^^^ 
 
 uA 
 
 LOS ANGiiLEb 
 LIBRARY
 
 INTRODUCTION TO 
 ECONOMIC STATISTICS
 
 INTRODUCTION TO 
 
 ECONOMIC STATISTICS 
 
 BY 
 GEORGE R. DAVIES, Ph. D. 
 
 PROFESSOR OF SOCIOLOGY, UNIVERSITY OF NORTH DAKOTA 
 
 t W i »f^ < jj*W\i **t 
 
 NEW YORK 
 
 THE CENTURY CO. 
 
 1922 
 
 91409
 
 Copyright, 1922, by 
 The Century Co. 
 
 Printed in U. S. A.
 
 "K 
 
 .%;: 
 
 . .5 
 
 , PREFACE 
 
 ' The application of statistical methods to special 
 
 fields such as demography, education, and economics 
 has in recent years advanced very rapidly. As a re- 
 sult a study of statistical principles must be confined 
 chiefly to one of the special fields if it is not to be 
 lost in the multiplicity of specific methods and illus- 
 trations. This text-book has been written with the in- 
 
 ►i terests of the student of economics in mind. 
 
 A common difficulty which the teacher of statistics 
 encounters is a lack of provision for laboratory work. 
 An attempt has here been made to supply this need 
 by furnishing illustrative problems, graphs, and data 
 which may be worked over by the student, and by add- 
 
 1^5] ing to each chapter a list of related exercises. The 
 
 ^ exercises will be found extensive enough so that the 
 teacher may select those which are adapted to his re- 
 quirements. The longer problems should be subdivided, 
 and the parts assigned to different members of the 
 class. Both the tables and the exercises may very well 
 ^ be supplemented by the use of data drawn from such 
 
 ^ sources as the Survey of Current Business, the Monthly 
 Labor Review, and the Statistical Abstract of the 
 United States (Superintendent of Documents, Govern- 
 ment Printing Office, Washington, D. C). The topics 
 covered in the text represent probably a maximum of 
 what can be mastered by a college class in a term. 
 Perhaps it may be found advisable to omit certain 
 topics, such as interpolating for quartiles, theories
 
 vi PREFACE 
 
 of price indexes, parabola trends, seasonal variations, 
 and the more complex methods of correlation. 
 
 Nearly all the material here presented has been ac- 
 cumulated from experience in the statistical laboratory 
 and class-room. Particular attention has been given 
 to the requirements in respect to fundamental theory 
 of the statistical departments of the larger banks and 
 business houses. Some of the recently developed 
 methods of handling business barometers have there- 
 fore been touched upon, and some attention has been 
 given to the theory of price and production indexes. 
 
 The book is an outgrowth of the undergraduate 
 course in statistics given by the writer at Princeton 
 University during the school year 1920-1921. This 
 course was modeled in its general features upon the 
 course given the preceding year by Professor J. H. 
 Williams, now of Harvard University. The writer 
 wishes to acknowledge his indebtedness to Professor 
 Williams for the general plan of the laboratory exer- 
 cises, as well as for many specific suggestions. Thanks 
 are also extended to Professors F. A. Fetter and E. 
 W. Kemmerer of Princeton for their interest and en- 
 couragement, and to Professor W. F. Willcox of Cor- 
 nell who read the first draft of the manuscript and 
 made several valuable suggestions for its revision. In- 
 debtedness is also acknowledged to the following for 
 their kind permission to reprint data: Mr. Roger W. 
 Babson, Professor Stanley E. Howard, Bureau of La- 
 bor Statistics, National City Bank, National Bureau of 
 Economic Research, National Industrial Conference 
 Board, Review of Economic Statistics, and the Quar- 
 terly Journal of the University of North Dakota.
 
 CONTENTS 
 
 CHAPTER PAGE 
 
 I Tabulation 3 
 
 II Types and Measures of Dispersion .... 20 
 
 III Indexes of Wages and Prices ...... 47 
 
 IV Quantity Indexes and Their Uses ..... 74 
 V Trends and Cycles 100 
 
 VI Correlation 131 
 
 Appendix I 
 
 Laboratory Material and References . . . 153 
 
 Appendix II 
 
 Tables of Powers and Roots 157 
 
 Appendix III 
 A Picture of the Progress of the United 
 
 States During 120 Years of National Life . 160 
 
 Index 161
 
 INTRODUCTION TO 
 ECONOMIC STATISTICS
 
 INTRODUCTION TO ECONOMIC 
 STATISTICS 
 
 CHAPTER I 
 
 TABULATION 
 
 The term ** statistics" when used to designate a 
 branch of study, implies an exposition of certain 
 methods employed in presenting and interpreting the 
 numerical aspects of a given subject. The science of 
 statistics consists, therefore, of principles and methods, 
 rather than of data. The principles are essentially 
 the same whether the application is made to biology, 
 demography, education, or economics. But the de- 
 tailed methods in these and other fields have in late 
 years become so specialized that it is hardly practi- 
 cable any longer to study statistics in the abstract. 
 The field of application here adopted is chiefly that of 
 general economics.^ Illustrations will be given of the 
 methods employed in organizing data, in computing 
 and employing indexes, and in measuring trends and 
 correlations. 
 
 * It should be noted that economic statistics are commonly distin- 
 guished from business statistics. The former subject studies general 
 market conditions, while the latter subject deals with the details of a 
 specific business establishment, and is therefore an adjunct of account- 
 ing. Business statistics vary so greatly from one establishment to an- 
 other that it is difficult to generalize from them. Their problems 
 consist largely of the application of statistical principles to specific 
 situations. For a discussion of the distinction see "The Scope of Busi- 
 ness Statistics," by R. P. Falkner, in the Quarterly Publications of 
 the American Statistical Association, June, 1918, pp. 24-29. 
 
 3
 
 4 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Preliminary Schedules. Statistical field work com- 
 monly begins with the preparation of schedules in 
 which to enter the desired data. These schedules may 
 be in the nature of questionnaires, or they may be mere 
 forms in w^hich to copy certain records. Since such 
 schedules will vary with the particular task in hand, 
 few rules can be laid down for their preparation. Ex- 
 perience will teach, however, that they must be very 
 carefully worked out in advance. In the first place, it 
 must be determined as precisely as possible what data 
 will be needed. If the schedule is in questionnaire 
 form, great care must be taken to make the questions 
 unambiguous and easily comprehended by those who 
 are to answer. Errors may often be checked by asking 
 a question in two ways, at different places in the list ; 
 as by calling for both the age and the date of birth. 
 
 After the preliminary schedules have been compiled, 
 the statistician begins to organize his data, so that con- 
 clusions may be deduced and presented in simplified 
 form. In so doing he will make use of the processes 
 of tabulation. These may best be explained by taking 
 an example. As such an example, we shall choose 
 certain wage schedules which have been recorded in 
 the Aldrich Report on ''Wholesale Prices, Wages, and 
 Transportation" (Senate Report No. 1394, dated 
 1893). The data here used will be found in Vol. IV, 
 pages 1463-1497, and refer to a Connecticut woolen 
 mill designated as Establishment No. 86. The wage 
 rolls as recorded in the report cover about half a 
 century, ending in 1891. They show the daily wages 
 paid to each class of workmen employed in the mill 
 in January and July of each year. We shall select
 
 TABULATION 5 
 
 for tabulation only the wages paid in July of 1870, 
 1880, and 1891. 
 
 The Primary or General Purpose Table. In tran- 
 scribing the selected wage schedules into a primary 
 table, it will be found advisable to edit the original 
 figures by making certain minor modifications. An in- 
 spection of the schedules will show that most of the 
 wage rates are expressed as multiples of five cents. 
 Of course, we may assume theoretically that the exact 
 economic values which are approximated in the actual 
 wages must form a continuous series, instead of one 
 having a regular interval. That is, if the wages could 
 be expressed as theoretically exact values, and if a 
 very large number of workers were involved, the rates 
 would be separated by intervals of only a small frac- 
 tion, of a cent. The case may be compared with the 
 measurement of the height of the individuals in a 
 large group. If the measurements are taken with 
 very precise instruments, the results will be expressed 
 in hundredths or perhaps thousandths of an inch. But 
 for practical purposes measurements to perhaps the 
 nearest quarter inch are sufficiently accurate. In the 
 same way when the employer made an offer of wages he 
 set his figure at a multiple of five cents, or in the case 
 of the larger wages at a multiple of twenty-five cents. 
 For his purpose, such an estimate of the market value 
 of labor was sufficiently accurate. The exceptions will 
 be found to be rates that were paid to an especially 
 large number of workers. In such cases differences 
 of a cent or two are of considerable consequence. 
 
 Continuous ajid Discrete Series. A series of meas- 
 urements or values occurring only at more or less reg-
 
 6 INTRODUCTION TO ECONOMIC STATISTICS 
 
 ular intervals is said to be discrete. Sometimes a series 
 A\all be naturally discrete, as when flowers are clas- 
 sitied by the number of petals, or when the spots in 
 a number of dice-throws are tabulated. But a series 
 which is theoretically continuous becomes artificially 
 discrete when a limit of accuracy is determined upon, 
 as when height is measured to the nearest quarter- 
 inch, or wages are expressed in multiples of five cents. 
 Our series of wage rates, as it stands in the records, 
 is therefore discrete at intervals of five cents, except 
 for a few items. 
 
 For purposes of classification it is desirable to have 
 our series of wage rates regularly discrete throughout. 
 Since it is not possible to break down the five cent 
 intervals to smaller ones, it will be necessary to modify 
 such rates as 67c. and $1.28 so as to classify them as 
 multiples of five. If there were only a few scattered 
 cases .of this sort affecting only a small number of 
 workers, they might merely be entered at the nearest 
 five cent intervals; that is, 67c. could be entered as 
 65c. and $1.28 as $1.30. But since the number of work- 
 ers at these irregular rates is exceptionally large, 
 such a procedure might result in too great a degree of 
 inaccuracy. We shall therefore apply a familiar arith- 
 metical device and break up the given number of work- 
 ers at each inconvenient rate into two groups, one 
 having a higher and one a lower rate than that stated, 
 but maintaining together the same average wage. The 
 procedure may be illustrated as follows : 
 
 {9 workers at 65c. 
 -f 
 6 workers at 70c.
 
 TABULATION 
 
 This result is obtained by taking 3/5 and 2/5 of the 
 fifteen workers (the fractional parts of the five cent 
 interval which lie between 67c. and the nearest mul- 
 tiples of 5) and placing the larger of the two results 
 at the rate expressed by the multiple of five nearest to 
 67c., that is, at 65c. The smaller of the two results is 
 
 TABLE I 
 
 WAGE EOLL IN A CONNECTICUT WOOLEN MILL 
 JULY CTF SPECIFIED YEARS 
 
 OCCUPATION 
 
 Burlers 
 
 Card cleaners. 
 
 Card tenders. . 
 
 Carpenters 
 
 Cloth Inspectors. 
 Drawers-in 
 
 Dressers. 
 
 Djera. 
 
 Firemen 
 
 Foremen-burlers . . . . 
 Fullers and giggers. 
 
 Hand«ra-in .... 
 Harness hands. 
 Loom fixers. . . 
 
 1870 
 
 NO. WAfiB 
 
 $ .80 
 
 .85 
 
 .70 
 
 1.10 
 
 1.25 
 
 .55 
 .60 
 .65 
 .80 
 2,75 
 
 1.25 
 
 1880 
 
 NO. WAGE 
 
 1.35 
 1.50 
 
 1.50 
 
 1.10 
 1.15 
 1.25 
 1.35 
 1.50 
 .40 
 
 1.50 
 1.10 
 1.15 
 
 11* 
 
 1 
 1 
 
 1* 
 1* 
 
 2 
 2 
 
 ^ .80 
 
 .85 
 
 1.15 
 
 .60 
 .65 
 
 2.75 
 1.60 
 1.90 
 1.95 
 
 1.50 
 1.55 
 
 1.25 
 1.30 
 
 1.05 
 1.20 
 1.25 
 
 .40 
 1.95 
 
 1891 
 
 NO. WAGE 
 
 14^ 
 
 $1.10 
 
 1 
 
 1.10 
 
 1 
 
 1.15 
 
 1 
 
 1.20 
 
 2 
 
 1.25 
 
 1 
 
 .75 
 
 1 
 
 .85 
 
 1 
 
 .90 
 
 1 
 
 1.00 
 
 1 
 
 2.75 
 
 1 
 
 1.90 
 
 1* 
 
 1.75 
 
 1* 
 
 1.85 
 
 4* 
 
 1.90 
 
 1 
 
 1.25 
 
 2 
 
 1.60 
 
 1 
 
 1.65 
 
 3 
 
 1.75 
 
 1 
 
 1.15 
 
 13 
 
 1.25 
 
 1 
 
 1.40 
 
 2 
 
 1.50 
 
 3 
 
 1.50 
 
 1 
 
 2.25 
 
 2 
 
 1.05 
 
 4 
 
 1.10 
 
 3 
 
 1.15 
 
 3 
 
 1.25 
 
 1 
 
 1.50 
 
 5» 
 
 .50 
 
 1 
 
 1.50 
 
 1 
 
 2.00 
 
 2 
 
 2.10 
 
 4 
 
 2.20
 
 8 INTRODUCTION TO ECONOMIC STATISTICS 
 
 TABLE I (Continued) 
 
 OCCUPATION 
 
 Machinists 
 
 Machinists — helpers 
 
 Number sewers 
 
 Overseers — Carding dept . . 
 
 " Dyehouse 
 
 " Finishing dept. 
 
 * * Fulling dept . . . 
 
 " Spinning dept. 
 
 " Spooling dep't. , 
 
 ** Weaving dept. 
 Piecers 
 
 Second hands. 
 Sewers 
 
 Shearers . 
 
 Sorters. . 
 Speckers. 
 
 Spinners — jack and mule. 
 Spoolers 
 
 Teamsters. 
 Twisters . . 
 Watchmen . 
 
 Weavers . 
 
 Weavers — pattern . 
 
 Winders 
 
 Yam carriers 
 Totals. . . 
 
 1870 
 
 NO. WAGE 
 
 1 
 1 
 
 1 
 
 2 
 1 
 
 1* 
 
 3 
 
 7 
 
 4* 
 3* 
 19 
 5 
 
 1^ 
 1 
 108 
 
 ^2.75 
 1.75 
 
 3.50 
 3.75 
 2.50 
 2.75 
 2.75 
 2.25 
 3.00 
 .75 
 .80 
 1.50 
 1.75 
 
 1.15 
 1.40 
 1.50 
 2.00 
 2.75 
 1.35 
 
 1.75 
 
 1.80 
 
 .60 
 
 1.50 
 
 1.50 
 
 1.05 
 1.10 
 1.30 
 1.35 
 
 1.00 
 1.25 
 
 1880 
 
 NO. WAGE 
 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 
 2 
 8 
 1 
 1 
 1* 
 
 2 
 1 
 
 1* 
 2* 
 1 
 
 5* 
 
 4* 
 
 1 
 
 7* 
 
 1 
 
 79 
 10' 
 
 2' 
 
 213 
 
 1891 
 
 $3.00 
 
 3.25 
 3.25 
 3.00 
 2.50 
 3.00 
 2.25 
 3.00 
 .70 
 .75 
 1.25 
 1.50 
 1.25 
 
 1.25 
 
 1.80 
 
 2.75 
 
 .90 
 
 .95 
 
 1.50 
 
 .65 
 
 .70 
 1.50 
 
 .90 
 1.50 
 
 1.20 
 1.40 
 1.45 
 
 .95 
 1.00 
 1.75 
 
 2 
 1 
 2* 
 2* 
 4 
 7 
 9* 
 14* 
 1 
 
 4* 
 1 
 1 
 1 
 
 89 
 
 60 
 
 3 
 
 2 
 
 4 
 
 1 
 
 1 
 
 12* 
 
 17' 
 
 1 
 
 361 
 
 NO. WAGE 
 
 $2.75 
 2.10 
 .90 
 4.00 
 4.25 
 3.50 
 2.50 
 2.75 
 2.50 
 3.00 
 
 1.75 
 
 1.00 
 1.25 
 1.35 
 
 1.70 
 2.75 
 .80 
 .85 
 1.25 
 1.30 
 .75 
 .80 
 1.60 
 .90 
 1.30 
 1.35 
 1.40 
 1.30 
 1.35 
 1.70 
 1.75 
 1.25 
 1.35 
 1.50 
 1.75 
 2.00 
 1.15 
 1.20 
 1.85 
 
 Female employes.
 
 TABULATION 9 
 
 placed at the rate of 70c. That the average wage has 
 not been changed by this operation is shown by the fact 
 that 
 
 15 X 67c. = 9 X 65c + 6 X 70c. 
 
 By the foregoing method all irregular wage rates 
 may now be reduced to approximately equivalent mul- 
 tiples of five cents. Of course, when a fraction appears 
 in the operation the nearest whole number is taken. 
 Thus modified, our wage schedules appear as shown 
 in Table I, which may be taken as an example of a 
 primary or general purpose table. 
 
 The Frequency Curve. The tabulation which we are 
 about to undertake has for its immediate object the 
 presentation of the frequency distribution of the wages 
 in question. Since the concept of a frequency distribu- 
 tion has a concise theoretical basis, it will be of ad- 
 vantage to turn briefly at this point to the theoretical 
 aspects of the subject. 
 
 If we take the square of a binomial, as ar -f 2ab -{- b-, 
 we have three classes of values as expressed by the 
 letters and their exponents, and these classes have 
 frequencies expressed by their coefficients, 1:2:1. If 
 instead of the second power we take the fourth power 
 of the binomial, we have five classes of values, having 
 frequencies respectively of 1 :4 :6 :4 :1. These frequen- 
 cies graphed as vertical blocks will form a figure such 
 as is outlined by the dotted line in Fig. 1. If instead 
 of the fourth power of the binomial we should take the 
 thousandth or millionth power, the steps in this blocked 
 frequency polygon would practically disappear, and 
 the figure would approach a smoothed bell-shaped 
 curve as indicated in the same figure. This theoretical
 
 10 INTRODUCTION TO ECONOMIC STATISTICS 
 
 distribution of classes of values, or something similar 
 to it, may be discovered to exist very generally in 
 natural and social phenomena, and is also the expres- 
 sion of what are known as the laws of chance. The 
 length of leaves on a given tree, the height of a group 
 of persons, the per cent net earnings of corporations, 
 
 IOOa 
 
 Figure 1. NormEil frequency curve (solid line), and an approxima- 
 tion to it (dotted line) based on the fourth power of a bionomial. 
 Horizontal scale in units of standard deviation from average (0). 
 Quartilo deviation (Q.D.), and average deviation (A.D.) 
 
 or the deviations from normal of a price index through 
 a series of years, will show when properly classified 
 and graphed an approximation to the bell-shaped fre- 
 quency curve. In order to discover whether this curve 
 is inherent in a given set of data, it is necessary first 
 that the data contain a considerable number of items, 
 and second that the classification be suitably adjusted 
 to the range and numbers. If, for example, the height 
 of a hundred persons were taken merely to the nearest
 
 TABULATION 11 
 
 foot, only two or three classes would appear. If, how^ 
 ever, the measurements were taken accurately to .01 
 inch, so many classes would appear that the fre- 
 quencies would be hopelessly scattered. But if we 
 made our measurements to the nearest inch, we would 
 obtain a series of frequencies somewhat like the fol- 
 lowing (the classes ranging from 60 to 73 inches in- 
 clusive) : 1:2:4:7:10:14:16:16:12:8:5:3:1:1. These fre- 
 quencies when graphed will give an approximation to 
 the bell-shaped curve. It will be necessary, therefore, 
 in tabulating our wage data to work out experimentally 
 the most suitable classification. 
 
 The Tally Sheet and Frequency Table. To facilitate 
 the classification of the wages selected for study, a tally 
 sheet is drawn up as shown in the first two columns of 
 Table II. We shall show here the details for only the 
 1891 figures, leaving the 1870 and 1880 figures to be 
 worked out by the*student. In studies where the items 
 must be entered singly, it is customary to use the famil- 
 iar ''four and cross" method of tallying (/>y/ =5), 
 but this is inappropriate when the items are already 
 partially grouped as they are here. In this case the 
 number of workers as shown in the wage roll is entered 
 in the appropriate line of the tally sheet, much as jour- 
 nal items are posted to a ledger. Each entry is sepa- 
 rated from adjacent ones by a dash. Each line is then 
 totaled, and the result entered under the five cent 
 column of ''Frequency Classes." A series of values 
 thus arranged according to magnitude is known as an 
 array. 
 
