Reports, Stecker, cost of living, 90 items and 
 rent. Formula 53. 
 
 Federal Reserve Bulletin, agricultural movements, 14 com- 
 modities, Formula 53. 
 
 Ibid., mineral production, 7 commodities, Formula 53. 
 Ibid., manufactured goods, 34 commodities, Formula 53. 
 Harvard Review of Economic Statistics, volume of pro- 
 duction (agriculture), 12 commodities, Formula 6023. 
 Ibid., volume of production (mining). Day, 9 commodi- 
 ties, Formula 6023. 
 
 Ibid., volume of production (manufacture). Day, 33 series. 
 Ibid., volume of production (last 3 combined), Day. 
 Ibid., Aberthaw, cost of reinforced concrete factory 
 building. 
 
 Summary of Business Conditions in the United States, 
 Am. Tel. & Tel. Co., construction costs, 15 principal 
 building materials and weighted average of wage rates. 
 Fred T. Ley & Co. (Springfield, Mass.), cost of building 
 construction. 
 
 American Writing Paper Company, paper production 
 costs, 5 materials and labor. Formula 1. 
 Federal Reserve Bulletin, foreign exchange rates, 18 
 leading currencies, Formula 29. (For other such indexes 
 — English, German, Swedish, Norwegian — see Federal 
 Reserve Bulletin, July, 1921, p. 794.) 
 Annalist, stocks, 25 railroads and 25 industrials. 
 New York Times, stocks, 50. 
 Wall Street Journal, stocks, 20 railroads. 
 Many other trade journals and newspapers carry index 
 numbers of stocks or bonds, or both. 
 
 For fuller information on many of the above index numbers, see Bulletin 
 284, United States Bureau of Labor Statistics ; International Labour Re- 
 view, pp. 52-75, July, 1922; and Emil Hofmann, Indexziffern im Inland 
 
438 THE MAKING OF INDEX NUMBERS 
 
 und im Amland, 127 pp. G. Braunsche Hofbuchdruckerei und Verlag, 
 Karlsruhe, 1921. 
 
 The above list is exclusive of index numbers of wages and of a great many 
 index numbers bearing on prices, the cost of living, etc. as between different 
 places. For information as to index numbers of wages the reader is referred 
 to the United States Bureau of Labor Statistics, the International Labour 
 Office, and the National Industrial Conference Board. For information 
 with regard to place to place index numbers, see also Report of an Enquiry 
 by the Board of Trade (British) into Working Class Rents, Hoiising and Retail 
 Prices, 1911. 
 
 In addition to the above specific index numbers various attempts have 
 been made to use index numbers of index numbers, or averages of aver- 
 ages. For example, George H. Wood ^ undertook to express the developn 
 ment of the consumption of the English population, and Nemnann- 
 Spallart to find a "measure of the variations in the economic and social 
 condition of nations" by "mean index numbers." ^ We might also 
 include under the rubric of index numbers the various trade barometers, 
 etc., which are in conmiercial use, such as Brookmire's, Babson's, the 
 Harvard Committee on Economic Research, the Alexander Hamilton 
 Institute, the American Institute of Finance, the Standard Statistics 
 Corporation, the London School of Economics, etc. 
 
 * George H. Wood, "Some Statistics of Working Class Progress since 1860." Journal 
 of the Royal Statistical Society, p. 639 et seq., esp. p. 654 et seq. 
 
 2 See Franz ^izek, Statistical Averages (translated by Warren M. Persons), New York, 
 1913, pp. 95-101. eq}. p. 100. 
 
APPENDIX II 
 
 THE INFLUENCE OF WEIGHTING 
 
 § 1. Introduction 
 
 The "best method of weighting" index numbers has long been the sub- 
 ject of debate. We have seen, however, that any method which is really 
 systematic, — whether it be I, II, III, IV, or one of the cross weight sys- 
 tems, — can be used to start with, provided the index number so obtained is 
 subsequently rectified. Rectification will take out the bias, however 
 great it may be to start with. Only freakish weighting is incorrigible. 
 
 Consequently, the whole subject of "the proper weighting" really dis- 
 appears in the result and plays no part in the main argument of this book. 
 But in view of the literature on the subject and in order to effect an ad- 
 justment between current ideas and the conclusions of this book, the sub- 
 ject is included, though relegated to this Appendix so as not to interrupt 
 the main com-se of reasoning in the text. In a few instances we shall need 
 to repeat slightly some of the observations in the text. 
 
 We began with a discussion of "simple" index numbers. These are 
 often loosely referred to as "unweighted" index nvunbers. More properly, 
 of course, they are evenly weighted index numbers, i.e. index numbers in 
 which every price relative has the same weight as every other. 
 
 We next noted (for all types of index numbers except the aggregative) 
 four methods of weighting by values, viz. I (by values of the commodities 
 in the base year) ; 7 V (by values in the given year) ; and II and III (by 
 the fictitious values found by multiplying the prices of one year by the 
 quantities of the other). And, for the aggregative type, we noted two 
 methods of weighting index numbers of prices by quantities, viz. I (by 
 quantities in the base year) and IV (by quantities in the given year). 
 
 Finally, in Chapter VIII, we used weights obtained by averaging the 
 weights of the opposite systems, I and IV, or II and III. These weights 
 were usually averaged geometrically but, in some cases, they were done 
 arithmetically and harmonically and might have been so done in aU. 
 
 We are now ready to answer, with some precision, the question : What 
 differences do different systems of weighting make in the resulting index 
 numbers? We have already, in Chapter V, seen that a biased system of 
 weighting makes a very considerable difference in the index number, — 
 substantially the same difference as does a biased type of index number. 
 Thus (for all except aggregatives) weightings /// and IV raise, while / 
 and II depress, any index number. We may here, for convenience, think 
 of this effect as measured relatively to a cross weight index number which 
 will lie about midway between the index numbers weighted I and II, on 
 the one hand, and the index numbers weighted /// and 77 on the other. 
 
 439 
 
440 
 
 THE MAKING OF INDEX NUMBERS 
 
 In the case of the arithmetic, geometric, and harmonic index numbers, 
 the upward bias of weighting /// and IV and the downward bias of I and 
 // amounted, in our example of 36 commodities for 1917 (on 1913 as base), 
 to about five per cent. 
 The reason for so large an influence of weighting was the bias itself — 
 
 Si'mp/e vs. Cross-Weighted 
 Index Numbers 
 (Prices) 
 
 *15 
 
 '14 
 
 US 
 
 te 
 
 77 
 
 78 
 
 Chart 66 P. Showing the difference which different weightings make 
 when uncomplicated by bias. The differences are very similar in the cases 
 of the arithmetic, harmonic, and geometric, but not very similar in the cases 
 of the median, mode, and aggregative. 
 
APPENDIX II 
 
 441 
 
 the fact, for instance, that by the weighting system IV the bigger a price 
 relative the more heavily it tends to be weighted and the smaller, the more 
 lightly. 
 
 But if we take systems of weighting in which the cards are not thus 
 stacked, i.e. systems devoid of bias, we shall find that differences in systems 
 of weighting, — even very wide differences, — make remarkably small 
 differences to the resulting index numbers. 
 
 The failure to distinguish between the effects of bias in the weighting 
 
 Simple vs. Cross-Weighted 
 Index Numbers 
 
 (Quantities) 
 
 '0 
 
 '/4 75 He '17 
 
 Chart 66Q. Analogous to Chart 66P. 
 
 '18 
 
442 THE MAKING OF INDEX NUMBERS 
 
 (which are important) and those of mere blind chance (which are usually 
 not very important) is responsible for much of the confusion on this sub- 
 ject and the existence of two apparently opposite opinions: one, that 
 weighting is, and the other, that it is not, important. 
 
 Simple vs. Cross-Weighted ^y:%04 
 
 Index Numbers ^-^ /^ 
 
 /v^ / 
 (Prices. Cont) // / 
 
 *i5 */4 '/S *t$ V7 Ud 
 
 Chart 67P. Showing the differences which different weightings make 
 when uncomplicated by bias to the factor antitheses of the index numbers 
 in Chart 66P. The differences correspond to those in Chart 66Q. 
 
APPENDIX II 443 
 
 § 2. Simple and Cross Weight Index Numbers Compared 
 
 The two unbiased systems of weighting which have been set forth in 
 this book are simple weighting (the weights being all equal) and cross 
 weighting (the weighting being averages of the weights under systems 
 / and IV, or II and 7/7). 
 
 The cross weight system is a careful and discriminating system of weight- 
 ing, every weight taking due account of all the data bearing on the case ; 
 while the simple is a careless and indiscriminate system which shuts its 
 eyes to all the differences among commodities. The weights in the two 
 systems — cross weight and simple — differ enormously, far more, in 
 fact, than the weights of 7 and IV, or of 77 and 777. 
 
 Simple vs. Cross-Weighted 
 Index Numbers 
 
 (Quanlities, Cont) ^foo4 
 
 .12 
 
 '10/4 
 
 ^1104 
 
 J 124 
 -22 
 
 ^//34 
 
 / __22=^- 32 
 
 
 .^ --../^ \ 
 
 52 
 
 .^'1154 
 
 \s% 
 
 YS 74 75 76 77 75 
 
 Chart 67Q. Analogous to Chart 67P. The differences correspond to 
 those in Chart 66P. 
 
444 THE MAKING OF INDEX NUMBERS 
 
 The cross weight formula for the arithmetic is 1003 (not 1103), and for 
 the harmonic 1013. The other cross weight formulae are 1123, 1133, 1143, 
 1153. Thus Formula 1003 Is a weighted arithmetic index number freed 
 of weight bias, but not freed of the (upward) type bias, inherent in the arith- 
 metic type. Likewise, 1013 is a weighted harmonic, freed of weight bias 
 but not freed of downward type bias. 
 
 We may now compare each of these six weighted index numbers, with 
 the corresponding one of the six simple or evenly weighted index numbers, 
 all twelve being free of weight bias. 
 
 Charts 66P and 66Q, 67P and 67Q compare the simple and cross weight 
 index numbers. 
 
 § 3. The Differences Haphazard 
 
 The first point which strikes us in these comparisons between simple 
 and cross weight index numbers is that there is no constant tendency for 
 one of the two to be above or below the other. The two curves inter- 
 twine, differing either way and about equally often. It is a matter of even 
 chance, not of bias. 
 
 § 4. The Differences among the Various Similar Types of Index Numbers 
 
 The second point which arrests attention is the remarkable similarity 
 in the influence of the different weighting in the case of the three chief types 
 of index numbers. That is, the difference between the simple and cross 
 weight arithmetics is practically the same as that between the simple and 
 cross weight harmonics, and as that between the simple and cross weight 
 geometries. 
 
 The other three types show peculiarities, though not always. The me- 
 dians usually behave somewhat similarly to the first three, the arithmetic, 
 harmonic, and geometric. But the modes are erratic compared with the 
 first three and with each other. The simple aggregative is very erratic, 
 while the cross weight aggregative is not. 
 
 § 5. The Differences Small 
 
 The third point which strikes us in making these comparisons is how 
 surprisingly small is the difference made by using the careful discriminating 
 cross weighting instead of the erratic simple weighting. This is aston- 
 ishing when we consider that the two sets of weights themselves differ 
 enormously. In the simple weighting all 36 commodities are equally 
 important while in the cross weighting (in the case of the price index 
 number in 1914, for instance) the highest weight (that for lumber) was 
 118 times as great as the lowest (that for skins) ; in 1915 the highest was 
 134 times the lowest; in 1916, it was 100 times; in 1917, 130 times; and 
 in 1918, 261 times. Yet, in spite of these enormous variations (and in 
 spite of the fact that there are only 36 commodities in the list), these 
 unbiased (simple and cross weighted) forms usually agree within five or 
 ten per cent. In fact, out of 60 comparisons between the simples and 
 cross weighted index numbers (for both prices and quantities), there are 
 
APPENDIX II 445 
 
 only 13 differences exceeding five per cent and only five over ten per cent. 
 In the case of the arithmetic, harmonic, and geometric, there is only one 
 instance of a discrepancy over eight per cent. This is for 1918 for the 
 harmonic where there is a discrepancy of over 30 per cent. 
 
 The reason for this large discrepancy is to be found in one commodity, 
 skins, the quantity of which fell between 1913 and 1918 tenfold. Although 
 this enormous fall is quite out of tune with the general movement of the 
 other 35 commodities, nevertheless it ought properly not to have much 
 influence on the average change of the 36 commodities because "skins" 
 was so insignificant a commodity. And in the weighted average this is the 
 case, since "skins" is given only ^j^inj of the total weight. But by the 
 simple weighting its influence is ^ of the total which is nearly a hundred 
 times the influence it should have. 
 
 Such a great change, as in the quantity of skins, is almost never met with 
 and when it does occur it is usually smothered up by the other commodi- 
 ties because of there being so many. In fact, it is smothered even in the 
 present case of only 36 commodities, except where the harmonic method is 
 used, which gives a special emphasis, as it were, to terms unusually small. 
 Probably such a freak effect would not be encountered once in a hundred 
 times in the ordinary course of using index numbers. 
 
 Professor Wesley C. Mitchell cites many actual examples ^ of the effect 
 of weighting as compared to simple index numbers. In general, the differ- 
 ences are less even than those here found, being seldom ten per cent, except 
 under the chaotic conditions created by the greeriback standard in 1862- 
 1878. Ordinarily the difference between the simple and the best weighted 
 index number of the Aldrich Senate Report was less than three per cent. 
 
 The influence of a change of weighting is, of course, different for dif- 
 ferent types of index numbers. In general, a given change in weighting 
 produces least effect in the mode, somewhat more in the median, and very 
 much more in the arithmetic, harmonic, and geometric. For the aggrega- 
 tive formula the process of weighting has a different meaning from what 
 it has for the rest, being a matter of quantities only. The effect of a change 
 in these quantities on the index number is small. It is about the same as 
 the effect of a change in the weighting of the arithmetic, harmonic, and 
 geometric, when only quantities are changed. Thus there is very little 
 difference between the aggregatives, 53 and 59, dependent on a change in 
 quantities only, — about the same difference as between the arithmetics 
 3 and 5, or 7 and 9, or the harmonics 13 and 15, or 17 and 19, geometries 
 23 and 25, or 27 and 29, in all of which cases the only change is in quan- 
 tities. 
 
 The effect of a change in weights is more spasmodic or irregular in the 
 cases of the mode and median than in that of the other four types. This 
 is true even when bias is involved. Thus there is no appreciable difference 
 between the modes, 43 and 49, and little between the medians, 33 and 39, 
 — and that little, spasmodic. There is much more difference between 
 the arithmetics, 3 and 9, or the harmonics, 13 and 19, or the geometries 
 23 and 29. 
 
 1 Bulletin S84, United States Bureau of Labor Statistics, pp. 61-62. 
 
446 
 
 THE MAKING OF INDEX NUMBERS 
 
 § 6. Bias More Disturbing than Chance 
 
 We have seen that in the case of the two unbiased weighting systems, 
 the simple and the cross, while the weights often vary a hundredfold, the 
 resulting index numbers seldom differ over five per cent. But the biased 
 forms, / and IV for instance, differ often eight or ten per cent, although 
 the weights never differ even as much as twofold. 
 
 This conclusion, that bias, even when weights vary little, is more dis- 
 turbing than chance, even when weights vary enormously, may be still 
 more definitely illustrated. If we take the 36 commodities at random, 
 i.e. regardless of their importance as to p's or q's — let us say, alphabeti- 
 cally, — and divide them into two groups of 18 each, and then multiply the 
 weights (quantities) of the first group by ten, the index number for 1917 
 (the year most likely to create a disturbance) becomes, by Formula 53, 
 175.20 per cent instead of 162.07, a difference of 8 per cent. Now observe 
 what happens if, instead of taking our two lists of 18 at random, we select 
 them so that the first 18 will be those which will have the greatest influence 
 in raising the result. When these hand picked 18 are increased tenfold 
 in importance the result is 201.33, exceeding 162.07 by 24 per cent. The 
 contrast in effects is shown in Table 59 : 
 
 TABLE 59. COMPARATIVE EFFECTS ON THE INDEX NUM- 
 BER FOR 1917 (BY FORMULA 3) OF INCREASING TEN- 
 FOLD THE WEIGHTS OF HALF OF THE 36 COMMODITIES 
 ACCORDING AS THE COMMODITIES ARE TAKEN AT RAN- 
 DOM, OR SELECTED TO MAKE THE LARGEST EFFECT 
 
 By using the true weights 
 
 By falsifying 18 weights tenfold at random 
 By falsifying 18 weights tenfold by selection 
 
 Index Number 
 
 162.07 
 175.20 
 201.33 
 
 Let us note the small effect in the index number of increasing tenfold 
 the weight (quantity) of skins, the commodity which shows the greatest 
 aberrations from the general course of prices of the 36 commodities. Tak- 
 ing Formula 1153, for instance, we find the following effects on the index 
 numbers of prices on 1913 as base : 
 
 TABLE 60. INDEX NUMBERS COMPUTED BY USING 
 DIFFERENT WEIGHTS FOR SKINS 
 
 - 
 Weight Used 
 
 ■ 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 True 
 
 Ten times true . . . 
 
 100 
 100 
 
 100.13 
 100.15 
 
 99.89 
 99.93 
 
 114.20 
 114.66 
 
 161.70 
 162.05 
 
 177.83 
 177.96 
 
APPENDIX II 
 
 447 
 
 The effect of even this enormous increase of the weight of the most erratic 
 commodity is negligible. 
 
 In this case the effect was small because skins had originally so small a 
 weight. I have, therefore, tried to find the commodity in whose case in- 
 creasing the weight would most affect the index number. This seems to be 
 hay which, though not as erratic as skins, has much more weight to start 
 with. We find, using the same formula, the following results : 
 
 TABLE 61. INDEX NUMBERS COMPUTED BY USING 
 DIFFERENT WEIGHTS FOR HAY 
 
 Weight Used 
 
 1913 
 
 B1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 True 
 
 Ten times true . . 
 
 100 
 100 
 
 100.13 
 103.75 
 
 99.89 
 101.27 
 
 114.20 
 104.29 
 
 161.70 
 159.71 
 
 177.83 
 184.04 
 
 This case is extreme (1) because the commodity is extreme, being chosen 
 for its big influence, (2) because its influence is magnified by the fact of 
 there being only 36 commodities, and (3) because the change in the weight 
 (tenfold) is extreme. Yet even under all these circumstances combined 
 the effect of the change in weight does not exceed 3.6 per cent except in 
 one instance when it reaches nearly ten per cent. 
 
 Such hand picked instances as those just described are not, of course, 
 fair or representative of the actual situations with which the computer of 
 index numbers has to deal. Ordinarily inaccuracy in weights will not pro- 
 duce appreciable effects because (1) any inaccuracy is not likely to be very 
 great, such as 100 per cent, much less tenfold ; (2) if it does happen to be 
 great it is not likely that, at the same time, the commodity to which it 
 attaches will be very important or very erratic, much less both important 
 and erratic ; (3) if some of these things do conspire, there is stUl a good 
 chance that opposite errors elsewhere will largely offset the effect ; (4) even 
 at the worst the effect is greatly reduced if a large number of commodities 
 is used ; the average commodity in a Ust of 100 commodities might deviate 
 from the general average 100 per cent without affecting the final result 
 by one per cent. 
 
 § 7. Errors in Weights Less Important than in Prices 
 
 Correct weights in an index number of prices are far less important than 
 correct prices. Chart 68 shows the index numbers by Formula 3 for 1914 
 and 1917, and shows (1) what it becomes if the weight of any one of the 
 36 commodities is doubled, the weights of the rest being unchanged, as 
 well as (2) what it becomes if the price, relatively to 1913, of any one is 
 doubled (the prices of the rest being unchanged). 
 
 It will be noted that the doubling of the weight does not greatly swerve 
 the 1914 figure from the original 99.93. The largest increase is produced 
 by doubling the weight of hay which raises the index number from this 
 
448 THE MAKING OF INDEX NUMBERS 
 
 99.93 to 100.54, or about half of one per cent, and the largest decrease 
 is produced by doubling the weight of bituminous coal or pig iron 
 which lowers the index number to 99.59. Doubling the price, on the 
 other hand, changes the figure very considerably, causing it to reach 115.07 
 when lumber is doubled in price. 
 
 The same contrast is exhibited in 1917. Doubling a weight changes 
 162.07 at most to 167.36 (in the case of bituminous coal) while doubUng a 
 price raises the 162.07 to 179.54 in the case of lumber. The average 
 
 Weighting Is Relativety Unimportant 
 
 igij Effect of Doubling a Price 
 
 o 
 
 Effect of Doubling a Weight 
 Effect of Doubling a Price 
 
 \5% 
 
 original 
 Indexna 
 
 Chart 68. Showing that if the weighting of (say) barley is doubled, 
 the index number for 1914 is slightly decreased and that for 1917 is slightly 
 increased, while if the price relative of barley is doubled the index number 
 is greatly increased in both cases. 
 
 change produced by doubling the weight is .15 for 1914 and 1.08 for 1917, 
 while the average change produced by doubling the price is 2.77 for 1914 
 and 4.49 for 1917. Reduced to percentages of the index numbers them- 
 selves, doubling a weight affects it on the average .15 per cent in 1914 and 
 .67 per cent in 1917 and doubling a price afifects it 2.78 per cent in 1914 
 and 2.77 per cent in 1917. 
 
 Thus the effect produced by doubling a price is, in 1914, 18 times the 
 effect produced by doubling a weight, and, in 1917, four times. These 
 figures measure the relative importance of accuracy in prices and accuracy 
 in weights. The latter is comparatively unimportant. Rough estimates 
 and even guesses in selecting weights are admissible but guess work in 
 selecting price data is dangerous. However, weighting increases in im- 
 
 ^ The simple arithmetic average change from the original figure disregarding direction 
 of change. 
 
APPENDIX II 449 
 
 portance with an increase in the dispersion of prices. In 1914, when the 
 dispersion was small, doubling a weight had less effect than in 1917 when 
 the dispersion was larger. A formula could be worked out connecting 
 dispersion with the effect of weighting, but it would be different for different 
 sorts of index numbers. 
 
 These results are representative. But it should be noted that, in ex- 
 ceptional cases, doubling the weight may produce an effect equal to, or 
 greater than, doubling the price. Thus, if an individual price relative is 
 almost zero (say one per cent) while the average of all is high (say 100 
 per cent), doubling the price relative from one to two per cent will evi- 
 dently produce only an infinitesimal effect on the average 100 — a mere 
 fraction of one per cent ^- while doubling the weight, if the commodity 
 already has a heavy weight, will pull the index number down a con- 
 siderable part of the 98 per cent deviation between that commodity's 
 low price relative and the high original average, 100. In practice, how- 
 . ever, such cases are rarely, if ever, met with. 
 
 §8. What Weights are Best?; 
 
 In view of what has been said it is clear that weights may be at fault 
 either because they are erratic or because they have a wrong bias. As 
 to the former, all will agree that "simple" weighting, being usually very 
 erratic, should be avoided whenever possible. As to bias, the matter is 
 not so simple. We must not jump to the conclusion that cross weights 
 are always best. They are best for the geometric, median, mode, and 
 aggregative ; but for the arithmetic, the best weighting is the biased weight- 
 ing I or //, and for the harmonic the best weighting is the biased weight- 
 ing III or IV, because the upward bias possessed by the arithmetic type 
 needs to be counteracted by a downward bias in the weighting, and the 
 downward bias of the harmonic needs an upward bias in the weighting. 
 
 It has usually been assumed that the problem of finding the best formula 
 for an index number consists of two separate problems : (1) to find the best 
 type, and (2) to find the best weighting. But these two problems cannot 
 be separated, for the weight which is best for one type is not best for an- 
 other. 
 
 What has been said applies to the primary formulae. The system of 
 weighting immediately sinks into insignificance when we cross these for- 
 mulae to rectify them. Even such an absurdly weighted formula as 9, 
 where upward biased weights are applied to exaggerate the already upward 
 biased type, the arithmetic, when rectified by crossing with the like doubly 
 biased harmonic, 13, yields an excellent and unbiased result-, 109. In 
 short, rectification will cure bad weighting if the badness is systematically 
 biased. 
 
 But if the fault is merely that the weighting is erratic, as in the case of 
 the simple index number, the rectification by Test 1 will be of little avail. 
 Rectification by Test 2 will help more, but not completely. In short, bias 
 can be neutrahzed by bias, but freakishness is nearly incorrigible. 
 
 Thus the simple Formulae 1, 11, 21, 31, 41, 51 are freakishly weighted. 
 Crossing Formulae 1 and 11 gives 101 which is practically identical with 
 
450 THE MAKING OF INDEX NUMBERS 
 
 21. Thus Formula 101 as well as 21, 31, 41, 51 are free from bias but not 
 from freakishness. Crossing each with the next following even numbered 
 formulse (t/iz. 102, 22, 32, 42, 52), their factor antitheses, we get 301, 321, 
 331, 341, 351, which are only slight improvements over the originals. 
 
 § 9. Summary 
 
 We may summarize the main points in this Appendix as follows : 
 
 (1) The greater the number of commodities in an index number of 
 prices the less is the index number affected by a change i,n weights, or in 
 price relatives. 
 
 (2) A change in a weight has far less influence than a change in a price 
 relative. 
 
 (3) The contrast between two index numbers having weights of oppo- 
 site bias is greater than that between the simple and the cross weight index 
 numbers, in spite of the fact that the variations in the size of the weights 
 are immensely greater in the latter. 
 
 (4) A biased type of formula may be largely corrected by using an 
 oppositely biased sort of weighting. 
 
 (5) Bias disappears by rectification. Freakishness does not. 
 
APPENDIX III 
 
 AN INDEX NUMBER AN AVERAGE OF RATIOS RATHER 
 THAN A RATIO OF AVERAGES 
 
 § 1. Introduction 
 
 An index number should be an average of ratios rather than a ratio of 
 averages. There are always these two ways of averaging the data from 
 which index numbers are constructed. Thus, for 36 commodities, we may 
 either (1) average the 36 figures for one of the two years taken by itself 
 and again average the 36 figures for the other year taken by itself, and then 
 obtain the ratio between these two averages, or (2) we may take each in- 
 dividual commodity and calculate its own special ratio, or relative, as be- 
 tween the two years and then average these 36 relatives. The first way 
 is to take one ratio of two averages ; the second is to take one average of 36 
 ratios. 
 
 As applied to prices, the first method tells us the change in the average of 
 various -prices of commodities ; the second tells us the average of the various 
 changes of prices. These two, though usually confused, are very distinct. 
 The latter is much the more essential concept ; the former, though it can 
 be computed, is apt, in general, to prove a delusion and a snare. The 
 reason is that an average of the prices of wheat, coal, cloth, lumber, etc., 
 is an average of incommensurables and therefore has no fixed numerical 
 value. It can be calculated, but the resulting figure depends arbitrarily 
 on the units we happen to choose. The index number is thus indetermi- 
 nate, yielding different results for every different kind of measure. If 
 wheat is $1 a bushel, coal $10 a ton, cloth $2 a yard, and lumber §20 a 
 
 thousand board feet we may say that 
 
 1 + 10 + 2 + 20, 
 
 = $8.25 is the 
 
 average price of these four commodities "per unit." Suppose the four 
 prices above mentioned to be the prices for 1913 and suppose the four prices 
 in 1918 to be different, as per the following table : 
 
 
 1913 
 
 1918 
 
 Wheat, per bushel 
 
 Coal, per ton 
 
 Cloth, per yard 
 
 Lumber, per thousand 
 
 $ 1 
 
 10 
 
 2 
 
 20 
 
 $ 2 
 
 10 
 
 3 
 
 50 
 
 Average price per unit ' 
 
 $ 8.25 
 
 $16.25 
 
 451 
 
452 THE MAKING OF INDEX NUMBERS 
 
 1 ft *?^ 
 
 The index number, as the ratio of these average prices, is ' or 197 per 
 
 8.25 
 
 cent. But as there are four entirely separate and incommensurable units, 
 any one of which can be changed without entailing change in the others, 
 it is clear that this "average price" is really an unstable compound. If 
 we choose to have coal measured by the hundredweight its price must be 
 regarded no longer as $10 but as 50 cents, and this without requiring any 
 corresponding change in the prices of wheat, cloth, or lumber. The "aver- 
 age price" then for 1913 becomes 1 + -50 + 2 + 20 ^ ^^ g^ ^^^ "unit." 
 
 13 87 
 The "average price" for 1918 becomes $13.87 giving — '—•, or 236 per cent 
 
 5.87 
 
 as the index number. 
 
 Thus, simply by changing at will the unit of measuring coal, even though 
 it is changed in both numerator and denominator, we change the index 
 number from 197 to 236 ! 
 
 When this method is applied to the case of the 36 commodities, their 
 average price in 1913 is found to be 6.636 and in 1918, 11.464, the ratio of 
 which is 172.76 per cent as compared with the " ideal." 
 
 The case above mentioned is really Formula 51 in our table. For the 
 formula for the average price of commodities in year "0" is evidently 
 po + Po + Po+--- ^j, _Po ^jjgfg ^ jg ^]jg number of commodities, while 
 n n 
 
 the corresponding average price for year " 1" is — ^, making the index 
 
 n 
 
 number for year "1" relatively to year "0" 
 
 n 
 
 n 
 
 But, cancelling n, this becomes — 2i, our Formula 51. This cancellation 
 
 assumes, of course, that the number of commodities averaged is the same 
 in both years. 
 
 Formula 51 and its derivatives 52, 151, 152, 251, 351 are the only for- 
 mulae in our list which have the incommensurable defect due to taking a 
 ratio of averages and are affected by a change in units of measurement. 
 I have included them, however, partly because 51 is actually used by Brad- 
 street and partly because 51 seemed, so far as any formula can be said to 
 do so, to fill in the otherwise vacant space for a "simple aggregative." 
 
 § 2. Some Ratios of Simple Averages Calculated 
 
 For the reader who is curious to see what the corresponding ratio of aver- 
 ages would be like for the various types the following notes are added. I 
 have gone through the calculations because I find even experienced workers 
 
APPENDIX III 453 
 
 in index numbers are confused on this subject and do not seem to realize 
 that the ratio of average method is untrustworthy. 
 
 The simple harmonic average of prices is — ^ for year "0" and 
 
 2(1) 
 
 .-— — for year "1." The index number in the sense of the ratio of these 
 averages is, therefore, 
 
 
 This formula, like 51, could be used if the units of measure were judi- 
 ciously selected, but there would be no object in using it. For our 36 
 commodities it gives as the ratio of the simple harmonic average of the 
 prices for 1918 to the corresponding average for 1913, 165.67 per cent. 
 
 The simple geometric average of prices is -\/p^ X p'o X p"o X . . . 
 for year "0" and the corresponding formula for year "1." The index 
 number is their ratio. Evidently this can be reduced to Formula 21 and 
 is the average of ratios. It gives the index number of prices for 1918 rela- 
 tively to 1913 as 180.12. In the case, therefore, of the simple geometric 
 we get the same result whether we take the ratio of averages or the average 
 of ratios, assuming the same number (n) in both years. 
 
 The simple median and simple mode of prices are even more absurd 
 than the simple arithmetic and the simple harmonic. Thus, for the 
 median, after arranging in the order of magnitude the prices of 1513 and 
 those of 1918, we find that the median price in 1913 lies between the price 
 of barley, which is .6263 per bushel, and rubber which is .8071 per pound, 
 and maybe taken as their (geometric) mean, .7110, while the median price in 
 1918 lies between the price of barley, which is 1.4611 per bushel, and wool, 
 which is 1.66 per pound, and may be taken as their (geometric) mean, 
 1.5574. The ratio of these two medians is 219.05 per cent, an absurd result. 
 
 There remains only the aggregative method. This method is scarcely 
 applicable for taking an average of prices or of quantities. It is certainly 
 not applicable at all to averaging quantities since a quantity is not a ratio, 
 and the aggregative method of averaging implies ratios, the numerators 
 of which are to be added together and the denominators likewise. As to 
 prices, if we choose to go back of the individual price, each price is resolv- 
 able into a ratio of a quantity of money to a quantity of a commodity sold 
 for that money and we can, of course, add together the money spent by a 
 specified group of people on all the commodities for the numerator and, for 
 the denominator, add together the number of bushels, tons, yards, board 
 feet, etc. But this procedure would be as impracticable as it would be use- 
 
454 THE MAKING OF INDEX NUMBERS 
 
 less and arbitrary. The result would be the same as that for the weighted 
 arithmetical average price method which follows next. 
 
 § 3. Some Ratios of Weighted Averages Calculated 
 The weighted arithmetic average of prices, if the weights be the quanti- 
 ties, gives ±i22£ as the average price per "unit in 1913." The numerator of 
 
 this fraction is, it is true, homogeneous ; it is not a sum of incommensur- 
 ables but a sum of money values. The denominator, however, is made up 
 of incommensurables. Consequently, the resulting average itself is de- 
 pendent for its particular numerical value on the accident of what par- 
 ticular units of measm-ement happen to be employed. 
 
 The " index number " for 1914 relative to 1913 then becomes 
 
 2gi 
 
 S90 
 
 In this, the Sgi and Sgo are, neither of them, homogeneous and, what 
 is here the vital point, they are not equal and so do not cancel out. Con- 
 sequently they vitiate the resulting index number which is likewise de- 
 pendent on the particular units chosen and, therefore, absurd as an index 
 number. It reduces to 
 
 SgoPo 2go 
 
 which is Formula 52 in our series, but one of the worst. 
 
 Let us now calculate the index number which the last formula represents 
 — the ratio of the weighted arithmetic average of prices in 1918 to 1913 
 for our 36 commodities. Taking the price quotations as they stand we 
 find the arithmetic average of the prices 
 
 for 1918 is ^^^ = 29186J05 ^ ^^^^^^ 
 2^6 57219.75 
 
 and for 1913 is ?'i^ = ^^^"^"^^^ = .308861 
 Sgo 42429.44 
 
 The index number is, therefore, the former divided by the latter, or 165.14 
 per cent. But this index number is built on quicksands. For no one 
 could complain if in our reckoning the quotation of cotton was made per 
 bale instead of per pound. To take an extreme illustration which will 
 show in an extreme degree the absurdity of the results obtained by this 
 formula by simply changing the unit of measurement, let us measure rubber 
 in grains instead of in pounds. Under these circumstances 
 
 2g5P5 ^ 29186.105 ^ Qjjgg^ 
 Sga 2517368.25 
 
APPENDIX III 455 
 
 Sgopo ^ 13104.818 ^ Q^gggg 
 Sgo 852913.64 
 
 and the index number is 75.46 per cent. 
 
 Which shall we choose, the 165.14 or the 75.46? Evidently an index 
 number so constructed would be indeterminate unless, as a part of its speci- 
 fications, we prescribe every unit of measiu-e to be used in its calculation ! 
 
 But if we alter the numerator by substituting go, q'o, etc, for 51, q'l, 
 etc., the formula becomes 
 
 SgoPi 
 
 Zgo 
 
 SgoPo 
 
 sgo 
 in which the Sgo may be canceled leaving 
 
 SgoPo 
 
 or Formula 53. 
 
 Or, we may alter the denominator by substituting gi, q\, etc., for go, q'o, 
 etc., in which case, after cancellation, we obtain Formula 54. In both 
 these cases the cancellation removes all traces of incommensurables. 
 
 It thus turns out that the best of our primary formulae {viz. 53 and 54) 
 may be regarded as ratios of price averages which, although they seem at 
 first to have the "incommensurable" defect, are really free of it; for the 
 incommensurables are the same in numerator and denominator and so 
 disappear in the final result. And such an index number as 53 or 54 is 
 not really a ratio of averages of prices of the two years. Only one of the 
 two figures (the denominator for Formula 53, for instance) can be claimed 
 as the true average price of the year referred to. The other had to be 
 altered in order to insure ultimate cancellation of the incommensurables. 
 If the method of averaging the prices were a good one in this case, it ought 
 to stand on its own feet for both years. 
 
 Let us next take the geometric. If the weights be pogo, p'og'o, etc., for 
 year "0" and pigi, p'lg'i, etc., for year "1," the ratio of the geometric aver- 
 ages of prices is 
 
 2pigi / ; — r~i 
 
 Vpo^isi p'xP 19 1 X . .. 
 
 Spogo 
 
 igo/ ; — ; — ; 
 
 VpoP"® p'oPoao X . . . 
 
 Taking as the units of commodities those quoted in the market, this 
 formula gives the index number for 1918 relatively to 1913 as 124.53 per 
 cent. But if we change lumber from M board feet to board feet the same 
 formula gives 71.14 per cent ! Like all the others, therefore, the geometric 
 ratio of averages has the fatal blight of incommensurability. To be freed 
 of it, it is necessary to alter the pg's in either the numerator or the denomi- 
 nator, or both, so as to make the two agree. In this way we can make 
 
456 THE MAKING OF INDEX NUMBERS 
 
 the method of averaging prices yield results given by the other method, 
 that of averaging ratios, and get the Formulae 23, 25, 27, 29, and 6023. 
 
 Thus we find only two cases where this defect of incommensurability 
 disappears, namely, (1) in the geometric average of prices with constant ^ 
 weights, the ratio of which yields our Formulae 21, 23, 25, 27, 29, 6023, 
 and (2) the arithmetic average of prices weighted by quantities (provided, 
 however, these quantities are taken as the same in both years) which yields 
 our 53, 54, and 6053. 
 
 All these derive their immunity from the incommensurable taint from 
 the fact that the incommensurable elements cancel out, so that they can 
 be reduced to an average of price ratios. Moreover, all except the ratio of 
 the simple geometric averages (which reduces to Formula 21) have to be 
 altered before they can be reduced to an average of ratios and even the 
 exception named presupposes the choice of the same niunber of commodities 
 in the years compared. 
 
 In short, all true index numbers are averages of ratios. A ratio of 
 averages, unless reducible to an average of ratios, is subject to a haphazard 
 change from every change of unit. In other words, it fails in the "com- 
 mensurabiUty test" (Appendix I, Note to Chapter XIII, § 9), the elemen- 
 tary requirement of every application of mathematics, namely, of 
 possessing homogeneity. 
 
 § 4. Cases Where Averages of Prices can Properly be Used 
 
 The only cases in which it is really justifiable to use the genuine method 
 of taking the ratio of averages is where the units are really or nearly com- 
 mensurable. Thus, it is entirely legitimate to obtain the index number of 
 various quotations of one special kind of commodity, such as salt, by taking 
 the average of its prices in different markets. In such a case the precau- 
 tion, so essential in the previous examples, of forcibly altering numerator 
 to suit denominator, or vice versa, does not need to be taken. The true 
 average for each year can be taken independently of the other years. An- 
 other case is where the commodities are of one general group, such as kinds 
 of coffee or fuels, e.g. coal and coke where the same unit, such as the ton, 
 is used for all so that there is no danger of changing one without, at the 
 same time, changing the others equally. 
 
 The most interesting practical examples, however, are the average wage 
 of different but similar kinds of labor and the average price of different 
 but similar kinds of securities, in which cases the objection of incommen- 
 surability applies but not very strongly. In the stock market the aver- 
 age price of stocks is taken, the "common unit," if it may be so called, 
 being the "par value." 
 
 § 5. Conclusion 
 
 It perhaps does not greatly matter if the general public thinks of a "price 
 level" as something which can be calculated for each year independently 
 of other years and, to suit this concept, it is possible by making prices in 
 
 ' Constant as between the two years in the index number, not necessarily as to a series 
 of years. 
 
APPENDIX III 457 
 
 "dollars' worth" of one year, instead of in pounds, yards, etc., to expound 
 the subject in such terms before an elementary class. But such a trans- 
 position of units covertly introduces price ratios. The method of taking 
 the ratio of average prices is too lame to walk alone and needs always to 
 lean on the other and fully trustworthy method of averaging the price 
 ratios. 
 
 We conclude, then, that while it is possible to calculate an index number 
 by first averaging prices for the two years and then taking the ratio of the 
 two averages, this procedure has one of two faults. Either it makes the 
 resulting index number dependent on the arbitrary choice of units of 
 measure, so creating "haphazard weighting," or it requires us to force or 
 falsify one of the two averages to make it match the other in order to 
 enable us to cancel out the "incommensurable" items; in the latter case, 
 the resultant formula turns out, after all, to be an average of ratios. In 
 short, the ratio of averages has either the fault of being haphazard or 
 the fault of being superfluous. 
 
APPENDIX IV 
 
 LANDMARKS IN THE HISTORY OF INDEX NUMBERS ^ 
 
 A complete history of index numbers remains to be written. Data 
 for it are contained in C. M. Walsh's Measxirement of General Exchange 
 Value, and are summarized in J. L. Laughlin's Principles of Money and in 
 Wesley C. Mitchell's Index Numbers of Wholesale Prices, Bulletin 173 of the 
 United States Bureau of Labor Statistics and its revision, Bulletin 284. 
 Here I shall be even more brief, setting forth merely the chief landmarks 
 in the history of index numbers. 
 
 In 1738 Dutot published the prices in the times of Louis XII and of 
 Louis XIV by the formula here numbered 51. That is, he merely com- 
 pared the sums of prices as quoted. In 1747, as pointed out by Professor 
 Willard Fisher,^ the Colony of Massachusetts created a tabular standard 
 for the payment of indebtedness as a means of escaping the effects of 
 the depreciation of paper money. The same device was re-enacted in 
 1780, the state issuing notes "Both Principal and Interest to be paid in 
 the then current Money of said State, in a greater or less Sum, according 
 as Five Bushels of CORN, Sixty-eight Pounds and four-seventh Parts of a 
 Pound of BEEF, Ten Pounds of SHEEP'S WOOL, and Sixteen Pounds of 
 SOLE LEATHER shall then cost, more or less than One Hundred and 
 Thirty Pounds current Money, at the then current Prices of said Arti- 
 cles." This is equivalent to Formula 9051, the aggregative, with arbi- 
 trarily chosen and constant weights. 
 
 In 1764 Carli in Italy used Formula 1, the simple arithmetic average, for 
 comparing the price levels of 1500 and 1750 as revealed by the prices of 
 grain, wine, and oil, to show the effect of the discovery of America on the 
 purchasing power of money. In 1798 the same formula was used, doubt- 
 less independently, by G. Shuckburgh Evelyn in England. In 1812 Ar- 
 thur Young introduced weighting into Shuckburgh's method, thus using 
 Formula 9001. He counted wheat five times, barley and oats twice, pro- 
 visions four times, day labor five times, and wool, coal, and iron, once each. 
 
 The price changes caused by the Napoleonic wars and the effects of 
 paper money led a few students to further studies in index numbers. In 
 1822 Lowe, and, in 1833, Scrope, both in England, proposed Formula 9051 ; 
 Scrope says the quantities should be "determined by the proportionate 
 consumption" of the various articles. Lowe proposed a "standard from 
 materials" reduced into tabular form which Scrope called "the tabular 
 standard." This meant the correction by means of an index number of 
 contracts to pay sums of money in the future. In 1853 J. Prince-Smith 
 introduced the use of algebraic formulae into this subject, although he did 
 not put much trust in index numbers. 
 
 ■ These " landmarks " are, of course, in addition to the detailed historical notes scattered 
 through the book, usually as the concluding sections of the various chapters. 
 
 2 "The Tabular Standard in Massachusetts," Quarterly Journal of Economics, May, 1913. 
 
 458 
 
APPENDIX IV 459 
 
 In 1863 Jevons in England used Formula 21, the simple geometric, and 
 in 1865, worked out index numbers for English prices back to 1782. He was 
 concerned chiefly in showing the "fall in the value of gold" caused by the 
 outpourings of the gold mines beginning in 1849. He endorsed and strongly 
 urged Scrope's proposal for a tabular standard of value. Jevons seems to 
 have been the first to have kindled in others an interest in the subject and 
 may perhaps be considered the father of index numbers. In 1864 Las- 
 peyres, who in Germany worked out index numbers for Hamburg by For- 
 mula 1, opposed Jevons' 21 and proposed 53. 
 
 In 1869 the London Economist began its publication of index numbers 
 for 22 commodities. This still continues and is the oldest of the current 
 series. It uses Formula 1, although the base number is 2200, instead of 
 100. Recently the number of commodities has been doubled. 
 
 In 1874 Paasche in Germany proposed Formula 54 and applied it to 
 22 commodities for the years 1868 to 1872. 
 
 The fall of world prices beginning in 1873, reversing the rise which so 
 interested Jevons, gave a new turn to the study of index numbers. In 
 1880 an Italian economist and statistician, Messedaglia, made a commence- 
 ment of studying the nature of averages in application to this subject, in 
 his II calcolo dei valori medii e le sue applicazioni statistiche. In 1881, 
 H. C. Burchard, Director of the United States Mint, constructed an index 
 number for the years 1824-1880. This seems to be the first index num- 
 ber for the United States. 
 
 In 1886 Sauerbeck presented a paper to the Royal Statistical Society, 
 and began his well-known series of index numbers still continued by the 
 Statist. He used Formula 1. In 1886 Soetbeer began his German series. 
 In 1887 and 1889 Edgeworth wrote the two "Memoranda" on index num- 
 bers for the British Association for the Advancement of Science, the most 
 thorough investigation of index numbers up to that time. He recom- 
 mended several forms of index numbers : the arithmetic average, both 
 weighted and simple, the simple median, and the simple geometric, ac- 
 cording to the object sought. In 1890 Westergaard argued for the geo- 
 metric mean, with simple or constant weighting {i.e. Formula 21 or 9021) 
 on the ground of fulfilling the Westergaard, or circular test. In 1893 
 Falkner in the Aldrich Report of the United States Senate published index 
 numbers from 1840 to 1891, using Formulae 1 and 9001. In 1897 Brad- 
 street's began publishing its index number, using Formula 51, the units 
 for the various commodities being all taken as one pound each. 
 
 The rise of prices beginning in 1896, and continuing beyond the World 
 War, gave still another stimulus to the study of index numbers. Beginning 
 about 1900, the whole world increasingly complained of the high cost of 
 living, and index numbers were increasingly used to measure the rising 
 tide of prices. In 1901 Walsh published his Measurement of General Ex- 
 change Value, the largest and best work, and the only general treatise on 
 the theory of the subject up to the present time. In 1901 Dun's index 
 number by Formula 53 began. In 1902 the United States Bureau of 
 Labor Statistics began its index number of wholesale prices. 
 
 The first index numbers were of wholesale prices and most index numbers 
 are such today. For a long time it was thought that goods at retail were 
 
460 THE MAKING OF INDEX NUMBERS 
 
 not sufficiently standardized as to quality to make retail index numbers 
 practicable. This difficulty has not been fully overcome. But index 
 numbers of retail prices of foods were begun in the United States in 1907, 
 and today index numbers of retail prices are very common in most coun- 
 tries. Index numbers of wages are not yet as fiUly developed as of retail 
 prices. 
 