 An inspection of the five cent frequencies shows that 
 we have discovered only a very rough approximation
 
 12 INTEODUCTION TO ECONOMIC STATISTICS 
 
 TABLE II 
 WAGES IN A CONNECTICUT WOOLEN MILL, JULY, 1891 
 
 
 
 Freqdency 
 
 
 
 
 Frequency 
 
 
 Daily 
 
 Tally 
 
 Classes— Inter- 
 
 
 Daily 
 
 Tally 
 
 Classes — Inter- 
 
 
 Wage 
 
 (No. of 
 
 vals of: 
 
 2 
 
 Wage 
 
 (No. of 
 
 vals OK : 
 
 2 
 
 $ 
 
 Workers) 
 
 
 
 $ 
 
 Workers) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5(J 
 
 15<^ 
 
 25^ 
 
 5Q<i 
 
 
 
 
 5(* 
 
 15^ 
 
 2.'-,(' 
 
 50<^ 
 
 
 .40 
 
 
 
 
 
 
 
 Carried over . 
 
 349 
 
 351 
 
 349 
 
 349 
 
 
 .45 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .50 
 
 6- 
 
 6 
 
 5 
 
 
 
 5 
 
 2.50 
 
 1-1- 
 
 2 
 
 
 
 
 351 
 
 .55 
 
 
 
 
 
 
 5 
 
 2.55 
 
 
 
 
 
 
 
 .60 
 
 
 
 
 5 
 
 
 5 
 
 2.60 
 
 
 
 
 
 2 
 
 
 
 .65 
 
 
 
 
 
 
 
 5 
 
 2.65 
 
 
 
 
 
 
 
 .70 
 
 
 
 
 
 42 
 
 5 
 
 2.70 
 
 
 
 
 
 6 
 
 
 .75 
 
 1-9- 
 
 10 
 
 
 
 
 15 
 
 2.75 
 
 1-1-1-1- 
 
 4 
 
 4 
 
 
 
 355 
 
 .80 
 
 2-14- 
 
 16 
 
 29 
 
 
 
 31 
 
 2.80 
 
 
 
 
 
 
 
 .85 
 
 1-2- 
 
 3 
 
 
 37 
 
 
 34 
 
 2.85 
 
 
 
 
 4 
 
 
 
 .90 
 
 1-3-4- 
 
 8 
 
 
 
 
 42 
 
 2.90 
 
 
 
 
 
 
 
 
 .95 
 
 
 
 10 
 
 
 
 
 42 
 
 2.95 
 
 
 
 
 
 
 
 1.00 
 
 1-1- 
 
 2 
 
 
 
 
 44 
 
 3.00 
 
 3- 
 
 3 
 
 
 
 
 358 
 
 1.05 
 
 2- 
 
 2 
 
 
 
 
 46 
 
 3.05 
 
 
 
 3 
 
 
 
 
 1.10 
 
 14-1-4- 
 
 19 
 
 38 
 
 58 
 
 
 65 
 
 3.10 
 
 
 
 
 3 
 
 
 
 1.15 
 
 1-1-3-12- 
 
 17 
 
 
 
 117 
 
 82 
 
 3.16 
 
 
 
 
 
 
 
 1.20 
 
 1-17- 
 
 18 
 
 
 
 
 100 
 
 3.20 
 
 
 
 
 
 
 3 
 
 
 1.25 
 
 ♦■1-1.3-3-2-4-3- 
 
 28 
 
 56 
 
 
 
 128 
 
 3.25 
 
 
 
 
 
 
 858 
 
 1.30 
 1.85 
 
 7-1-2- 
 8-1-8-2- 
 
 10 
 19 
 
 
 59 
 
 
 13^ 
 157 
 
 3.30 
 3.35 
 
 
 
 
 
 
 
 
 
 1.40 
 
 1-1- 
 
 2 
 
 21 
 
 
 
 159 
 
 3.40 
 
 
 
 
 
 
 
 1.45 
 
 
 
 
 
 
 159 
 
 3.45 
 
 
 
 
 
 
 
 1.50 
 
 2-3-1-1-4- 
 
 11 
 
 
 
 
 170 
 
 3.50 
 
 1- 
 
 1 
 
 1 
 
 
 
 359 
 
 1.55 
 
 
 
 14 
 
 
 
 170 
 
 3.55 
 
 
 
 
 
 
 
 1.60 
 
 2-1- 
 
 3 
 
 
 106 
 
 
 173 
 
 3.60 
 
 
 
 
 1 
 
 
 
 1.85 
 
 1- 
 
 1 
 
 
 
 
 174 
 
 8.65 
 
 
 
 
 
 
 
 
 1.70 
 
 2-89- 
 
 91 
 
 169 
 
 
 180 
 
 265 
 
 3.70 
 
 
 
 
 
 1 
 
 
 1.75 
 
 1-3-2-60-1- 
 
 67 
 
 
 
 
 332 
 
 3.75 
 
 
 
 
 
 
 359 
 
 1.80 
 
 
 
 
 
 
 332 
 
 3.80 
 
 
 
 
 
 
 
 
 1.85 
 
 1-1- 
 
 2 
 
 7 
 
 74 
 
 
 334 
 
 3.85 
 
 
 
 
 
 
 
 
 1.90 
 
 1-4- 
 
 5 
 
 
 
 
 339 
 
 3.90 
 
 
 
 
 
 
 
 1.95 
 
 
 
 
 
 
 339 
 
 3.95 
 
 
 
 1 
 
 
 
 
 2.00 
 
 1-1- 
 
 2 
 
 2 
 
 
 
 
 341 
 
 4.00 
 
 1- 
 
 1 
 
 
 
 
 360 
 
 2.05 
 
 
 
 
 
 
 341 
 
 4.05 
 
 
 
 
 
 
 
 2.10 
 
 2-1- 
 
 3 
 
 
 9 
 
 
 844 
 
 4.10 
 
 
 
 
 
 1 
 
 
 
 2.16 
 
 
 
 7 
 
 
 
 344 
 
 4.15 
 
 
 
 
 
 
 
 2.20 
 
 4- 
 
 4 
 
 
 
 10 
 
 348 
 
 4.20 
 
 
 
 
 
 2 
 
 
 2.J5 
 
 1- 
 
 1 
 
 
 
 
 349 
 
 4.25 
 
 1- 
 
 1 
 
 1 
 
 
 
 361 
 
 2.30 
 
 
 
 1 
 
 
 
 349 
 
 
 
 
 
 
 
 
 2.S5 
 
 
 
 
 1 
 
 
 349 
 
 
 
 
 
 1 
 
 
 
 2.40 
 
 
 
 
 
 
 349 
 
 
 
 
 
 
 
 
 2.46 
 
 
 
 2 
 
 
 
 349 
 
 
 
 
 
 
 
 
 
 Totals 
 
 349 
 
 351 
 
 349 
 
 349 
 
 
 
 Totals 
 
 361 
 
 361 
 
 361 
 
 361 
 
 
 to the theoretical frequency curve. We therefore ex- 
 periment with larger groupings to see if we can thus 
 obtain more distinctive results. Classes at fifteen, 
 twenty-five, and fifty cent intervals are shown in the 
 designated columns. These classes are found by 
 adding the five cent frequencies falling within the 
 limits indicated by the horizontal bars. Obviously,
 
 TABULATION 
 
 13 
 
 several variations of these groupings could be made 
 by beginning at different points in the scale; but the 
 arrangement here chosen, which gives regular classes 
 back to the zero point, is the most natural one to take. 
 Comparing the different groupings, we see that the 
 twenty-five and fifty cent intervals give results as 
 smooth as we are likely to get. Since a classification 
 with larger intervals promises to be too indefinite, it 
 is useless to carry our frequency classifications 
 further. 
 
 The Derived or Special Purpose Table. In order to 
 give a summarized presentation of the fifty cent fre- 
 quencies. Table III has been drawn up. In this table 
 the classes are designated by stating the upper and 
 lower limits, as $.50 to $.95, inclusive. If the class 
 interval is so arranged that the mid-point falls at a 
 round number, the class may be designated by this 
 number. Thus the twenty-five cent frequencies could 
 
 TABLE III 
 WAGES IN A CONNECTICUT WOOLEN MILL * 
 
 
 NUMBER 
 
 AND PERCENTAGE 
 
 OF WORKERS DISTRIBUTED 
 
 
 ACCORDING TO DAILY WAGES, JULY 
 
 WAGE PER DAY 
 
 
 
 
 
 DOLLARS 
 
 18 
 
 70 
 
 1880 
 
 1891 
 
 
 NO. 
 
 % 
 
 NO. 
 
 % 
 
 NO. 
 
 % 
 
 to .45 
 
 2 
 
 1.8 
 
 2 
 
 0.9 
 
 
 
 
 
 .50 to .95 
 
 18 
 
 16.7 
 
 58 
 
 27.2 
 
 42 
 
 11.6 
 
 1.00 to 1.45 
 
 54 
 
 50.0 
 
 126 
 
 59.2 
 
 117 
 
 32.4 
 
 1.50 to 1.95 
 
 22 
 
 20.4 
 
 17 
 
 8.0 
 
 180 
 
 49.8 
 
 2.00 to 2.45 
 
 3 
 
 2.8 
 
 1 
 
 0.5 
 
 10 
 
 2.8 
 
 2.50 to 2.95 
 
 6 
 
 5.6 
 
 3 
 
 1.4 
 
 6 
 
 1.7 
 
 3.00 to 3.45 
 
 1 
 
 0.9 
 
 6 
 
 2.8 
 
 3 
 
 0.8 
 
 3.50 to 3.95 
 
 2 
 
 1.8 
 
 
 
 
 
 1 
 
 0.3 
 
 4.00 to 4.45 
 
 
 
 
 
 
 
 
 
 2 
 
 0.6 
 
 Total 
 
 108 
 
 100 
 
 213 
 
 100 
 
 361 
 
 100 
 
 • Aldrich Report, pp. 1463 ff.
 
 14 INTRODUCTION TO ECONOMIC STATISTICS 
 
 be tabulated according to the nearest quarter dollar. 
 The usual method is, however, to state the limits, as 
 here shown. 
 
 In contrast with Table I, which presented in orderly 
 form practically all the detailed data of our study, this 
 table is a derived or special purpose table. It aims to 
 present only certain features of the wage schedules, 
 and therefore purposely omits details. It is in the 
 nature of a generalization, condensing the original 
 facts into as brief a compass as is practicable. 
 
 The preparation of such a table usually calls for 
 both consideration and skill. The bracketing system 
 used in the headings, or captions, is obvious — the wider 
 blocks bracket and designate the smaller ones imme- 
 diately beneath. Which set of subdivisions are entered 
 as captions, and which are entered in the stub to the 
 left, is usually determined by the exigencies of space. 
 The arrangement of the details will depend upon the 
 nature of the table. In census tables, for example, 
 the current date is placed in the first column, to the 
 left, because of its greater importance, thus reversing 
 the chronological order. In such tables, also, totals 
 will be given the place of prominence at the top, 
 directly beneath the caption. In an elaborate table 
 percentage columns will be placed together, or in a 
 separate table, to allow of easy comparison. Correla- 
 tive items in the caption or stub should be arranged 
 in some logical order, whether by magnitude as in the 
 case of the frequency classes, chronologically as the 
 successive wage distributions, geographically as in the 
 case of a census list of states, by order of origin, or 
 merely alphabetically.
 
 TABULATION 
 
 15 
 
 In computing the percentage columns, each fre- 
 quency is divided by the total. The work will be suf- 
 ficiently accurate if computed on a slide rule or string 
 chart. The percentage totals will not necessarily come 
 to exactly one hundred per cent, because of the inac- 
 curacies involved in cutting off decimals. If it is de- 
 sired, however, they may be brought to the proper 
 
 60. 
 
 /S70 — 
 
 /S3 
 
 /S9/ = 
 
 / m 2.00 doo 
 
 Figure 2. Frequency polygons 
 
 total merely by turning one or two items that stand 
 at or near five in the first decimal dropped. Thus 
 7.55 may be written 7.6 or 7.5, according to which is 
 needed to make up the total of one hundred; or 7.56 
 might even be written 7.5 if no better can be done. A 
 sufficient degree of accuracy can always be secured by 
 extending the number of decimals retained. But, in 
 general, the special purpose table should not exhibit 
 meticulous accuracy. Decimals should be shortened
 
 16 INTRODUCTION TO ECONOMIC STATISTICS 
 
 or dropped, fractions avoided, and large numbers 
 rounded. 
 
 The Frequency Polygon. The percentage columns 
 of Table III may now be graphically presented by 
 ''frequency polygons," as shown in Figure 2. The 
 percentage columns are here used in preference to 
 the absolute numbers because they reduce the three 
 polygons to the same scale. It will be seen that in 
 drawing the frequency polygons the points represent- 
 ing the percentage frequencies are plotted directly 
 above the mid-point of each class, respectively. These 
 points are then joined, the individual years being dis- 
 tinguished by different kinds of lines. The graph 
 brings out very well the general advance in minimum, 
 maximum, and mean Avages that occurred in 1891 in 
 the mill under consideration. The data for a single 
 year may also be graphed as a ''rectangular histo- 
 gram," as illustrated in the next chapter (Fig. 3, p. 25; 
 solid line). Both the frequency polygon and the rec- 
 tangular histogram are frequency curves approxi- 
 mately expressed. 
 
 Difficult Features of Tabulation. Before leaving the 
 subject of tabulation, a few general suggestions may 
 be made. In the tabulation considered in this chapter, 
 the problem of interpreting the term "wages" has 
 been solved for us by the Aldrich Report. But if we 
 had undertaken the task of filling in the original 
 schedules by actual field work, we should have been 
 faced with the difficulty of drawing a somewhat arti- 
 rficial line distinguishing wages from salaries, and 
 perhaps from commissions and other direct or indirect 
 income. From the standpoint of economic theory, of 
 course, salaries are generically wages. But in practice
 
 TABULATION 17 
 
 payments for relatively responsible and skilled work, 
 contracted usually on the basis of a considerable period 
 of time, and carrying some degree of stability of 
 tenure, are classed as salaries. They are excluded 
 from wage schedules as being presumably determined 
 less directly by supply and demand considerations. 
 Likewise most statistical units, however precise and 
 simple they may appear at first glance, usually present 
 many difficulties when they are applied to real condi- 
 tions. Precisely what, for example, should be included 
 in a tabulation as a book, a farm, an accident, a ton- 
 mile? Almost any unit that may be chosen will be 
 found to call for careful discrimination, and an exam- 
 ination of current usage. 
 
 A further difficulty is encountered when data con- 
 cerning given units are being gathered and compared 
 over a certain period of time, or from different con- 
 temporaneous environments. It often happens that 
 the definition of the unit varies at different times or 
 in different places ; or perhaps the basis of estimating 
 the frequencies may be altered, or comparisons may 
 be invalidated by changing conditions. In our study of 
 the Connecticut mill we may evidently assume that 
 the basis of the classification has not changed materi- 
 ally through the period studied, since the occupations 
 classed as wage-earning are specified. BT;t we might 
 modify our interpretation of the change in wage levels 
 upon observing that processes of work had altered, 
 that the work-day was shortening, that child labor 
 legislation was affecting the personnel, that the per- 
 centage of female workers shifted from 28% to 19%, 
 and then to 32%, or that the cost of living had fallen. 
 Thus it is always necessary in comparative studies to
 
 18 INTRODUCTION TO ECONOMIC STATISTICS 
 
 consider carefully both the environmental conditions 
 and the statistical units employed. 
 
 The statistician who is at all ingenious will discover 
 many short cuts to lighten the work of tabulation. A 
 method which is often useful is that of entering the 
 original data on 3 X 5 or 4 X 6 cards. Suppose, for 
 example, that we wished to classify the students of a 
 given college according to their entrance grades, the 
 class of school from which entering, fraternity mem- 
 bership, and scholastic standing during their college 
 course. A numbered card could be prepared for each 
 student, and the desired data entered. The cards could 
 then be sorted as desired, the sub-totals could be de- 
 termined, or the data listed. If, however, such work 
 is to be done on an extended scale, a tabulating ma- 
 chine will be required. Such a machine automatically 
 sorts and counts special cards on which the required 
 data have been recorded by a keyed punch. The larger 
 business houses are using machine tabulators increas- 
 ingly; and of course extensive compilations like a cen- 
 sus are prepared principally by machines. 
 
 Library Work. The subject of tabulation has been 
 extensively treated by writers on statistics. Day's 
 article, cited below, will prove to be very valuable to 
 the student. It may be found reprinted in Secrist's 
 ** Readings," together with another excellent article 
 on the same subject. Chapter IV of the same book is 
 especially pertinent to the subject of wage tabulations. 
 Chapter VII of Rugg's text-book presents a concise 
 description of the frequency curve. Machine tabula- 
 tion is described in the circulars of the Tabulating 
 Machine Company, of New York.
 
 TABULATION 19 
 
 REFERENCES 
 
 Bailey and Cummings, Statistics, Chapters I-V. 
 
 Bowley, Arthur L., Elements of Statistics, Chapter IV. 
 
 Day, E. E., "Standardization of the Construction of Statis- 
 tical Tables," Quarterly Publications pf the American 
 Statistical Association, March, 1920, pp. 59-66. 
 
 Koren, John, A History of Statistics. 
 
 Rugg, H. 0., Statistical Methods Applied to Education, Chap- 
 ter VII. 
 
 Secrist, Horace, An Introduction to Statistical Methods, 
 Chapters I-V. 
 
 Secrist, Horace, Readings and Problems in Statistical Meth- 
 ods, Chapters I-V. 
 
 Yule, G. U., An Introduction to the Theory of Statistics, 
 Chapter VI. 
 
 EXERCISES 1 
 
 1. Draw up tally sheets and frequency classifications for the 
 years 1870 and 1880. 
 
 2. Tabulate the twenty-five cent classes, showing both abso- 
 lute and percentage figures, for the years 1870, 1880, and 
 1891. Draw frequency polygons from the percentage 
 data. 
 
 3. Similarly tabulate and graph the fifteen cent classes. 
 
 4. Graph the five cent classes for the years 1870, 1880, and 
 1891. 
 
 5. Draw rectangular histograms of the fifty cent frequencies 
 for 1870 and 1880. Compare the relative advantages of 
 the frequency polygon and the rectangular histogram. 
 
 6. Classify and tabulate separately the female workers for 
 the years 1870, 1880, and 1891. (Starred items— see foot- 
 note. Table I.) 
 
 7. Obtaining data from the Aldrich Report, study the wages 
 paid in Establishment No. 86 in July of 1875 and 1885. 
 Classify, tabulate, and graph as for the other years 
 studied. 
 
 8. Obtain the average scholarship grades for a selected group 
 of students (100 or more), classify these grades, tabulate, 
 and draw a frequency polygon. 
 
 9. Toss two coins twenty-five times, keeping a record of the 
 number of heads thrown at each toss. Classify and tabu- 
 late the results, and draw a frequency polygon. Toss 
 four coins fifty times, making similar records. What prin- 
 ciple is illustrated? 
 
 * Before beginning a notebook the student abould read Appendix I,
 
 CHAPTER II 
 
 TYPES AND MEASURES OF DISPERSION 
 
 After the frequency distribution of a given array 
 has been presented in suitable form, there remains the 
 task of finding simple numerical measures by which 
 it may be summed up for purposes of ready descrip- 
 tion and comparison. The two features thus to be 
 expressed are the typical wage and the degree of dis- 
 persion or ** spread" about the type. As a wage type, 
 and a base from which dispersion may be measured, 
 the common average, or arithmetic mean, will immedi- 
 ately suggest itself. In the case of the wage rolls given 
 in the preceding chapter, the average may be most con- 
 veniently found by multiplying the wages, as tabulated 
 at five cent intervals, by their respective frequencies. 
 The sum of these products, divided by the number of 
 workers, is the average. It is the weighted average of 
 the class values at five cent intervals, since these values 
 are emphasized in accordance with their frequencies. 
 We shall find that weighted averages are sometimes 
 taken in which the weights are derived estimates of 
 the importance which should be attached to the values, 
 respectively; but in this case the weighting amounts 
 simply to a summing up of the original wages. The 
 
 formula for the weighted average is — ^rj— (summa- 
 tion of the frequencies times the class values, divided 
 
 20
 
 TYPES AND MEASUEES OF DISPERSION 21 
 
 by the number of items). The accompanying table 
 (Table IV), derived from Table II, shows the process 
 of finding the average wage for the year 1891. 
 
 TABLE IV 
 WAGE EOLL AND AVERAGE WAGE 
 
 CONNECTICUT MILL, JULY, 1891 
 
 Single Wage 
 
 No. of Workers 
 
 Total Wage 
 
 $ .50 
 
 5 
 
 $2.50 
 
 .75 
 
 10 
 
 7.50 
 
 .80 
 
 16 
 
 12.80 
 
 .85 
 
 3 
 
 2.55 
 
 .90 
 
 8 
 
 7.20 
 
 1.00 
 
 2 
 
 2.00 
 
 1.05 
 
 2 
 
 2.10 
 
 1.10 
 
 19 
 
 20.90 
 
 1.15 
 
 17 
 
 19.55 
 
 1.20 
 
 18 
 
 21.60 
 
 1.25 
 
 28 
 
 35.00 
 
 1.30 
 
 10 
 
 13.00 
 
 1.35 
 
 19 
 
 25.65 
 
 1.40 
 
 2 
 
 2.80 
 
 1.50 
 
 11 
 
 16.50 
 
 1.60 
 
 3 
 
 4.80 
 
 1.65 
 
 1 
 
 1.65 
 
 1.70 
 
 91 
 
 154.70 
 
 1.75 
 
 67 
 
 117.25 
 
 1.85 
 
 2 
 
 3.70 
 
 1.90 
 
 5 
 
 9.50 
 
 2.00 
 
 2 
 
 4.00 
 
 2.10 
 
 3 
 
 6.30 
 
 2.20 
 
 4 
 
 8.80 
 
 2.25 
 
 1 
 
 2.25 
 
 2.50 
 
 2 
 
 5.00 
 
 2.75 
 
 4 
 
 11.00 
 
 3.00 
 
 3 
 
 9.00 
 
 3.50 
 
 1 
 
 3.50 
 
 4.00 
 
 1 
 
 4.00 
 
 4.25 
 
 1 
 
 4.25 
 
 Total 
 
 361 
 
 $541.35 
 
 
 1.49958 
 
 Average 
 
 
 $1.50 
 
 The Mode. In addition to the arithmetic mean, there 
 are other types which the statistician uses in summar-
 
 22 INTEODUCTION TO ECONOMIC STATISTICS 
 
 izing an array and measuring dispersion. One of these 
 is the mode. The mode is applicable, however, only 
 to frequency distributions which conform in their gen- 
 eral outlines to the theoretical frequency curve. It is 
 the value which lies at the point of greatest frequency, 
 and in the normal curve is therefore identical with 
 the average. In the graph of a frequency distribution 
 it is easily recognizable as the value indicated on the 
 horizontal scale at the point directly under the highest 
 point of the curve. (See Figure 3, page 25.) 
 
 The mode is particularly useful in connection with 
 those frequency curves which, though conforming in 
 general outlines to the theoretical, are extended more 
 on the one side than the other. Such curves are said to 
 be skewed. The wage data studied in the preceding 
 chapter gives curves which are somewhat skewed to 
 the right, so that a small secondary mode sometimes 
 appears. But in their original five cent frequencies 
 they do not give a smooth enough curve to allow of 
 a very definite mode. When, however, such curves are 
 strongly skewed, and are yet passably smooth, the 
 mode is preferable to the average as a type of the 
 array.^ Suppose, for example, that in a wage array a 
 few very large salaries are included. In such a case 
 the average may fall between the wages and the 
 salaries, at a point where the frequencies are small. 
 The mode, on the other hand, states the wage or salary 
 most frequently paid. It is not affected by the skewed 
 extreme of the curve; that is, by the relatively small 
 number of large salaries. 
 
 ' Frequency curves that are strongly skewed to the right will some- 
 times appear normal if transferred to semi-logarithmic paper, the
 
 TYPES AND MEASURES OF DISPERSION 23 
 
 Determining the Mode. In frequency distributions 
 that are irregular the mode may often be approxi- 
 mately determined by a study of the larger frequency 
 groupings. Let us take as an illustration the twenty- 
 five cent frequency classes derived from the 1891 wage 
 data. These are shown in the third column of Table V, 
 the first and second columns being the data from which 
 they are derived, as given in Table II. The latter part 
 of the series is omitted, however, since it cannot affect 
 the position of the mode. In the fourth and succeed- 
 ing columns, variations of the twenty-five cent fre- 
 quencies are formed by beginning the classes at differ- 
 ent points in the scale. In each case the mode should 
 lie somewhere within the class having the largest 
 frequency ; that is, it should lie in the following classes : 
 $1.50 — $1.70 
 
 1.55 — 1.75 
 
 1.60 — 1.80 
 
 1.65 — 1.85 
 
 1.70 — 1.90 
 Since the only point common to these five classes is 
 $1.70, this sum may be regarded as the mode. It will 
 be seen, however, that this is the same value which 
 would be taken as the mode on the basis of the five cent 
 classes. The same result would in this case also be 
 obtained by using the fifty cent classes. Ordinarily, 
 this method is applied only to the largest classes which 
 it is practicable to use in the frequency classification. 
 It may give a quite different result from that which 
 is obtained from the smallest classes. 
 
 logarithmic scale being used as the base line. When this is the case, 
 the distribution is normal on the basis of the geometric rather than 
 the arithmetic mean (see page 94).
 
 24 INTEODUCTION TO ECONOMIC STATISTICS 
 
 TABLE V 
 
 THE MODE 
 
 WAGE EOLL, CONNECTICUT MILL, JULY, 1891 
 
 V 
 
 F 
 
 FREQUENCIES IN 25c CLASSES 
 
 
 
 
 
 
 VARIOUS GROUPINGS 
 
 
 SUMMARY 
 
 $ .40 
 
 
 
 
 5 
 
 
 
 
 5 
 
 .45 
 
 
 
 
 
 5 
 
 
 
 5 
 
 .50 
 
 5 
 
 
 
 
 5 
 
 
 5 
 
 .55 
 
 
 
 
 
 
 
 5 
 
 5 
 
 .60 
 
 
 
 5 
 
 
 
 
 
 5 
 
 .65 
 
 
 
 
 10 
 
 
 
 
 10 
 
 .70 
 
 
 
 
 
 26 
 
 
 
 26 
 
 .75 
 
 10 
 
 
 
 
 29 
 
 
 29 
 
 .80 
 
 16 
 
 
 
 
 
 37 
 
 37 
 
 .85 
 
 3 
 
 37 
 
 
 
 
 
 37 
 
 .90 
 
 8 
 
 
 29 
 
 
 
 
 29 
 
 .95 
 
 
 
 
 
 15 
 
 
 
 15 
 
 1.00 
 
 2 
 
 
 
 
 31 
 
 
 31 
 
 1.05 
 
 2 
 
 
 
 
 
 40 
 
 40 
 
 1.10 
 
 19 
 
 58 
 
 
 
 
 
 58 
 
 1.15 
 
 17 
 
 
 84 
 
 
 
 
 84 
 
 1.20 
 
 18 
 
 
 
 92 
 
 
 
 92 
 
 1.25 
 
 28 
 
 
 
 
 92 
 
 
 92 
 
 1.30 
 
 10 
 
 
 
 
 
 77 
 
 77 
 
 1.35 
 
 19 
 
 59 
 
 
 
 
 
 59 
 
 1.40 
 
 2 
 
 
 42 
 
 
 
 
 42 
 
 1.45 
 
 
 
 
 
 32 
 
 
 
 32 
 
 1.50 
 
 11 
 
 
 
 
 16 
 
 
 16 
 
 1.55 
 
 
 
 
 
 
 15 
 
 15 
 
 1.60 
 
 3 
 
 106M 
 
 
 
 
 
 106 
 
 1.65 
 
 1 
 
 
 162M 
 
 
 
 
 162 
 
 1.70 
 
 91 
 
 
 
 162M 
 
 
 
 162M 
 
 1.75 
 
 67 
 
 
 
 
 161M 
 
 
 161 
 
 1.80 
 
 
 
 
 
 
 
 165M 
 
 165 
 
 1.85 
 
 2 
 
 74 
 
 ' 
 
 
 
 74 
 
 1.90 
 
 5 
 
 
 9 
 
 
 
 
 9 
 
 1.95 
 
 
 
 
 
 9 
 
 
 
 9 
 
 2.00 
 
 2 
 
 
 
 
 10 
 
 
 10 
 
 2.05 
 
 
 
 
 
 
 
 5 
 
 5 
 
 2.10 
 
 3 
 
 9 
 
 
 
 
 
 9 
 
 2.15 
 
 
 
 
 8 
 
 
 
 
 8 
 
 2.20 
 
 4 
 
 
 
 8 
 
 
 
 8 
 
 2.25 
 
 1 
 
 
 
 
 5 
 
 
 5 
 
 2.30 
 
 
 
 
 
 
 
 5 
 
 5 
 
 2.35 
 
 
 
 1 
 
 
 
 
 
 1 
 
 2.40 
 
 
 
 
 
 
 
 
 
 2.45 
 
 
 
 
 
 
 
 
 
 etc. 
 
 
 
 
 
 

 
 TYPES AND MEASURES OF DISPERSION 25 
 
 As applied to the given wages, however, the fore- 
 going method of locating the mode is objectionable. 
 The $1.70 frequency is relatively so large that in each 
 classification it determines the mode without allowing 
 due weight to the large frequencies a little further 
 down the scale. The latter might be considered a sec- 
 ondary mode, but we shall here assume that the use of 
 more extensive data would result in a single mode. 
 We may therefore illustrate the use of a method which 
 
 200A 
 
 I 
 
 \ 
 
 / 
 / 
 / 
 
 J— 
 
 
 
 tVa^e: 
 
 JO 
 
 1.00 
 
 —I — 
 I.SO 
 
 H»l I ■% M ,1 
 
 — I — 
 Z.00 
 
 2.J0 
 
 jno J-^O 4-00 4.S0 
 
 Figure 3. Eectangiilar histogram of wages in a Connecticut mill, 
 1891, fifty cent classes (solid line), and smoothed frequencies (broken 
 line). 
 
 is applicable to irregular frequencies, or to data pre- 
 sented in only a few large classes. It will, of course, 
 be understood that with such limited data no very de- 
 pendable result can be obtained. 
 
 In using this method, we shall not only approxi- 
 mately determine the mode, but draw the smoothed 
 curve as well. The method is illustrated in Figure 3. 
 The frequencies are first represented by a rectan- 
 gular histogram. In drawing the histogram the ver- 
 tical lines should theoretically be drawn at a point
 
 26 INTRODUCTION TO ECONOMIC STATISTICS 
 
 midway between the two adjacent class limits; for 
 example, the line separating the first and second 
 classes falls at .475.^ It is evident that the mode lies 
 in the class $1.50 to $1.95, but it is desirable to locate 
 it somewhat precisely within the class. This may be 
 done by dividing the class interval into two parts pro- 
 portional inversely to the adjacent frequencies.^ In so 
 doing the class limits are considered $1,475 and $1,975, 
 as drawn. The division may readily be constructed 
 geometrically, or it may be computed as follows ; 
 10 
 
 $1,475 H X $.50 = $1.51 
 
 117 + 10 
 
 The formula for this operation is, 
 
 •pi 
 
 M - Li + ^ X C 
 
 Fm+Fn 
 
 in which 
 
 M = the mode. 
 
 Lj = the lower limit of the modal class. 
 
 Fm and Fn = frequencies adjacent to the one contain- 
 ing the mode, in the order named. 
 C = the class interval. 
 Smoothing the Frequencies. After the mode has 
 been determined, the histogram may be smoothed into 
 a frequency curve. This curve is drawn to conform 
 as closely as possible to the theoretical bell-shaped 
 curve; and yet to maintain, frequency by frequency, 
 the same area as the original rectangular figure. Thus 
 in the drawing A^ = Aj + ^3? ^i = ^2, and Ci = Cj. 
 The curve culminates approximately at the mode as 
 
 * The clasa limits should be so placed that the items in the modal class 
 average close to the mid-point of the class. 
 
 ' It is sometimes iireferable to take the average of the items in the 
 modal and adjacent classes, or to include more classes.
 
 TYPES AND MEASURES OF DISPERSION 27 
 
 previously determined, but the height is merely esti- 
 mated with reference to the required area. As thus 
 drawn, the curve presents an estimate of the probable 
 distribution of the economic values expressed by wage 
 rolls of the type studied. The irregularity of the data, 
 however, makes it far from typical. 
 
 The Median. Another type often used to represent 
 a given array is the median. The median is the value of 
 the middle item in an array. The number of this item 
 
 N + l^ 
 is found by the formula, ; and its value may be 
 
 2 
 
 determined by reference to the summation column of 
 the frequency table. For example, in the 1891 wage 
 data the median item is number 181, and its value as 
 determined by means of the summation column is 
 $1.70. That is, the 181st item falls within the $1.70 
 class. In case the median item should prove to be 
 fractional, and should fall between two frequencies, the 
 median would not be precisely determined. Suppose, 
 for example, that the median item had been number 
 174^. In such a case the median would lie between 
 the limits $1.65 and $1.70. 
 
 Comparison of the Three Types. Theoretically, in a 
 frequency curve having very small class intervals the 
 median value is indicated at the foot of a perpendic- 
 ular line which bisects the area of the curve. The aver- 
 age, on the other hand, would lie at the foot of a similar 
 perpendicular which would balance as an axis the 
 weight of the two sides, supposing the area of the curve 
 
 * The unit is added to counterbalance the space from to 1. Or, the 
 formula may be considered as an expression of the average of the 
 extreme ordinals of the array.
 
 28 INTRODUCTION TO ECONOMIC STATISTICS 
 
 to have been cut out of a material of uniform weight. 
 When skewness is regular, the mode, median, and 
 average are located on the value scale in the order 
 named, the intervals separating them being in the ratio 
 of about two to one. In the normal curve the three 
 types are identical.^ 
 
 The use of the average, median, and mode as types 
 
 SALARIES PAID IN REPRESENTATIVE UNIVERSITIES AND 
 
 COLLEGES IN THE UNITED STATES IN 1919-20. 
 
 Public institutioTis 
 
 TITLE OF POSITION 
 
 President or chancellor. 
 
 Dean or director , 
 
 Professor 
 
 Associate professor. . . . , 
 Assistant professor. . . . 
 
 Instructor 
 
 Assistant , 
 
 NUMBER OP 
 PERSONS 
 
 77 
 
 367 
 
 2,460 
 
 822 
 
 1,705 
 
 2,138 
 
 855 
 
 MINIMUM 
 SALARY 
 
 $2,500 
 1,200 
 
 300 
 300 
 500 
 300 
 75 
 
 MAXIMUM 
 SALARY 
 
 $12,500 
 
 10,000 
 10,000 
 4,000 
 4,000 
 3,100 
 2,500 
 
 President or chancellor. 
 
 Dean or director , 
 
 Professor 
 
 Associate professor. . . . 
 Assistant professor. . . . 
 
 Instructor 
 
 Assistant , 
 
 AVERAGE 
 SALARY 
 
 $6,647 
 3,819 
 3,126 
 2,514 
 2,053 
 1,552 
 801 
 
 MEDIAN 
 SALARY 
 
 $6,000 
 3,500 
 
 3,000 
 2,500 
 2,000 
 1,500 
 750 
 
 MOST 
 
 FREQUENT 
 
 SALARY 
 
 $6,000 
 
 3,000 
 3,000 
 3,000 
 1,800 
 1,500 
 1,200 
 
 * Two other forms of the average are sometimes used in statistical 
 work. One is the geometric mean. This may be found by averag- 
 ing the logarithms of the numbers instead of the numbers themselves. 
 It ia the Jith root of the product of the numbers. Some statisticians 
 advocate the use of the geometric mean in finding the average periodic 
 change in prices. Considered merely as an average of prices apart from 
 the use of weights, the geometric mean is logically correct because it 
 measures ratios of divergence rather than absolute amounts. The other 
 type of average is the harmonic mean, which has occasionally been 
 applied to the same purpose. For two numbers, a and b, it is computed 
 
 2 ab 
 by the formula — —r-- In arithmetic this is the formula which is used to 
 a -f b 
 
 find an average rate of travel when two rates for two equal distances are 
 
 given. In general, it may be described as the reciprocal of the average 
 
 of the reciprocals of the given numbers.
 
 TYPES AND MEASURES OF DISPERSION 29 
 
 may be illustrated by the foregoing table which was 
 issued by the United States Bureau of Education and 
 reprinted in the Monthly Labor Review of January, 
 1921. In this table the term ''most frequent salary" 
 signifies the mode. 
 
 Quartile Deviation. The types we have now consid- 
 ered are used as the basis for measuring dispersion, 
 though the average is more commonly employed than 
 the other two. The simplest measure of dispersion is 
 related to the median, and is called the quartile devia- 
 tion. This is found by computing the value of the first 
 and third quartiles of an array. The quartiles are 
 analogous to and include the median, being located at 
 
 the quarter divisions of the array. The location of the 
 
 I -J 
 
 first quartile is found by the formula — j — -, and of 
 
 the third quartile by the formula j — . The values 
 
 of these items are determined by reference to the fre- 
 quency table in the same manner as the median value 
 was determined. The second quartile is, of course, 
 identical with the median. The quartile range is found 
 by subtracting the value of the first quartile from the 
 value of the third, and the quartile deviation is half 
 of this difference. It may be seen by reference to 
 Figure 1 that the quartile deviation as thus found is 
 simply the average distance between the median and 
 the adjacent quartiles, as measured on the base line. 
 In the 1891 wage roll the first quartile is item No. 90yo, 
 and its value is $1.20. The third quartile is No. 271i/2» 
 and its value is $1.75. The quartile range is therefore 
 $.55, and the quartile deviation is $.28. This means
 
 30 INTRODUCTION TO ECONOMIC STATISTICS 
 
 that half the workers receive wages that fall within a 
 range averaging twenty-eight cents above and below 
 the median ; that is, between $1.20 and $1.75.^ 
 
 For purposes of comparison the quartile deviation 
 should usually be reduced to a percentage basis. This 
 is done by dividing it by a value regarded as typical of 
 the array. Since the quartile deviation is related to 
 the median, it would appear logical to take this type 
 as a base. But it is customary to take instead a point 
 lying midway between the first and third quartile 
 values ; that is, the average of the two. In a perfectly 
 regular curve this value would naturally be identical 
 with the median. The reason for taking this base is 
 obviously that it is the point from which the quartile 
 deviation is assumed to be directly measured. The 
 
 Q3+Q1 
 formula for it is (third quartile plus first 
 
 2 
 
 quartile, divided by two). The formula for the quartile 
 
 Q3-Q1 
 deviation is . The latter divided by the 
 
 2 
 
 former is , which is therefore the formula for 
 
 Q3+Q1 
 the coefficient of quartile deviation. 
 
 Interpolation. Before leaving the quartiles, a method 
 of locating them by interpolation between the class 
 intervals should be described. We will illustrate the 
 method by applying it to the 1891 wage data, though 
 
 * The quartile deviation is also called the "probable error" of a 
 frequency distribution. The term is derived from the ' ' Theory of 
 Errors, ' ' and connotes the central range of the distribution within 
 which an item added to tho series will have an even chance of fall- 
 ing.
 
 TYPES AND MEASUEES OF DISPERSION 31 
 
 in fact the five cent classes give quartile values precise 
 enough for most purposes. The type of problem to 
 which the method is best adapted is one in which an 
 array as given is classified only in a few large groups. 
 But supposing that it is desirable to know the quartile 
 values very precisely in the 1891 wage data, we may 
 find them as described below. 
 
 In assuming that we may interpolate at any point 
 between the items of the original frequency classifica- 
 tion, we are evidently regarding the series as con- 
 tinuous rather than discrete. In the case of the wage 
 roll we shall be dealing, then, with the theoretical 
 economic values underlying the wages as paid. The 
 actual frequencies are therefore to be considered as 
 indicating proportionate numbers, which may be in- 
 creased indefinitely as in the case of any multiple ratio. 
 The original discrete five cent classes are now to be 
 considered as having continuous class intervals; for 
 example, a fifty cent wage is taken to indicate an 
 economic value within the limits $.475 and $.525. ' The 
 frequencies are assumed to be equally spaced between 
 these limits. 
 
 When about to interpolate, we locate the first 
 
 N N + 1 
 
 quartile by the formula — , instead of as in the 
 
 4 4 
 
 previous case. The second and third quartiles are 
 two and three times this number, respectively. The 
 reason for omitting the unit in the formula, and for 
 ignoring it also in an analogous formula for sub-divid- 
 ing the class, is that our hypothesis of a continuous 
 scale renders the unit of negligible value. It is as if
 
 32 INTRODUCTION TO ECONOMIC STATISTICS 
 
 the items were regarded as being groups of thousands 
 or millions; that is, as if they were indefinitely sub- 
 divisible. Having located the quartile items, we find 
 the class in which they fall by inspection of the fre- 
 quency table, as before. We next find the position of 
 the quartile within the class; that is, the fraction of 
 the interval that it is advanced beyond the lower limit. 
 The corresponding value is then determined. The 
 process is the same as that used in interpolating in 
 logarithmic or other tables. 
 