 In 1911, in my Purchasing Power of Money, I included a chapter and a 
 long Appendix on index numbers. In 1912 Knibbs, the Statistician of 
 Australia, urged Formula 53 on various grounds, especially ease of com- 
 putation, and discussed the subject mathematically. In 1915 Mitchell 
 published his thoroughgoing monograph on index numbers of wholesale 
 prices, already mentioned. Bulletin 173 of the United States Bureau of 
 Labor Statistics (revised as Bulletin 284, 1921). 
 
 In 1918 the National Industrial Conference Board published an index 
 number of the cost of living. In 1919 the United States Bureau of Labor 
 Statistics published an index number of the cost of living, including not 
 only foods, which had hitherto been almost the only retail items used in 
 index numbers, but substantially everything else. 
 
 Thus, since the begiiining of the present century, index numbers have 
 spread very fast. In the United States we now have among others the 
 index numbers of the United States Bureau of Labor Statistics, of the 
 Federal Reserve Board, of Dun's, of Bradstreet's, of Gibson, of the Times 
 Annalist, of Babson, of the National Industrial Conference Board, of the 
 Harvard Committee on Economic Research, and of the Massachusetts 
 Special Commission on the Necessaries of Life. A list, as nearly complete 
 as possible, of the index numbers, both discontinued and current, of all 
 countries has already been given in Appendix I (Note to Chapter XVII, 
 §14.) 
 
 It will be noticed that index numbers are a very recent contrivance. 
 That is, although we may push back the date of their invention a century 
 and three quarters, their current use did not begin till 1869 at the 
 earliest, and not in a general way till after 1900. In fact, it may be 
 said that their use is only seriously beginning today. 
 
 As stated in the text, in England the wages of over three million laborers 
 have been periodically, adjusted by means of an index number. 
 
APPENDIX V 
 
 LIST OF FORMULAE FOR INDEX NUMBERS 
 (For Reference) 
 
 § 1. Key to the Principal Algebraic Notations 
 
 po and qo represent price and quantity of a commodity at time "0" and 
 
 Pi and qi at time "1" 
 p'o and q'o represent price and quantity of another commodity at time "0" 
 
 and p'\ and q\ at time "1" 
 p"o and q"f) represent price and quantity of another commodity at time "0 " 
 
 and p"\ and q'\ at time " 1 " 
 p"'o and q"'o represent price and quantity of another commodity at time " " 
 and p"'i and q"'i at time "1" 
 etc., etc. 
 
 — , ^^ ^77^, etc. are price relatives the average of which is Pm 
 
 Po p p 
 
 —, ^, ^-TT' ^*c- ^^6 quantity relatives the average of which is Qoi 
 go go go 
 
 F is abbreviation for -2iil 
 2pogo 
 
 § 2. Key to Numbering of Formulae of Index Numbers 
 
 PRIMARY FORMULA (1-99) 
 
 Formula 
 No. 
 
 
 Formula 
 No. 
 
 
 1 
 
 Simple Arithmetic 
 
 2 
 
 Factor Antithesis of 1 
 
 31 
 
 7 
 9 
 
 Weighted I Arithmetic 
 Weighted II Arithmetic 
 Weighted III Arithmetic 
 Weighted IV Arithmetic 
 
 42 
 61 
 8 
 10 
 
 Factor Antithesis of 3 
 Factor Antithesis of 5 
 Factor Antithesis of 7 
 Factor Antithesis of 9 
 
 11 
 
 Simple Harmonic 
 
 12 
 
 Factor Antithesis of 1 1 
 
 13 
 15 
 171 
 19" 
 
 Weighted I Harmonic 
 Weighted II Harmonic 
 Weighted III Harmonic 
 Weighted IV Harmonic 
 
 14 
 16 
 
 18 2 
 201 
 
 Factor Antithesis of 13 
 Factor Antithesis of 15 
 Factor Antithesis of 17 
 Factor Antithesis of 19 
 
 * Reduces to Formula 63. 
 
 2 Reduces to Formula 54 
 
 461 
 
462 
 
 THE MAKING OF INDEX NUMBERS 
 
 Formula 
 No. 
 
 
 Formula 
 
 No. 
 
 
 211 
 
 Simple Geometric 
 
 22 2 
 
 Factor Antithesis of 21 
 
 23 
 25 
 27 
 29 
 
 Weighted I Geometric 
 Weighted II Geometric 
 Weighted III Geometric 
 Weighted IV Geometric 
 
 24 
 26 
 28 
 30 
 
 Factor Antithesis of 23 
 Factor Antithesis of 25 
 Factor Antithesis of 27 
 Factor Antithesis of 29 
 
 31 « 
 
 Simple Median 
 
 32* 
 
 Factor Antithesis of 31 
 
 33 
 35 
 37 
 39 
 
 Weighted I Median 
 Weighted II Median 
 Weighted III Median 
 Weighted IV Median 
 
 34 
 36 
 38 
 40 
 
 Factor Antithesis of 33 
 Factor Antithesis of 35 
 Factor Antithesis of 37 
 Factor Antithesis of 39 
 
 41 « 
 
 Simple Mode 
 
 42 « 
 
 Factor Antithesis of 41 
 
 43 
 45 
 47 
 49 
 
 Weighted I Mode 
 Weighted II Mode 
 Weighted III Mode 
 Weighted IV Mode 
 
 44 
 46 
 
 48 
 50 
 
 Factor Antithesis of 43 
 Factor Antithesis of 45 
 Factor Antithesis of 47 
 Factor Antithesis of 49 
 
 51 7 
 
 Simple Aggregative 
 
 528 
 
 Factor Antithesis of 51 
 
 63 
 69 » 
 
 Weighted I Aggregative 
 Weighted IV Aggregative 
 
 54 
 60"" 
 
 Factor Antithesis of 53 
 Factor Antithesis of 59 
 
 1 Same as Formula 121. 
 ' Same as Formula 122. 
 ' Same as Formula 131. 
 * Same as Formula 132. 
 
 ' Same as Formula 141. 
 
 • Same as Formula 142. 
 ^ Same as Formula 151. 
 
 * Same as Formula 152. 
 
 ' Same as Formula 54. 
 '" Same as Formula 53. 
 
 CROSS FORMULA FULFILLING TEST 1 (100-199) 
 (All Crossings of Fonnulae are by Geometric Mean) 
 
 101 Cross between 1 and 11 
 
 103 ■ Cross between 3» ^13 
 
 105 ' Cross between 5^\^1S 
 
 107 Cross between 7^/\>17 
 
 109 Cross between U *19 
 
 121 Cross between 21 and 21 
 
 123 Cross between 23 
 125 Cross between 25 1 
 27 J 
 29 
 
 102 Factor Antithesis of 101 and cross between 2 and 12 
 
 104 1 Factor Antithesis 
 106 1 Factor Antithesis 
 108 Factor Antithesis 
 110 Factor Antithesis 
 
 of 103 
 of 105 
 of 107 
 of 109 
 
 and cross between 
 and cross between 
 and cross between 8 
 and cross between 10' 
 
 122 Factor Antithesis of 121 and cross between 22 and 22 
 
 124 Factor Antithesis 
 126 Factor Antithesis 
 
 of 123 and cross between 24 
 of 125 and cross between 26 \ 
 28/ 
 30 
 
 131 Cross between 31 and 31 
 
 132 Factor Antithesis of 131 and cross between 32 and 32 
 
 133 Cross between 33 
 135 Crosa between 35 1 
 
 37. 
 
 39 
 
 134 Factor Antithesis 
 136 Factor Antithesis 
 
 of 133 and cross between 34 
 of 135 and crosa between 36 1 
 38/ 
 40 
 
 ' Reduces to Formula 353. 
 
APPENDIX V 
 
 463 
 
 141 Cross between 41 and 41 
 
 142 Factor Antithesis of 141 and cross between 42 and 42 
 
 143 Cross between 43 1 
 
 145 Cross between 45 1 1 
 
 47/ f 
 
 49 J 
 
 144 Factor Antithesis of 143 and cross between 44 
 146 Factor Antithesis of 145 and cross between 46 \ 
 
 48/ ■ 
 50 
 
 151 Cross between 51 and 51 
 
 152 Factor Antithesis of 151 and cross between 52 and 52 
 
 1531 Cross between 53 \ 
 59/ 
 
 1541 Factor Antithesis of 153 and cross between 54 \ 
 
 60/ 
 
 1 Reduces to Formula 353. 
 
 CROSS FORMULA FULFILLING TEST 2 (200-299) 
 
 201 
 
 Cross between 1 and 2 
 
 2313 
 
 Cross between 31 and 32 
 
 203' 
 2051 
 207 
 209 
 
 Cross between 3 and 4 
 Cross between 5 and 6 
 Cross between 7 and 8 
 Cross between 9 and 10 
 
 233 
 
 235 
 237 
 239 
 
 Cross between 33 and 34 
 Cross between 35 and 36 
 Cross between 37 and 38 
 Cross between 39 and 40 
 
 211 
 
 Cross between 11 and 12 
 
 241* 
 
 Cross between 41 and 42 
 
 213 
 215 
 2171 
 2191 
 
 Cross between 13 and 14 
 Cross between 15 and 16 
 Cross between 17 and 18 
 Cross between 19 and 20 
 
 243 
 245 
 247 
 249 
 
 Cross between 43 and 44 
 Cross between 45 and 46 
 Cross between 47 and 48 
 Cross between 49 and 50 
 
 2212 
 
 Cross between 21 and 22 
 
 2516 
 
 Cross between 51 and 52 
 
 223 
 225 
 227 
 229 
 
 Cross between 23 and 24 
 Cross between 25 and 26 
 Cross between 27 and 28 
 Cross between 29 and 30 
 
 253 « 
 259 « 
 
 Cross between 53 and 54 
 Cross between 59 and 60 
 
 ' Reduces to Formula 353. 
 * Same as Formula 321. 
 
 8 Same as Formula 331. 
 * Same as Formula 341. 
 
 ' Same as Formula 351. 
 * Reduces to Formula 353. 
 
 CROSS FORMUL.^ FULFILLING BOTH TESTS (300-399) 
 
 301 
 
 Cross between 1,11; 2, 12 
 
 also between 101 and 102 
 
 also between 201 and 211 
 
 3031 
 
 Cross between 3, 19 ; 4, 20 
 
 also between 103 and 104 
 
 also between 203 and 219 
 
 3051 
 
 Cross between 5, 17 ; 6, 18 
 
 also between 105 and 106 
 
 also between 205 and 217 
 
 307 
 
 Cross between 7, 15 ; 8, 16 
 
 also between 107 and 108 
 
 also between 207 and 215 
 
 309 
 
 Cross between 9, 13 ; 10, 14 
 
 also between 109 and 110 
 
 also between 209 and 213 
 
 321 
 
 Cross between 21, 21 ; 22, 22 
 
 also between 121 and 122 
 
 also between 221 and 221 
 
 323 
 
 Cross between 23, 29 ; 24, 30 
 
 also between 123 and 124 
 
 also between 223 and 229 
 
 ^ Reduces to Formula 353. 
 
464 
 
 THE MAKING OF INDEX NUMBERS 
 
 325 
 
 Cross between 25, 27 ; 26, 28 
 
 also between 125 and 126 
 
 also between 225 and 227 
 
 331 
 
 Cross between 31, 31 ; 32, 32 
 
 also between 131 and 132 
 
 also between 231 and 231 
 
 333 
 
 Cross between 33, 39 ; 34, 40 
 
 also between 133 and 134 
 
 also between 233 and 239 
 
 335 
 
 Cross between 35, 37 ; 36, 38 
 
 also between 135 and 136 
 
 also between 235 and 237 
 
 341 
 
 Cross between 41, 41 ; 42, 42 
 
 also between 141 and 142 
 
 also between 241 and 241 
 
 343 
 
 Cross between 43, 49 ; 44, 50 
 
 also between 143 and 144 
 
 also between 243 and 249 
 
 345 
 
 Cross between 45, 47 ; 46, 48 
 
 also between 145 and 146 
 
 also between 245 and 247 
 
 351 
 
 Cross between 51, 51 ; 52, 52 
 
 also between 151 and 152 
 
 also between 251 and 251 
 
 353 
 
 Cross between 53, 59 ; 54, 60 
 
 also between 153 and 154 
 
 also between 253 and 259 
 
 
 The foregoing formulae constitute the "main series"; the following, the 
 
 "supplementary series." 
 
 CROSS WEIGHT FORMULAE (1000-1999) 
 
 (Cross by Geometric Mean) 
 
 (1003 and 1013 do not fulfill Test 1 ; all 1100-1199 fuIfiU Test 1 and 1300- 
 1399 fulfiU both tests) 
 
 1003 
 
 Cross weight from 3 and 9 ; also from 5 and 7 
 
 1004 
 
 Factor Antithesis of 1003 
 
 1013 
 
 Cross weight from 13 and 19 ; also from 15 and 17 
 
 1014 
 
 Factor Antithesis of 1013 
 
 1103 
 
 Cross between 1003 and 1013 
 
 1104 
 
 Factor Antithesis of 1103 
 
 1123 
 
 Cross weight from 23 and 29 ; also from 25 and 27 
 
 1124 
 
 Factor Antithesis of 1123 
 
 1133 
 
 Cross weight from 33 and 39 ; also from 35 and 37 
 
 1134 
 
 Factor Antithesis of 1133 
 
 1143 
 
 Cross weight from 43 and 49 ; also from 45 and 47 
 
 1144 
 
 Factor Antithesis of 1143 
 
 1153 
 
 Cross weight from 53 and 59 
 
 1154 
 
 Factor Antithesis of 1 153 
 
 1303 
 1323 
 1333 
 1343 
 1353 
 
 Cross between 1103 and 1104 
 Cross between 1123 and 1124 
 Cross between 1133 and 1134 
 Cross between 1143 and 1144 
 Cross between 1153 and 1154 
 
 
 
APPENDIX V 
 
 465 
 
 CROSS WEIGHT FORMULA (2000-4999) 
 (Other than by Geometric Cross) 
 
 2153 
 2353 
 
 Cross -weight (arithmetically) from 53 and 54 
 Cross between 2153 and 2154 
 
 2154 
 
 Factor Antithesis of 2153 
 
 3153 
 3353 
 
 Cross weight (harmonically) from 53 and 54 
 Cross between 3153 and 3154 
 
 3154 
 
 Factor Antithesis of 3153 
 
 4153 
 4353 
 
 Cross weight (Lehr's) from 53 and 54 
 Cross between 4153 and 4154 
 
 4154 
 
 Factor Antithesis of 4153 
 
 MISCELLANEOUS FORMULAE (5000-9999) 
 
 Crosses of Cross Formulae (5000-5999) 
 
 5307 
 
 Cross between 307 and 309 
 
 5323 
 
 Cross between 323 and 325 
 
 5333 
 
 Cross between 333 and 335 
 
 5343 
 
 Cross between 343 and 345 
 
 Broadened Base Formula (6000-6999) 
 
 6023 
 6053 
 
 Like 23 except that base is average over two or more years 
 Like 53 except that base is average over two or more years 
 
 Blend (7000-7999) 
 
 7053 Average of 353's reckoned for every year 
 
 Arithmetic and Harmonic Averages of Formulae (8000-8999) 
 
 8053 
 8054 
 
 83531 
 
 Simple arithmetic average of 53 and 54 
 Simple harmonic average of 53 and 54 
 (also factor antithesis of 8053) 
 Cross of 8053 and 8054 
 
 Round Weight Formulae (9000-9999) ^ 
 
 9051 Calculated like 51 after judicious shifts of decimal points of the 36 quotations. 
 
 1 Reduces to Formula 353. 
 
 2 For 9001, 9011, 9021, 9031, and 9041, none of which are calculated in this book, see § 3 
 of this Appendix, Table 62. 
 
466 
 
 THE MAKING OF INDEX NUMBERS 
 
 § 3. TABLE 62. FORMULiE FOR INDEX NUMBERS 
 
 (F is abbreviation for -Ell^) 
 2po3o 
 Arithmetic Types 
 
 Symbols for Identtfication 
 
 Formula 
 
 
 No. 
 
 Letter 
 
 Name 
 
 
 1 
 
 A 
 
 Simple 
 
 2^ 
 Po 
 
 n 
 
 Carli 
 Schuckbvirg- 
 
 Evelyn 
 Economist 
 Sauerbeck, 
 
 Statist 
 Most others 
 
 2 
 
 
 
 V: '' 
 n 
 
 
 3* 
 
 A7 
 
 Weighted / 
 
 Zpo9o — 
 Spo?o 
 
 U. S. Bur. Labor 
 
 Statistics 
 
 4t 
 
 
 
 230P0 — ■ 
 Sg-oPo 
 
 
 5t 
 
 All 
 
 Weighted II 
 
 2po?i - 
 S/Jo^i 
 
 
 6* 
 
 
 
 ^1 
 290P1 — 
 
 290P1 
 
 
 7 
 
 A III 
 
 Weighted /// 
 
 Spi?o- 
 2pi5o 
 
 
 8 
 
 
 
 5i 
 S91P0 — 
 
 Sgipo 
 
 
 9 
 
 AIV 
 
 Weighted IV 
 
 2pi9i 
 
 Palgrave 
 
 10 
 
 
 
 2gipi 
 
 
 * Reduces to 63. 
 
 t Reduces to 54. 
 
APPENDIX V 
 
 467 
 
 
 
 TABLE 62 (<C<mtinued) 
 Harmonic Types 
 
 
 Symbols for Identification 
 
 FOBMULA 
 
 
 No. 
 
 Letter 
 
 Name 
 
 Approved bt 
 
 11 
 
 H 
 
 Simple 
 
 n 
 
 1 Vi 
 
 Coggeshall 
 
 12 
 
 
 
 V : " 
 
 
 13 
 
 H/ 
 
 Weighted / 
 
 Spogo 
 
 2pogo- 
 
 
 14 
 
 
 
 y . 2g0P0 
 
 2?oPo — 
 
 
 15 
 
 mi 
 
 Weighted // 
 
 2po?i 
 2Pogi - 
 
 
 16 
 
 
 
 . 2goPi 
 2goPi - 
 
 
 17* 
 
 mil 
 
 Weighted/// 
 
 
 
 18t 
 
 
 
 Sgipo - 
 
 
 19t 
 
 H/7 
 
 Weighted /F 
 
 2pigi 
 
 
 20* 
 
 
 
 2?iPi- 
 
 
 
 
 * Reduces t 
 
 53. t Reduces to 54. 
 
 
468 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62 (Continued) 
 Geometric Types 
 
 Symbols for Identification 
 
 FORMUI^ 
 
 
 No. 
 
 Letter 
 
 Name 
 
 
 21* 
 
 G 
 
 G/ 
 
 Simple 
 
 ^JEl . PJL ... 
 ' Po p'o 
 
 Jevons 
 
 Westergaard 
 
 Flux 
 
 22t 
 
 
 ylqo q'o 
 
 Nicholson 
 Walsh 
 
 23 
 
 Weighted I 
 
 ^Poqol/pi^^Polo/p'lXP'oQ'o^^^ 
 
 
 24 
 
 G/7 
 
 
 _ ^qoPoj/qAQoPo/q'Aq'oP'o _ 
 
 
 25 
 
 Weighted II 
 
 SpoW/pAPoaYp'iV'"^''... 
 
 
 26 
 
 
 _ ^qopil / qi^yopi / q\\q'op'i^^^ 
 
 
 27 
 
 GUI 
 
 Weighted III 
 
 2Pigol/pA PiQo/p'APWo^^^ 
 
 
 28 
 
 
 
 V ^ ^'^xl(-Y'^°(—Y'^'''- 
 
 
 29 
 
 GIV 
 
 Weighted IV 
 
 2pi9i//pi\p»9'/p'iy''^'',.. 
 
 Federal Reserve 
 Board 
 
 30 
 
 
 
 V ^ ^''f'//'£iY'^Y—Y '''''••■ 
 
 
 * Same as 121. 
 
 t Same aa 122. 
 
APPENDIX V 
 
 469 
 
 TABLE 62 (Continued) 
 Median Types 
 
 Symbols for Identification 
 
 Formula 
 
 
 No. 
 
 Letter 
 
 Name 
 
 
 31* 
 
 Me 
 
 Simple 
 
 Middle term of price 
 relatives 
 
 Edgeworth 
 Mitchell 
 
 32t 
 
 
 
 V -r- (Middle term of 
 
 quantity relatives) 
 
 
 33 
 
 Me/ 
 
 Weighted / 
 
 Mid-weight term of 
 price relatives 
 
 
 34 
 
 
 
 V -i- (Mid-weight term of 
 quantity relatives) 
 
 
 35 
 
 Me II 
 
 Weighted II 
 
 Mid-weight term of 
 price relatives 
 
 
 36 
 
 
 
 V -r- (Mid-weight term of 
 quantity relatives) 
 
 
 37 
 
 Me III 
 
 Weighted /// 
 
 Mid-weight term of 
 price relatives 
 
 
 38 
 
 
 
 V -5- (Mid-weight term of 
 quantity relatives) 
 
 
 39 
 
 Me IV 
 
 Weighted IV 
 
 Mid-weight term of 
 price relatives 
 
 
 40 
 
 
 
 V -5- (Mid-weight term of 
 quantity relatives) 
 
 
 * Same as 131. 
 
 t Same as 132. 
 
470 THE MAKING OF INDEX NUMBERS 
 
 
 
 TABLE 62 (Continued) 
 Mode Types 
 
 
 Stmbols fob Identification 
 
 Formula 
 
 Approved bt 
 
 No. 
 
 Letter 
 
 Name 
 
 41* 
 
 Mo 
 
 Simple 
 
 Commonest price 
 relative 
 
 
 42t 
 
 
 
 V -i- (Commonest quantity- 
 relative) 
 
 
 43 
 
 Mo/ 
 
 Weighted / 
 
 Weightiest price 
 relative 
 
 
 44 
 
 
 
 V -7- (Weightiest quantity- 
 relative) 
 
 
 45 
 
 Mo// 
 
 Weighted // 
 
 Weightiest price 
 relative 
 
 
 46 
 
 
 
 V -r- (Weightiest quantity- 
 relative) 
 
 
 47 
 
 Mo III 
 
 Weighted /// 
 
 Weightiest price 
 relative 
 
 
 48 
 
 
 
 V -i- (Weightiest quantity 
 relative) 
 
 
 49 
 
 Mo IV 
 
 Weighted IV 
 
 Weightiest price 
 relative 
 
 
 50 
 
 
 
 V -i- (Weightiest quantity 
 relative) 
 
 
 
 * 
 
 Same as 141. 
 
 t Same aa 142. 
 
 
APPENDIX V 
 
 471 
 
 TABLE 62 iContinued) 
 Aggregative Types 
 
 Symbols for Identification 
 
 Formula 
 
 
 No. 
 
 Letter 
 
 Name 
 
 
 51* 
 
 Ag 
 
 Simple 
 
 2pi 
 Spo 
 
 Bradstreet 
 Dutot 
 
 52t 
 
 
 
 S30 
 
 Drobisch 
 Rawson- 
 Rawson 
 
 53 
 
 Ag7 
 
 Weighted I 
 
 Spo3o 
 
 Dun 
 Fisher 
 Knibbs 
 Laspeyrea 
 Scrope 
 
 U. S. Bur. Lab. 
 Stat. 
 
 54 
 
 ^ 
 
 
 SgoPo 
 
 Fisher 
 
 Paasche 
 
 Scrope 
 
 59t 
 
 Ag/F 
 
 Weighted IV 
 
 2pigi 
 
 Spogi 
 
 
 60 § 
 
 
 
 
 
 * Same as 151. 
 t Same as 152. 
 
 t Same as 54. 
 § Same as 53. 
 