 In the 1891 wage data, the first quartile is located at 
 item 901/4. This item falls in the class having theo- 
 retically a lower limit of $1,175 and an upper limit of 
 $1,225. The preceding class ends with the 82nd item, 
 and the quartile is therefore advanced 8^/4 items in its 
 own class of 18 items. This advance is 8^1 -^ 18, or .46 
 of the class interval. This fraction of the class interval 
 of $.05, is $.023, which, added to the lower limit of the 
 class, gives the quartile value of $1,198. Read to the 
 nearest cent, this value happens to be the same as that 
 obtained without interpolation. 
 
 The process may be summed up as follows : 
 I 
 
 Q = L, + — . C 
 F 
 in which, 
 
 Q = Quartile value 
 
 Lj = Lower limit of class containing quartile 
 
 I = Quartile item minus last item of preceding 
 
 class ^ 
 F = Frequency of class containing quartile 
 C = Interval of same class 
 
 *"Item" here refers to the number, not the value.
 
 TYPES AND MEASURES OF DISPERSION 33 
 
 Quartile Dispersion. A statement of the quartiles 
 and the highest and lowest wage paid serves to give a 
 fairly good idea of a frequency distribution even with- 
 out any further computation of precise measures of 
 dispersion. In Table VI such a statement is presented 
 
 EANGE AND TREND OF WAGES, 1870-1891, IN A CONNECTICUT 
 WOOLEN MILL 
 
 4.oo~- 
 
 J.OO. 
 Z.00. 
 
 /.so. 
 
 1.00. 
 
 .so. 
 
 ;5o. 
 
 /i=/7j^/?esT; Q,-G^=^uarr/fe5, L = /on^esT wage 
 
 H 
 
 1370 
 
 mo 
 
 mo 
 
 Figure 4. Semi-logarithmic, or ratio paper 
 
 for the wage data of 1870, 1880, and 1891, together 
 with the quartile deviations and their coefficients. The 
 table includes the interpolated values, although, as has 
 been intimated, their computation is hardly worth 
 while here except as an illustration of the method. In 
 Figure 4 the discrete quartiles and limits are shown
 
 34 INTRODUCTION TO ECONOMIC STATISTICS 
 
 graphed upon semi-logarithmic paper. The vertical 
 scale of this paper is similar to the scale of a slide- 
 rule, hence the ratio of periodic change may be com- 
 pared by means of the slant of the lines connecting 
 the values. To be complete, however, such grauhic 
 
 TABLE VI 
 
 RANGE OF DAILY WAGES IN A CONNECTICUT MILL— SCALE 
 TAKEN BOTH AS DISCRETE AND CONTINUOUS 
 
 WAGE 
 
 1870 
 
 1880 
 
 1891 
 
 ARRAY 
 
 DIS- 
 CRETE 
 
 CONTINU- 
 OUS 
 
 DIS- 
 CRETE 
 
 CONTINU- 
 OUS 
 
 DIS- 
 CRETE 
 
 CONTINU- 
 OUS 
 
 Lower Limit 
 
 $ .40 
 1.10 
 1.30 
 
 1.50 
 3.75 
 
 $ .38 
 
 1.09 
 1.30 
 1.51 
 
 3.78 
 
 $ .40 
 
 .95 
 1.20 
 1.25 
 3.25 
 
 $ .38 
 
 .93 
 
 1.19 
 
 1.24 
 
 3.28 
 
 $ .50 
 1.20 
 1.70 
 1.75 
 4.25 
 
 $ .48 
 
 1st Quartile 
 
 2nd Quartile 
 
 1.20 
 L68 
 
 3rd Quartile 
 
 1.73 
 
 Higher Limit 
 
 4.28 
 
 Q. Deviation 
 
 .20 
 15% 
 
 .21 
 16% 
 
 .15 
 
 14% 
 
 .16 
 
 14% 
 
 .28 
 19% 
 
 .27 
 
 — coefficient 
 
 18% 
 
 representation should show at least annual data. It 
 might also very well show decile points; that is, the 
 wages occurring at the tenths instead of the quarters 
 of each array. 
 
 The Ogive. A convenient method of presenting an 
 array and at the same time of graphically determining 
 the quartiles, is shown in Figure 5. The construction 
 is based upon the assumption of a continuous series of 
 values, and parallels the procedure of finding the quar- 
 tile values by interpolation. The frequencies are 
 plotted from the summation column of the original 
 five cent classes. For convenience of interpretation, a 
 dot marks the entry as it would be made on the graph 
 if the series were taken as discrete. A slanting Une is 
 drawn across each class interval, beginning with the 
 summation total of the preceding class, and ending
 
 TYPES AND MEASURES OF DISPERSION 35 
 
 with the summation total of the given class. The given 
 frequency is thus represented as distributed evenly- 
 through the class. The resulting figure is known as 
 an ogive. To find the quartiles, the vertical scale rep- 
 resenting the whole array is divided into four equal 
 parts, and horizontal lines are drawn from the quartile 
 division points until the ogive is intersected. From 
 
 
 
 
 
 
 
 JSO. 
 
 0^ .- 
 
 
 r 
 
 /%<fj /// a CMnecf/ci/fMi/l,/S9/ 
 
 HO. 
 
 
 
 
 %zoo. 
 
 ^^ _^ 
 
 
 
 i 
 
 
 t 
 
 
 
 Y 
 
 
 
 Q, 
 
 
 1 \ 
 
 
 
 ^ 
 
 
 j 
 
 
 50. 
 
 
 J 
 
 
 
 
 
 ^ 
 
 r . 
 
 <v' 
 
 jf? 
 
 
 Hiye: 
 
 ^ .JO 1.00 1.50 ZOO 2.50 5.00 5.50 4.00 
 
 Figure 5. Cumulative curve, or ogive 
 
 the points of intersection perpendiculars are drawn 
 to the base line. The foot of each perpendicular marks 
 upon the horizontal scale one of the quartile values. 
 The deciles may be found by dividing the horizontal 
 scale into tenths, and proceeding as before. This 
 graphic process, worked out on large sheets of cross- 
 section paper, is usually the most convenient method 
 of finding the quartiles or deciles.^ 
 
 * A more complex form of the ogive has recently been introduced 
 for testing the regularity of a frequency distribution. This ogive is
 
 36 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Average and Standard Deviation. We shall now 
 consider the more commonly used mathematical 
 measures of dispersion, — the average deviation and the 
 standard deviation. The former is coming to be fairly- 
 well known as applied to economic data. The latter is, 
 however, generally favored by the mathematician, but 
 its chief statistical use at present lies in connection 
 with the measurement of correlation, a subject which 
 will be taken up later. 
 
 The principles involved in average and standard 
 deviation may best be illustrated by taking a very 
 simple example. Suppose that four workers are em- 
 ployed at daily wages of $2.00, $6.00, $7.00 and $9.00, 
 respectively. The average wage is $6.00. The first 
 wage differs from the average by $4.00, the second is 
 at the average, the third differs by $1.00, and the 
 fourth differs by $3.00. The sum of these differences 
 is $8.00, or an average of $2.00 for each wage. The 
 average deviation (A. D.) is therefore $2.00, which may 
 be taken as a measure of the ''spread" of the wages.^ 
 The standard deviation (u) is computed by squaring 
 the deviations, averaging the squares, and finding the 
 square root of this result. In each case a coefficient 
 may be found by dividing by the average wage. The 
 computations are written out in the following form : 
 
 drawn upon so-called probability paper, and is constructed from the sum- 
 mated percentage frequencies. The vertical scale of the probability 
 paper is so graduated that a normal curve will form a straight line 
 diagonally across the paper. The divergence of a given distribution 
 from normal may be estimated by its departure from a straight line. 
 The paper for this graph, as well as for other statistical work, may be 
 obtained from the Codex Rook Company of New York, or from other 
 publishers of statistical material. 
 
 ^ The total spread, or range, from the highest to the lowest wage is 
 sometimes given as an inexpert measure of disiJcrsion. J3ut it is of 
 little value because the wage limits are set by single items which 
 have only a haphazard relation to the rest of the array. The average 
 and standard deviations, however, take account of all the items.
 
 TYPES AND MEASURES OF DISPERSION 37 
 
 AVERAGE DEVIATION j 
 
 STANDARD DEVIATION 
 
 V 
 
 
 D 
 
 V 
 
 D 
 
 D» 
 
 $2 
 6 
 7 
 9 
 
 
 $4 
 
 1 
 3 
 
 ) "s 
 
 $2 
 6 
 7 
 9 
 
 $-4 
 
 1 
 3 
 
 16 
 
 1 
 9 
 
 4 ) 24 
 
 4 
 
 4) 24 
 
 
 
 4 ) 26 
 
 A = 6 
 
 A. D. 
 Coef. 
 
 = 2 
 =^=33% 
 
 A = 6 
 
 Coef. 
 
 a' = 6.5 
 a =2.55 
 
 -'f -43% 
 
 A practical application of average deviation may be 
 cited from Dewing, ''Corporation Finance," Vol. III. 
 The writer states that the earnings of corporations pro- 
 ducing inexpensive necessities, directly consumed, are 
 most regular; while the earnings of corporations pro- 
 ducing expensive indirect goods are least regular. He 
 illustrates the two types of corporations by the 
 Diamond Match Company and the American Locomo- 
 tive Company, computing the average deviations of 
 the net earnings of the two companies. The coefficient 
 in the first case is 7.1%, and in the second case 50%. 
 The dispersion might have been measured in other 
 ways, as by the standard deviation, but the quartile 
 measure would not be applicable to such a small 
 number of unclassified deviations. 
 
 The quartile, average, and standard deviations do 
 not give the same results, as they measure progres- 
 sively larger portions of the frequency curve (see Fig. 
 1, p. 10). But in regular distributions, a comparison 
 will be the same whichever measure is used to make 
 the comparison. In an irregular distribution which 
 has a few extreme items, the standard deviation will 
 give an exceptionally large result, since the process of 
 squaring the deviations emphasizes these extremes. 
 
 91409
 
 38 INTEODUCTION TO ECONOMIC STATISTICS 
 
 A Short-cut Method. The work of finding the aver- 
 age or standard deviation is often rather tedious, par- 
 ticularly when the average is expressed as a decimal. 
 In finding the former, however, it is not important that 
 the average should be expressed very precisely, since 
 the slight error involved in cutting short a decimal is 
 minimized in the process of the w^ork. And in finding 
 the standard deviation, a short-cut process may be 
 used. In this process a convenient average is assumed, 
 and a correction is made later. The method of making 
 the correction may be illustrated by the simple wage 
 scale of four items previously used. The average 
 wage, from which the deviations are to be measured, 
 will be assumed to be $7. Needless to say, this assump- 
 tion is not here advantageous, though it would have 
 been if the average had been, let us say, $6.75. 
 When the deviations are measured from the assumed 
 average of seven, they give an algebraic sum of — 4, 
 which results from the fact that the ''correction" 
 appears once in each deviation. Hence the algebraic 
 sum of the deviations, divided by the number of items, 
 will give the ''correction," — the term being taken to 
 mean the sum which must be added to the assumed 
 average to make it the exact average. The correction 
 (K) when found is added to the assumed average, and 
 its square is subtracted from the average squared 
 deviation. By so doing, whatever error may have been 
 involved in assuming an average is eliminated. Other- 
 wise, the work is as before. The computation is set 
 down as indicated at the top of the next page. 
 
 Deviation Computed from Frequency Tables. The 
 illustration that has been considered thus far in the
 
 TYPES AND MEASURES OF DISPERSION 39 
 
 V 
 2 
 6 
 
 = 7 
 9 
 
 D 
 
 —5 
 
 —1 
 
 
 
 2 
 
 D2 
 
 25 
 1 
 
 4 
 
 
 4)— 4 
 
 K = — 1 
 
 Ax= 7 
 
 4)30 
 
 7.5 
 
 K2 = l 
 
 
 A= 6 
 
 a2 = 6.5 
 C7 =2.55 
 
 Hoof. - ^-^^ 
 
 43% 
 
 discussion of average and standard deviation, has been 
 simplified by the fact that the array is not classified 
 into frequency groups. When computed from a fre- 
 quency table, the average and standard deviations 
 require a somewhat more complex process, though in 
 fact the principles involved are precisely those already 
 explained. The one point to be observed is that the 
 vilass values must be multiplied by their respective 
 frequencies in order that all items may be taken into 
 account. The method is shown in Table VII, where the 
 wage data of 1891 are again taken up.^ In these com- 
 putations, the class values are taken at approximately 
 the mid-points of the class intervals. This is the usual 
 procedure, though a small error is introduced by so 
 doing. An exact computation would require that the 
 actual average value of each class, as determined by 
 dividing the total wages of the class by its frequencies, 
 should be substituted. 
 
 ^A complex graph of a frequency distribution, known as the Lorenz 
 curve, may be described in connection with the data of Table VII. 
 This graph is based upon the F and FV columns, as shown under
 
 40 INTRODUCTION TO ECONOMIC STATISTICS 
 
 TABLE VII 
 
 AVERAGE DEVIATION AND STANDARD DEVIATION 
 
 WAGE ROLL IN A CONNECTICUT MILL, JULY, 1891 
 
 AVERAGE DEVIATION 
 
 V 
 $ .75 
 1.25 
 1.75 
 2.25 
 2.75 
 3.25 
 3.75 
 4.25 
 
 F 
 
 FV 
 
 42 
 
 $31.50 
 
 117 
 
 146.25 
 
 180 
 
 315.00 
 
 10 
 
 22.50 
 
 6 
 
 16.50 
 
 3 
 
 9.75 
 
 1 
 
 3.75 
 
 2 
 
 8.50 
 
 361 
 
 ) 553.75 
 
 A = 1.53 
 
 D 
 
 
 FD 
 
 5 .78 
 
 
 $32.76 
 
 .28 
 
 
 32.76 
 
 .22 
 
 
 39.60 
 
 .72 
 
 
 7.20 
 
 1.22 
 
 
 7.32 
 
 1.72 
 
 
 5.16 
 
 2.22 
 
 
 2.22 
 
 2.72 
 
 361) 
 
 5.44 
 
 
 132.46 
 
 
 A.D. 
 
 = .367 
 .367 
 
 Coef . = 
 
 1.53 
 
 24% 
 
 Average Deviation. The two columns are reduced to percentages and 
 summated, giving the following results: 
 Upper limit 
 
 of class F (2) FV (2) 
 
 $1.00 11.6% 5.7% 
 
 1.50 44.0 32.1 
 
 2.00 93.9 89,0 
 
 2.50 96.7 93.1 
 
 3.00 98.3 96.1 
 
 3.50 99.2 97.8 
 
 4.00 99.4 98.5 
 
 4.50 100.0 100.0 
 
 The two summated columns are then plotted as coordinates, the first on 
 the horizontal &eale and the second on the vertical scale. If the wages 
 were all alike, a direct diagonal would result, while disparity of wages 
 registers in the concavity of the line. The use of the five cent classes 
 would give a more accurate representation. The curve has been often 
 used for presenting a comparison of the distribution of wealth or income 
 in difi;ercnt countries. 
 
 20 4a 60 to too
 
 TYPES AND MEASURES OF DISPERSION 41 
 
 STANDAED DEVIATION (Assumed Average = $1.75) ^ 
 
 V 
 
 F 
 
 D 
 
 FD 
 
 
 FD» 
 
 ! .75 
 
 42 
 
 $-1.00 
 
 $-42.00 
 
 
 $42.00 
 
 1.25 
 
 117 
 
 -.50 
 
 -58.50 
 
 
 29.25 
 
 1.75 
 
 180 
 
 
 
 
 
 
 
 
 2.25 
 
 10 
 
 .50 
 
 5.00 
 
 
 2.50 
 
 2.75 
 
 6 
 
 1.00 
 
 6.00 
 
 
 6.00 
 
 3.25 
 
 3 
 
 1.50 
 
 4.50 
 
 
 6.75 
 
 3.75 
 
 1 
 
 2.00 
 
 2.00 
 
 
 4.00 
 
 4.25 
 
 2 
 361 
 
 2.50 
 
 5.00 
 
 36] 
 
 12.50 
 
 
 361)-78.00 
 
 1)103.00 
 
 
 
 
 K = -.216 
 
 
 .2853 
 
 
 
 
 Ax = 1.75 
 A = 1.534 
 
 
 = .0467 
 
 
 = .2386 
 
 7= .49 
 49 
 
 Formulas. The formulas for average and standard 
 deviation are as follows: 
 
 A. D. = — r^ (the deviations here considered pos- 
 itive) 
 S.D. (a)=j/:^_K2 
 
 in which 
 
 F = Frequencies 
 
 D = Deviations (from assumed average if followed 
 
 by a correction) 
 N = Total number of items in array. 
 K = Correction for error in assumed average = 
 SFD 
 N 
 If an average of zero is assumed, the second formula 
 becomes : 
 
 s.D.^f-v!_(-vy 
 
 *A column showing D^ is often included, but in most cases it may bo 
 omitted and FD^ obtained by multiplying D x FD.
 
 42 INTRODUCTION TO ECONOMIC STATISTICS 
 
 In some cases, particularly where a calculating ma- 
 chine is used, this modification of the formula will be 
 fouiid desirable. It calls for the computation of only 
 the columns FV and FV-. Its use may be illustrated 
 by reference to Table IV, p. 21. If the first and third 
 columns are multiplied across and totaled, a result of 
 $890.50 \yi[l be obtained. This is SFV^. Dividing by 
 
 (2FV\2 
 -KJ 1 
 
 gives $.218, the square root of which is $.47. This is 
 the standard deviation, obtained somewhat more accu- 
 rately than before, since smaller class intervals are 
 taken. The modified formula will often be found 
 useful in connection with time series, where no fre- 
 quencies are involved. 
 
 Summary. In Table VIII a final summary is made 
 of the dispersion of wages in the Connecticut mill here 
 studied. It is, of course, obvious that the use of a 
 variety of measures is for purposes of illustration 
 only. In practical work of this sort only one measure 
 would be used, probably either the quartile or the 
 average deviation. Skewness would doubtless be com- 
 pared merely by an inspection of the frequency poly- 
 gons. Our more exhaustive study, however, gives a 
 very precise picture of the dispersion of wages. On 
 the whole, the "spread" lessens somewhat after 1870, 
 though the change is not great enough to render the 
 data incomparable. It will be noted that the relative 
 skewness of the curves changes but little. Since, then, 
 the dispersion of wages does not materially change, the 
 average wage may be safely taken as an index of the 
 periodic wage level in the given mill.
 
 TYPES AND MEASURES OF DISPERSION 43 
 
 TABLE VIII 
 
 DISPEESION OF WAGES, CONNECTICUT MILL, JULY, 1870, 1880, 
 
 AND 1891 
 
 MEASURE 
 
 Quartile Deviation 
 
 Average Deviation* 
 
 " " (exact).. 
 Standard Deviation* 
 
 " " (exact).. 
 Skewness* 
 
 1870 1880 1891 
 
 $ .21 
 .45 
 .41 
 .62 
 .61 
 .71 
 
 $ .16 
 .28 
 .24 
 .50 
 .46 
 .64 
 
 .27 
 .37 
 .35 
 .49 
 .47 
 .55 
 
 1870 1880 1891 
 
 16% 
 31 
 30 
 44 
 44 
 114 
 
 14% 
 22 
 20 
 40 
 38 
 128 
 
 18% 
 24 
 23 
 32 
 31 
 113 
 
 * Computed from fifty cent classes ; V = mid-point of class. 
 
 Measurement of Skewness. Tlie average deviation, 
 as we have seen, uses the first power of the deviations, 
 while the standard deviation uses the second power. 
 If in an analogous way the third power is used, a 
 measure of skewness is obtained. A mathematical 
 measure of this sort is, however, seldom required in 
 economic statistics. The relative degree of skewness 
 may be roughly determined by comparing the outlines 
 of frequency curves, or by noting the position of the 
 modes relative to the averages or medians. But if an 
 accurate measure is desired, the following formula 
 should be employed : 
 
 Skewness 
 
 f 
 
 N 
 
 The measure may be reduced to a coefficient by divid- 
 ing by the standard deviation. 
 
 Library Work. The subjects of types and disper- 
 sion are treated in great detail in several of the 
 standard text-books, such as Bowley's and Yule's. The
 
 44 INTRODUCTION TO ECONOMIC STATISTICS 
 
 most complete work on the former subject is that of 
 Zizek, cited below. For an exposition of graphic repre- 
 sentation, the student should not fail to consult Brin- 
 ton's work, and the Statistical Atlas of the United 
 States published in connection with the census. The 
 ratio chart (semi-logarithmic, or *'arith-log," paper) 
 is well treated in an article by Irving Fisher, as well 
 as in an article by J. A. Field reprinted in Secrist's 
 "Readings." Whipple's text-book gives an explana- 
 tion of the use of probabiHty paper as a means of 
 presenting and testing frequency curves. King's text- 
 book, page 156, explains the Lorenz curve. 
 