472 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62' (Continued) 
 
 Arithmetic and Harmonic Crosses 
 (fulfilling Test 1) 
 
 Symbols for Identification 
 
 Formula. 
 
 Approved by 
 
 No. 
 
 Name 
 
 
 Cross of : 
 
 
 
 
 Simples 
 
 
 101 
 
 Vl X 11 
 
 
 
 
 
 
 102 
 
 V2 X 12t 
 
 
 103* 
 
 Weighted 
 A 7 & H 77 
 
 
 
 
 V3 X 19 
 
 
 
 
 
 
 104* 
 
 V4 X 20t 
 
 
 105* 
 
 Weighted 
 A 77 & H 777 
 
 
 
 
 V5 X 17 
 
 
 
 
 
 
 106* 
 
 V6 X 18t 
 
 
 107 
 
 Weighted 
 A 777 & H 77 
 
 
 
 
 V7 X 15 
 
 
 
 
 
 
 108 
 
 V8 X 16t 
 
 
 109 
 
 Weighted 
 A 7F&H7 
 
 
 
 
 V9 X 13 
 
 
 
 
 
 
 110 
 
 VlO X 14t 
 
 
 * Reduces to 353. 
 
 t Also the factor antithesis of the immediately preceding formula, i.e. V -r said pre- 
 ceding formula with p's and q's interchanged. 
 
APPENDIX V 
 
 473 
 
 TABLE 62 {Continued) 
 
 Geometric Crosses 
 (fulfilling Test 1) 
 
 Symbols for Identification 
 
 Formula 
 
 
 No. 
 
 Name 
 
 Approved by 
 
 
 Cross of : 
 
 
 
 121* 
 
 Simples 
 
 V21 X 21 
 
 
 
 
 
 
 122 1 
 
 V22 X 22| 
 
 
 123 
 
 Weighted 
 G/&G/F 
 
 
 
 
 V23 X 29 
 
 
 
 
 
 
 124 
 
 V24 X 30$ 
 
 
 125 
 
 Weighted 
 G // & G /// 
 
 
 
 
 V25 X 27 
 
 
 
 
 
 
 126 
 
 V26 X 28| 
 
 
 * Reduces to 21. t Reduces to 22. 
 
 t Also the factor antithesis of the immediately preceding formula, i.e. V -i- said pre- 
 ceding formula with p'a and g's interchanged. 
 
 Median Crosses 
 (fulfilling Test 1) 
 
 Symbols fob Identification 
 
 Formula 
 
 Approved by 
 
 No. 
 
 Name 
 
 
 
 Cross of : 
 
 
 
 
 Simples 
 
 
 131* 
 
 V31 X 31 
 
 
 
 
 
 
 132 1 
 
 V32 X 32$ 
 
 
 133 
 
 Weighted 
 
 Me 7 & Me IV 
 
 
 
 
 V33 X 39 
 
 
 
 
 
 
 134 
 
 V34 X 40$ 
 
 
 135 
 
 Weighted 
 
 Me // & Me III 
 
 
 
 
 V35 X 37 
 
 
 136 
 
 
 V36 X 38$ 
 
 
 * Reduces to 31. t Reduces to 32. 
 
 t Also the factor antithesis of the immediately preceding formula, i.e. V -r said pre- 
 ceding formula with p'a and s's interchanged. 
 
474 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62 (Continued) 
 
 Mode Crosses 
 (fulfilling Test 1) 
 
 Symbols for Identification 
 
 FORMtTLA 
 
 
 No. 
 
 Name 
 
 
 
 Cross of : 
 
 
 
 
 Simples 
 
 
 141* 
 
 V41 X 41 
 
 
 
 
 
 
 142 1 
 
 V42 X 42 1 
 
 
 143 
 
 Weighted 
 
 Mo / & Mo IV 
 
 
 
 
 V43 X 49 
 
 
 
 
 
 
 144 
 
 V44 X 50 1 
 
 
 145 
 
 Weighted 
 
 Mo II & Mo /// 
 
 
 
 
 V45 X 47 
 
 
 
 
 
 
 146 
 
 V46 X 48 1 
 
 
 * Reduces to 41. t Reduces to 42. 
 
 t Also the factor antithesis of the immediately preceding formula, i.e. V -j- said pre- 
 ceding formula with p's and g'a interchanged. 
 
 
 Aggregative Crosses 
 (fulfilling Test 1) 
 
 
 Symbols fob Identification 
 
 Formula 
 
 
 No. 
 
 Name 
 
 
 
 Cross of : 
 
 
 
 
 Simples 
 
 
 151* 
 
 V5I X 51 
 
 
 
 
 
 
 152 1 
 
 V52 X 52 § 
 
 
 153 J 
 
 Weighted 
 
 Ag / & Ag IV 
 
 
 
 
 V53 X 59 
 
 
 
 
 
 
 154T 
 
 V54 X 60 § 
 
 
 * Reduces to 51. t Reduces to 52. J Reduces to 353. 
 
 J Also the factor antithesis of the immediately preceding formula, i.e. V -5- said pre- 
 ceding formula with p'a and ^'s interchanged. 
 
APPENDIX V 
 
 475 
 
 TABLE 62 (Continued) 
 
 Akithmetic Crosses 
 (fulfilling Test 2) 
 
 Symbols for Identification 
 
 Formula 
 
 
 No. 
 
 Name 
 
 Approved bt 
 
 
 Cross of : 
 
 
 
 201 
 
 Simple 
 and its 
 Fact. Antith. 
 
 
 
 Vl X2 
 
 
 203* 
 
 Weighted A / 
 and its 
 Fact. Antith. 
 
 
 
 
 V3 X4 
 
 
 205* 
 
 Weighted A // 
 and its 
 Fact. Antith. 
 
 
 
 
 V5 X6 
 
 
 207 
 
 Weighted A III 
 and its 
 Fact. Antith. 
 
 
 
 
 V7 X8 
 
 
 209 
 
 Weighted A IV 
 and its 
 Fact. Antith. 
 
 
 
 
 V9 X 10 
 
 
 * Reduces to 353. 
 
476 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62 {Cmtinued) 
 
 
 Harmonic Crosses 
 (fulfiUing Test 2) 
 
 
 Symbols fob Identification 
 
 FoRMUIiA 
 
 ApPBOVla) BY 
 
 No. 
 
 Name 
 
 
 Cross of : 
 
 
 
 211 
 
 Simple 
 and its 
 Fact. Antith. 
 
 
 
 Vll X 12 
 
 
 213 
 
 Weighted H / and 
 
 its 
 
 Fact. Antith. 
 
 
 
 
 Vl3 X 14 
 
 
 215 
 
 Weighted H // 
 and its 
 Fact. Antith. 
 
 
 
 
 Vl5 X 16 
 
 
 217* 
 
 Weighted H /// 
 and its 
 Fact. Antith. 
 
 
 
 
 Vl7 X 18 
 
 
 219* 
 
 Weighted H IV 
 and its 
 Fact. Antith. 
 
 
 
 
 Vl9 X 20 
 
 
 
 *R< 
 
 iduces to 353. 
 
 
APPENDIX V 
 
 477 
 
 TABLE 62 (Continued) 
 
 Geometric Crosses 
 (fulfilling Test 2) 
 
 Symbols for Identification 
 
 Formula. 
 
 Approved by 
 
 No. 
 
 Name 
 
 
 Cross of : 
 
 
 
 221* 
 
 Simple 
 and its 
 Fact. Antith. 
 
 
 
 V21 X 22 
 
 
 223 
 
 Weighted G / 
 and its 
 Fact. Antith. 
 
 
 
 
 V23 X 24 
 
 
 225 
 
 Weighted G // 
 and its 
 Fact. Antith. 
 
 
 
 
 V25 X 26 
 
 
 227 
 
 Weighted G /// 
 and its 
 Fact. Antith. 
 
 
 
 
 V27 X 28 
 
 
 229 
 
 Weighted G IV 
 and its 
 Fact. Antith. 
 
 
 
 
 V29 X 30 
 
 
 * Same as 321. 
 
478 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62 (Cmtinued) 
 
 Median Crosses 
 (fulfilling Test 2) 
 
 Symbols for Identification 
 
 Formula 
 
 Approved by 
 
 No. 
 
 Name 
 
 
 Cross of : 
 
 
 
 231* 
 
 Simple 
 and its 
 Fact. Antith. 
 
 
 
 V31 X 32 
 
 
 233 
 
 Weighted Me / 
 and its 
 Fact. Antith. 
 
 
 
 
 V33 X 34 
 
 
 235 
 
 Weighted Me // 
 and its 
 Fact. Antith. 
 
 
 
 
 V35 X 36 
 
 
 237 
 
 Weighted Me III 
 and its 
 Fact. Antith. 
 
 
 
 
 V37 X 38 
 
 
 239 
 
 Weighted Me /7 
 and its 
 Fact. Antith. 
 
 
 
 
 V39 X 40 
 
 
 * Same as 331. 
 
APPENDIX V 
 
 479 
 
 TABLE 62 {Continued) 
 
 Mode Crosses 
 (fuimiing Test 2) 
 
 Symbols for Identification 
 
 FOKMULA 
 
 
 No. 
 
 Name 
 
 
 
 Cross of : 
 
 
 
 241* 
 
 Simple 
 and its 
 Fact. Antith. 
 
 
 
 V41 X 42 
 
 
 243 
 
 Weighted Mo / 
 and its 
 Fact. Antith. 
 
 
 
 
 V43 X 44 
 
 
 245 
 
 Weighted Mo // 
 and its 
 Fact. Antith. 
 
 
 
 
 V45 X 46 
 
 
 247 
 
 Weighted Mo III 
 and its 
 Fact. Antith. 
 
 
 
 
 V47 X 48 
 
 
 249 
 
 Weighted Mo 77 
 and its 
 Fact. Antith. 
 
 
 
 
 V49 X 50 
 
 
 * Same as 341. 
 
 Aggregative Crosses 
 (fulfiUing Test 2) 
 
 Symbols for Identification 
 
 Formula 
 
 
 No. 
 
 Name 
 
 
 
 Cross of : 
 
 
 
 251* 
 
 Simple 
 and its 
 Fact. Antith. 
 
 
 
 V51 X 52 
 
 
 253 1 
 
 Weighted Ag I 
 and its 
 Fact. Antith. 
 
 
 
 
 V53 X 54 
 
 
 259 1 
 
 Weighted Ag IV 
 and its 
 Fact. Antith. 
 
 
 
 
 V59 X 60 
 
 
 * Same as 351. 
 
 t Reduces to 353. 
 
480 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62 {Continued) 
 
 Arithmetic and Harmonic Crosses 
 (fulfilling both tests) 
 
 STMB0L8 FOR Identification 
 
 No. 
 
 301 
 
 303^ 
 
 305^ 
 
 307 
 
 309 
 
 Name 
 
 Crosses of ; 
 
 Simples A & H 
 and their 
 Fact. Antith. 
 
 Weighted A / & H 77 
 and their 
 Fact. Antith, 
 
 Weighted All &B. Ill 
 and their 
 Fact. Antith. 
 
 Weighted A 77/ & H 77 
 and their 
 Fact. Antith. 
 
 Weighted A 7F & H 7 
 and their 
 Fact. Antith. 
 
 Formula 
 
 </l X 2 X 11 X 12 or 
 VlOl X 102" or 
 V201 X 211 
 
 ■s/S X 4 X 19 X 20 or 
 Vl03 X 104 or 
 V203 X 219 
 
 -^5 X 6 X 17 X 18 or 
 Vl05 X 106 or 
 V205 X 217 
 
 a/7 X 8 X 15 X 16 or 
 Vl07 X 108 or 
 V207 X 215 
 
 ■\/9 X 10 X 13 X 14 or 
 Vl09 X no" or 
 V209 X 213 
 
 Approved by 
 
 Geometric Crosses 
 (fulfilling both tests) 
 
 321 1 
 
 323 
 
 325 
 
 Simple G 
 and its 
 Fact. Antith. 
 
 Weighted G 7 & G 7F 
 and their 
 Fact. Antith. 
 
 Weighted G 77 & G 777 
 and their 
 Fact. Antith. 
 
 -v/21 X 22 X 21 X 22 
 or Vl21 X 122 or 
 V221 X 221 
 
 \/23 X 24 X 29 X 30 
 or Vl23 X 1 24 or 
 V223 X 229 
 
 \/25 X 26 X 27 X 28 
 or Vl25 X 1 26 or 
 V225 X 227 
 
 * Reduces to 353. 
 
 t Reduces to 221. 
 
APPENDIX V 
 
 481 
 
 TABLE 62 {Continued) 
 
 Median Crosses 
 (fulfilling both tests) 
 
 Symbols for Identification 
 
 No. 
 
 331^ 
 
 333 
 
 335 
 
 Name 
 
 Crosses of : 
 
 Simple Me 
 and its 
 Fact. Antith. 
 
 Weighted Me I & Me IV 
 and their 
 Fact. Antith. 
 
 Weighted Me // & Me III 
 and their 
 Fact. Antith. 
 
 Formula 
 
 V31 X 32 X 31 X 32 
 or Vl31 X 132 or 
 V231 X 231 
 
 
 a/33 X 34 X 39 X 40 
 or ^^133 X 134 or 
 V233 X 239 
 
 
 </35 
 or 
 
 X 36 X 37 X 38 
 Vl35 X 136 or 
 
 V235 X 237 
 
 Approved by 
 
 Mode Crosses 
 (fulfilling both tests) 
 
 
 Simple Mo 
 and its 
 Fact. Antith. 
 
 
 
 341 1 
 
 -\/41 X 42 X 41 X 42 
 or Vl41 X 142 or 
 V241 X 241 
 
 
 
 Weighted Mo 7 & Mo /F 
 and their 
 Fact. Antith. 
 
 
 
 343 
 
 </43 X 44 X 49 X 50 
 or Vl43 X 144 or 
 V243 X 249 
 
 
 
 Weighted Mo II & Mo III 
 and their 
 Fact. Antith. 
 
 
 
 345 
 
 </45 X 46 X 47 X 48 
 or Vl45 X 146 or 
 V245 X 247 
 
 
 * Reduces to 231. 
 
 t Reduces to 241. 
 
482 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62 {Continued) 
 
 Aggregative Crosses 
 (fulfilling both tests) 
 
 Symbols for Identification 
 
 Formula 
 
 Approved by 
 
 No. 
 
 Name 
 
 
 Crosses of : 
 
 
 
 
 Simple Ag 
 and its 
 Fact. Antith. 
 
 
 351* 
 
 \/51 X 52 X 51 X 52 
 or V'l51 X 152 or 
 V251 X 251 
 
 
 
 Weighted Ag/&Ag/F 
 and their 
 Fact. Antith. 
 
 "Ideal" 
 
 Fisher 
 
 353 1 
 
 \ Spogo 2po?i 
 
 Pigou 
 Walsh 
 Allyn Young 
 
 * Reduces to 251. 
 
 t Same as 103, 104, 105, 106, 153, 154, 203, 205, 217, 219, 253, 259, 303, 305. 
 
 The foregoing formulae constitute the "main series"; the following, the 
 'supplementary series." 
 
 Cross-weight Arithmetics and Harmonics 
 
 (fulfilling neither test) 
 
 No. 
 
 Name 
 
 Formula 
 
 Approved by 
 
 
 Derived by : 
 
 /Vi\ 
 
 
 1003 
 
 Crossing 
 weights 
 of 3&9 
 or of 5 & 7 
 
 
 
 ^VmoMiI^^J 
 
 
 
 ^'^Poqo PiQi 
 
 
 
 Fact. Antith. 
 of 1003 
 
 /qi\ 
 
 
 1004 
 
 
 
 
 Z^qoPa qiPi 
 
 
 
 of 13 & 19 
 or of 15 & 17 
 
 
 
 1013 
 
 sVpo^o PiQx 
 
 
 
 2Vpo3o p,qx h!l\ 
 
 
 
 Fact. Antith. 
 of 1013 
 
 
 
 1014 
 
 SV^opo qiPi 
 sVgoPo qiPi (^ 
 
 
APPENDIX V 
 
 TABLE 62 ^Continued) 
 
 483 
 
 No. 
 
 Name 
 
 Formula 
 
 
 
 Crosses op Preceding (fulfilling Test 1) 
 
 
 
 Cross of : 
 
 
 
 
 Cross-weights 
 A&H 
 
 
 1103 
 
 V1003 X 1013 
 
 
 1104 
 
 Fact. Antith. 
 of 1103 
 
 
 
 
 V1004 X 1014 
 
 
 Cross-weight Geometrics, Medians, Modes, and Aggregatives 
 (fulfiUing Test 1) 
 
 No. 
 
 Name 
 
 Formula 
 
 Approved by 
 
 
 Derived by : 
 
 
 Walsh 
 
 
 Crossing 
 weights 
 of 23 & 29 
 or of 25 & 27 
 
 
 1123 
 
 SVpogo Pl9l//pi Wpogo Pig, 
 
 
 1124 
 
 Fact. Antith. 
 
 of 
 
 1123 
 
 
 
 SVgopo QWij/qi\Vqopo qipi 
 
 
 1133 
 
 of 33 & 39 
 or of 35 & 37 
 
 Mid cross-weight term of price 
 relatives 
 
 
 1134 
 
 Fact. Antith. 
 
 of 
 
 1133 
 
 „ . Mid cross-weight term of 
 quantity relatives 
 
 
 1143 
 
 of 43 & 49 
 or of 45 & 47 
 
 Weightiest cross-weight price 
 relative 
 
 
 1144 
 
 Fact. Antith. 
 
 of 
 
 1143 
 
 „ . Weightiest cross-weight 
 quantity relative 
 
 
 1153 
 
 of 53 & 59 
 
 2Vgogi Po 
 
 Scrope 
 Walsh 
 
 1154 
 
 Fact. Antith. 
 
 of 
 
 1153 
 
 . ^Vpopi qi 
 ^Vpopi go 
 
 Walsh 
 
484 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62 {Continued) 
 
 Crosses of Preceding Cross-weight Formula 
 (fulfilling Test 1 and Test 2) 
 
 Symbols for Identification 
 
 
 No. 
 
 Formula 
 
 
 
 
 
 1303 
 
 V1103 X 1104 
 
 
 
 
 
 1323 
 
 V1123 X 1124 
 
 
 
 
 
 1333 
 
 V1133 X 1134 
 
 
 
 
 
 1343 
 
 V1143 X 1144 
 
 
 
 
 
 1353 
 
 V1153 X 1154 
 
 
 Cross-weight Aggregatives, Miscellaneous 
 
 Symbols foh Identification 
 
 Formula 
 
 Approved by 
 
 No. 
 
 Name 
 
 
 
 (fulfilling Test 1) 
 
 
 2153* 
 
 Arithmetically 
 crossed weight 
 aggregative 
 
 Z^«+^'pi 
 
 Edgeworth 
 Fisher 
 
 
 2^» + ^>o 
 
 Jj 
 
 Marshall 
 Walsh 
 
 2154* 
 
 Fact. Antith. 
 
 of 
 
 2153 
 
 Y • ^ 
 
 Walsh 
 
 
 
 
 2353^ 
 
 Cross of 
 
 preceding 
 
 two 
 
 (fulfilling Tests 1 and 2) 
 
 V2153 X 2154 
 
 * As to alternative forms, see Note " Alternative Forms of Certain Formulae " at end 
 of table. 
 
APPENDIX V 
 
 TABLE 62 (Continued) 
 
 485 
 
 Symbols fok Identification 
 
 Formula 
 
 Approved bt 
 
 No. 
 
 Name 
 
 
 
 (fulfilling Test 1) 
 
 
 3153* 
 
 Harmonically 
 crossed weight 
 aggregative 
 
 / 2 \ 
 
 
 
 \qo gi/ 
 
 
 
 / ^ \ 
 
 
 
 \?o qif 
 
 
 3154 
 
 Fact. Antith. 
 
 of 
 
 3153 
 
 I ^ \ 
 
 
 
 2(L+L)?i 
 
 TT . \P0 Pi/ 
 
 
 
 / M 
 
 
 
 2 1 ^ L Uo 
 
 \Po Pi/ 
 
 
 (fulfilling Tests 1 and 2) 
 
 Cross of 
 
 preceding 
 
 two 
 
 V3153 X 3154 
 
 (fulfilling Test 1) 
 
 Weighted 
 arithmetically 
 crossed weight 
 aggregative 
 
 Po + Pi 
 
 ^Pogo+Pigip^ 
 
 Po + Pi 
 
 Fact. Antith. 
 
 of 
 
 4153 
 
 F ^ 
 
 ■^ goPo + glPlg ^ 
 
 go + gl 
 ^ goPo + glPl „ 
 
 2. 1 qa 
 
 go + gi 
 
 Lehr 
 
 (fulfilling Tests 1 and 2) 
 
 Cross of 
 
 preceding 
 
 two 
 
 V4153 X 4154 
 
 * As to alternative forms, see Note " Alternative Forms of Certain Formulae " at end of table. 
 
486 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 62 {Continued) 
 
 Crosses of Crosses 
 (fulfilling Tests 1 and 2) 
 
 No. 
 
 Formula 
 
 Approved by 
 
 
 
 
 5307 
 
 V307 X 309 
 
 * 
 
 
 
 
 5323 
 
 V323 X 325 
 
 
 
 
 
 5333 
 
 V333 X 335 
 
 
 
 
 
 5343 
 
 V343 X 345 
 
 
 Geometric and Aggregative Broadened Base Formula 
 (fulfilling neither test) 
 
 Symbols for Identification 
 
 FOEMUIiA 
 
 Approved by 
 
 No. 
 
 Name 
 
 6023 
 
 Geometric 
 
 Same as 23 
 
 Day 
 
 
 Broadened base 
 
 after substituting 
 
 Persons 
 
 
 1913-14 
 
 for "0," 0-1, or '13-'14 
 
 
 6023 
 
 Same 
 
 Same as 23 
 
 Dav 
 
 
 1913-16 
 
 after substituting 
 for "0," 0-1-2-3, or 
 '13-'14-'15-'16 
 
 Persons 
 
 6023 
 
 Same 
 
 Same as 23 
 
 Day 
 
 
 1913 and 1918 
 
 after substituting 
 for "0," and 5, or 
 '13 and '18 
 
 Persons 
 
 6053 
 
 Aggregative 
 
 Same as 53 
 
 
 
 Broadened base 
 
 after substituting 
 
 
 
 1913-14 
 
 for "0," 0-1, or '13-' 14 
 
 
 6053 
 
 Same 
 
 Same as 53 
 
 
 
 1913-16 
 
 after substituting 
 for "0," 0-1-2-3, or 
 '13-' 14^' 15-' 16 
 
 
 6053 
 
 Same 
 
 Same as 53 
 
 
 
 1913-18 
 
 after substituting 
 
 for "0," 0-1-2-3-4-5, or 
 
 '13-' 14-' 15-' 16-' 17-' 18 
 
 
APPENDIX V 
 
 487 
 
 TABLE 62 (Continued) 
 
 Arithmetic and Harmonic Means of Aggregative Index Numbers 
 
 (fulfilling neither test) 
 
 No. 
 
 Name 
 
 Formula 
 
 Approved by 
 
 7053 
 
 Arithmetic mean 
 of ideal formula 
 on different base 
 years 
 
 353 ('13) + 353 ('14) + 353 ('15) + 
 353 ('16) + 353 ('17) + 353 ('18) 
 -j-6 
 
 
 8053 
 
 Arithmetic mean 
 of aggregative 
 
 53 + 54 _ Zpo^o ' SpoQ'i 
 
 Sidgwick 
 Drobisch 
 
 
 2 2 
 
 
 8054 
 
 Fact. Antith. 
 
 of 
 
 8053 
 
 'SqiPo , Sg-ipi 
 y . 2goPo SgoPi _ 2 
 
 
 
 2 Spoqo Spogi 
 
 
 
 
 
 
 8353* 
 
 V8053 X 8054 
 
 
 * Reduces to 353. 
 