 REFERENCES 
 
 Bowley, Arthur L., Elements of Statistics, Chapters V-VII. 
 
 Brinton, W. C, Graphic Methods for Presenting Facts. 
 
 Fisher, Irving, "The Ratio Chart," Quarterly Publications 
 of the American Statistical Association, June, 1917, pp. 
 577-601. 
 
 King, W. I., Elements of Statistical Method, Chapters XII- 
 XIV. 
 
 Marshall, Wm. C, Graphical Methods, Chapters I-III. 
 
 Secrist, Horace, Readings and P?-ohlems in Statistical Meth- 
 ods, pp. 282-305. 
 
 Whipple, G. C, Vital Statistics, Chapter XII. 
 
 Yule, G. U., An Introduction to the Theory of Statistics, 
 Chapters VII and VIII. 
 
 Zizek, Franz, Statistical Averages. 
 
 EXERCISES 
 
 1. Using the five cent frequencies and classes, find the aver- 
 age wage for 1870 and 1880. 
 
 2. Using 25c class intervals, determine the mode for 1870 
 and 1880, following the process illustrated on page 24. 
 
 3. Using 50c class intervals, similarly determine the mode 
 for 1870 and 1880. 
 
 4. Using the 50c frequencies, determine by a mathematical
 
 TYPES AND MEASURES OF DISPERSION 45 
 
 formula the position of the mode in the 1870 and 1880 
 data. Draw rectangular histograms and smooth them. 
 
 5. Explain why different values for the mode are obtained 
 in the two preceding exercises. Which results are the 
 more valid? Why? 
 
 6. Find the quartile items and their values in the 1870 and 
 1880 wage data by interpolating in the 5c classes. Com- 
 pute the quartile deviations and coefficients. 
 
 7. Draw ogives of the wage data for 1870 and 1880 — 5e 
 frequencies — showing the quartile values, 
 
 8. Summate the percentage frequencies from Table III, page 
 13, and plot on probability paper. 
 
 9. Find the average deviation and coefficient for the 1870 
 and 1880 wage data, 50c classes. 
 
 10. Find the standard deviation and coefficient for the 1870 
 and 1880 wage data, 50c classes, using the method in- 
 volving an assumed average and correction. 
 
 11. Using data prepared in Exercise 9, draw Lorenz curves 
 of wage distributions in 1870 and 1880. Draw a similar 
 curve from the data of Table IV, page 21. 
 
 12. Apply the modified formula for standard deviation (page 
 41) to the five cent wage data for 1870 and 1880. 
 
 13. During a certain period the rate of bank discount was 
 as follows : 
 
 Rate per cent 
 
 21/2 
 
 3 
 
 No. 
 
 of days 
 174 
 408 
 
 31/2 
 4 
 
 
 132 
 165 
 
 41/2 
 5 
 
 
 36 
 37 
 
 51/2 
 6 
 
 
 20 
 26 
 
 7 
 
 
 2 
 
 (a) Compute the average rate of discount, taking the 
 number of days as the frequencies. 
 
 (b) In what classes (rate per cent) do the quartiles 
 fall? 
 
 (c) What rate per cent may be taken as the mode? 
 Why? 
 
 (No interpolation is required in the above problem.) 
 14. Find the coefficients of average deviation, standard devia- 
 tion, and skewness for the following frequency distribu-
 
 46 INTRODUCTION TO ECONOMIC STATISTICS 
 
 tion. Locate the quartiles by interpolation and cheek the 
 results by means of an ogive. 
 
 V F 
 
 $1 1 
 
 2 3 
 
 3 2 
 
 4 2 
 
 5 1 
 
 6 1 
 
 15. The following table shows the increase in the cost of 
 living for ten cities from December, 1914, to December, 
 1920. Classify these percentages to the nearest multiple, 
 of five, and find the average deviation. (Bureau of 
 Labor Statistics' data.) 
 
 Boston 97.4 
 
 Buffalo 101.7 
 
 Chicago 93.3 
 
 Cleveland 104. 
 
 Detroit 118.6 
 
 Los Angeles 96.7 
 
 New York 101.4 
 
 Philadelphia 100.7 
 
 San Francisco 85.1 
 
 Seattle 94.1 
 
 16. Apply the arithmetic formula for determining the mode 
 
 to the following three time series. The months may be 
 treated as if they were class intervals, and the per- 
 centages as if they were frequencies. Graph each series 
 and construct a smoothed curve : 
 
 Percentage of crops harvested monthly in the United 
 States, as reported by Department of Agriculture. 
 
 Month Wheat Corn Cotton 
 
 May 0.5 — — 
 
 June 22.0 0.1 — 
 
 July 42.3 0.1 1.4 
 
 Aug 28.4 1.5 11.5 
 
 Sept 6.5 15.8 31.6 
 
 Oct 0.3 28.3 34.4 
 
 Nov — 43.3 16.0 
 
 Dec — 10.9 4.7 
 
 Jan. -April — — 0.4
 
 CHAPTER III 
 
 INDEXES OF WAGES AND PRICES 
 
 The Nature of Indexes. A large part of statistical 
 work concerns itself with the making and interpreting 
 of indexes. By an index is meant a number, whether 
 absolute or relative, which is used in comparisons to 
 measure a given condition. Used collectively, the term 
 implies a series of such indexes, forming a multiple 
 ratio. Practically all indexes are compiled by a 
 process of sampling. Thus, though it is impossible to 
 record any large proportion of actual wages and prices, 
 yet it is possible to estimate changes in the wage or 
 price level by the use of well-selected samples. Just 
 what may be regarded as sufficiently complete data in 
 sampling cannot be determined precisely, but must be 
 judged largely on the basis of experience. Straws 
 show which way the wind blows, and likewise the price 
 of a product in a single locality will often accurately 
 reflect the trend of a world market. There is no 
 dependable uniformity, however. Some prices respond 
 quickly and universally to changes in supply or de- 
 mand, while others move slowly and irregularly. With 
 respect to wages, it is commonly observed that the 
 market is somewhat slow in its movements. In an 
 industrial center like New England, however, the 
 market should be fairly responsive. One might never- 
 theless hesitate to take the wages at a single mill as 
 
 an index of wages for the whole country; but com- 
 
 47
 
 48 INTRODUCTION TO ECONOMIC STATISTICS 
 
 parisons will show that such an index would, in fact, 
 have some dei^ree of reliability. 
 
 Indexes of Wages. The wage averages considered 
 in the preceding chapter will be taken provisionally as 
 indexes of the wage level in the United States. Accord- 
 ing to these indexes, daily wages in 1870 stood at $1.38, 
 they fell by 1880 to $1.21, but climbed by 1891 to $1.50. 
 The changes may be presented more clearly, however, 
 if the figures are reduced to another form. Since in- 
 dexes are used as ratios, they may be multiplied or 
 divided through by any factor to suit given require- 
 ments. If in this case they are divided by $1.38, the 
 wage in 1870, they are said to be reduced to a base of 
 1870, since the index for that date will become 100.^ 
 Expressed literally, the result is 100%, but the per 
 cent sign is usually dropped as being unnecessary in 
 a ratio. The index numbers now read: 
 
 Year. Wage. 
 
 1870 100 
 
 1880 88 
 
 1891 109 
 
 Indexes of Real Wages. In a study of changes in 
 the wage level, a further factor of great importance 
 must be taken into consideration. This factor is the 
 cost of living. Changes in the cost of living affect the 
 prosperity of wage earners inversely. Hence ''real 
 wages" — a term denoting the purchasing power of 
 wages — will be measured by money wages divided by 
 
 * The base which is theoretically best to use in deriving an index from 
 absolute numbers, is an average of the numbers. Its advantages are, 
 first, that it is a stable value from which to measure the items, and 
 second, that each index is made to suggest its relative position in the 
 series. Applied to the given wage data, the base becomes the average 
 wage of 99c, and the indexes become 101, 89, and 110 — each expressing a 
 percentage of the average.
 
 INDEXES OF WAGES AND PRICES 49 
 
 prices. Whether absolute or relative numbers are 
 taken to measure wages and prices, the quotients may 
 be regarded as comprising an index of real wages, and 
 may be reduced to any desired base. If wages and 
 prices are expressed as indexes having the same base, 
 then the resulting index of real wages will also have 
 this base. 
 
 Various Wage Indexes. The accuracy of our provi- 
 sional wage index may be tested and the study ex- 
 jtended, by the introduction of wage data of a more 
 general character. These data will be taken from three 
 sources: (1) ''The Movement of Wages in the Cotton 
 Manufacturing Industry of New England," by Pro- 
 fessor Stanley E. Howard; (2) The Aldrich Report, 
 and (3) the publications of the Bureau of Labor Sta- 
 tistics, of the Department of Labor at Washington. 
 The first of these sources gives a carefully prepared 
 index of weekly wages in the Massachusetts cotton 
 manufacturing industry from 1860 to 1914. It is de- 
 rived in part from the Aldrich Report, and uses the 
 principles of tabulation and measurement already 
 explained. The second gives a general index of wages 
 down to 1891, based on wages from many industries, 
 and covering various sections of the country. The 
 third source furnishes an index of hour rates in the 
 United States from 1840 to 1920. Hour rates, of 
 course, are not entirely satisfactory as a basis for an 
 index of actual earnings because of the gradual reduc- 
 tion in the length of the working day. This reduction 
 has, however, been offset by increased over-time pay 
 at higher rates, by a gain in leisure hours, and prob- 
 ably by more regular employment. And as a matter 
 of fact, the index of hour rates will be found to con-
 
 50 INTRODUCTION TO ECONOMIC STATISTICS 
 
 form somewhat closely to the index of weekly wages. 
 As measuring the cyclical changes in wages, both in- 
 dexes give practically the same results. 
 
 In Table IX the wage indexes from the sources just 
 mentioned are shown for the years 1870, 1880, and 
 1891, together mth the indexes already derived. By 
 means of price indexes — the nature of which will be 
 considered later — nominal wages are reduced to real 
 wages. The indexes of both nominal and real wages 
 
 TABLE IX 
 
 INDEXES OF NOMINAIi AND KEAL WAGES, JULY, 1870, 1880, 
 
 AND 1891 
 
 SOURCE OF DATA 
 
 Connecticut Mill 
 
 Nominal wages (aver- 
 age) 
 
 Prices 
 
 Real wages 
 
 Mass. Cotton Mills 
 
 Nominal wages (base, 
 1860) 
 
 Prices 
 
 Real wages 
 
 U. S.-Aldrich Repprt 
 
 Nominal wage (base, 
 1860) 
 
 Prices (base, 1860) . . . . 
 
 Real wages 
 
 U. S. Bureau of Lab. Stat. 
 
 Nominal wage (base, 
 1913) 
 
 Prices (base, 1913) . . . . 
 
 Real wages 
 
 PRIMARY INDEXES 
 
 1870 1880 
 
 1.37.5 
 1.47 
 .935 
 
 166 
 
 140 
 118 
 
 167.1 
 144.4 
 116 
 
 67 
 
 147 
 
 46 
 
 1.207 
 
 1.09 
 
 1.11 
 
 154 
 105 
 147 
 
 143.0 
 104.9 
 136 
 
 60 
 
 109 
 
 55 
 
 1891 
 
 1.50 
 
 .82 
 
 1.83 
 
 172 
 
 79 
 
 217 
 
 168.6 
 94.4 
 179 
 
 69 
 
 82 
 84 
 
 DERIVED INDEXES 
 
 1870 
 
 100 
 
 100 
 
 100 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 1880 
 
 118 
 
 93 
 125 
 
 86 
 118 
 
 90 
 
 121 
 
 1891 
 
 109 
 196 
 
 104 
 184 
 
 101 
 154 
 
 103 
 
 185 
 
 are next reduced to a base of 1870, in which form they 
 may be readily compared. It will be seen that the 
 indexes of real wages in the Connecticut mill, based 
 on very slender data though they are, do not ditfer 
 markedly from the others.
 
 INDEXES OF WAGES AND PRICES 51 
 
 A more complete statement of the results of Pro- 
 fessor Howard's study, and of the Bureau of Labor 
 Statistics' index, is presented in Table X. As in the 
 
 TABLE X 
 INDEXES OF WAGES AND PRICES, 1870-1920 
 
 
 MASSACHUSETTS 
 
 UNITED STATES 
 
 Year 
 
 
 
 
 
 
 
 
 Wages 
 
 Wholesale 
 
 Real 
 
 Wages 
 
 Wholesale 
 
 Real 
 
 
 (Weekly) 
 
 Prices 
 
 Wages 
 
 (Hr. Rates) 
 
 Prices 
 
 Wageg 
 
 1870 
 
 166 
 
 140 
 
 50 
 
 67 
 
 147 
 
 46 
 
 1871 
 
 177 
 
 130 
 
 58 
 
 68 
 
 136 
 
 50 
 
 1872 
 
 183 
 
 135 
 
 58 
 
 69 
 
 141 
 
 49 
 
 1873 
 
 178 
 
 134 
 
 56 
 
 69 
 
 140 
 
 49 
 
 1874 
 
 163 
 
 128 
 
 54 
 
 67 
 
 133 
 
 50 
 
 1875 
 
 150 
 
 120 
 
 53 
 
 67 
 
 125 
 
 64 
 
 1876 
 
 145 
 
 111 
 
 55 
 
 64 
 
 116 
 
 55 
 
 1877 
 
 142 
 
 108 
 
 55 
 
 61 
 
 113 
 
 54 
 
 1878 
 
 145 
 
 98 
 
 63 
 
 60 
 
 102 
 
 59 
 
 1879 
 
 144 
 
 94 
 
 65 
 
 59 
 
 98 
 
 60 
 
 1880 
 
 154 
 
 105 
 
 62 
 
 60 
 
 109 
 
 55 
 
 1881 
 
 149 
 
 102 
 
 62 
 
 62 
 
 106 
 
 58 
 
 1882 
 
 157 
 
 103 
 
 64 
 
 63 
 
 108 
 
 58 
 
 1883 
 
 158 
 
 98 
 
 69 
 
 64 
 
 102 
 
 63 
 
 1884 
 
 155 
 
 89 
 
 74 
 
 64 
 
 92 
 
 70 
 
 1885 
 
 150 
 
 82 
 
 77 
 
 64 
 
 86 
 
 74 
 
 1886 
 
 153 
 
 80 
 
 80 
 
 64 
 
 84 
 
 76 
 
 1887 
 
 160 
 
 81 
 
 83 
 
 67 
 
 84 
 
 80 
 
 1888 
 
 164 
 
 84 
 
 83 
 
 67 
 
 87 
 
 77 
 
 1889 
 
 169 
 
 81 
 
 89 
 
 68 
 
 84 
 
 81 
 
 1890 
 
 173 
 
 80 
 
 91 
 
 69 
 
 81 
 
 85 
 
 1891 
 
 172 
 
 79 
 
 92 
 
 69 
 
 82 
 
 84 
 
 1892 
 
 172 
 
 75 
 
 97 
 
 69 
 
 76 
 
 91 
 
 1893 
 
 180 
 
 75 
 
 102 
 
 69 
 
 77 
 
 90 
 
 1894 
 
 168 
 
 68 
 
 104 
 
 67 
 
 69 
 
 97 
 
 1895 
 
 165 
 
 66 
 
 105 
 
 68 
 
 70 
 
 97 
 
 1896 
 
 175 
 
 64 
 
 116 
 
 69 
 
 66 
 
 105 
 
 1897 
 
 174 
 
 64 
 
 116 
 
 69 
 
 67 
 
 103 
 
 1898 
 
 171 
 
 66 
 
 110 
 
 69 
 
 69 
 
 100 
 
 1899 
 
 164 
 
 72 
 
 97 
 
 70 
 
 74 
 
 95 
 
 1900 
 
 189 
 
 78 
 
 102 
 
 73 
 
 80 
 
 91 
 
 1901 
 
 190 
 
 77 
 
 104 
 
 74 
 
 79 
 
 94 
 
 1902 
 
 191 
 
 80 
 
 101 
 
 77 
 
 85 
 
 91 
 
 1903 
 
 197 
 
 81 
 
 103 
 
 80 
 
 85 
 
 94 
 
 1904 
 
 196 
 
 80 
 
 103 
 
 80 
 
 86 
 
 92 
 
 1905 
 
 200 
 
 82 
 
 103 
 
 82 
 
 85 
 
 96 
 
 1906 
 
 216 
 
 87 
 
 105 
 
 85 
 
 88 
 
 97 
 
 1907 
 
 240 
 
 92 
 
 HI 
 
 89 
 
 94 
 
 95 
 
 1908 
 
 228 
 
 87 
 
 111 
 
 89 
 
 91 
 
 98 
 
 1909 
 
 211 
 
 90 
 
 99 
 
 90 
 
 97 
 
 93 
 
 1910 
 
 209 
 
 93 
 
 95 
 
 93 
 
 99 
 
 94 
 
 1911 
 
 207 
 
 92 
 
 96 
 
 95 
 
 95 
 
 100 
 
 1912 
 
 223 
 
 95 
 
 99 
 
 97 
 
 101 
 
 98 
 
 1913 
 
 227 
 
 96 
 
 100 
 
 100 
 
 100 
 
 100 
 
 1914 
 
 229 
 
 95 
 
 102 
 
 102 
 
 100 
 
 102 
 
 1915 
 
 (Base 
 
 (Base 
 
 
 103 
 
 101 
 
 
 1916 
 
 1860) 
 
 1860) 
 
 
 111 
 
 124 
 
 
 1917 
 
 
 
 
 128 
 
 176 
 
 
 1918 
 
 
 
 
 162 
 
 196 
 
 
 1919 
 
 
 
 
 184 
 
 (Spring) 
 
 212 
 
 
 1920 
 
 
 
 
 234 
 
 243 
 
 
 
 1 
 
 
 (Slimmer) 
 

 
 52 INTRODUCTION TO ECONOMIC STATISTICS 
 
 preceding table, an index of real wages is derived by 
 the use of an index of wholesale prices. The price in- 
 dex shown for Massachusetts is merely an adaptation 
 of the Bureau of Labor Statistics' data for the United 
 States. The index of real wages for Massachusetts 
 has been changed from a base of 1860, as first derived, 
 to a base of 1913, in order to allow of comparisons 
 with the corresponding index for the United States. 
 Both indexes point to the fact that real wages have 
 about doubled in the interval from 1870 to 1914, but 
 that the greater part of this increase came before 1890. 
 Wholesale and Retail Prices. A question may be 
 raised regarding the validity of using an index of 
 wholesale prices as a measure of changes in the cost 
 of living. Unfortunately, no adequate index of retail 
 prices covering the years here studied is available. An 
 index of wholesale prices has therefore been substi- 
 tuted. It is a well established fact, however, that 
 wholesale prices swing in the same direction and at 
 nearly the same time as retail prices, but that they 
 move somewhat more extremely. In the course of the 
 usual moderate cyclic changes, therefore, the former 
 will parallel the latter closely enough to serve as a 
 substitute. But when the price swings are extremely 
 low, the substitution will doubtless give an exagger- 
 ated rise in real wages. Such is evidently the case in 
 the decade from 1890 to 1900, when prices fell to the 
 lowest point in the century. The apparent rise in real 
 wages at that time should therefore be discounted, 
 though just how much, cannot be accurately deter- 
 mined. The opposite result is obtained during the 
 great upswing of prices from 1914 to 1920. For these
 
 INDEXES OF WAGES AND PRICES 53 
 
 years, however, adequate data on the cost of living 
 are obtainable. In Table XI estimates derived from 
 such data are used in the place of wholesale prices, 
 and real wages are then computed.^ 
 
 TABLE XI 
 
 INDEXES OF WAGES AND COST OF LIVING 
 UNITED STATES, 1913-1920 
 
 (Estimated from Bureau of Labor Statistics' data) 
 
 Year 
 
 Wages 
 
 Cost of Living 
 
 Real Wages 
 
 1913 
 
 100 
 
 100 
 
 100 
 
 1914 
 
 102 
 
 100 
 
 102 
 
 1915 
 
 103 
 
 100 
 
 103 
 
 1916 
 
 111 
 
 110 
 
 101 
 
 1917 
 
 128 
 
 134 
 
 95 
 
 1918 
 
 162 
 
 154 
 
 105 
 
 1919 
 
 200 
 
 180 
 
 111 
 
 1920 
 
 225 
 
 211 
 
 107 
 
 Indexes of the Cost of Living. The computation of 
 an index of the cost of living involves many difficulties, 
 both theoretical and practical. To begin with, the units 
 employed often vary in quality, and are difficult to 
 standardize. Hence the first step in the computation 
 is the dramng up of a selected list of articles of staple 
 grades. These articles must be sufficient in number 
 
 ^ It is not intended that the wage data here studied shall be taken 
 as a final measurement of the course of real wages. They were com- 
 piled chiefly with the purpose in mind of illustrating certain methods of 
 work. But it is the opinion of the writer, based on a study of such 
 material as is available, that they represent a passably good estimate. 
 Other studies, however, have purported to show a decline in real 
 wages since the last decade of the nineteenth century. An interesting 
 study of this sort appears in the American Economic Review, Septem- 
 ber, 1921. In this study the cost of living is assumed to be measured 
 by retail food prices. But this is a very questionable measure, inas- 
 much as retail food prices have risen about as rapidly as wholesale 
 prices since the period of agricultural depression. This abnormal rise, 
 of course, makes real wages by comparison appear to fall. The study 
 also uses an index of hour rates which purports to show a slower 
 rise than is shown by the index published by the Bureau of Labor 
 Statistics. It should be noted that the data here discussed do not cover 
 wages of government employes.
 