 All Types of Index Numbers with Constant Weights 
 
 Symbols fob Identification 
 
 Formula 
 
 
 No. 
 
 Name 
 
 
 
 Weighted by 
 
 arbitrary 
 
 constants 
 
 2y, — where the w's 
 Po are arbitrary 
 
 Dun 
 
 9001 
 
 Arithmetic 
 
 Falkner 
 
 Zw constant weights 
 
 Ar. Young 
 
 9011 
 
 Harmonic 
 
 Sw; where the w's 
 
 
 
 p^ are arbitrary 
 ^"' — constant weights 
 
 
 
 Geometric 
 
 
 
 9021 
 
 2w//pA«) where the w's 
 \ \ / ' ' ■ ^^® arbitrary 
 ^° constant weights 
 
 
 9031 
 
 Median 
 
 Mid-weight term of price relatives 
 
 
 9041 
 
 Mode 
 
 Weightiest price relative 
 
 
 9051 
 
 Aggregative 
 
 Xio pi where the w's 
 Xw po ^^^ arbitrary 
 
 constant weights 
 
 Lowe 
 
488 THE MAKING OF INDEX NUMBERS 
 
 ALTERNATIVE FORMS OF CERTAIN FORMULA 
 
 Many formulse may be changed into forms other than those given in the 
 foregoing table. The footnotes to the table indicate some transformations 
 such as of Formula 3 into Formula 53. There are many others. Thus we 
 may derive at least five alternative forms for Formula 2153, five for 2154 
 two for 2353, five for 3153, seven for 4153. In most of these cases, the 
 form easiest to calculate is not that given in the table. Thus the most 
 easily calculated form of 2153 is 
 
 that of 2154 is 
 
 and of 3153 
 
 2(5i + 9o)Po 
 
 1 I 5Mo 
 
 Spogo 
 
 I I ^Pogi 
 
 Spigi 
 
 gogi 
 
 go + gi 
 
 2po 
 
 gogi 
 go+gi 
 
APPENDIX VI 
 
 NUMERICAL DATA AND EXAMPLES 
 
 s 1. the data for the 36 commodities, prices and quantities 
 
 Table 63. Prices of the 36 Commodities, 1913-1918 
 
 No. 
 
 COMMODITT 
 
 po 
 
 Pi 
 
 P2 
 
 P3 
 
 P4 
 
 P6 
 
 
 
 1913 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 1 
 
 Bacon 
 
 .1236 
 
 .1295 
 
 .1129 
 
 .1462 
 
 .2382 
 
 .2612 
 
 2 
 
 Barley . . 
 
 
 
 .6263 
 
 .6204 
 
 .7103 
 
 .8750 
 
 1.3232 
 
 1.4611 
 
 3 
 
 Beef . . 
 
 
 
 .1295 
 
 .1364 
 
 .1289 
 
 .1382 
 
 .1672 
 
 .2213 
 
 4 
 
 Butter . . 
 
 
 
 .2969 
 
 .2731 
 
 .2743 
 
 .3179 
 
 .4034 
 
 .4857 
 
 5 
 
 Cattle . . 
 
 
 
 12.0396 
 
 11.9208 
 
 12.1354 
 
 12.4375 
 
 15.6354 
 
 18.8646 
 
 6 
 
 Cement 
 
 
 
 1.5800 
 
 1.5800 
 
 1.4525 
 
 1.68SS 
 
 2.0942 
 
 2.6465 
 
 7 
 
 Coal, anth. 
 
 
 
 5.0636 
 
 5.0592 
 
 5.0464 
 
 5.2906 
 
 5.6218 
 
 6.5089 
 
 8 
 
 Coal, bit. . 
 
 
 
 1.2700 
 
 1.1700 
 
 1.0400 
 
 2.0700 
 
 3.5800 
 
 2.4000 
 
 9 
 
 Coffee . . 
 
 
 
 .1113 
 
 .0816 
 
 .0745 
 
 .0924 
 
 .0929 
 
 .0935 
 
 10 
 
 Coke . . 
 
 
 
 3.0300 
 
 2.3200 
 
 2.4200 
 
 4.7800 
 
 10.6600 
 
 7.0000 
 
 11 
 
 Copper . . 
 
 
 
 .1533 
 
 .1318 
 
 .1676 
 
 .2651 
 
 .2764 
 
 .2468 
 
 12 
 
 Cotton . . 
 
 
 
 .1279 
 
 .1121 
 
 .1015 
 
 .1447 
 
 .2350 
 
 .3178 
 
 13 
 
 Eggs . . 
 
 
 
 .2468 
 
 .2660 
 
 .2597 
 
 .2945 
 
 .4015 
 
 .4827 
 
 14 
 
 Hay . . . 
 
 
 
 11.2500 
 
 12.3182 
 
 11.6250 
 
 10.0625 
 
 17.6042 
 
 21.8958 
 
 15 
 
 Hides . . 
 
 
 
 .1727 
 
 .1842 
 
 .2076 
 
 .2391 
 
 .2828 
 
 .2144 
 
 16 
 
 Hogs . . 
 
 
 
 8.3654 
 
 8.3608 
 
 7.1313 
 
 9.6459 
 
 15.7047 
 
 17.5995 
 
 17 
 
 Iron bars . 
 
 
 
 1.5100 
 
 1.2000 
 
 1.3700 
 
 2.5700 
 
 4.0600 
 
 3.5000 
 
 18 
 
 Iron, pig . 
 
 
 
 14.9025 
 
 13.3900 
 
 13.5758 
 
 18.6708 
 
 38.8082 
 
 36.5340 
 
 19 
 
 Lead (white) 
 
 
 
 .0676 
 
 .0675 
 
 .0698 
 
 .0927 
 
 .1121 
 
 .1271 
 
 20 
 
 Lead . . 
 
 
 
 .0437 
 
 .0386 
 
 .0467 
 
 .0686 
 
 .0879 
 
 .0741 
 
 21 
 
 Lumber 
 
 
 
 90.3974 
 
 90.9904 
 
 90.5000 
 
 91.9000 
 
 105.0400 
 
 121.0455 
 
 22 
 
 Mutton 
 
 
 
 .1025 
 
 .1010 
 
 .1073 
 
 .1250 
 
 .1664 
 
 .1982 
 
 23 
 
 Petroleum 
 
 
 
 .1233 
 
 .1200 
 
 .1208 
 
 .1217 
 
 .1242 
 
 .1695 
 
 24 
 
 Pork . . 
 
 
 
 .1486 
 
 .1543 
 
 .1429 
 
 .1618 
 
 .2435 
 
 .2495 
 
 25 
 
 Rubber . 
 
 
 
 .8071 
 
 .6158 
 
 .6573 
 
 .6694 
 
 .6477 
 
 .5490 
 
 26 
 
 Silk . . . 
 
 
 
 3.9083 
 
 4.0573 
 
 3.6365 
 
 5.4458 
 
 5.9957 
 
 6.9770 
 
 27 
 
 Silver . . 
 
 
 
 .5980 
 
 .5481 
 
 .4969 
 
 .6566 
 
 .8142 
 
 .9676 
 
 28 
 
 Skins . , 
 
 
 
 2.5833 
 
 2.6250 
 
 2.7188 
 
 4.1729 
 
 5.5208 
 
 5.5625 
 
 29 
 
 Steel rails . 
 
 
 
 28.0000 
 
 28.0000 
 
 28.0000 
 
 31.3333 
 
 38.0000 
 
 54.0000 
 
 30 
 
 Tin, pig . 
 
 
 
 44.3200 
 
 35.7000 
 
 38.6600 
 
 43.4800 
 
 61.6500 
 
 87.1042 
 
 31 
 
 Tin plate . 
 
 
 
 3.5583 
 
 3.3688 
 
 3.2417 
 
 5.1250 
 
 9.1250 
 
 7.7300 
 
 32 
 
 Wheat . . 
 
 
 
 .9131 
 
 1.0412 
 
 1.3443 
 
 1.4165 
 
 2.3211 
 
 2.2352 
 
 33 
 
 Wool . . 
 
 
 
 .5883 
 
 .5975 
 
 .7375 
 
 .7900 
 
 1.2841 
 
 1.6600 
 
 34 
 
 Lime . . 
 
 
 
 1.2500 
 
 1.2500 
 
 1.2396 
 
 1.4050 
 
 1.7604 
 
 2.3000 
 
 35 
 
 Lard . . 
 
 
 
 .1101 
 
 .1037 
 
 .0940 
 
 .1347 
 
 .2170 
 
 .2603 
 
 36 
 
 Oats . . 
 
 
 
 .3758 
 
 .4191 
 
 .4958 
 
 .4552 
 
 .6372 
 
 .7747 
 
 489 
 
490 THE MAKING OF INDEX NUMBERS 
 
 Table 64. Quantities Marketed op the 36 Commodities, 1913-1918 
 (in millions of units) 
 
 No. 
 
 Commodity 
 
 qo 
 1913 
 
 01 
 
 1914 
 
 92 
 
 1915 
 
 53 
 
 1916 
 
 qt 
 1917 
 
 95 
 
 1918 
 
 1 
 
 Bacon, lb 
 
 1077. 
 
 1069. 
 
 1869. 
 
 1481. 
 
 1187. 
 
 1498. 
 
 2 
 
 Barley, bu. . 
 
 
 178.2 
 
 195. 
 
 228.9 
 
 182.3 
 
 209. 
 
 256.4 
 
 3 
 
 Beef, lb. . . 
 
 
 6589. 
 
 6522. 
 
 6820. 
 
 7134. 
 
 8417. 
 
 10244. 
 
 4 
 
 Butter, lb. 
 
 
 1757. 
 
 1780. 
 
 1800. 
 
 1820. 
 
 1842. 
 
 1916. 
 
 5 
 
 Cattle, owt. . 
 
 
 69.8 
 
 67.6 
 
 71.5 
 
 S3.1 
 
 103.5 
 
 118.3 
 
 6 
 
 Cement, bbl. . 
 
 
 85.8 
 
 84.4 
 
 84.4 
 
 92. 
 
 88.1 
 
 69.4 
 
 7 
 
 Coal, anth., ton 
 
 
 6.9 
 
 6.86 
 
 6.78 
 
 6.75 
 
 7.83 
 
 7.69 
 
 8 
 
 Coal, bit., ton 
 
 
 477. 
 
 424. 
 
 443. 
 
 502. 
 
 552. 
 
 583. 
 
 9 
 
 Coffee, lb. . . 
 
 
 863. 
 
 1002. 
 
 1119. 
 
 1201. 
 
 1320. 
 
 1144. 
 
 10 
 
 Coke, short ton 
 
 
 46.3 
 
 34.6 
 
 41.6 
 
 54.5 
 
 56.7 
 
 55. 
 
 11 
 
 Copper, lb. . 
 
 
 812.3 
 
 620.5 
 
 1043.5 
 
 1429.8 
 
 1316.5 
 
 1648.3 
 
 12 
 
 Cotton, lb. . 
 
 
 2785. 
 
 2820. 
 
 2838. 
 
 3235. 
 
 3423. 
 
 3298. 
 
 13 
 
 Eggs, doz. 
 
 
 1722. 
 
 1759. 
 
 1791. 
 
 1828. 
 
 1882. 
 
 1908. 
 
 14 
 
 Hay, ton . . 
 
 
 79.2 
 
 83. 
 
 103, 
 
 111. 
 
 94.9 
 
 89.8 
 
 15 
 
 Hides, lb. . . 
 
 
 672. 
 
 924. 
 
 1227. 
 
 1212. 
 
 1113. 
 
 663. 
 
 16 
 
 Hogs, cwt. . 
 
 
 68.4 
 
 65.1 
 
 76.8 
 
 8R.2 
 
 67.8 
 
 82.4 
 
 17 
 
 Iron bar, cwt. 
 
 
 79.2 
 
 50.4 
 
 82.6 
 
 132.4 
 
 133. 
 
 132. 
 
 18 
 
 Iron, pig, ton 
 
 
 31. 
 
 23.3 
 
 29.9 
 
 39.4 
 
 38.7 
 
 38.1 
 
 19 
 
 Lead (white), lb 
 
 
 286. 
 
 318. 
 
 312. 
 
 268. 
 
 230. 
 
 216. 
 
 20 
 
 Lead, lb. . . 
 
 
 823.7 
 
 1025.6 
 
 1014.1 
 
 1104.5 
 
 1099.8 
 
 1083. 
 
 21 
 
 Lumber, M bd. i 
 
 t. . 
 
 21.8 
 
 20.7 
 
 20.5 
 
 22.3 
 
 21.2 
 
 19.2 
 
 22 
 
 Mutton, lb. . 
 
 
 732. 
 
 734. 
 
 629. 
 
 618. 
 
 474. 
 
 513. 
 
 23 
 
 Petroleum, gal. 
 
 
 10400. 
 
 11200. 
 
 11840. 
 
 12640. 
 
 14880. 
 
 15680. 
 
 24 
 
 Pork, lb. . . 
 
 
 9211. 
 
 8871. 
 
 9912. 
 
 10524. 
 
 8427. 
 
 11426. 
 
 25 
 
 Rubber, lb. . 
 
 
 115.8 
 
 136.6 
 
 231.4 
 
 258.8 
 
 375.9 
 
 351.5 
 
 26 
 
 Silk. lb. . . 
 
 
 19.1 
 
 19.1 
 
 20. 
 
 24.4 
 
 29.4 
 
 27.1 
 
 27 
 
 Silver, oz. . . 
 
 
 146.1 
 
 144. 
 
 173.4 
 
 139.3 
 
 133.6 
 
 140.7 
 
 28 
 
 Skins, skin 
 
 
 6.7 
 
 5.9 
 
 4.3 
 
 5.6 
 
 2.7 
 
 .7 
 
 29 
 
 Steel rails, ton 
 
 
 3.5 
 
 1.95 
 
 2.2 
 
 2.86 
 
 2.94 
 
 2.37 
 
 30 
 
 Tin, pig, cwt. 
 
 
 1.04 
 
 .95 
 
 1.16 
 
 1.43 
 
 1.56 
 
 1.59 
 
 31 
 
 Tin, plate, cwt. 
 
 
 15.3 
 
 17.3 
 
 19.7 
 
 22.8 
 
 29.5 
 
 28. 
 
 32 
 
 Wheat, bu. . 
 
 
 555. 
 
 654. 
 
 588. 
 
 642. 
 
 605. 
 
 562. 
 
 33 
 
 Wool, lb. . . 
 
 
 448. 
 
 550. 
 
 699. 
 
 737. 
 
 707. 
 
 752. 
 
 34 
 
 Lime, bbl., 300 1 
 
 b. '. 
 
 23.3 
 
 22.5 
 
 25. 
 
 27.1 
 
 24. 
 
 20.2 
 
 35 
 
 Lard, lb. . . 
 
 
 1100. 
 
 955. 
 
 1050. 
 
 1141. 
 
 927. 
 
 1107. 
 
 36 
 
 Oata, bu. . . 
 
 
 1122. 
 
 1240. 
 
 1360. 
 
 1480. 
 
 1587. 
 
 1538. 
 
 §2. EXAMPLES, IN TABULAR FORM, SHOWING HOW TO CALCULATE INDEX 
 NUMBERS BY THE NINE MOST PRACTICAL FORMULA 
 
 The following nine model examples may be of assistance to the reader 
 who desires practical and specific directions for calculating an index num- 
 ber. They include all of the eight formulae mentioned in Chapter XVII, 
 § 8, as the formulae most recommended for practical use, together with 
 8053, a makeshift for 353. Formulse 53, 54, and 8053, are given first and 
 axe followed by the others in the same order as in Chapter XVII, § 8. 
 
APPENDIX VI 491 
 
 In each case the data used are those for the 36 commodities as given on 
 the two preceding pages. 
 
 Formula 53, Laspeyres', Aggregative I, Poi =■ — 
 
 2pogo 
 
 (For discussion see pp. 56-60, 131-2, 237-40) 
 
 Computation of 2po3o 
 
 Per Unit Million Units 
 
 1 (Bacon); po = $0.1236 ; ^o = 1077. ; po?o =.1236 X 1077 =133.117 
 
 2 (Barley) ; p'o = .6263 ; g'o = 178.2; p'og'o = .6263 X 178.2 = 111.607 
 
 3 (Beef) p"o3"o = . 1295 X 6589 =853.276 
 
 4 .2969X1757 =521.653 
 
 36 .3758X1122 =421.648 
 
 (adding) Spogo = 13104.818 
 
 Computation of Spi^o 
 
 1 pi = .1295; go = 1077; pigo = • 1295 X 1077 =139.47 
 
 2 p'ig'o = .6204X 178.2 = 110.56 
 
 3 .1364X6589 =898.74 
 
 36 .4191X1122 =470.23 
 
 (adding) Spigo = 13095.78 
 
 Whence 
 
 P„j =IM2.= 13095.78 ^ gg gg ^^^^ ^ ^^^^^ numbcr for 1914 
 Zpogo 13104.818 
 Likewise 
 
 p ^ 2p2g0 ^ 13061.84 ^ ggg^ „ „ ^ „ „ „ jgjg 
 
 Spogo 13104.818 
 Likewise 
 p„3 = ^P^g° = 14950.13 ^ ^^^Qg „ „ ^ „ „ „ jgjg 
 
 Spogo 13104.818 
 Likewise 
 
 Pn, = ^P^g" = 21238.49 ^ 162.07 " " = " " " 1917 
 
 Spogo 13104.818 
 Likewise 
 
 p _ ^P&Qo _ 23308.95 _ ..-yo- ,, ,, _ »> j> n igig 
 
 °' ~ Spogo ~ 13104.818 
 The above is by the fixed base system. 
 
492 THE MAKING OF INDEX NUMBERS 
 
 93 per cent 
 
 By the chain system, we have 
 
 
 Poi = 
 
 13095.78 
 13104.818 
 
 99.93 
 
 Pl2 = 
 
 13059.052 
 13033.034 
 
 100.20 
 
 P23 = 
 
 16233.560 _ 
 14280.976 
 
 113.67 
 
 P34 = 
 
 25388.869 
 17789.440 
 
 142.72 
 
 P48 = 
 
 27690.677 
 
 109.92 
 
 25191.136 
 
 Whence, by successive multiplication 
 
 Poi = 99.93 
 
 = 99.93 per cent = index number for 1914 
 P01P12 = 99.93 X 100.20 
 
 = 100.13 per cent = index number for 1915 
 P01P12P23 = 99.93 X 100.20 X 113.67 
 
 = 113.82 per cent = index number for 1916 
 P01P12P23P34 = 99.93 X 100.20 X 113.67 X 142.72 
 
 = 162.44 per cent = index number for 1917 
 P01PX2P23P34P45 = 99.93 X 100.20 X 113.67 X 142.72 X 109.92 
 = 178.56 per cent = index number for 1918 
 
 Formula 54, Paasche's, Aggregative IV, Poi = 
 
 2pogi 
 
 (For discussion see pages cited for Formula 53, especially pp. 131-2) 
 Computation of Spi^i 
 
 1 pi = .1295 gi = 1069. pi^i = .1295 X 1069. = 138.436 
 
 2 p'lq'i = .6204 X 195. = 120.978 
 
 3 .1364 X 6522. = 889.601 
 
 36 519.684 
 
 (adding) Spi^i 13033.034 
 Computation of Spo5i 
 
 1 pogi = .1236 X 1069. = 132.13 
 
 2 .6263 X 195. = 122.13 
 
 36 ■ 465.99 
 
 (adding) Spogi 12991.81 
 
 Whence 
 
 Poi = ^^^ = 13033.034 ^ jQQ 22 per cent = index number for 1914 
 Spogi 12991.81 
 
APPENDIX VI 493 
 
 D 2^292 14280.976 ,nn in i. -1 , , . 
 
 Po2 = -—- = TTT^^r^TTrr- = 100.10 per cent = index number for 1915 
 Spog2 14266.81 
 
 p ^ 17789.440 ^ jj^ 3 „ „ ^ „ „ „ 
 
 15557.52 
 
 Po4 = 161.05 " " = " " " 1917 
 
 PoB = 177.43 " " = " " " 1918 
 
 The chain figures in this and subsequent examples may be derived, as 
 in the previous example, by linking. Thus P01P12P23 = 100.32 X 100.01 
 X 114.45 = 114.83 per cent = index number for 1916. 
 
 Formula 8053, Poi = ^^^^ + ^^^^ = ^^°g" ^P°g^ 
 2 2 
 
 (For discussio.n see pp. 174-7) 
 
 „ 99.93 + 100.32 mn 10 • j i. r -.ni. 
 
 Poi = ■ = 100.12 = mdex number for 1914 
 
 P02 = 99.89 = " " " 1915 
 
 etc. 
 
 Formula 353, "Ideal," Poi = V(53) X (54) = yl^Ml x ^Ml 
 
 V Spogo 2po3i 
 
 (For discussion see pp. 220-9, 234-42) 
 
 Poi = V99.93 X 100.32 = 100.12 = index number for 1914 
 P02 = 99.89 = " " " 1915 
 
 etc. 
 
 The square root may be extracted "by hand," by logarithms, or (most 
 quickly), by a calculating machine, in which case the total time required 
 to calculate the five figures (fixed base) is 14.3 hours. But it is seldom, 
 if ever, necessary actually to extract the square root because the two figures 
 under the radical are always so close together that the preceding Formula 
 8053 (requiring 14. 1 hours) can be used instead. 
 
 The results of 8053 and 353 agree to the second decimal place, provided 
 53 and 54 do not differ by more than 1 per cent, which is usually the case. 
 Whether or not they so differ can always be seen at a glance. In case they 
 do differ by more than 1 per cent and the calculator still wishes to avoid 
 the process of root extraction he can almost as quickly get the result by 
 "trial and error," using 8053 as a basis. 
 
 Thus, let 53 = 101.22 per cent and 54 = 104.26 per cent. Their dif- 
 ference 3.04 exceeds 1 per cent (which would be 1.0122), We find 8053 = 
 
 101.22 + 104.26 ^ 102.74. We know that the geometric mean, which 
 
 2 
 we seek, is slightly smaller. We therefore try 102.73 by comparing its 
 square ([102.73p = 105.535 per cent) with what it should be {i.e. 101.22 X 
 
494 THE MAKING OF INDEX NUMBERS 
 
 104.26 = 105.532 per cent). Here the square is slightly too great but is 
 nearer than the square of 102.72, which is 105.514 per cent. Therefore 
 102.73 is the result sought. 
 A second and more systematic method of avoiding root extraction is 
 
 to calculate 8053 = 102.74 and 8054 = — — ^ — — = — ? — 
 
 + 7^77 TTT-7r + 
 
 (53) (54) 101.22 104.26 
 = 102.72. The geometric mean of these two is necessarily 353 * ; but 
 these two (8053 and 8054) will always be within 1 per cent of each other, 
 (even if the original 53 and 54 differ by as much as 25 per cent), so 
 that their arithmetic mean (here 102.73 per cent) will always be accurate 
 to the second decimal place. 
 