 54 INTRODUCTION TO ECONOMIC STATISTICS 
 
 and importance to represent fairly all the necessities 
 commonly purchased by the average working-class 
 family. After this has been done, the prices of these 
 articles as sold in representative stores must be tabu- 
 lated. If the index is to cover a considerable territory, 
 data must be gathered from a number of localities. 
 The common average of the prices of each article at a 
 given date is found. Under certain conditions, as 
 when the extreme items are likely to be in error, the 
 median is preferable in place of the common average. 
 Since the work of collecting and tabulating prices 
 requires elaborate organization, it is done on a large 
 scale by only a few agencies. One of these is the 
 National Industrial Conference Board. ^ Another is a 
 commission established by the Legislature of the State 
 
 ^ The National Industrial Conference Board is affiliated with the Na- 
 tional Association of Manufacturers and other similar organizations, 
 and has its headquarters in New York. It publishes among other things 
 an excellent index of increases in the cost of living in the United 
 States. This index is issued promptly each mouth, and is the best 
 available source by which current changes may be measured. By per- 
 mission, the indexes for the year 1920 are here reprinted. The base is 
 July, 1914, and the figures show the per cent rise. 
 
 Month, 
 
 Cost of 
 
 
 
 
 Fuel and 
 
 
 1920 
 
 Living 
 
 Food 
 
 Shelter 
 
 Clothing 
 
 Light 
 
 Sundries 
 
 January 
 
 90.2 
 
 97 
 
 43 
 
 170 
 
 49 
 
 77 
 
 February 
 
 93.5 
 
 101 
 
 45 
 
 177 
 
 49 
 
 78 
 
 March 
 
 94.8 
 
 100 
 
 49 
 
 177 
 
 49 
 
 83 
 
 April 
 
 96.6 
 
 100 
 
 50 
 
 188 
 
 51 
 
 83 
 
 May 
 
 101.6 
 
 111 
 
 51 
 
 187 
 
 55 
 
 83 
 
 June 
 
 103.0 
 
 115 
 
 51 
 
 176 
 
 61 
 
 85 
 
 July 
 
 104.5 
 
 119 
 
 58 
 
 166 
 
 66 
 
 85 
 
 August 
 
 103.2 
 
 119 
 
 58 
 
 155 
 
 69 
 
 85 
 
 September 
 
 99.4 
 
 107 
 
 59 
 
 155 
 
 78 
 
 83 
 
 October 
 
 97.3 
 
 103 
 
 59 
 
 148 
 
 83 
 
 90 
 
 November 
 
 93.1 
 
 93 
 
 66 
 
 128 
 
 100 
 
 92 
 
 December 
 
 90.0 
 
 93 
 
 66 
 
 105 
 
 100 
 
 92 
 
 Of these indexes the only one which shows a further rise up to October, 
 1921, is that for shelter. This registers 71% during March to June, 
 inclusive, and 69% during the four months following. Fuel and Light 
 and Sundries begin to decline after .lanuary. Food shows an increase 
 for four months following a minimum of 45% in June, but this change 
 ia reflected in only a minor degree in the aggregate index. The apex 
 of the post-war boom is, then, according to this index, in July, 1920,
 
 INDEXES OF WAGES AND PRICES 55 
 
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 56 INTRODUCTION TO ECONOMIC STATISTICS 
 
 of Massachusetts. But doubtless the best known and 
 most authoritative work is done by the Bureau of 
 Labor Statistics. The methods used in combining 
 average prices into a single index may be illustrated 
 by the Bureau of Labor Statistics' index of the food 
 cost of living. 
 
 An Index of Food Prices. In computing an index 
 of the food cost of living, the Bureau of Labor Statis- 
 tics has listed 22 important articles of food, as shown 
 in Table XII. The use of this list was begun in Jan- 
 uary, 1913, and was continued to January, 1921, when 
 a revised list of 43 articles was substituted. Prices on 
 these articles were compiled monthly from a number 
 of important cities selected from the various terri- 
 torial divisions of the United States. The averages 
 of these prices for the year 1913 and for the month of 
 June, 1920, are shown in Table XII, together with the 
 annual consumption per workingman's family as ascer- 
 tained by a special investigation, which is here as- 
 sumed to apply directly to 1913. From these data the 
 increase in the food cost of living from the pre-war 
 level to the climax of prices in 1920 may be computed. 
 
 The Aggregate Method. The two standard methods 
 of making the computation are shown in Table XII. 
 The first and simplest of these is known as the aggre- 
 gate method, and is illustrated in the first five columns 
 of the table. The process consists merely in finding 
 the total cost of a year's supply of the given articles 
 at the two contrasted dates (columns 3 and 5.) The 
 total cost in June, 1920 ($479.44), is then divided by 
 the total cost in 1913 ($215.89). The result shows that 
 the former cost was 222% as compared with 100% in 
 the base year, 1913 — an increase of 122%. While the
 
 INDEXES OF WAGES AND PRICES 57 
 
 totals do not actually represent more than about two- 
 thirds of the average annual food cost per family, yet 
 they doubtless are representative enough to give a 
 close approximation to the correct percentage increase. 
 
 The Proportional Expenditure Method. A second 
 and more complex process, known as the proportional 
 expenditure method, is illustrated in columns 6, 7, and 
 8. By this method the relative price in June, 1920, as 
 compared with 1913 is found (column 6). By so doing, 
 all prices, whether large or small, are obviously put 
 on the same basis, since each is based on 100% in 1913. 
 These relative prices are next averaged by a process of 
 weighting similar to that explained in Chapter II in 
 connection with Table IV. The weights, as shown in 
 column 7, measure the relative importance in a work- 
 ingman's budget of an annual supply of each article. 
 To illustrate, the first weight is obtained by dividing 
 $8,128 by $215.89. This gives approximately 3.8%, but 
 both the decimal point and the per cent sign are 
 dropped as being unnecessary in a ratio. In the same 
 way the second weight, 33, is obtained by dividing 
 $7,136 by $215.89, and so on for the remainder of the 
 weights. The accuracy of the work may be checked 
 by means of the total, which should come to approxi- 
 mately 1000. In the computation of such weights, it 
 is usually unnecessary to strive for extreme accuracy ; 
 as, for example, by carrying the figures to several 
 decimal places. A considerable margin of error may 
 be present in the weights without materially affecting 
 the weighted average. 
 
 A Comparison of the Two Methods. It will be ob- 
 served that the result by the proportional expenditure 
 method is exactly the same in this instance as that
 
 58 INTRODUCTION TO ECONOMIC STATISTICS 
 
 obtained by the aggregate method. But this would not 
 always be the case. Where there is a difference, pref- 
 erence is likely to be given to the result obtained by the 
 former method. The jid vantage claimed for this 
 method arises from the fact that it is not practicable 
 to make frequent revisions of the consumption esti- 
 mates. When an estimate has become somewhat out 
 of date, its inaccuracies may have considerable effect 
 upon the results obtained by the aggregate method. 
 But it is assumed that the proportional expenditure 
 weights, based as they are upon both the prices and 
 the consumption of a specific period, will remain rela- 
 tively accurate for a longer period than will the con- 
 sumption estimates taken alone. Hence their use is 
 thought to give somewhat more dependable results. 
 In practice it will be found that the proportional ex- 
 penditure method is not as tedious as it seems in the 
 illustration. Tlie weights, when once- obtained, may 
 be used unchanged for a long period; and the finding 
 of the relative prices does not usually mean additional 
 work, since they are in any case desirable for purposes 
 of comparison. 
 
 The formuia for the price index by the aggregate 
 method is : 
 
 p _2pnqx 
 
 X n — XT' 
 
 ^Poqx 
 in which 
 
 Pn = price index for the relative period 
 
 Pn = prices for the relative period 
 
 q, = quantities consumed during a given period, 
 
 preferably the base period 
 Po = prices for the base period 
 The process may be described briefly as a comparison
 
 INDEXES OF WAGES AND PRICES 59 
 
 of two price averages, both obtained by woighiiiig for 
 quantities. 
 
 The formula for a price index by (he proportional 
 expenditure method is :^ 
 
 SP^XpxQx 
 JTn — po 
 
 SpxqT" 
 
 This process may be described as an average of rela- 
 tive prices weighted for the vahie eoiisuuied during 
 some specified period, often prior to the beginning 
 of the interval over which the price comparison is be- 
 ing made. In connection with both formulas, it must 
 be understood that the factors are taken distributively ; 
 that is, only prices and weights belonging to the same 
 article are multiplied.'^* 
 
 An inspection of the fornmlas will show why the 
 two results obtained in Table XII are alike. As is 
 often the case, the year indicated by the subscript x, 
 to which the quantities and weights are assumed to 
 belong, is identical with the base year indicated by the 
 
 ' The formula as thus stated docs not indicato tlio reduction of tlio 
 proj)ortional oxpendituro weijrlits to ])c'rfi>ntagos. Tlio valuo is, of 
 course, unaircctod by audi loduction; and for j)urj)oflcs o*' lator com- 
 parison tlio form as jj^iven is proforablo. 
 
 'Use is somotimofl made of tho Imrmoiiic mean In Hiidin^j tlie average 
 of the relatives. This averaj:je is found 1/y lakinp the reciprocals oJ 
 the relatives, comjjutinfj their averajje, and then takiii}^ the reciprocal 
 of this averaj^'e. it may easily bo seen that the effect of takiiifj^ the 
 reciprocals of the relatives is to reverse the base; that is, tlu! later year 
 becomes the bast< instead of the earlier year. If vveij^hts nrc^ usi-d it is 
 therefore preferable that they be derived from the data of the lat.i^r 
 year in (indinjf the harmonic mean. Takin;j the reciprocal of the 
 averajife again reverses the bases. Rut the index obtained by the usual 
 direct method will not be quite the same as that obtained by the 
 harmonic mean. This will ordinarily be the case whether the weif^hta 
 are shifted to the later base, or not; or indee<l whether any weights 
 are employed or not. The same distinction between the arithmetic and 
 the harmonic mean is well brought out by taking a flimple average 
 of a given set of prices, and comparing it with tho harmonica mean of 
 the same prices. The latter is the recii)rocal of the average of the 
 quantities of each commodity purchasable for ono dollar.
 
 60 INTRODUCTION TO ECONOMIC STATISTICS 
 
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 CJOJrH,-,rH(MCgca.-((MCOeO(MOOOOOOONir5 
 
 ^^^j^jQjOrOjOj^jQ^jQ^ -^^ ^ ^ jQ j2 ^ jQ ^ 
 00000'*<'OlCTtlOOlot~tDlOlO"*t^mC<I05l^r-) 
 
 -3^ 
 
 ^ w 
 
 ^-s^^ 
 
 ° S '^ S sr-^ a> jrf c3 3 pj <v, d 
 -^ "^ ■"■■-' — — ^ - '-J .i; o 
 
 2 u 
 
 5 a 
 
 CQ«fCiO(l|PHmW^-l^I^HWo;^WI^OP^A^02OE-^ 
 
 n ^ c- " ^ _j *-* "^ --w^- CU (— H dJ 
 
 d - • 
 
 3
 
 INDEXES OF WAGES AND PRICES 61 
 
 subscript o. The numerator of the second formula 
 therefore reduces to the same form as that of the first 
 formula, and the denominators also have the same 
 value. But the assumption that the quantities as given 
 apply directly to the year 1913 was adopted merely to 
 simplify the table for purposes of illustration. As a 
 matter of fact, these quantities were obtained by an in- 
 vestigation made in 1918, and were not put into use 
 until the beginning of 1921. The consumption esti- 
 mates actually used by the Bureau of Labor Statistics 
 from 1913 to 1920, inclusive, are shown in Table 12-A, 
 together with the annual retail food prices for the 
 same period, and for June, 1921. It will be seen that 
 these estimates go back to the year 1901. 
 
 Limitations. Some of the limitations of the fore- 
 going methods of computing changes in the cost of liv- 
 ing may be easily seen. It is evident that the assum- 
 ing of an unvarying consumption may sometimes result 
 in material inaccuracies. If there is a very uneven 
 rise in prices, buyers will begin to substitute the 
 cheaper article for the more expensive, wherever this 
 can be properly done. The effect on the one hand 
 will be to moderate the unevenness of price changes, 
 but on the other hand it will make material altera- 
 tions in the budget. But the frequent revision of con- 
 sumption data, and the complexity of methods of com- 
 putation, would involve more labor than at present 
 seems practicable to allow. Besides, the revision of 
 consumption data raises another problem. Changing 
 quantities may in part reflect rising standards of com- 
 fort whereas the term, ''cost of living," implies pro- 
 vision only for those necessities which are required to 
 maintain the social efficiency of the family. Strictly
 
 62 INTRODUCTION TO ECONOMIC STATISTICS 
 
 speaking, the measurement of changes in such costs 
 would call for the dietician 's units of protein, carbohy- 
 drates, fats, and calories, and would require very de- 
 tailed analyses. Hence, as far as the cost of living 
 is concerned, we are driven back to the comparatively 
 simple methods actually in use. The results obtained 
 by such methods must, of course, always be taken as 
 approximations only. 
 
 Combining the Partial Indexes. In measuring 
 changes in the entire cost of living, the Bureau of 
 Labor Statistics makes an index for several groups of 
 commodities, in addition to food. The process of com- 
 bining these partial indexes into a single measure is an 
 application of the proportional expenditure method. 
 The budgetary studies of 1918 furnish the weights 
 now in use. The partial indexes in June, 1920, the 
 weights, and the process of finding a single index, may 
 be seen in the following table : 
 
 Index Weight 
 
 Item of June, 1920 (percent 
 
 Expenditure (base, 1913) of budget) Extension 
 
 Food 219.0 38.2 8365.80 
 
 Clothing 287.5 16.6 4772.50 
 
 Housing 134.9 13.4 1807.66 
 
 Fuel and light 171.9 5.3 911.07 
 
 Furniture and 
 
 furnishings 292.7 5.1 1492.77 
 
 Miscellaneous 201.4 21.3 4289.82 
 
 99.9 ) 21639.62 
 
 216.5 
 
 Indexes of Wholesale Prices. Though neither the 
 
 aggregate nor the proportional expenditure method is 
 
 theoretically perfect, yet one or the other is used in 
 
 practically all the price indexes in common use. In
 
 INDEXES OF WAGES AND PRICES 63 
 
 adapting the latter method to wholesale prices, the 
 weights are usually made to reflect the proportional 
 value produced, rather than family consumption, but 
 the principle is essentially the same. Applying this 
 method, the Bureau of Labor Statistics compiles an 
 elaborate monthly index of wholesale prices. The 
 number of items tabulated, the base used, and other 
 details of the work have been varied from time to 
 time, but the partial results have been combined into a 
 single index having the same base. For several years 
 prior to the war, the base used was an average of the 
 decade 1890-99, which was a period of unusually low 
 prices. But in more recent years, 1913 has been em- 
 ployed because it may be considered representative 
 of normal conditions immediately preceding the war. 
 As compiled for January, 1920, the index is derived 
 from a weighted average of 327 price quotations. 
 The items are classified under nine headings, each 
 having its own partial index. ^ This is the most com- 
 
 ^ Annual group index numbers for the years 1890-1920 will be found 
 in the Monthly Labor Beview, Feb., 1921, page 45. The figures for 
 recent dates are as follows: 
 
 GEOUP INDEX NUMBERS— UNITED STATES— BUREAU OF 
 
 LABOR STATISTICS 
 
 For the Years 1913-1920, and Sept., 1920 and 1921 
 
 
 
 
 •a 
 
 
 •o 
 
 
 
 
 
 
 
 ■a 
 
 a 
 .So 
 
 1| 
 
 P bo 
 
 
 3Sa 
 
 
 
 .2 C 
 
 aj2 
 
 1913 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 1914 
 
 103 
 
 103 
 
 98 
 
 96 
 
 87 
 
 97 
 
 101 
 
 99 
 
 99 
 
 1915 
 
 105 
 
 104 
 
 100 
 
 93 
 
 97 
 
 94 
 
 114 
 
 99 
 
 99 
 
 1916 
 
 122 
 
 126 
 
 i-ZH 
 
 119 
 
 148 
 
 101 
 
 159 
 
 115 
 
 120 
 
 1917 
 
 189 
 
 176 
 
 181 
 
 175 
 
 208 
 
 124 
 
 198 
 
 144 
 
 155 
 
 1918 
 
 220 
 
 189 
 
 239 
 
 163 
 
 181 
 
 151 
 
 221 
 
 196 
 
 193 
 
 1919 
 
 234 
 
 210 
 
 261 
 
 173 
 
 161 
 
 192 
 
 179 
 
 236 
 
 217 
 
 1920 
 
 218 
 
 239 
 
 302 
 
 238 
 
 186 
 
 308 
 
 210 
 
 366 
 
 236 
 
 Sept. 
 
 
 
 
 
 
 
 
 
 
 1920 
 
 210 
 
 223 
 
 278 
 
 284 
 
 192 
 
 318 
 
 222 
 
 371 
 
 239 
 
 Sept. 
 
 
 
 
 
 
 
 
 
 
 1921 
 
 122 
 
 146 
 
 187 
 
 178 
 
 120 
 
 193 
 
 162 
 
 223 
 
 146
 
 64 INTRODUCTION TO ECONOMIC STATISTICS 
 
 prehensive study of wholesale prices published, but 
 for business uses it has the disadvantage of appearing 
 two or three months late. 
 
 One of the most widely used commercial indexes of 
 wholesale prices is Bradstreet's. This index is based 
 on about one hundred important articles, for which 
 prices are easily available in the principal business 
 centers. It is published promptly each month, and 
 is therefore valuable to the business man who desires 
 to keep in touch with the immediate trend of the mar- 
 ket. The aggregate method of computation is em- 
 ployed, and the result is given merely as a sum of 
 money, which may be reduced to any desired base. 
 The lack of a definite system of weights is a defect 
 of this index, but it is partially compensated for by a 
 careful selection of items, and a repetition of important 
 articles by a quotation for more than one grade. 
 
 Dun's monthly index of wholesale prices is perhaps 
 not as widely known as Bradstreet's, but it appears to 
 be more scientifically compiled. It is based upon ap- 
 proximately 300 price quotations, which are grouped 
 under seven heads, and finally combined into a single 
 index, the result being stated in dollars. Prices are 
 weighted in accordance with estimated per capita con- 
 sumption. Just how the weights are applied is not 
 apparent. Commercial houses, as a rule, do not pub- 
 lish the details of their statistical work. 
 
 The Federal Reserve Board has begun the publica- 
 tion of a monthly index of wholesale prices, based on 
 about 90 commodities. It is intended for use particu- 
 larly in international comparisons, but its value is 
 not limited to this field.' Still another index intended 
 
 ^ The Federal Eeserve Board classifies its data so as to present indexes
 
 INDEXES OF WAGES AND PRICES 65 
 
 to measure wholesale price movements is published by 
 the Babson Statistical Organization. This index is 
 compiled monthly from quotations on ten basic com- 
 modities, and in spite of its narrow base, serves a use- 
 ful purpose. A number of other less complete indexes 
 might be mentioned, some of which appear weekly. 
 One of these is the Annalist's weekly index of whole- 
 sale food, a weighted average of 25 prices. It is obvious 
 that there may be many price indexes giving somewhat 
 different results, yet each one valid in its own sphere. 
 The purpose a given index is to serve will always con- 
 trol the selection of items and the weighting. 
 