 Formula 216S, Edgeworth-Marshall's Aggregative, Poi = ^ ■ 
 
 2(go + qi)Po 
 
 (For discussion see pp.. 194-5, 401-7, 428-30) 
 
 This is usually t a sufficiently accurate makeshift for 353 and requires 9.6 
 hours as against 14.1 hours for 8053 and 14.3 hours for 353. 
 Computation of 2 (go + gOpi 
 
 1 (go + gi)pi = (1077. + 1069. ) X .1295 = 277.9070 
 
 2 ( 178.2 + 195.0) X .6204 = 231.5333 
 
 36 = 989.9142 
 
 (adding) 2(go + qi)Pi = 26128.814 
 
 (similarly) 2(go + qdpo = 26096.628 
 
 Whence Poi = '- = 100.12 per cent = index number for 1914. 
 
 26096.628 
 
 Likewise Poa = 99.89 per cent = index number for 1915. 
 
 etc. 
 
 Formula 605S (for discussion see pp. 312-3, 318-20) (assuming 1913- 
 1914 the "broadened base") is derived exactly as 2153 above except that 
 go + gi is retained throughout all five computations instead of changing to 
 go + g2 in computing P02, etc. If 1913-'14-'15 is the broadened base, 
 go + gi + 92 is so used. 
 
 Formula 53 has already been exemplified. 
 
 Formula 9051, ^^^ (for discussion see pp. 198, 327-8, 348) is like 53 
 Swpo 
 
 except that the id's replace the g's and are round numbers (1, 10, 100, 
 
 etc.). These factors merely shift the decimal points of the p's so that 
 
 Formula 9051 is really Formula 51 with such shifts, each shift being the 
 
 best round guess at the proper factor. 
 
 *See Appendix I (Note to Chapter IX, J 1). 
 tSee Appendix I (Note to Chapter XV. §2). 
 
APPENDIX VI 495 
 
 1 pi = .1295; w = 1000; wpi = 1000 X .1295 = 129.5 
 
 2 p'i = .6204; 100 X .6204 = 62.04 
 
 3 .1364 1364. 
 
 36 419.1 
 
 (adding) Zwpi = 12697.242 
 
 Likewise Zwpo = 12487.4043 
 
 Whence p„ = 12697.242 ^ 
 
 12487.4043 
 Similarly P02 = 103.10 
 
 etc. 
 
 Formula 21, Simple Geometric, Poi = -^P^P'^P"^ — 
 
 ^Pop'op"o ... 
 
 (For discussion see pp. 33-5, 211-2, 260-4) 
 
 1 log Pi = log .1295 
 
 = 1.11227 
 
 
 2 log p'l = log .6204 
 
 = 1.79267 
 
 
 3 log .1364 
 
 = 1.13481 
 
 
 4 
 
 1.43632 
 
 
 
 36 
 
 1.62232 
 
 
 (adding) Slogpi 
 
 2.13755 
 
 
 Similarly S log po 
 
 2.81385 
 
 
 (subtracting) 
 
 1.32370 = 35.32370 
 
 -36 
 
 (dividing by n = 36) 
 
 .98121 
 
 - 1= 1.98121 
 
 which is the log of Poi = 95.77 per cent 
 
 
 Similarly P02 = 96.79 " " 
 
 
 etc. 
 
 
 
 Avoiding logarithms. The many users of index numbers who wish to 
 avoid logarithms and geometric means, such as Formula 21, may use the 
 
 formula ^ '^ ^ — -. This is practically coincident with Formula 101 and 
 
 so with 21. 
 
 A somewhat similar remark applies when the problem is how best, with- 
 out recourse to logarithms, to utilize rough weights in averaging two or 
 more price relatives, or two or more index numbers already supplied. Sup- 
 pose, for instance, we wish to calculate an index number for "the general 
 level of prices" by combining existing index numbers of (1) wholesale com- 
 modity prices, (2) retail commodity prices, (3) prices of shares on the Stock 
 Exchange, and (4) wages, assuming that the separate index numbers of 
 (1), (2), (3), (4) are, respectively, 200, 150, 250, 125, and that their rough 
 
496 THE MAKING OF INDEX NUMBERS 
 
 weights (representing, say, their roughly estimated values in exchange 
 during a series of years) are 10, 5, 3, 1. The arithmetic formula 
 
 10 X 2.00 + 5 X 1.50 + 3 X 2.50 + 1 X 1.25 ^ ^ g^^g 
 10 + 5 + 3 + 1 
 
 (practically Formula 1003) would be improper, having an appreciable 
 upv/ard bias because the 200, 150, 250, 125 disperse widely ; the harmonic 
 formula 
 
 l« + ^+^ + l =1.8387 
 
 10 X -^+ 5 X -^+ 3 X ^— + 1 X — ^ 
 2.00 1.50 2.50 1.25 
 
 (practically Formula 1013) would be improper for the opposite reason; 
 the geometric formula 
 
 ^/ (2.00)10 X (1.50)5 X (2.50)3 X (1.25) 
 
 would be the best, but requires logarithms; the aggregative is im- 
 practicable, since our weights, which are values, cannot be translated 
 into quantities. We have recourse, then, to an average of the first two 
 above — what is practically Formula 1103, i.e. we take the above arith- 
 metic and harmonic averages, namely 1.9079 and 1.8387, and average them 
 arithmetically, obtaining 1.8733. Or, instead of resting content with this 
 result, we could (though it would seldom if ever be worth while) proceed 
 another step by also averaging the 1.9079 and 1.8387 harmonically and 
 then taking the arithmetic average of the two results (1.8733 and 1.8727), 
 which is 1.8730, and so on, if desired, to any number of stages, thereby ap- 
 proximating the geometric mean of 1.9079 and 1.8387 as closely as we wish. 
 
 Formula 31, Simple Median, mid-term among the price relatives, — , — , ... 
 
 Po p'o 
 
 (For discussion see pp. 35-6, 209-12, 260-4) 
 
 1 Pi x= .1295 
 Po 
 
 2 PJ = 
 p'o 
 
 Rearranging these 36 price relatives in the order of their magnitudes, we 
 
 find 
 
 lowest price relative (coffee) 73.32 per cent 
 
 next lowest price relative (rubber) 76.30 
 
 .1236 
 
 — 
 
 104.77 
 
 per 
 
 cent 
 
 .6204 
 .6263 
 
 = 
 
 99.06 
 
 »» 
 
 >> 
 
 n >> 
 
 18th (barley) 99.06 
 
 19th (white lead) 99.85 
 
 highest (wheat) 114.03 " 
 
APPENDIX VI 497 
 
 The median lies between the two middlemost terms, the 18th and 19th, 
 99.06 and 99.85, and is most simply taken as their arithmetic mean (al- 
 though most properly their geometric mean) Poi = 99.45 
 
 SimUarly P02 = 98.57 
 
 etc. 
 
 A little time may be saved by not rearranging the order of terms but 
 crossing off from the original list any pair of terms, one very high and one 
 very low so as to make sure that they are on opposite sides of the median ; 
 then likewise erase another pair of extreme terms, i.e. two which surely 
 lie astride of the median, and so on until so few terms are left that the me- 
 dian is obvious. 
 
 Another practical index number, calculated partly by Formula 53 and 
 partly by Formula 9051, is described on p. 346. Formula 1 (simple arith- 
 metic) is exemplified on pp. 15-24 but is not recommended for practical 
 use. Formula 3 (base weighted arithmetic) is best reduced to Formula 53 
 before calculating. 
 
APPENDIX VII 
 
 TABLE 65. INDEX NUMBERS BY 134 FORMULA FOR PRICES 
 BY THE FIXED BASE SYSTEM AND (IN NOTEWORTHY 
 CASES) THE CHAIN SYSTEM 
 
 (1913 = 100) 
 
 Although only the specified Price indexes are here given, Quantity indexes'as well as 
 Price indexes — both fixed base and chain — have been computed for all the 134 formulae 
 and are utilized in the charts. 
 
 PRIMARY FORMULA (1-99) 
 
 Those for which figures are given conform to neither test. 
 
 Arithmetic 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 in Accuract, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 1 
 
 Fixed 
 Chain 
 
 96.32 
 96.32 
 
 98.03 
 97.94 
 
 123.68 
 125.33 
 
 175.79 
 
 175.65 
 
 186.70 
 193.42 
 
 3rd in speed 
 3rd in sim- 
 plicity 
 
 2 
 
 Fixed 
 Chain 
 
 100.18 
 WO.IS 
 
 95.93 
 
 95.47 
 
 109.71 
 
 107.83 
 
 152.75 
 
 152.42 
 
 177.13 
 
 177.69 
 
 15th in speed 
 
 (3) 
 
 
 Same as 53 (necessarily) 
 
 
 
 (4) 
 
 
 Same as 54 (necessarily) 
 
 
 
 (5) 
 
 
 Same as 54 (necessarily) 
 
 
 
 (6) 
 
 
 Same as 53 (necessarily) 
 
 
 
 7 
 
 Fixed 
 
 100.55 
 
 101.77 
 
 117.77 
 
 180.53 
 
 186.98 
 
 
 8 
 
 Fixed 
 
 99.02 
 
 97.36 
 
 111.45 
 
 152.42 
 
 167.06 
 
 
 9 
 
 Fixed 
 Chain 
 
 100.93 
 100.93 
 
 102.33 
 102.10 
 
 118.29 
 
 122. 41 
 
 180.72 
 I8O.4O 
 
 187.18 
 205.56 
 
 
 10 
 
 Fixed 
 
 98.70 
 
 96.97 
 
 111.10 
 
 154.96 
 
 169.27 
 
 
 ♦ As revised in Chapter XVI, § 9. 
 
 498 
 
APPENDIX VII 
 
 499 
 
 TABLE 65 (Continued) 
 Harmonic 
 
 Identi- 
 fication 
 
 NuMBEB 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 11 
 
 Fixed 
 Chain 
 
 95.19 
 
 95.19 
 
 95.58 
 95.64 
 
 119.12 
 117.71 
 
 157.88 
 158.47 
 
 171.79 
 167.76 
 
 3rd in speed 
 9th in sim- 
 plicity 
 
 12 
 
 Fixed 
 Chain 
 
 103.48 
 103.48 
 
 101.31 
 101.97 
 
 115.35 
 
 117.72 
 
 172.11 
 
 172.55 
 
 243.67 
 217.65 
 
 15th in speed 
 
 13 
 
 Fixed 
 Chain 
 
 99.26 
 99.26 
 
 97.84 
 98.45 
 
 111.01 
 
 108.19 
 
 147.19 
 148.14 
 
 168.59 
 157.78 
 
 8th in speed 
 
 14 
 
 Fixed 
 
 101.81 
 
 102.41 
 
 116.80 
 
 168.37 
 
 189.80 
 
 
 15 
 
 Fixed 
 
 99.65 
 
 98.11 
 
 111.02 
 
 144.97 
 
 166.85 
 
 
 16 
 
 Fixed 
 
 101.34 
 
 101.98 
 
 116.63 
 
 168.60 
 
 189.38 
 
 
 (17) 
 
 
 Same as 53 (necessarily) 
 
 
 
 (18) 
 
 
 Same as 54 (necessarily) 
 
 
 
 (19) 
 
 
 Same as 54 (necessarily) 
 
 
 
 (20) 
 
 
 Same as 53 (necessarily) 
 
 
 
 * As revised in Chapter XVI, § 9. 
 
500 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 {Continued) 
 Geometric 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 to Circular 
 Test 
 
 21 
 
 
 Same as 121 (necessarily) 
 
 
 
 22 
 
 
 Same as 122 (necessarily) 
 
 
 
 23 
 
 Fixed 
 Chain 
 
 99.61 
 99.61 
 
 98.72 
 99.28 
 
 111.45 
 110.91 
 
 154.08 
 155.03 
 
 173.30 
 166.93 
 
 17th in speed 
 18th in sim- 
 plicity 
 
 24 
 
 Fixed 
 
 101.02 
 
 101.32 
 
 115.64 
 
 164.85 
 
 182.84 
 
 
 25 
 
 Fixed 
 
 99.99 
 
 99.07 
 
 112.58 
 
 152.45 
 
 172.37 
 
 
 26 
 
 Fixed 
 
 100.60 
 
 100.88 
 
 115.42 
 
 165.37 
 
 182.61 
 
 
 27 
 
 Fixed 
 
 100.25 
 
 100.67 
 
 115.82 
 
 170.82 
 
 182.45 
 
 
 28 
 
 Fixed 
 
 99.65 
 
 98.82 
 
 112.98 
 
 157.09 
 
 172.27 
 
 
 29 
 
 Fixed 
 
 100.63 
 
 101.17 
 
 116.26 
 
 170.44 
 
 182.41 
 
 
 30 
 
 Fixed 
 
 99.29 
 
 98.41 
 
 112.67 
 
 158.70 
 
 173.60 
 
 
 * As revised in Chapter XVI, § 9. 
 
APPENDIX VII 
 
 501 
 
 TABLE 65 {Continued) 
 Median 
 
 Identi- 
 fication 
 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity or Formula, 
 and Conformity 
 TO Circular 
 Test 
 
 31 
 
 
 Same as 131 (necessarily) 
 
 
 
 32 
 
 
 Same as 132 (necessarily) 
 
 
 
 33 
 
 Fixed 
 Chain 
 
 100.34 
 100.34 
 
 99.39 
 99.70 
 
 107.17 
 106.80 
 
 156.12 
 
 150.22 
 
 169.14 
 173.34 
 
 16th in speed 
 
 34 
 
 Fixed 
 
 101.20 
 
 104.66 
 
 117.57 
 
 165.53 
 
 181.97 
 
 
 35 
 
 Fixed 
 
 100.48 
 
 99.41 
 
 107.37 
 
 160.18 
 
 169.14 
 
 
 36 
 
 Fixed 
 
 100.97 
 
 104.01 
 
 117.62 
 
 165.49 
 
 182.16 
 
 
 37 
 
 Fixed 
 
 100.61 
 
 99.65 
 
 108.77 
 
 163.84 
 
 188.25 
 
 
 38 
 
 Fixed 
 
 100.57 
 
 102.07 
 
 116.74 
 
 157.84 
 
 179.74 
 
 
 39 
 
 Fixed 
 
 100.75 
 
 99.97 
 
 109.08 
 
 163.84 
 
 178.12 
 
 
 40 
 
 Fixed 
 
 100.52 
 
 101.78 
 
 116.85 
 
 159.90 
 
 180.33 
 
 
 * As revised in Chapter XVI, § 9. 
 
502 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 {Continued) 
 Mode 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 in Accuracy, 
 Speed, Simplio- 
 itt of Formula, 
 
 AND CoNFORMITT 
 
 TO Circular 
 Test 
 
 41 
 
 
 Same as 141 (necessarily) 
 
 
 
 42 
 
 
 Same as 142 (necessarily) 
 
 
 
 43 
 
 Fixed 
 
 101. 
 
 100. 
 
 108. 
 
 164. 
 
 168. 
 
 
 44 
 
 Fixed 
 
 103. 
 
 106. 
 
 132. 
 
 196. 
 
 180. 
 
 
 45 
 
 
 Same figures as for 43 
 
 
 
 46 
 
 
 Same figures as for 44 
 
 
 
 47 
 
 
 Same figures as for 43 
 
 
 
 48 
 
 
 Same figures as for 44 
 
 
 
 49 
 
 
 Same figures as for 43 
 
 
 
 50 
 
 
 Same figures as for 44 
 
 
 
 * As revised in Chapter XVI, § 9. 
 
APPENDIX VII 
 
 503 
 
 TABLE 65 (Continued) 
 Aggregative 
 
 Identi- 
 fication 
 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 and Conformity 
 TO Circular 
 Test 
 
 51 
 
 
 Same as 151 (necessarily) 
 
 
 
 52 
 
 
 Same as 152 (necessarily) 
 
 
 
 53 1 
 
 Fixed 
 Chain 
 
 99.93 
 
 99.93 
 
 99.67 
 100.13 
 
 114.08 
 
 113.82 
 
 162.07 
 
 162.44 
 
 177.87 
 178.56 
 
 4th in speed 
 5tb. in sim- 
 phcity 
 
 54t 
 
 Fixed 
 Chain 
 
 100.32 
 
 100.33 
 
 100.10 
 100.33 
 
 114.35 
 
 114.83 
 
 161.05 
 
 162.02 
 
 177.43 
 
 178.43 
 
 13th in speed 
 6th in sim- 
 plicity 
 
 59 
 
 
 Same as 54 (necessarily) 
 
 
 
 60 
 
 
 Same as 53 (necessarily) 
 
 
 
 * As revised in Chapter XVI, § 9. 
 t 53 =3, 6, 17, 20, 60. 
 K 54 = 4, 5, 18, 19. 59. 
 
504 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 {Continued) 
 
 CROSS FORMULAE (100-199) 
 
 Those for which figures are given fulfill Test 1 only. 
 
 Arithmetic and Harmonic Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of Fihst 
 20 in Accuracy, 
 Speed, Simplic- 
 ity OP Formula, 
 and Conformity 
 TO Circular 
 Test 
 
 101 
 
 Fixed 
 Chain 
 
 95.75 
 95.75 
 
 96.80 
 96.78 
 
 121.38 
 121.46 
 
 166.60 
 166.84 
 
 179.09 
 180.13 
 
 9th in speed 
 
 102 
 
 Fixed 
 
 101.81 
 
 98.58 
 
 112.50 
 
 162.14 
 
 207.75 
 
 
 103 
 
 
 Same as 353 (necessarily) 
 
 
 
 104 
 
 
 Same as 353 (necessarily) 
 
 
 
 105 
 
 
 Same as 353 (necessarily) 
 
 
 
 106 
 
 
 Same as 353 (necessarily) 
 
 
 
 107 
 
 Fixed 
 
 100.10 
 
 99.92 
 
 114.35 
 
 161.78 
 
 176.63 
 
 
 108 
 
 Fixed 
 
 100.17 
 
 99.64 
 
 114.01 
 
 160.31 
 
 177.87 
 
 
 109 
 
 Fixed 
 
 100.09 
 
 100.06 
 
 114.59 
 
 163.10 
 
 177.64 
 
 
 110 
 
 Fixed 
 Chain 
 
 100.24 
 
 100.24 
 
 99.65 
 100.18 
 
 113.91 
 
 114.14 
 
 161.53 
 
 162.06 
 
 179.24 
 178.52 
 
 
 * As revised Iq Chapter XVI, § 9. 
 
APPENDIX VII 
 
 505 
 
 TABLE 65 {Continued) 
 Geometric Crosses 
 
 Identi- 
 fication 
 
 Number 
 
 121 
 (21) 
 
 122 
 
 (22) 
 
 123 
 
 124 
 
 125 
 
 126 
 
 Base 
 
 Fixed 
 Chain 
 
 Fixed 
 Chain 
 
 Fixed 
 Chain 
 
 Fixed 
 Chain 
 
 Fixed 
 Chain 
 
 Fixed 
 Chain 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 
 20 IN AcCtTRACT, 
 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 to Circular 
 
 Test 
 
 95.77 96.79 121.37 166.65 180.12 
 Same as Fixed Base (necessarily) 
 
 101.71 98.62 112.60 161.88 194.14 
 Same as Fixed Base (necessarily) 
 
 100.12 
 100.12 
 
 100.16 
 100.16 
 
 100.12 
 
 100.12 
 
 100.12 
 100.13 
 
 99.94 
 
 100.24 
 
 99.85 
 
 100.23 
 
 99.87 
 
 100.24 
 
 99.85 
 100.22 
 
 113.83 
 
 114.63 
 
 114.25 
 
 114-26 
 
 114.19 
 
 114.33 
 
 114.20 
 114-56 
 
 162.05 
 
 162.75 
 
 161.74 
 
 162. IS 
 
 161.37 
 
 162.18 
 
 161.18 
 
 162.54 
 
 177.80 
 178.87 
 
 178.16 
 178.50 
 
 177.34 
 
 178.36 
 
 177.36 
 178.81 
 
 6th in speed 
 10th in sim- 
 plicity 
 1st in con- 
 formity 
 
 18th in speed 
 1st in con- 
 formity 
 
 15th in ac- 
 curacy 
 
 17th in ac- 
 curacy 
 
 14th in ac- 
 curacy 
 
 16th in ac- 
 curacy 
 
 * As revised in Chapter XVI, § 9. 
 
506 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 ^Continued) 
 Median Crosses 
 
 Identi- 
 fication 
 
 Number 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 in Accuract 
 Speed, Simplic- 
 ity OF Formula, 
 and Conformity 
 TO Circular 
 Test 
 
 131 
 
 (31) 
 
 Fixed 
 Chain 
 
 99.45 
 99.45 
 
 98.57 
 99.33 
 
 118.81 
 117.50 
 
 163.81 
 155.86 
 
 190.92 
 180.07 
 
 10th in speed 
 4th in sim- 
 plicity 
 
 132 
 
 (32) 
 
 Fixed 
 
 100.11 
 
 102.20 
 
 116.01 
 
 162.15 
 
 183.54 
 
 
 133 
 
 Fixed 
 
 100.54 
 
 99.68 
 
 108.12 
 
 159.93 
 
 173.57 
 
 
 134 
 
 Fixed 
 
 100.86 
 
 103.21 
 
 117.21 
 
 162.69 
 
 181.15 
 
 
 135 
 
 Fixed 
 
 100.54 
 
 99.53 
 
 108.07 
 
 162.00 
 
 178.44 
 
 
 136 
 
 Fixed 
 
 100.77 
 
 103.04 
 
 117.18 
 
 161.62 
 
 180.95 
 
 
 ♦ As revised in Chapter XVI, § 9. 
 
 Mode Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity op Formula, 
 AND Conformity 
 to Circular 
 Test 
 
 141 
 
 (41) 
 
 Fixed 
 Chain 
 
 98. 
 98. 
 
 98. 
 95. 
 
 108. 
 104. 
 
 135. 
 131. 
 
 190. 
 151. 
 
 12th in speed 
 
 142 
 
 (42) 
 
 Fixed 
 
 104. 
 
 108. 
 
 125. 
 
 167. 
 
 183. 
 
 
 143 
 
 Same figures as for 43 
 
 
 144 
 
 Same figures as for 44 
 
 
 145 
 
 Same figures as for 43 
 
 
 146 
 
 Same figures as for 44 
 
 
 "' As revised in Chapter XVI, § 9. 
 
APPENDIX VII 
 
 507 
 
 Identi- 
 fication 
 Numb EH 
 
 151 
 
 (51) 
 
 152 
 
 (52) 
 
 153 
 
 154 
 
 Base 
 
 Fixed 
 Chain 
 
 Fixed 
 Chain 
 
 TABLE 65 {Continued) 
 Aggregative Crosses 
 
 1914 
 
 1916 
 
 1916 
 
 95.88 96.29 107.70 146.90 172.76 
 Same as Fixed Base (necessarily) 
 
 1917 
 
 1918 
 
 97.12 I 97.18 I 114.55 158.65 | 165.15 
 Same as Fixed Base (necessarily) 
 
 Same as 353 (necessarily) 
 
 Same as 353 (necessarily) 
 
 Ranks* of First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 1st in speed 
 1st in sim- 
 plicity 
 1st in con- 
 formity 
 
 5th in speed 
 20th in sim- 
 plicity 
 1st in con- 
 formity 
 
 * As revised in Chapter XVI, § 9. 
 
 CROSS FORMULA (200-299) 
 
 Those for which figures are given conform to Test 2 only. 
 
 Arithmetic Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1S18 
 
 Ranks* of First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 201 
 
 Fixed 
 
 98.23 
 
 96-97 
 
 116.43 
 
 163.87 
 
 181.85 
 
 
 203 
 
 
 Same as 353 (necessarily) 
 
 
 205 
 
 
 Same as 353 (necessarily) 
 
 
 207 
 
 Fixed 
 
 99.78 
 
 99.54 
 
 114.56 
 
 165.88 
 
 176.74 
 
 
 209 
 
 Fixed 
 
 99.81 
 
 99.61 
 
 114.63 
 
 167.35 
 
 178.00 
 
 
 * As revised in Chapter XVI, § 9. 
 
508 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 (Continued) 
 Harmonic Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity OP Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 211 
 
 Fixed 
 
 99.24 
 
 98.40 
 
 117.22 
 
 164.84 
 
 204.60 
 
 
 213 
 
 Fixed 
 
 100.53 
 
 100.10 
 
 113.87 
 
 157.42 
 
 178.88 
 
 
 215 
 
 Fixed 
 
 100.49 
 
 100.03 
 
 113.79 
 
 156.34 
 
 177.76 
 
 
 217 
 
 
 Same as S53 (necessarily) 
 
 
 219 
 
 
 Same as 353 (necessarily) 
 
 
 * As revised in Chapter XVI, § 9. 
 
 Geometric Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity of Formula, 
 AND Conformity 
 to Circular 
 Test 
 
 221 
 
 
 Same as 321 (necessarily) 
 
 
 223 
 
 Fixed 
 
 100.31 
 
 100.01 
 
 113.52 
 
 159.37 
 
 178.01 
 
 
 225 
 
 Fixed 
 
 100.29 
 
 99.97 
 
 113.99 
 
 158.78 
 
 177.42 
 
 
 227 
 
 Fixed 
 
 99.95 
 
 99.74 
 
 114.39 
 
 163.81 
 
 177.29 
 
 
 229 
 
 Fixed 
 
 99.96 
 
 99.78 
 
 114.45 
 
 164.47 
 
 177.95 
 
 
 ♦ As revised in Chapter XVI, S 9- 
 
APPENDIX VII 
 
 509 
 
 TABLE 65 (Continued) 
 Median Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity or Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 231 
 
 
 Same as 331 (necessarily) 
 
 
 
 233 
 
 Fixed 
 
 100.77 
 
 101.99 
 
 112.27 
 
 160.76 
 
 175.44 
 
 
 235 
 
 Fixed 
 
 100.72 
 
 101.69 
 
 112.38 
 
 162.81 
 
 175.53 
 
 
 237 
 
 Fixed 
 
 100.59 
 
 100.85 
 
 112.69 
 
 160.81 
 
 183.94 
 
 
 239 
 
 Fixed 
 
 100.63 
 
 100.87 
 
 112.90 
 
 161.86 
 
 179.22 
 
 
 * As revised in Chapter XVI, § 9. 
 
 Mode Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity OP Formula, 
 and Conformity 
 to Circular 
 Test 
 
 241 
 
 
 Same as 341 (necessarily) 
 
 
 
 243 
 
 Fixed 
 
 102. 
 
 103. 
 
 119. 
 
 179. 
 
 174. 
 
 
 245 
 
 
 Same figures as for 243 
 
 
 
 247 
 
 
 Same figures as for 243 
 
 
 
 249 
 
 
 Same figures as for 243 
 
 
 
 * As revised in Chapter XVI, § 9. 
 
510 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 (Continued) 
 Aggregative Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 and conformitt 
 to Circular 
 Test 
 
 ^ 251 
 
 
 Same as 351 (necessarily) 
 
 
 
 253 
 
 
 Same as 353 (necessarily) 
 
 
 
 259 
 
 
 Same as 353 (necessarily) 
 
 
 
 * As revised in Chapter XVI, § 9. 
 
 CROSS FORMULAE (300-399) 
 
 Fulfilling both tests 
 
 Arithmetic and Harmonic Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 and Conformity 
 to Circular 
 Test 
 
 301 
 
 Fixed 
 
 98.73 
 
 97.68 
 
 116.82 
 
 164.35 
 
 192.89 
 
 
 303 
 
 
 Same as 353 (necessarily) 
 
 
 
 305 
 
 
 Same as 353 (necessarily) 
 
 
 
 307 
 
 Fixed 
 
 100.13 
 
 99.78 
 
 114.17 
 
 161.04 
 
 177.25 
 
 
 309 
 
 Fixed 
 
 100.17 
 
 99.85 
 
 114.25 
 
 162.31 
 
 178.44 
 
 
 * As revised in Chapter XVI, § 9. 
 
APPENDIX VII 
 
 511 
 
 TABLE 65 (Continued) 
 Geometkic Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 321 
 
 (221) 
 
 Fixed 
 Chain 
 
 98.70 
 Sam 
 
 97.70 116.91 1 164.25 187.00 
 e as Fixed Base (necessarily) 
 
 1st in con- 
 formity 
 
 323 
 
 Fixed 
 Chain 
 
 100.13 
 100.13 
 
 99.89 
 100.23 
 
 113.99 
 11445 
 
 161.90 
 162.47 
 
 177.98 
 178.69 
 
 9th in ac- 
 curacy 
 10th in con- 
 formity 
 
 325 
 
 Fixed 
 Chain 
 
 100.12 
 100.12 
 
 99.85 
 100.23 
 
 114.19 
 114-45 
 
 161.28 
 162.36 
 
 177.35 
 
 178.58 
 
 8th in ac- 
 curacy 
 
 9th in con- 
 formity 
 
 * As revised in Chapter XVI, § 9. 
 
 Median Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 331 
 
 (231) 
 
 Fixed 
 
 99.78 
 
 100.37 
 
 117.40 
 
 162.98 
 
 187.19 
 
 
 333 
 
 Fixed 
 
 100.70 
 
 101.43 
 
 112.59 
 
 161.31 
 
 177.32 
 
 
 335 
 
 Fixed 
 
 100.65 
 
 101.27 
 
 112.53 
 
 161.81 
 
 179.69 
 
 
 * As revised in Chapter XVI, § 9. 
 
512 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 (Continued) 
 Mode Crosses 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 341 
 
 (241) 
 
 Fixed 
 
 100.96 
 
 102.88 
 
 116.19 
 
 150.15 
 
 186.47 
 
 
 343 
 
 
 Same figures as for 243 
 
 
 
 345 
 
 
 Same figures as for 243 
 
 
 
 * As revised in Chapter XVI, § 9. 
 
 Aggregative Crosses 
 
 
 
 
 
 
 
 
 Ranks* of First 
 
 
 
 
 
 
 
 
 20 IN Accuracy, 
 
 Identi- 
 
 
 
 
 
 
 
 Speed, Simplic- 
 
 fication 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 ity op Formula, 
 
 Number 
 
 
 
 
 
 
 
 AND Conformity 
 TO Circular 
 Test 
 
 351 
 
 Fixed 
 
 96.50 
 
 96.73 
 
 111.07 
 
 152.66 
 
 168.91 
 
 11th in speed 
 
 (251) 
 
 
 Same as 
 
 Fixed Base (necessarily) 
 
 
 1st in con- 
 
 
 
 
 
 
 
 
 formity 
 
 353 1 
 
 Fixed 
 
 100.12 
 
 99.89 
 
 114.21 
 
 161.56 
 
 177.65 
 
 1st in ac- 
 curacy 
 
 
 Chain 
 
 100.12 
 
 100.23 
 
 114.32 
 
 162.23 
 
 178.4d 
 
 17th in sim- 
 plicity 
 2nd in con- 
 formity 
 
 * As revised in Chapter XVI, § 9. 
 
 t 353 = 103, 104, 105, 106, 153, 154, 203, 205, 217, 219, 253, 259, 303, 305. 
 
APPENDIX VII 
 
 513 
 
 TABLE 65 (Continued) 
 CROSS WEIGHT FORMULA (1000-4999) 
 Ckoss Weight Aeithmetic and Harmonic 
 1000-1099 fulfill neither test. 
 
 Identi- 
 fication 
 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 IN Accuracy, 
 Speed, Simpmc- 
 itt op Formula, 
 and conpormitt 
 TO Circular 
 Test 
 
 1003 
 
 Fixed 
 
 100.45 
 
 100.93 
 
 116.02 
 
 170.81 
 
 182.54 
 
 
 1004 
 
 Fixed 
 
 99.47 
 
 98.60 
 
 112.84 
 
 158.01 
 
 173.03 
 
 
 1013 
 
 Fixed 
 
 99.81 
 
 98.91 
 
 112.53 
 
 153.51 
 
 173.02 
 
 
 1014 
 
 Fixed 
 
 100.83 
 
 101.10 
 
 115.54 
 
 165.24 
 
 182.94 
 
 
 Crosses op Preceding 
 1100-1199 fulfill Test 1 only. 
 
 1103 
 
 Fixed 
 
 100.13 
 
 99.91 
 
 114.26 
 
 161.93 
 
 177.72 
 
 1104 Fixed 100.15 99.84 114.18 161.58 177.92 
 
 * As revised in Chapter XVI, § 9. 
 
514 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 (Continued) 
 Ceoss Weight Geometkic, Median, Mode, Aggregative 
 
 Identi- 
 fication 
 
 Number 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 in Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 1123 
 
 Fixed 
 Chain 
 
 100.14 
 100.14 
 
 99.89 
 100.24 
 
 114.17 
 
 114.24 
 
 161.62 
 
 162.06 
 
 177.87 
 178.40 
 
 18th in ac- 
 curacy 
 
 1124 
 
 Fixed 
 Chain 
 
 100.12 
 100.12 
 
 99.91 
 
 100.24 
 
 114.28 
 115.05 
 
 161.78 
 
 163.36 
 
 177.73 
 
 179.70 
 
 19th in ac- 
 curacy 
 
 1133 
 
 Fixed 
 
 100.52 
 
 99.57 
 
 108.39 
 
 162.63 
 
 170.85 
 
 
 1134 
 
 Fixed 
 
 100.75 
 
 103.33 
 
 117.53 
 
 162.59 
 
 182.15 
 
 
 1143 
 
 
 Same figures as for 43 
 
 
 
 1144 
 
 
 Same figures as for 44 
 
 
 
 1153 
 
 Fixed 
 Chain 
 
 100.13 
 100.13 
 
 99.89 
 100.23 
 
 114.20 
 114.30 
 
 161.70 
 162.21 
 
 177.83 
 178.37 
 
 12th in ac- 
 curacy 
 
 14th in sim- 
 pUcity 
 
 1154 
 
 Fixed 
 
 100.12 
 
 99.90 
 
 114.24 
 
 161.73 
 
 177.76 
 
 13th in ac- 
 curacy 
 
 * As revised in Chapter XVI, S 9. 
 
APPENDIX VII 515 
 
 TABLE 65 (Continued) 
 
 Crosses or Cross Weight Formula, All Ttpes (1300-1399) 
 
 Fulfilling both tests 
 
 Identi- 
 fication 
 
 Number 
 
 Basb 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 1303 
 
 Fixed 
 
 100.14 
 
 99.88 
 
 114.22 
 
 161.75 
 
 177.82 
 
 
 1323 
 
 Fixed 
 Chain 
 
 100.13 
 100.13 
 
 99.90 
 100.24 
 
 114.23 
 
 114-65 
 
 161.70 
 
 162.71 
 
 177.80 
 179.05 
 
 5th in ac- 
 curacy 
 
 6th in con- 
 formity 
 
 1333 
 
 Fixed 
 
 100.63 
 
 101.43 
 
 112.87 
 
 162.61 
 
 176.41 
 
 
 1343 
 
 
 Same figures as for 243 
 
 
 
 1353 
 
 Fixed 
 Chain 
 
 100.13 
 
 100.13 
 
 99.89 
 100.23 
 
 114.22 
 
 114.33 
 
 161.71 
 
 162.27 
 
 177.79 
 
 178.45 
 
 4th in ac- 
 curacy 
 
 5th in con- 
 formity 
 
 * As revised in Chapter XVI, i 9. 
 
516 
 
 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 (Continued) 
 
 Other Cross Weight Formul.^ (2000-4999) 
 
 2100-2199 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1916 
 
 1916 
 
 1917 
 
 1918 
 
 Ranks* of First 
 20 IN Accuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 and Conformity 
 
 TO ClKCULAB 
 
 Test 
 
 2153 
 
 Fixed 
 Chain 
 
 100.12 
 
 100.12 
 
 99.89 
 100.23 
 
 114.23 
 
 114.34 
 
 161.52 
 
 162.25 
 
 177.63 
 
 178.52 
 
 10th in ac- 
 curacy 
 
 14th in speed 
 8th in sim- 
 pHcity 
 
 11th in con- 
 formity 
 
 2154 
 
 Fixed 
 Chain 
 
 100.14 
 
 100.14 
 
 99.90 
 
 100.24 
 
 114.21 
 
 114.31 
 
 101.69 
 
 162.38 
 
 177.72 
 178.65 
 
 11th in ac- 
 curacy 
 
 
 
 
 2300-2399 
 
 
 2353 
 
 Fixed 
 Chain 
 
 100.13 
 100.13 
 
 99.89 
 100.23 
 
 114.22 
 
 114.32 
 
 161.60 
 102.31 
 
 177.67 
 178.58 
 
 2nd in ac- 
 curacy 
 
 3rd in con- 
 formity 
 
 
 
 
 3100-3199 
 
 
 3153 
 
 Fixed 
 
 100.15 
 
 99.88 
 
 114.23 
 
 162.11 
 
 176.94 
 
 
 3154 
 
 Fixed 
 
 100.12 
 
 99.92 
 
 114.28 
 
 161.77 
 
 177.78 
 
 
 
 
 
 3300-3399 
 
 
 3353 
 
 Fixed 
 Chain 
 
 100.14 
 100.14 
 
 99.90 
 
 100.24 
 
 114.35 
 
 114.28 
 
 161.94 
 
 162.14 
 
 177.36 
 
 178.39 
 
 20th in ac- 
 curacy 
 
 
 
 
 4100-4199 
 
 
 4153 
 
 Fixed 
 
 Chain 
 
 100.12 
 
 100.12 
 
 99.97 
 
 100.25 
 
 114.44 
 
 114.55 
 
 162.40 
 
 162.45 
 
 178.26 
 178.79 
 
 
 4154 
 
 Fixed 
 Chain 
 
 100.14 
 100.14 
 
 99.88 
 100.24 
 
 114.08 
 
 114.20 
 
 161.16 
 
 161.96 
 
 176.79 
 
 178.14 
 
 
 
 
 
 4300-4399 
 
 
 4353 
 
 Fixed 
 
 100.13 
 
 99.92 
 
 114.26 
 
 161.78 
 
 177.52 
 
 
 *A3 revised in Chapter XVI, § 9. 
 
APPENDIX VII 
 
 517 
 
 TABLE 65 (Continved) 
 
 MISCELLANEOUS FORMULA (5000-9999) 
 
 Crosses op Cross Formula (5000-5999) 
 
 Identi- 
 fication 
 Number 
 
 Base 
 
 1914 
 
 1915 
 
 1913 
 
 1917 
 
 1918 
 
 Ranks* op First 
 20 inaccuracy, 
 Speed, Simplic- 
 ity OF Formula, 
 AND Conformity 
 TO Circular 
 Test 
 
 5307 
 
 Fixed 
 
 100.15 
 
 99.82 
 
 114.21 
 
 161.67 
 
 177.84 
 
 
 5323 
 
 Fixed 
 Chain 
 
 100.13 
 100.13 
 
 99.87 
 100.23 
 
 114.09 
 IU.45 
 
 161.59 
 
 162.42 
 
 177.67 
 178.64 
 
 3rd in ac- 
 curacy 
 
 4th in con- 
 formity 
 
 5333 
 
 Fixed 
 
 100.68 
 
 101.35 
 
 112.56 
 
 161.56 
 
 178.50 
 
 
 5343 
 
 
 Same figures as for 243 
 
 
 
 Broadened Base Formtjl^ (6000-6999) 
 
 
 6023 
 ('13-'I4) 
 
 100.12 
 
 99.50 
 
 112.25 
 
 153.53 
 
 173.45 
 
 19th in sim- 
 plicity 
 1st in con- 
 formity 
 
 6023 
 ('13-'16) 
 
 99.93 
 
 99.88 
 
 113.61 
 
 156.61 
 
 175.32 
 
 ditto 
 
 6023 
 ('13 & '18) 
 
 99.45 
 
 99.12 
 
 114.23 
 
 159.93 
 
 179.54 
 
 ditto 
 
 6053 
 ('13-'14) 
 
 100.12 
 
 100.09 
 
 113.89 
 
 161.26 
 
 177.73 
 
 7th in speed 
 7th in sim- 
 plicity 
 1st in con- 
 formity 
 
 6053 
 ('13-'16) 
 
 100.02 
 
 100.04 
 
 113.99 
 
 161.88 
 
 178.24 
 
 ditto 
 
 6053 
 ('13-' 18) 
 
 99.79 
 
 99.85 
 
 114.04 
 
 161.59 
 
 177.88 
 
 ditto 
 
 * As revised in Chapter XVI, § 9. 
 
518 THE MAKING OF INDEX NUMBERS 
 
 TABLE 65 (Continued) 
 
 
 
 
 
 
 
 
 Ranks* of First 
 
 
 
 
 
 
 
 
 20 IN Accuracy, 
 
 Identi- 
 
 
 
 
 
 
 
 Speed, Simplic- 
 
 fication 
 
 Base 
 
 1914 
 
 1915 
 
 1916 
 
 1917 
 
 1918 
 
 ity OF Formula, 
 
 Number 
 
 
 
 
 
 
 
 AND Conformity 
 TO Circular 
 Test 
 
 Average of 353 by Six Bases (7000-7999) 
 
 7053 
 
 100.09 99.96 114.03 161.53 177.90 
 
 Arithmetic and Harmonic Means of Aggregatives (8000-8999) 
 
 8053 
 
 Fixed 
 Chain 
 
 100.12 
 
 100.12 
 
 99.89 
 100.23 
 
 114.21 
 114.S3 
 
 161.56 
 
 162.24 
 
 177.65 
 178.50 
 
 6th in ac- 
 curacy 
 15th in sim- 
 
 pHcity 
 7th in con- 
 formity 
 
 8054 
 
 Fixed 
 Chain 
 
 100.12 
 100.12 
 
 99.89 
 100.23 
 
 114.21 
 114.32 
 
 161.56 
 
 162.23 
 
 177.65 
 178.49 
 
 7th in ac- 
 curacy 
 16th in sim- 
 
 pHcity 
 8th in con- 
 formity 
 
 8353 
 
 
 (cross of above) = 353 
 
 
 
 Round Weight Formula (9000-9999) 
 
 900 It 
 
 
 
 
 
 
 
 11th in sim- 
 pHcity 
 
 9011t 
 
 
 
 
 
 
 
 12th in sim- 
 pUcity 
 
 902 It 
 
 
 
 
 
 
 
 13th in sim- 
 plicity 
 1st in con- 
 formity 
 
 9051 
 
 Fixed 
 
 101.68 
 
 103.10 
 
 113.63 
 
 160.37 
 
 182.07 
 
 2nd in speed 
 2nd in sim- 
 plicity 
 1st in con- 
 formity 
 
 * As revised in Chapter XVI, § 9. t Not calculated. See footnote to Table 47, p. 348. 
 
APPENDIX VIII 
 
 SELECTED BIBLIOGRAPHY 
 
 1863. William Stanley Jevons. Investigations in Our Currency and 
 Finance. Sections II-IV, pp. 13-150. London, 1909. (Reprints of 
 various articles published in 1863, etc.) 
 
 1887-1889. F. Y. Edgeworth. Reports of the Committee (of the British' 
 Association for the Advancement of Science) appointed for the purpose 
 of investigating the best methods of ascertaining and measuring va- 
 riations in the value of the monetary standard. In Reports of the As- 
 sociation pubUshed in 1888, pp. 254-301; 1889, pp. 188-219; 1890, 
 pp. 133-64. 
 
 1901. Correa Moylan Walsh. The Measurement of General Exchange- 
 Value. 580 pp. Macmillan, 1901. 
 
 1903. H. Fountain. " Memorandum on the Construction of Index Num- 
 bers of Prices," from Report on Wholesale and Retail Prices in the 
 United Kingdom in 1902, House of Commons Paper No. 321 of 1903, 
 pp. 429-52. Darling & Son, 1903. 
 
 1911. Irving Fisher. The Purchasing Power of Money, pp. 198-234, 
 pp. 385-430. Macmillan, 1911. 
 
 1912. G. H. Knibbs. Prices, Price Indexes, and Cost of Living in Aus- 
 tralia. Commonwealth Bureau of Census and Statistics, Labour and 
 Industrial Branch, Report No. 1, Appendix. McCarron, Bird & Co., 
 Melbourne, December, 1912. 
 
 1915. Wesley C. Mitchell. Index Numbers of Wholesale Prices in the 
 United States and Foreign Countries. U. S. Bureau of Labor Statistics, 
 Bulletin 284, October, 1921. (Revision of Bulletin 173, July, 1915). 
 
 1916. Frederick R. Macaulay. " Making and Using of Index Numbers." 
 American Economic Review, pp. 203-9, March, 1916. 
 
 1916. Wesley C. Mitchell. " A Critique of Index Numbers of the Prices 
 of Stocks." Journal of Political Economy, pp. 625-93, July, 1916. 
 
 1918. G. H. Knibbs. Price Indexes, Their Nature and Limitations, the 
 Technique of Computing Them, and Their Application in Ascertain- 
 ing the Purchasing Power of Money. Commonwealth Bureau of 
 Census and Statistics, Labour and Industrial Branch, Report No. 9, 
 Appendix. McCarron, Bird & Co., Melbourne, 1918. 
 
 1919. A. L. Bowley. " The Measurement of Changes in the Cost of 
 Living." Journal of the Royal Statistical Society, pp. 343-61, May, 
 1919. 
 
 1920. A. C. Pigou. The Economics of Welfare, pp. 69-90. Macmillan, 
 1920. 
 
 1921. G. E. Bamett. " Index Numbers of the Total Cost of Living." 
 Quarterly Journal of Economics, pp. 240-63, February, 1921. 
 
 519 
 
520 THE MAKING OF INDEX NUMBERS 
 
 1921. Irving Fisher. " The Best Form of Index Number." Quarterly 
 
 Publication of the American Statistical Association, pp. 533-51, March, 
 
 1921. 
 1921. A. W. Flux. " The Measurement of Price Changes." Journal of 
 
 the Royal Statistical Society, pp. 167-215, March, 1921. 
 1921. Correa Moylan Walsh. The Problem of Estimation. 139 pp. 
 
 P. S. King, London, 1921. 
 1921. Warren M. Persons. " Fisher's Formula for Index Numbers." 
 
 Review of Economic Statistics, pp. 103-13, May, 1921. 
 1921. Allyn A. Young. " The Measurement of Changes of the General 
 
 Price Level." Quarterly Journal of Economics, pp. 557-73, August, 
 
 1921. 
 1921. Truman L. Kelley. " Certain Properties of Index Nmnbers." 
 
 Quarterly Publication of the American Statistical Association, pp. 
 
 826-41, September, 1921. 
 1921. Lucien March. " Les modes de mesure du mouvement general 
 
 des prix." Metron, pp. 57-91, September, 1921. 
 
 (For completer references see the bibliographies issued from time to 
 time by the Library of Congress.) 
 
INDEX 
 
 References to pages where technical terms are defined or explained, have 
 been set in boldface type. 
 