 As a means of comparing some of the indexes just 
 mentioned, Table XIII is given, showing several whole- 
 sale price indexes for the year 1913, and by months 
 for 1920. The data for 1920 are of interest because 
 they contain the peak of prices for the war and post- 
 war cycle. ^ 
 
 TABLE XIII 
 INDEXES OF WHOLESALE PEICES 
 
 
 Bureau 
 
 Brad- 
 street 's 
 
 
 Federal 
 
 
 Period 
 
 of Labor 
 
 Dun's 
 
 Reserve 
 
 Babson 's 
 
 
 Statistics 
 
 
 Board 
 
 
 1913 
 
 100 
 
 $9.2115 
 
 $120.8865 
 
 100 
 
 $1.26 
 
 1920 
 
 
 
 
 
 
 Jan. 
 
 248 
 
 20.3638 
 
 247.394 
 
 242 
 
 3.30 
 
 Feb. 
 
 249 
 
 20.8690 
 
 253.748 
 
 242 
 
 3.44 
 
 Mar. 
 
 253 
 
 20.7950 
 
 253.016 
 
 248 
 
 3.59 
 
 Apr. 
 
 265 
 
 20.7124 
 
 257.901 
 
 263 
 
 3.61 
 
 May 
 
 272 
 
 20.7341 
 
 263.332 
 
 264 
 
 3.66 
 
 June 
 
 269 
 
 19.8752 
 
 262.149 
 
 258 
 
 3.71 
 
 July 
 
 262 
 
 19.3528 
 
 260.414 
 
 250 
 
 3.60 
 
 Aug. 
 
 250 
 
 18.8273 
 
 252.288 
 
 234 
 
 3.43 
 
 Sept. 
 
 242 
 
 17.9746 
 
 248.257 
 
 226 
 
 3.39 
 
 Oct. 
 
 225 
 
 16.9094 
 
 237.341 
 
 208 
 
 3.25 
 
 Nov. 
 
 207 
 
 15.6750 
 
 227.188 
 
 190 
 
 2.98 
 
 Dec. 
 
 189 
 
 13.6263 
 
 211.628 
 
 171 
 
 2.75 
 
 for goods imported and exported, and producers' and consumers' goods, 
 as well as for certain commodity groups. These indexes are published 
 monthly in the Federal Beserve Bulletin. 
 
 ^ In comparing these data, it should be noted that the commercial 
 indexes (Bradstreet 's. Dun's, and Babson 's) are based on price quota- 
 tions for the first of each month.
 
 66 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Other Indexes. In addition to indexes of commodity 
 prices, many other indexes, measuring various phases 
 of business activity, are published. Speculative and 
 investment activities are measured by indexes of stock 
 and bond prices. A good index of stock prices is dif- 
 ficult to make because of the continually changing 
 status of the corporations issuing the stocks. The 
 indexes now in common use are based on quotations of 
 standard shares listed on the New York Stock Ex- 
 change. Railroad and industrial shares are usually 
 compiled separately, and the two sets combined into a 
 composite index. Financial and productive activities 
 in the general markets are measured by a variety of 
 data, such as the number of shares traded on the New 
 York Stock Exchange, bank clearings in New York 
 and in the country as a whole, bank deposits, money 
 in circulation, gold movements, the interest rate, the 
 production of pig iron, the number of building per- 
 mits issued in leading cities, the number and extent of 
 business failures, and the balance of foreign trade. 
 Labor conditions are measured by wage and employ- 
 ment data, examples of which are the reports of the 
 New York State Industrial Commission. Retail trade 
 is indicated by reports of department store sales. 
 Most of these various indexes as they appear in the 
 financial papers, consist of mere statements of periodi- 
 cal figures. Certain phases of their statistical elabora- 
 tion will be considered later under the subject of 
 
 trends. 
 
 REFERENCES 
 
 Bamett, George E., "Index Numbers of the Total Cost of 
 Living," Quarterly Journal of Economics, February, 1921, 
 pp. 240-263.
 
 INDEXES OF WAGES AND PRICES 67 
 
 Bowley, Arthur L., "The Measurement of Changes in the 
 Cost of Living" (and discussion), Journal of the Royal 
 Statistical Societij, May, 1919, pp. 343-372. 
 
 Fisher, Irving, Stahilizing the Dollar, Chapters I-III. 
 
 Howard, Stanley E., The Movement of Wages in the Cotton 
 Manufacturing Industry of New England. 
 
 Meeker, Royal, ''Some Features of the Statistical Work of 
 the Bureau of Labor Statistics," Quarterly Publications of 
 the American Statistical Association, March, 1915, pp. 431- 
 441. 
 
 Mitchell, Wesley C, Index Numbers of Wholesale Prices in 
 the United States and Foreign Countries, Bulletin No. 173 
 (Whole Number), Bureau of Labor Statistics, July, 1915. 
 
 Secrist, Horace, An Introduction to Statistical Methods, 
 Chapters IX and X. 
 
 Secrist, Horace, Readings and Problems in Statistical 
 Methods, Chapter VIII. 
 
 Stewart, Walter W., "Prices During the War," Quarterly 
 Publications of the American Statistical Association, Sep- 
 tember, 1920, pp. 305-313. 
 
 EXERCISES 
 
 1. Reduce the three indexes shown on page 53 to a base 
 consisting of the average of each series, respectively. 
 
 2. Reduce Bradstreet 's. Dun's and Babson's indexes of 
 wholesale prices for 1920 (page 65) to a 1913 base. 
 Compare the divergence of these indexes, and the others 
 given in the same table, for the month of May by the 
 use of the coefficient of average deviation. 
 
 3. (a) Plot on the same sheet of semi-logarithmic paper the 
 two indexes of real wages given in Table X, page 51. 
 (b) Plot together on the same chart the indexes of 
 wages and cost of living shown in Table XI, page 53. 
 
 4. Assuming that the ratio of American to European com- 
 modity prices — both price levels being stated in indexes 
 having 1913 as a base — measures the relative deprecia- 
 tion of European currencies, find the theoretical value 
 of these currencies for September, 1921, using the data 
 given below {Federal Reserve Bulletin, Nov., 1921).
 
 68 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Express the market prices of exchange as percentages 
 of the theoretical values. 
 
 Exchange in 
 Index of New York. 
 
 Wholesale Prices (Cables, per 
 Country September, 1921 cent of par) 
 
 United States 143 — 
 
 United Kingdom ... 191 76.5 
 
 France 344 37.7 
 
 Italy 580 21.8 
 
 Germany 1777 4.0 
 
 Sweden 182 81.3 
 
 Norway 287 48.0 
 
 Denmark 224 65.9 
 
 5. The following table gives a monthly index of wholesale 
 prices obtaining in the United Kingdom ("Statist" in- 
 dex) and the average monthly price in New York of 
 sterling exchange (cables), for the year 1920. Using the 
 Federal Reserve Board index (page 65) as a measure 
 of the price level in the United States, find the theoretical 
 value of the pound sterling (par, $4.8665) each month 
 in terms of American money. Compare the results thus 
 obtained with the cost of sterling exchange by graphing 
 both series. (The graph should be drawn so as to show 
 the zero line at the base. Why?) 
 
 Statist Sterling 
 
 index cables 
 
 1920 (Base, 1913) (New York) 
 
 January 288 $3.68 
 
 February 306 3.39 
 
 March 307 3.72 
 
 April 313 3.93 
 
 May 305 3.85 
 
 June 300 3.95 
 
 July 299 3.86 
 
 August 298 3.63 
 
 September 292 3.52 
 
 October 282 3.47 
 
 November 263 3.43 
 
 December 243 3.63
 
 INDEXES OF WAGES AND PRICES 69 
 
 6. From the data of Table XII-A, page 60, find index num- 
 bers of the food cost of living, 1913-1920, inclusive, and 
 June, 1921 ; base, 1913. Compare the results with the fol- 
 lowing index published by the Bureau of Labor Statistics : 
 
 Index of 
 Year food costs 
 
 1913 100 
 
 1914 102 
 
 1915 101 
 
 1916 114 
 
 1917 146 
 
 1918 168 
 
 1919 186 
 
 1920 203 
 
 June, 1921 144* 
 
 * Based on 43 articles. 
 
 7. From the average prices given below, find the relative 
 prices in 1918 as compared with 1913. Find also the 
 weighted average of these "relatives," applying the 
 weights given. 
 
 1913 1918 Weights 
 
 Wlieat, bushel $1.04 $2.31 4 
 
 Corn, bushel 71 1.84 10 
 
 Cotton, pound 13 .32 5 
 
 Iron, ton 14.90 36.52 3 
 
 Copper, pound 16 .25 1 
 
 8. On the basis of the following prices of five articles of 
 food, and the relative importance of each in the family 
 budget, find the increase in the food cost of living be- 
 tween the dates given : 
 
 Price in Price in 
 
 Article 1913 1920 Importance 
 
 Steak, pound $0.22 $0.40 70 
 
 Milk, quart 09 .17 140 
 
 Bread, pound 06 .12 140 
 
 Butter, pound 38 .70 115 
 
 Sugar, pound 06 .20 40 
 
 9. Compute a set of proportional expenditure weights on 
 the basis of 1901 consumption and 1913 prices. Apply 
 these weights to the relatives given in column 6, Table 
 XII, page 55.
 
 70 INTRODUCTION TO ECONOMIC STATISTICS 
 
 10. From the following table, compute an index of agricul- 
 tural real wages having 1913 as its base. Compare the 
 results with the corresponding index of hour rates in 
 the United States by graphing both on the same chart 
 for the years in which farm wages are given. (The base 
 line of the graph should show each year from 1875 to 
 1920. Why?) 
 
 WAGES OF CERTAIN CLASSES OF MALE FARM LABOR BY THE 
 MONTH WITHOUT BOARD 
 
 ("Monthly Labor Review," July, 1920, and March, 1921.) 
 
 Year Wage 
 
 1875 19.87 
 
 1879 16.42 
 
 1882 18.94 
 
 1885 17.97 
 
 1888 18.24 
 
 1890 18.33 
 
 1892 18.60 
 
 1893 19.10 
 
 1894 17.74 
 
 1895 17.69 
 
 1898 19.38 
 
 1899 20.23 
 
 1902 22.14 
 
 1910 27.50 
 
 1911 28.77 
 
 1912 29.58 
 
 1913 30.31 
 
 1914 29.88 
 
 1915 30.15 
 
 1916 32.83 
 
 1917 40.43 
 
 1918 48.80 
 
 1919 56.29 
 
 1920 64.95 
 
 11. The Bureau of Labor Statistics found the increases in 
 the cost of living for various classes of commodities in 
 the United States, from 1913 to the year and month 
 indicated to have been as follows:
 
 INDEXES OF WAGES AND PRICES 71 
 
 Per Cent of Increase 
 
 Item of Dec. Dec. Dec. Dec. Dee. Dec. Dec. 
 
 Expenditure 1914 1915 1916 1917 1918 1919 1920 
 
 Food 5.0 5.0 26.0 57.0 87.0 97.0 78.0 
 
 Clothing 1.0 4.7 20.0 49.1 105.3 168.7 158.5 
 
 Housing 0.0 1.5 2.3 .1 9.2 25.3 51.1 
 
 Fuel and light 1.0 1.0 8.4 24.1 47.9 56.8 94.9 
 
 Furniture and 
 
 furnishings 4,0 10.6 27.8 50.6 113.6 163.5 185.4 
 
 Miacellaneous 3.0 7.4 13.3 40.5 65.8 90.2 108.2 
 
 Using the weights given on page 62, find the total in- 
 crease in the cost of living, and the index of the same, 
 for the periods named in the foregoing table. Compare 
 the results with those given in the Monthly Labor Re- 
 view, November, 1921, page 83. 
 12. Using price quotations obtained locally, and from cata- 
 logs from which local purchases are commonly made, 
 find the increase in the cost of living between two given 
 dates, preferably 1913 or 1914 and the present time. In 
 finding the food index, make use of the proportional ex- 
 penditure weights given in Table XII, page 55. For 
 the other groups of items, make use of the following pro- 
 portional expenditure weights quoted from Massachusetts 
 House Report, No. 1500. Items for which quotations are 
 not available may be dropped, or substitutions may be 
 made. 
 
 WEIGHTINGS IN THE CLOTHING INDEX 
 MEN'S 
 
 Overcoat! 
 
 Suit \ 39 
 
 Trousers J 
 
 Shoes 15 
 
 Hats 4 
 
 Gloves 6 
 
 Socks 4 
 
 Shirts 6 
 
 Collars 2 
 
 Underwear 6 
 
 Night Garments 2 
 
 Total 84
 
 72 INTRODUCTION TO ECONOMIC STATISTICS 
 
 WOMEN 'S 
 
 Suit ] 
 
 Topcoat y • • 27 
 
 Street DressJ 
 
 Underwear 5 
 
 "Waists 
 Kimono 
 
 House Dress |^ 18 
 
 Aprons 
 Night Gown 
 Underskirt 
 
 Shoes : 12 
 
 Gloves 3 
 
 Hosiery 2 
 
 Corsets 4 
 
 Hats 9 
 
 Total 80 
 
 SHELTER INDEX 
 
 Obtain rentals of several representative homes for the 
 
 two dates required, and find the average per cent in- 
 crease. 
 
 WEIGHTINGS IN THE FUEL INDEX 
 
 Coal 10 
 
 Kerosene 1 
 
 Gas 2 
 
 Electricity 2 
 
 Total 15 
 
 WEIGHTINGS IN THE SUNDRY INDEX 
 
 Ice 15 
 
 Carfare 15 
 
 Entertainment 25 
 
 Medicine 25 
 
 Insurance 50 
 
 Church 30 
 
 Tobacco, etc 20 
 
 Reading 10 
 
 House furnishings 45 
 
 Organizations 25 
 
 Total 260
 
 INDEXES OF WAGES AND PRICES 73 
 
 Combine the various partial indexes by means of thp 
 following proportional expenditure weightsf 
 
 Food 43 J 
 
 Shelter * ' ' jy"™ 
 
 Clothing ' ' * ' 232 
 
 Fuel and light *.*.'.'.*.*.'."* 56 
 
 Sundries .'!.*!!!!' 20*4
 
 CHAPTER IV 
 QUANTITY INDEXES AND THEIR USES 
 
 Value and Quantity Indexes. Students of general 
 market conditions are interested in changes in the 
 volume of production, since such changes have an inti- 
 mate relation to prices, wages, and business activity. 
 The attention of statisticians has therefore been re- 
 cently turned to the development of indexes of pro- 
 duction. Such indexes may be of two sorts; (1) an 
 index of value production, measuring the number of 
 dollars' worth of goods created in a given period of 
 time; and (2) an index of physical production, meas- 
 uring the same output primarily in such physical units 
 as pounds, bushels, and yards. Somewhat akin to the 
 latter is an index of the physical volume of trade, 
 which attempts to measure the total number of physical 
 units traded in a given period. Because of the sea- 
 sonal nature of a large part of industry, indexes of 
 production are generally based on the year as the unit 
 of time.^ 
 
 Indexes of value production do not call for an ex- 
 tended discussion, since they are merely inventories of 
 representative annual output at the average current 
 prices. The process of finding them may be illus- 
 
 * One of the most practical uses of production index numbers, how- 
 ever, requires monthly or weekly data, corrected for seasonal variations. 
 These data are beginning to be used in connection with business 
 barometrics, as will be noted in the next chapter. 
 
 74
 
 QUANTITY INDEXES AND THEIR USES 75 
 
 trated by multiplying each year's output of wheat, 
 corn, cotton, pig iron, and copper, as shown in Table 
 XIV, by their respective prices as shown in Table 
 XIV- A. The resulting annual totals may be taken pro- 
 visionally as an index of value production of raw 
 materials, and may be reduced to any desired base. 
 Such indexes are not, however, of much use in them- 
 selves, except as they may be employed in connec- 
 tion with the computation of physical production and 
 price indexes. 
 
 If the total value production for a given year, as 
 measured by an adequate index, shows an increase over 
 the preceding year, this increase may evidently be at- 
 tributed either to a growing volume of physical pro- 
 duction, or to rising prices, or to a combination of both 
 factors. It therefore follows that suitable indexes of 
 general prices and of physical production, multiplied 
 across year by year, must give an index of value pro- 
 duction. This fundamental principle may be expressed 
 by the formula : ^ 
 
 PnQn = Vn 
 
 in which 
 
 Pn = the price index for a given year 
 
 Qn = the physical production index for the same 
 year 
 
 Vn = the value production index for the same year 
 The formula may also be written : 
 
 Vn Vn 
 
 Pn = 7^, and Qn = p- 
 
 vjn Xn 
 
 An Index of Physical Production. The statistical 
 
 * In applying this formula, it is preferable that the two given 
 indexes should be reduced to the same base, but this is not mathe- 
 matically essential. It is not the absolute value of the derived index 
 numbers that is important, but only their ratios to each other.
 
 TABLE XIV » 
 PRODUCTION OF SPECIFIED COMMODITIES, U. S., 1870-1920 
 
 Yeae 
 
 Wheat 
 
 Corn 
 
 Cotton 
 
 Pig Iron 
 
 Copper 
 (mil- 
 lions op 
 
 POUNDS) 
 
 (millions 
 
 (millions 
 
 (millions 
 
 (millions 
 
 
 OP bushels) 
 
 OF bushels" 
 
 OF bales) 
 
 OF tons) 
 
 1870 
 
 236 
 
 1,094 
 
 4.352 
 
 1.665 
 
 28 
 
 1871 
 
 231 
 
 992 
 
 2.974 
 
 1.707 
 
 29 
 
 1872 
 
 250 
 
 1,093 
 
 3.931 
 
 2.549 
 
 28 
 
 1873 
 
 281 
 
 932 
 
 4.170 
 
 2.561 
 
 35 
 
 1874 
 
 308 
 
 850 
 
 3.833 
 
 2.401 
 
 39 
 
 1875 
 
 292 
 
 1,321 
 
 4.632 
 
 2.024 
 
 40 
 
 1876 
 
 289 
 
 1,284 
 
 4.474 
 
 1.869 
 
 43 
 
 1877 
 
 364 
 
 1,343 
 
 4.774 
 
 2.067 
 
 47 
 
 1878 
 
 420 
 
 1,388 
 
 5.074 
 
 2.301 
 
 48 
 
 1879 
 
 449 
 
 1,548 
 
 5.755 
 
 2.742 
 
 52 
 
 1880 
 
 499 
 
 1,717 
 
 6.606 
 
 3.835 
 
 60 
 
 1881 
 
 383 
 
 1,195 
 
 5.456 
 
 4.144 
 
 72 
 
 1882 
 
 504 
 
 1,617 
 
 6.950 
 
 4.623 
 
 91 
 
 1883 
 
 421 
 
 1,551 
 
 5.713 
 
 4.596 
 
 116 
 
 1884 
 
 513 
 
 1,796 
 
 5.6S2 
 
 4.098 
 
 145 
 
 1885 
 
 357 
 
 1,936 
 
 6.576 
 
 4.045 
 
 166 
 
 1886 
 
 457 
 
 1,665 
 
 6.505 
 
 5.683 
 
 158 
 
 1887 
 
 456 
 
 1,456 
 
 7.047 
 
 6.417 
 
 181 
 
 1888 
 
 416 
 
 1,988 
 
 6.938 
 
 6.490 
 
 226 
 
 1889 
 
 491 
 
 2,113 
 
 7.473 
 
 7.604 
 
 227 
 
 1890 
 
 402 
 
 1,490 
 
 8.653 
 
 9.203 
 
 260 
 
 1891 
 
 612 
 
 2,060 
 
 9.035 
 
 8.280 
 
 284 
 
 1892 
 
 516 
 
 1,628 
 
 6.700 
 
 9.157 
 
 345 
 
 1893 
 
 396 
 
 1,619 
 
 7.493 
 
 7.125 
 
 329 
 
 1894 
 
 460 
 
 1,213 
 
 9.901 
 
 6.658 
 
 354 
 
 1895 
 
 467 
 
 2,151 
 
 7.161 
 
 9.446 
 
 381 
 
 1896 
 
 428 
 
 2,284 
 
 8.533 
 
 8.623 
 
 460 
 
 1897 
 
 530 
 
 1,903 
 
 10.898 
 
 9.653 
 
 494 
 
 1898 
 
 675 
 
 1,924 
 
 11.189 
 
 11.774 
 
 527 
 
 1899 
 
 5_47 
 
 2,078 
 
 9.393 
 
 13.621 
 
 569 
 
 '1900 . 
 1901 
 
 522 
 
 2n^05 
 
 10.102 
 
 13.789 
 
 606 
 
 748 
 
 1,523 
 
 8.583 
 
 16.878 
 
 602 
 
 1902 
 
 670 
 
 2,524 
 
 10.588 
 
 17.821 
 
 660 
 
 1903 
 
 638 
 
 2,244 
 
 9.820 
 
 18.009 
 
 698 
 
 1904 
 
 552 
 
 2,467 
 
 13,451 
 
 16.497 
 
 813 
 
 1905 
 
 693 
 
 2,708 
 
 10.495 
 
 22.992 
 
 889 
 
 1906 
 
 735 
 
 2,927 
 
 12.983 
 
 25.307 
 
 918 
 
 1907 
 
 634 
 
 2,592 
 
 11.058 
 
 25.781 
 
 869 
 
 1908 
 
 665 
 
 2,669 
 
 13.086 
 
 15.936 
 
 943 
 
 1909 
 
 737 
 
 2,772 
 
 10.073 
 
 25.795 
 
 1,093 
 
 1910 
 
 635 
 
 2,886 
 
 11.568 
 
 27.304 
 
 1,080 
 
 1911 
 
 621 
 
 2,531 
 
 15.553 
 
 23.650 
 
 1,097 
 
 1912 
 
 730 
 
 3,125 
 
 13.489 
 
 29.727 
 
 1,243 
 
 1913 
 
 763 
 
 2,447 
 
 13.983 
 
 30.966 
 
 1,224 
 
 1914 
 
 891 
 
 2,673 
 
 15.906 
 
 23.332 
 
 1,150 
 
 1915 
 
 1,026 
 
 2,995 
 
 11.068 
 
 29.916 
 
 1,388 
 
 1916 
 
 636 
 
 2,567 
 
 11.364 
 
 39.435 
 
 1,928 
 
 1917 
 
 637 
 
 3,065 
 
 11.302 
 
 38.621 
 
 . 1,890 
 
 1918 
 
 921 
 
 2,503 
 
 12.041 
 
 39.055 
 
 1,994 
 
 1919 
 
 941 
 
 2,917 
 
 11.421 
 
 31.015 
 
 1,289 
 
 1920 
 
 787 
 
 3,232 
 
 13.366 
 
 36.415 
 
 1,345 
 
 ^This table and the one following are taken with some modifications 
 from Babson 's Business Barometers, by permission of the author. 
 