 Aberthaw Index, cited, 368. 
 Accuracy of index numbers, 330-349. 
 Aggregative, the word, 15, 371; fixed 
 
 base and chain methods agree for 
 
 simple, 373. 
 Aggregative average, simple, 39-40; 
 
 peculiarities of the, 378-379. 
 Aggregative formulae, systems of 
 
 weighting for, 56-57; cross weight, 
 
 187; list of, 201; comments on, 
 
 234-237. 
 Aldrich Senate Report, cited, 333, 445. 
 Alexander Hamilton Institute, cited, 
 
 438. 
 Algebraic notations, key to, 461. 
 American Institute of Finance, cited, 
 
 438. 
 American Writing Paper Co. Index, 
 
 cited, 368. 
 Antitheses, rectifying formulae by 
 
 crossing time, 136-142; rectifying 
 
 formulae by crossing factor, 142-144, 
 
 396-397; fourfold relationship of, 
 
 144-145. 
 Antithesis, time, 118; factor, 118; 
 
 numerical and graphic illustrations 
 
 of time, 119-120; numerical and 
 
 graphic illustrations of factor, 125- 
 
 130. 
 Arithmetic average, simple, 15-23; 
 
 among the worst of index numbers, 
 
 29-30; lies above geometric, 375- 
 
 377. 
 Arithmetic formulae, cross weight 
 
 harmonic and, 187-189; list of 
 
 harmonic and, 199. 
 Arithmetic forward by arithmetic 
 
 backward exceeds unity, 383-384. 
 Attributes of index number, 8-9. 
 Australian Bureau of Census and 
 
 Statistics, cited, 363. 
 Average, index number defined as an, 
 
 3; a simple, 4-6; a weighted, 6-8; 
 
 note on definition of word, 373-375' 
 
 See under Aggregative, Arithmetic, 
 
 Geometric, Harmonic, Median, and 
 Mode. 
 Averaging, 136; of various individual 
 quotations for one commodity, 
 317-318. 
 
 Babson, cited, 438, 460. 
 
 Barnett, G. E., cited, 519. 
 
 Base, fixed, 15-18; chain, 18-22. 
 
 Base number, 18, 371. 
 
 Base year, 19. 
 
 Base year values, weighting by, 
 compared with weighting by given 
 year values, 45-53. 
 
 Bell, Charles A., statement by, on 
 method of splicing employed by 
 U.S. Bureau of Labor Statistics, 
 427-428. 
 
 Bias, 86; single, 86; in arithmetic and 
 harmonic types of formulae, 86-88; 
 weight and type, 91-94; double, 102- 
 105; rela\ion between dispersion and, 
 108-111, 387-395; errors and, gener- 
 ally relative, 116-117; use of term by 
 diiferent statisticians, 117; formulae 
 characterized by, capable of rectifi- 
 cation, 266; tables of deviation and, 
 390, 392, 393; of Formulae 53 and 54 
 slight, 410-412; of 6023 and 23 as 
 affected by price-quantity corre- 
 lation, 428; more disturbing than 
 chance, in weighting, 446-447. 
 
 Bibliography on index numbers, 519- 
 520. 
 
 Blending, 305; substitutes for, 306- 
 308. 
 
 Bowley, A. L., simple median average 
 approved by, 36; use of term "bias" 
 by, 117; cited, 519. 
 
 Bradstreet, cited, 207, 333, 460; simple 
 aggregative approved by, 459, 471. 
 
 British Board of Trade, cited, 332, 333, 
 438. 
 
 British Imperial Statistical Confer- 
 ence, resolution passed by, on 
 
522 
 
 INDEX 
 
 methods of constructing index num- 
 bers, 240-241. 
 
 Broadened base system, 312-313. 
 
 Brookmire, cited, 438. 
 
 Burchard, H. C, index number con- 
 structed by, 459. 
 
 Calculation of formulae, speed of, 321- 
 329. 
 
 Calculation of weighted median and 
 mode, 377-378. 
 
 Canadian Department of Labor, cited, 
 332, 334. 
 
 Carli, G. R., simple arithmetic average 
 approved by, 29, 458, 466. 
 
 Chain base system, 18-22; for simple 
 geometric fixed base system agrees 
 with, 371-372; for simple aggregative 
 fixed base system agrees with, 373. 
 
 Circular test, 270-271; illustration of 
 non-fulfillment, by case of three 
 unlike countries, 271-272; can be 
 fulfilled only if weights are constant, 
 274-276; question as to how near to 
 fulfillment in actual cases, 276 ff. ; 
 the circular gap, or deviation from 
 fulfilling circular test of Formula 353, 
 278-288; status of all formulae rela- 
 tively to, 288-292 ; reduction of, to a 
 triangular test, 295; note on alge- 
 braic expression of, 413; conforma- 
 tion of simple or constant weighted 
 geometric to, 413, 416; formula 
 satisfying, for three dates will satisfy 
 for four, etc., 426-427. 
 
 Circular (test) gap, 277-280; tabu- 
 lation of, for Formula 353, 280-283; 
 discussion of, of Formula 353, 283- 
 287; comparison of, of 134 different 
 formulae, 287-288; meaning of 
 "equal and opposite," 418. 
 
 Coggeshall, F., harmonic index number 
 approved by, 33, 467. 
 
 Commensurability, as test of index 
 number of prices, 420-426. 
 
 Commodity reversal tests, 63-64. 
 See under Tests. 
 
 Cross between two factor antitheses 
 fulfills Test 2, 396-397. 
 
 Cross formula, 185, 407. 
 
 Cross references between "Purchasing 
 Power of Money" and this book, 
 419. 
 
 Cross weight formula, 185. 
 
 Crossing of formulae. 136-183. 
 
 Crossing of weights possible geometri- 
 cally, arithmetically, harmonically, 
 401-407. 
 
 Davies, George R., Formula 353 
 approved by, 242. 
 
 Day, E. E., studies by, 14 n.; quantity 
 figures worked out by, 110; Formula 
 6023 approved by, 486; cited, 253, 
 254, 313, 314, 316, 317, 326, 328, 
 342, 343, 384. 
 
 Determinateness, as test of index num- 
 ber of prices, 420-423. 
 
 Deviation, standard, no; tables of, 
 337, 391; tables of bias and, 390, 
 392, 393. 
 
 Dispersion, 108; dependence of bias on, 
 108-111; charts showing, measured 
 by standard deviations, 290-294; 
 notes on bias and, in formulae, 387— 
 390; "skewness" of, 408-410. 
 
 Dispersion index, tables showing, 
 compared with standard deviation, 
 392, 393. 
 
 Drobisch, M. W., use of factor an- 
 titheses by, 134; cross formula sug- 
 gested by, 196; Formula 52 approved 
 by, 471; Formula 8053 approved by, 
 487. 
 
 Dun, Formula 53 approved by, 471; 
 Formula 9001 approved by, 487; 
 cited, 336, 460. 
 
 Dutot, simple aggregative index num- 
 ber approved by, 40, 458, 471. 
 
 Economist (London), simple arithmetic 
 approved by, 459, 466; cited, 29, 333. 
 
 Edgeworth, F. Y., simple median 
 approved by, 36, 262, 469; cross 
 weight aggregative proposed by, 
 196; "probability" system of 
 weighting of, 379-380; Formula 
 2153 approved by, 484; recommen- 
 dations of, with regard to index 
 numbers, 459; cited, 255, 320, 365, 
 366, 408, 519. 
 
 Entry, as test of index number of 
 prices, 420^23. 
 
 Erratic index numbers, 11 2-1 16. 
 
 Errors, joint. See Joint errors. 
 
 Errors, probable. See Probable error. 
 
 Factor antithesis. See Antithesis. 
 Factor reversal tests, 72. See under 
 Tests. 
 
INDEX 
 
 523 
 
 Fairness, a requirement in index 
 numbers, 9, 10, 62. 
 
 Falkner, R. P., Formula 9001 ap- 
 proved by, 459, 487. 
 
 Federal Reserve Board, geometric 
 weighted by given year values ap- 
 proved by, 468; cited, 460. 
 
 Fisher, Irving, cited, 25, 82, 242, 520; 
 The Rate of Interest, cited, 63 n.; 
 Purchasing Power of Money, cited, 
 82, 381, 519; relation of present book 
 to Appendix on Index Numbers in 
 Purchasing Power of Money, 418- 
 426; Formula 53 approved by, 471; 
 Formula 54 approved by, 471; ideal 
 index number approved by, 482; 
 Formula 2153 approved by, 484. 
 
 Fisher, Willard, cited, 458. 
 
 Fixed base system, 15-18; for simple 
 geometric chain system agrees with, 
 371-372; for simple aggregative chain 
 system agrees with, 373. 
 
 Flux, A. W., cited, 111, 296, 320, 366, 
 520; simple geometric approved by, 
 468. 
 
 Formulse, classification of, in six types, 
 15; time reversal tests as finders of, 
 118-135; rectifying, by crossing 
 them, 136-183; rectifying, by 
 crossing their weights, 184-196; 
 lists of, 170-174; main series of, 
 184, 197; supplementary series of, 
 184; seven classes of, 202; compari- 
 son of, with view to selecting the 
 best, 206-212; comparison of other, 
 with the "ideal" (Formula 353), 
 243-269; eight most practical, 361- 
 362; method for comparing with 
 "ideal," 412-413; key to numbering 
 of, 461-465; table of, for index num- 
 bers, 466-488; alternative forms of 
 certain, 488. 
 
 Fountain, H., cited, 519. 
 
 Freakishness, of median and mode, 
 1 1 2-1 1 &, 209-211; of simple aggre- 
 gative, 207-209; lessening, by in- 
 creasing number of commodities, 
 216-218; formulse rendered wholly 
 unreliable by, 266. 
 
 Geometric average, simple, 33-35; 
 cross weight, 186; fist of formulae, 
 200; comparison of the simple, and 
 the simple median, 260-264; fixed 
 base and chain methods agree for, 
 
 371-372; lies between arithmetic 
 
 above and harmonic below, 375-377. 
 Gibson, Thomas, cited, 460. 
 Given year values, weighting by, 
 
 compared with weighting by base 
 
 year values, 45-53. 
 
 Haphazard, applied to weighting, 207; 
 index numbers found to be, 218; 
 differences between simple and cross 
 weight index numbers are, 444. See 
 Freakishness. 
 
 Harmonic average, 30; the simple, 30- 
 33; hes below geometric, 375-377. 
 
 Harmonic formulse, cross weight arith- 
 metic and, 187-191; list of, 199. 
 
 Harvard Committee on Economic 
 Research, cited, 53, 438, 460. 
 
 Hastings, Hudson, cited, 429. 
 
 Historical notes, on methods of 
 weighting, 59-60; on reversal tests, 
 82; on biased index numbers, 117; 
 on tests as finders of formulse, 134- 
 135; on crossing of formulse, 183; on 
 crossing of weights, 196; on Formula 
 353, 240-242; on circular test, 295- 
 296; on fixed base, broadened base, 
 and chain systems, 320; on usp of 
 index numbers, 458-460. 
 
 Hofmann, Emil, cited, 437. 
 
 Holt & Co. Index, cited, 368. 
 
 Hybrid weighting, 53-56. 
 
 Ideal blend, 305-306. 
 
 Ideal index number (Formula 353), 
 220-225; probable error of, 225-229; 
 history of, 240-242; Formula 2153 
 close to, 428-430. 
 
 Index numbers, 3; simple arithmetic, 
 4-5; weighted arithmetic, 6-7; 
 attributes of, 8-9; fairness of, 9-10, 
 62; six types of, compared, 11 ff. ; 
 simple harmonic, 30-33; simple 
 geometric, 33-35; simple median, 
 35-36; simple mode, 36-39; simple 
 aggregative, 39-40; comparison of 
 six simple forms of, 41-42; calcu- 
 lation of, by different methods of 
 weighting, 43-56; only two systems 
 of weighting for aggregative type of, 
 56-57; relation of weighted aggre- 
 gative to weighted arithmetic and 
 weighted harmonic, 60, 379; reversal 
 tests of, 62-82; joint errors between, 
 83-86; erratic and freakish, 112-116; 
 
524 
 
 INDEX 
 
 rectificationof formulae, by crossing, 
 136-183; rectifying formulse by cross- 
 ing their weights, 184-196; the best 
 simple, 206-212; finding the very 
 best, 213-242; comparison of all, 
 with Formula 353, 243-269; results 
 of comparisons among 134 varieties, 
 266-269; so-called circular test of, 
 270-296; blending apparently in- 
 consistent results, 297-320; influ- 
 ence of assortment and number of 
 samples, 331-340; future uses of, 
 367-369; Hst of discontinued, 432- 
 433; list of current, 433-438; in- 
 fluence of wBighting on, 439-457; 
 averages of ratios rather than ratio 
 of averages, 451-457; landmarks in 
 history of, 458-460; list of formulce 
 for, 462-488; examples showing how 
 to calculate, 490^97; tables of, 
 by 134 formulEe, 498-518; bibliog- 
 raphy on subject of, 519-520. 
 
 Institute of Finance, American, cited, 
 438. 
 
 International Labour Office, cited, 438. 
 
 International Labour Review, cited, 437. 
 
 Jevons, W. S., simple geometric ap- 
 proved by, 35, 459, 468; cited, 139, 
 296, 519. 
 
 Joint errors between index numbers, 
 83-86; expressible by product or 
 quotient, 88-90. 
 
 Kelley, Truman L., "Certain Proper- 
 ties of Index Numbers" by, cited, 
 331, 334, 340, 424, 520; method 
 proposed by, of measuring probable 
 error of index number, 430-431. 
 
 Kemmerer, E. W., cited, 368. 
 
 Key, to principal algebraic notations, 
 461; to numbering of formulse of 
 index numbers, 461-465. 
 
 Knibbs, G. H., weighted aggregative 
 formula approved by, 59, 240, 460, 
 471; cited, 230, 366, 371, 519. 
 
 Laspeyres, E., weighted aggregative 
 formula approved by, 59, 459, 471; 
 formula of, in relation to factor 
 antithesis, 131-132; cited, 60, 161, 
 168, 169, 240, 255, 320, 387, 412. 
 
 Laughlin, J. L., cited, 458. 
 
 Lehr, J., cited, 134, 196, 255, 326; 
 Formula 2154 approved by, 485. 
 
 Linking,''process of, 22. " 
 
 London School of Economics, cited, 
 
 438. 
 Lowe, Joseph, Formula 9051 approved 
 
 by, 458, 487. 
 
 Macalister, "Law of the Geometric 
 Mean," cited, 231 n. 
 
 Macaulay, F. R., theorem of, relative 
 to so-called circular test, 292-293; 
 cited, 241, 366, 426, 519. 
 
 Main series of formulae, 184, 197. 
 
 March, Lucien, cited, 296, 520. 
 
 Marshall, Alfred, cross weight aggre- 
 gative approved by, 196, 484; chain 
 base system suggested by, 320; 
 cited, 255, 365, 366. 
 
 Massachusetts Commission on the 
 Necessaries of Life, cited, 460. 
 
 Median average, simple, 35-36; freak- 
 ishness of, 210-211; compared with 
 simple geometric, 260-264; calcu- 
 lation of weighted, 377-378. 
 
 Median formulae, cross weight, 186; 
 list of, 200; comments on, 258-260. 
 
 Meeker, Royal, cited, 240, 366. 
 
 Messedaglia, A., cited, 459. 
 
 Method for comparing other formulae 
 with "ideal," 412^13. 
 
 Mitchell, Wesley C, data collected by, 
 14; use of simple median average by, 
 36, 469; cited, 38, 39, 216, 232, 233, 
 295, 331, 332, 3.34, 335, 336, 366, 
 371, 408, 426, 445, 458, 460, 519. 
 
 Mode, simple, 36-39; method of 
 finding the simple, 372-373; calcula- 
 tion of weighted, 377-378; if above 
 geometric forward, below it back- 
 ward, 407. 
 
 Modes, cross weight, 186; Hst of for- 
 mulae in group of, 201; comments 
 on, 258-260. 
 
 National Industrial Conference Board, 
 
 cited. 438, 460. 
 Neumann-Spallart, cited, 438. 
 Nicholson, J. S., cited, 134; Formula 
 
 22 approved by, 468. 
 Numbering of formulae, system of, 
 
 142, 461-465. 
 
 Ogburn, W. F., theorem of, relative to 
 so-called circular test, 292-293; 
 formula of, for Macaulay's Theorem, 
 426. 
 
INDEX 
 
 525 
 
 Paasche, H., weighted aggregative 
 formula of, 60; formula of, in relation 
 to factor antithesis, 131-132; For- 
 mula 5U approved by, 459, 471; 
 cited, 161, 168, 169, 240, 255, 387. 
 
 Palgrave, R. H. Inglis, arithmetic 
 weighted by given year values ap- 
 proved by, 466; cited, 102, 103, 111. 
 
 Pearl, Raymond, cited, 382. 
 
 Percentaging, i6. 
 
 Persons, W. M., studies by, 14 n. ; 
 quantity figures worked out by, 
 110; index of crops, 236-239; refer- 
 ence by, to "Fisher's Index Num- 
 ber," 242; defense of Day's index 
 number by, 316; Formula 6023 
 approved by, 486; cited, 313, 314, 
 316, 326, 328, 331, 336, 340, 343, 
 366, 384, 410, 411, 430, 486, 520. 
 
 Pierson, N. G., objections to index 
 numbers quoted, .1; time reversal 
 test first used by, 82; cited, 49, 117, 
 224. 
 
 Pigou, A. C, mention of Formula 353 
 by, 241-242; ideal index number 
 approved by, 482; cited, 366, 519. 
 
 Price relative, 3, 16. 
 
 Prices, dispersion of individual, 11—14; 
 errors in weights less important than 
 in, 447-449. 
 
 Probability system of weighting, 
 379-381. 
 
 Probable error of index number, 341; 
 Kelley's method of measuring, 430- 
 431; of Formula 353, 225-229; deri- 
 vation of, of 13 formulae, 407-408. 
 
 Product, joint error expressible by, 
 88-90. 
 
 Proportionality, as test of index num- 
 ber of prices, 420-423. 
 
 Purchasing power, an index number of, 
 377. 
 
 Quantities, dispersion of individual 
 
 prices and, 11-14. 
 Quantity relatives, dispersion of, 110- 
 
 111. 
 Quartets, 145; arranging of formulae 
 
 in, 145-147; list of, 164-170. 
 Quotient, joint error expressible by, 
 
 88-90. 
 
 Ratio, price, 3, 78; quantity, 72-73. 
 
 78; value, 74, 78. 
 Ratio chart method of plotting, 25-27. 
 
 Ratios, index number average of, 
 rather than ratio of averages, 451- 
 457. 
 
 Rawson-Rawson, use of factor antith- 
 eses by, 134; Formula 52 approved 
 by, 471; cited, 368. 
 
 Reciprocals, use of, in calculating 
 simple harmonic average, 30. 
 
 Rectification of formulae, 136; by 
 crossing time antitheses, 140-142; 
 by crossing factor antitheses, 142- 
 144; of simple arithmetic and har- 
 monic by both tests, 145-149; by 
 crossing weights, 184r-196; order of, 
 398-399. 
 
 Reversal tests. See Tests. 
 
 Samples, use of, in measuring price 
 movements, 330-331; influence of 
 assortment of, 331-334; number of, 
 336-340. 
 
 Sauerbeck, A., simple arithmetic 
 approved by, 459, 466; cited. 111, 
 117, 317, 342, 345, 349, 395. 
 
 Schuckburgh-Evelyn, G., simple arith- 
 metic approved by, 458, 466. 
 
 Scope of our conclusions, 381-383. 
 
 Scrope, G. Poulett, cross weight aggre- 
 gative formula approved by, 196; 
 Formula 53 approved by, 471; For- 
 mula 54 approved by, 471; Formula 
 1153 approved by, 483; Formula 
 9051 approved by, 458. 
 
 Sidgwick, H., Formula 8053 approved 
 by, 196, 487. 
 
 "Skewness" of dispersion, question 
 concerning, 408-410. 
 
 Speed of calculation of index numbers, 
 321-329. 
 
 Splicing, 310-312; as applied to aggre- 
 gative index numbers, 427-428. 
 
 Standard deviation. See Deviation. 
 
 Standard Statistics Corporation, cited, 
 438. 
 
 Statist (London), simple arithmetic ap- 
 proved by, 466; cited, 29, 342, 345, 
 349. 
 
 Supplementary series of formulae, 184. 
 
 Tests, reversal, 62-63; commodity 
 reversal, 63-64; time reversal, 64- 
 65; time reversal, illustrated nu- 
 merically and graphically, and ex- 
 pressed algebraically, 65-72; factor 
 reversal, 72; simple arithmetic index 
 
526 
 
 INDEX 
 
 number tested by factor reversal, 
 72-76; factor reversal, illustrated 
 graphically, 76-77 ; error revealed by 
 factor reversal, 77-79 ; factor reversal 
 analogous to other reversal, 79-82; 
 reversal, as finders of formulae, 118- 
 135; rectifying formulae by, 136-183; 
 importance of conformity to, by first- 
 class index numbers, 268; so-called 
 circular, 270-295 (see Circular test) ; 
 triangular, 295. 
 
 Time antithesis. See Antithesis. 
 
 Time reversal test. See Tests. 
 
 Time studies for calculating index 
 numbers, 321-325. 
 
 Times Annalist, cited, 460. 
 
 United States Bureau of Labor Statis- 
 tics, aggregative weighted by base 
 year values approved by, 471; cited, 
 53, 59, 240, 335, 341, 342, 344, 346, 
 363, 369, 437, 438. 
 
 United States Bureau of Standards, 
 cited, 225. 
 
 Value, ratio. See Ratio, value. 
 
 Wages, adjustment of, by index num- 
 bers, 368, 460. 
 
 Walras, L., cited, 296. 
 
 Walsh, C. M., cited, 29, 35, 40, 59, 60, 
 121, 207, 255, 326, 328, 366, 408, 458, 
 459, 519, 520; on use of simple mode 
 average, 39; on aggregative form of 
 index number, 42; importance of 
 time reversal test recognized by, 82; 
 idea of type bias expressed by, 117; 
 use of time reversal test by, 134; 
 cross weight aggregative formula 
 approved by, 196; reference by, to 
 Formula 353, 241, 242; on the so- 
 called circular test, 295-296; For- 
 
 mula 22 approved by, 468; Formula 
 1123 approved by, 483; ideal index 
 number approved by, 482; Formulae 
 1153 and 1154 approved by, 483; 
 Formulae 2153 and 2154 approved 
 by, 484. 
 
 War Industries Board, weighted index 
 number of, 44, 342-343; cited, 14, 
 216, 262, 333, 334, 336, 339, 340, 
 344, 410. 
 
 Weighting, 6-8; just basis for, 43-45; 
 by base year values or by given year 
 values, 45-53; two intermediate 
 (hybrid) systems of, 53-56; only 
 two systems of, for aggregative type 
 of index number, 56-57; history of, 
 59-60; additional systems of, 61; 
 bias in, 91-94; influence of, 439 ff.; 
 simple and cross, compared, 443- 
 444; errors in, less important than 
 in prices, 447-449; the best system 
 of, 449-450. 
 
 Westergaard, idea of circular test 
 propounded by, 295; simple geo- 
 metric approved by, 459, 468. 
 
 Withdrawal or entry, as test of index 
 number of prices, 420-423. 
 
 Wood, George H., "Some Statistics 
 of Working Class Progress since 
 1860" by, 438. 
 
 Young, Allyn A., estimate of Formula 
 353 by, 242; probability system of 
 weighting of, 379-380; " ideal " index 
 number approved by, 482; cited, 
 366, 520. 
 
 Young, Arthur, Formula 9001 ap- 
 proved by, 458, 487; cited, 43, 45. 
 
 Zizek, Franz, Statistical Averages by, 
 438. 
 
 347/ 
 X279 
 
"^rl. ^ 
 
^^-^ -^"c^. 
 
 ■J- \ 
 
 cV . - - 
 
 '^A. 
 
 ^<^ 
 
 ^ 
 
 
 \.^ '), - XV. ,^\\^' 
 
 
 =^i$»i^; ^.# -^^^K