 7G
 
 TABLE XIV— A 
 
 AVERAGE PRICES OF SPECIFIED COMMODITIES, U S 
 
 (EASTERN MARKETS), 1870-1920 ' 
 
 TEAR 
 
 WHEAT 
 
 CORN 
 
 COTTON 
 
 PIG IRON 
 
 COPPER 
 
 
 PER BU. 
 
 PER BU. 
 
 PER BALE 
 
 PER TON 
 
 PER LB. 
 
 1870 
 
 1.30 
 
 1.02 
 
 119.50 
 
 33.23 
 
 0.211 
 
 1871 
 
 1.60 
 
 .77 
 
 84.50 
 
 35.08 
 
 .241 
 
 1872 
 
 1.62 
 
 .70 
 
 110.50 
 
 48.94 
 
 .355 
 
 1873 
 
 1.76 
 
 .63 
 
 100.50 
 
 42.79 
 
 ,280 
 
 1874 
 
 1.39 
 
 .86 
 
 89.50 
 
 30.19 
 
 ,220 
 
 1875 
 
 1.33 
 
 .84 
 
 77.00 
 
 25.53 
 
 ,226 
 
 1876 
 
 1.35 
 
 .628 
 
 64.50 
 
 20.75 
 
 .210 
 
 1877 
 
 1.63 
 
 .593 
 
 59. 
 
 19.25 
 
 ,190 
 
 1878 
 
 1.24 
 
 ,535 
 
 56. 
 
 17.05 
 
 ,165 
 
 1879 
 
 1.24 
 
 .47 
 
 54. 
 
 22.82 
 
 .186 
 
 1880 
 
 1.30 
 
 .55 
 
 57.50 
 
 29.86 
 
 .214 
 
 1881 
 
 1.30 
 
 .62 
 
 60. 
 
 22.54 
 
 .181 
 
 1882 
 
 1.32 
 
 .77 
 
 57.50 
 
 23.20 
 
 .191 
 
 1883 
 
 1.17 
 
 .64 
 
 59. 
 
 19.62 
 
 .165 
 
 1884 
 
 1.00 
 
 .615 
 
 54.50 
 
 16.80 
 
 .110 
 
 1885 
 
 .94 
 
 .51 
 
 52. 
 
 15.20 
 
 .108 
 
 1886 
 
 .888 
 
 .52 
 
 46. 
 
 16.77 
 
 .110 
 
 1887 
 
 .88 
 
 .488 
 
 51. 
 
 20.05 
 
 .138 
 
 1888 
 
 .94 
 
 .593 
 
 50. 
 
 16.82 
 
 ,167 
 
 1889 
 
 .91 
 
 .438 
 
 53. 
 
 14.35 
 
 ,134 
 
 1890 
 
 .92 
 
 .485 
 
 55. 
 
 15.10 
 
 ,156 
 
 1891 
 
 1.05 
 
 .675 
 
 43. 
 
 13.78 
 
 .127 
 
 1892 
 
 .908 
 
 .54 
 
 38.50 
 
 12.74 
 
 .115 
 
 1893 
 
 .739 
 
 .499 
 
 42.50 
 
 11.42 
 
 .107 
 
 1894 
 
 .611 
 
 .509 
 
 34.50 
 
 9.93 
 
 .095 
 
 1895 
 
 .669 
 
 .477 
 
 37. 
 
 10.86 
 
 .105 
 
 1896 
 
 .781 
 
 .340 
 
 39.50 
 
 10.29 
 
 .109 
 
 1897 
 
 .954 
 
 .319 
 
 35. 
 
 9.42 
 
 .113 
 
 1898 
 
 .952 
 
 .376 
 
 29.50 
 
 9.46 
 
 .120 
 
 1899 
 
 .794 
 
 .413 
 
 34. 
 
 16.58 
 
 .177 
 
 1900 
 
 .804 
 
 .453 
 
 46. 
 
 17.04 
 
 .166 
 
 1901 
 
 .803 
 
 .567 
 
 43.50 
 
 13.61 
 
 .161 
 
 1902 
 
 .836 
 
 .684 
 
 45. 
 
 20.00 
 
 .116 
 
 1903 
 
 .853 
 
 .572 
 
 55.50 
 
 17.08 
 
 ,132 
 
 1904 
 
 1.107 
 
 .594 
 
 58.50 
 
 12.73 
 
 .128 
 
 1905 
 
 1.028 
 
 .593 
 
 49. 
 
 15.57 
 
 .156 
 
 1906 
 
 .865 
 
 .560 
 
 57.50 
 
 16.70 
 
 ,193 
 
 1907 
 
 .963 
 
 .640 
 
 60.50 
 
 23,10 
 
 ,200 
 
 1908 
 
 1.049 
 
 .786 
 
 53. 
 
 15.54 
 
 ,132 
 
 1909 
 
 1.263 
 
 .767 
 
 63. 
 
 16.12 
 
 ,131 
 
 1910 
 
 1.118 
 
 .668 
 
 75.50 
 
 15.16 
 
 ,129 
 
 1911 
 
 .963 
 
 .711 
 
 65. 
 
 13.67 
 
 .125 
 
 1912 
 
 1.091 
 
 .711 
 
 57.50 
 
 14,93 
 
 ,164 
 
 1913 
 
 1.041 
 
 .711 
 
 64. 
 
 14.90 
 
 .155 
 
 1914 
 
 1.094 
 
 .793 
 
 55.50 
 
 13.41 
 
 ,133 
 
 1915 
 
 1.291 
 
 .837 
 
 50.50 
 
 13.58 
 
 .174 
 
 1916 
 
 1.468 
 
 .929 
 
 72. 
 
 18.67 
 
 .272 
 
 1917 
 
 2.346 
 
 1.776 
 
 117.50 
 
 40.07 
 
 .272 
 
 1918 
 
 2.31 
 
 1.840 
 
 158.50 
 
 36.52 
 
 ,247 
 
 1919 
 
 2.34 
 
 1.771 
 
 161.50 
 
 32.16 
 
 .192 
 
 1920 
 
 2.65 
 
 1.669 
 
 173. 
 
 44.03 
 
 .175 
 
 77
 
 78 INTRODUCTION TO ECONOMIC STATISTICS 
 
 process of developing an index of physical production 
 may be illustrated by the use of the limited data shown 
 in Table XIV. In aggregating bushels, bales, tons, 
 and pounds for any given year, it will be hardly admis- 
 sible to add the units as tabulated. "While the numbers 
 might be taken abstractly and thus combined into an 
 index, yet such a process would give as much weight 
 to a bushel or a pound as to a bale or a ton. It is true 
 that we might here reduce all our units to pounds, but 
 even if this were done a pound of copper should be 
 stressed more than a pound of corn or of iron, because 
 of its greater importance in the markets. The simplest 
 way to give each item of production its proper place in 
 the total will be to remeasure it in terms of a standard 
 value. That is, we may take as the physical unit the 
 amount of each commodity that can be bought for a 
 dollar at a standard price. The number of such units 
 for each year may then be summated as an index of 
 physical production.^ Expressed algebraically, the 
 process is : 
 
 Qn = Sp^qn 
 
 In so far as the complete index is concerned, this is 
 equivalent to averaging the quantities as originally 
 tabulated, weighting them for standard price (pm). 
 
 A Standard Price. The term ' * standard price ' ' has 
 been used to imply a price which may be regarded as 
 representative of a given commodity for the whole in- 
 
 * It can be shown, as follows, that the product of price and quantity 
 can properly be taken as physical units: 
 Let p = price per pound in dollars 
 and n = number of pounds in a given output. 
 Then 1/p = number of pounds purchasable for one dollar (the new 
 
 physical unit), 
 and n -i- 1/p = np, the number of now physical units in the output.
 
 QUANTITY INDEXES AND THEIR USES 79 
 
 terval under consideration. Since the standard price 
 is to be used virtually as a weight, it need not be deter- 
 mined with very great accuracy. But from the point 
 of view of the 'Hheory of errors," it should be the 
 average price during the whole interval of time which 
 is covered by the series of quantity indexes dependent 
 upon it. It follows, therefore, that in periods of rap- 
 idly changing prices, somewhat different comparative 
 results may be obtained by varying the interval of 
 time over which the comparisons are made. This may 
 seem anomalous, but it arises inevitably from the fact 
 that the concept of aggregate quantity requires a unit 
 dependent upon value ; namely, a dollar's worth. Since 
 value is unstable, the unit of quantity is also unstable. 
 In practice, however, this instability is usually insig- 
 nificant. 
 
 The application of the foregoing principles to the 
 data of Table XIV requires the finding of a standard 
 price for each of the commodities tabulated. As a 
 base for this price, a period of about twenty years 
 prior to the war has been chosen, the purpose being to 
 exclude war prices as extreme, and to emphasize the 
 later rather than the earlier part of the half century 
 studied.^ The averages have been taken approxi- 
 mately, and have been slightly modified by the use of 
 data not here cited. They are as follows : 
 
 ^ Because of the shifting of relative prices, it might be theoretically 
 preferable to subdivide the half century under consideration into at 
 least three different periods (e. g., 1870-1896, 1896-1914, and 1914- 
 1920), and to derive different sets of weights for use in each period. 
 The series of indexes could be brought together in the overlapping 
 years by a simple adjustment of the bases. But the slight gain in 
 accuracy thus made would not be worth while here, since in any case 
 the results obtained from such meager data cannot be considered as 
 anything more than approximations.
 
 80 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Wheat, per bushel $1.04 
 
 Corn, per bushel 64 
 
 Cotton, per bale 56.00 
 
 Pig Iron, per ton 16.00 
 
 Copper, per pound 16 
 
 In applying these prices, it should be remembered that 
 they are in effect weights, and so may be treated as a 
 multiple ratio. They may therefore be changed by 
 division into the more convenient form 13 :8 :700 :200 :2. 
 Physical Production in the United States. After the 
 standard prices have been decided upon, each annual 
 item of production is multiplied by its appropriate 
 weight. Two sub-totals are taken for each year, one 
 for crops and one for minerals. These sub-totals con- 
 stitute two provisional index series. The completed 
 indexes are reduced to a base of 1913. A further step 
 may be taken by noting the fact, as shown in various 
 statistical studies, that an index of mineral production 
 runs very close to an index of manufactures. By suit- 
 able weighting, the indexes of crops and iitinerals may 
 therefore be combined so as to include^ in effect, an 
 estimate for the value added by manufacturing. By 
 a comparison of aggregate values and a little experi- 
 mentation, the requisite weights may be placed at six 
 for crops and four for minerals, manufactures being 
 theoretically included in the latter. This process of 
 weighting and combining is admittedly very crude, 
 and would not be expected ordinarily to give more 
 than a rough approximation to a comprehensive index. 
 But it happens in this case that the five commodities 
 on which the work is based are very dependable and 
 typical as far as production is concerned. The index
 
 TABLE XV 
 
 INDEXES OF PHYSICAL PEODUCTION, UNITED STATES, 
 
 1870-1920 
 
 YEAR 
 
 CROPS 
 
 MINERALS 
 
 GEN ER All 
 
 AGGREGATE 
 
 PER CAPITA 
 
 1870 
 
 38 
 
 5 
 
 25 
 
 61 
 
 1871 
 
 33 
 
 5 
 
 22 
 
 53 
 
 1872 
 
 38 
 
 7 
 
 25 
 
 60 
 
 1873 
 
 36 
 
 7 
 
 24 
 
 56 
 
 1874 
 
 34 
 
 6 
 
 23 
 
 52 
 
 1875 
 
 45 
 
 6 
 
 29 
 
 64 
 
 1876 
 
 44 
 
 5 
 
 28 
 
 60 
 
 1877 
 
 48 
 
 6 
 
 31 
 
 65 
 
 1878 
 
 51 
 
 6 
 
 33 
 
 67 
 
 1879 
 
 57 
 
 8 
 
 37 
 
 73 
 
 1880 
 
 63 
 
 10 
 
 42 
 
 81 
 
 1881 
 
 47 
 
 11 
 
 33 
 
 61 
 
 1882 
 
 62 
 
 13 
 
 42 
 
 78 
 
 1883 
 
 56 
 
 13 
 
 39 
 
 69 
 
 1884 
 
 64 
 
 13 
 
 43 
 
 76 
 
 1885 
 
 63 
 
 13 
 
 43 
 
 74 
 
 1886 
 
 61 
 
 17 
 
 43 
 
 72 
 
 1887 
 
 57 
 
 19 
 
 42 
 
 69 
 
 1888 
 
 67 
 
 20 
 
 48 
 
 77 
 
 1889 
 
 73 
 
 23 
 
 53 
 
 83 
 
 1890 
 
 59 
 
 27 
 
 46 
 
 71 
 
 1891 
 
 78 
 
 26 
 
 57 
 
 86 
 
 1892 
 
 62 
 
 29 
 
 49 
 
 72 
 
 1893 
 
 59 
 
 24 
 
 45 
 
 66 
 
 1894 
 
 58 
 
 24 
 
 44 
 
 63 
 
 1895 
 
 72 
 
 31 
 
 55 
 
 77 
 
 1896 
 
 76 
 
 31 
 
 58 
 
 79 
 
 1897 
 
 76 
 
 34 
 
 59 
 
 79 
 
 1898 
 
 81 
 
 39 
 
 65 
 
 85 
 
 1899 
 
 77 
 
 45 
 
 64 
 
 83 
 
 1900 
 
 78 
 
 46 
 
 65 
 
 83 
 
 1901 
 
 71 
 
 51 
 
 63 
 
 78 
 
 1902 
 
 92 
 
 57 
 
 78 
 
 95 
 
 1903 
 
 84 
 
 58 
 
 74 
 
 88 
 
 1904 
 
 92 
 
 57 
 
 78 
 
 91 
 
 1905 
 
 97 
 
 74 
 
 88 
 
 100 
 
 1906 
 
 107 
 
 80 
 
 96 
 
 108 
 
 1907 
 
 93 
 
 80 
 
 88 
 
 97 
 
 1908 
 
 100 
 
 59 
 
 83 
 
 90 
 
 1909 
 
 99 
 
 85 
 
 93 
 
 99 
 
 1910 
 
 100 
 
 88 
 
 96 
 
 100 
 
 1911 
 
 100 
 
 80 
 
 92 
 
 95 
 
 1912 
 
 112 
 
 98 
 
 106 
 
 108 
 
 1913 
 
 100 
 
 100 
 
 100 
 
 100 
 
 1914 
 
 112 
 
 81 
 
 100 
 
 98 
 
 1915 
 
 115 
 
 101 
 
 109 
 
 106 
 
 1916 
 
 94 
 
 136 
 
 110 
 
 106 
 
 1917 
 
 104 
 
 133 
 
 115 
 
 109 
 
 1918 
 
 102 
 
 137 
 
 116 
 
 108 
 
 1919 
 
 111 
 
 102 
 
 107 
 
 99 
 
 1920 
 
 115 
 
 110 
 
 113 1 103 
 
 81
 
 82 INTRODUCTION TO ECONOMIC STATISTICS 
 
 obtained from them in fact cheeks remarkably well 
 with Stewart's (1890-1920) and Day's (1899-1920) in- 
 dexes of physical production. The results, put into 
 index form, are sho^vn in Table XV. 
 
 Price Indexes. The theory of price indexes ^ may be 
 advantageously reviewed in the light of the principles 
 discussed in this chapter. It will be apparent that a 
 price index, to be theoretically precise, must conform 
 to the equation, 
 
 Since this equation is valid whether the indexes sub- 
 stituted in it have been reduced to a common base or 
 expressed merely in dollars and ''dollar's worths," it 
 may be written, by the use of formulas previously con- 
 sidered : 
 
 p _ ^Pnq° 
 
 X n — "^T ' 
 
 ^Pmqn 
 
 The index numbers obtained by the direct use of this 
 formula will have as their base approximately an aver- 
 age index, as determined by the use of average prices 
 in the denominator. If, however, the indexes of value 
 and quantity have first been reduced to a given base, 
 then the price indexes will have that base. 
 
 * The theories of price and quantity indexes discussed in this book 
 involve the use of the same list of items from one date to another. If 
 a change in the number or character of the items is to be made, as the 
 change from 22 to 43 items in the Bureau of Labor Statistics' index 
 of retail food prices, it is assumed that a new series is begun, and is 
 connected with the old merely by an adjustment of the new base so 
 as to make the indexes at the two overlapping dates agree. The difficult 
 theoretical problem of constructing an index based upon continually 
 shifting lists of commodities is not considered, since it has little imme- 
 diate practical value. Production indexes, however, to be quite accurate, 
 ought to be gradually broadened to take account of the growing diversi- 
 fication of industry. Hence data such as freight and canal tonnage, 
 which reflect this increasing diversification, are useful in measuring pro- 
 duction.
 
 QUANTITY INDEXES AND THEIR USES 83 
 
 For the purpose of making a comparison with the 
 proportional expenditure method, the formula may 
 be modified by twice inserting the factor pm in such a 
 way that it will cancel from each term of the numerator 
 summation, as follows : 
 
 vP° v/ 
 
 ■p _ P'" 
 
 J- n -^ 
 
 ^Pmqn 
 
 The formula as thus written indicates relatives based 
 upon standard prices, and an averaging of these rela- 
 tives with weights consisting of contemporary physical 
 quantities measured in units of a dollar's worth at 
 standard prices. Briefly stated, the formula means 
 relatives weighted for "dollar's worths." Of course 
 in actual work the simpler form previously stated 
 ^ould be used; the latter form is given merely for 
 comparison and interpretation. Since this method of 
 finding a price index involves a standardization of 
 quantity units throughout a given period, it may be 
 called the method of standard quantities. 
 
 An Approximate Method. The averaging of prices by 
 the use of quantity weights suggests an approximate 
 method for finding price indexes which may here be 
 stated by way of further comparison. This method 
 uses actual rather than relative prices, and is weighted 
 by the use of the average physical quantities (qm) 
 produced, consumed, or traded during the period of 
 time in question. It is analogous to the method used 
 in finding quantity indexes. Its formula is : 
 
 Pn — Sp^qm 
 The series of indexes so derived may be reduced to any 
 desired base. The formula is a convenient one, but
 
 84 INTRODUCTION TO ECONOMIC STATISTICS 
 
 it is not theoretically valid, and therefore will not give 
 very dependable results. It fails to meet the test im- 
 plied in the formula : 
 
 PnQn = Vn 
 
 Theoretical Difficulties. Certain theoretical difficul- 
 ties encountered in developing a method of finding a 
 price index may be revealed by approaching the prob- 
 lem from another angle. Suppose, first, that we had to 
 deal merely with one commodity having a value for a 
 given year of four dollars, and for the succeeding 
 year of five dollars. If the first year is taken as a 
 base, the price index for the second year is 125. If 
 the second year is taken as a base, the price index for 
 the first year is 80. These two results are consistent, 
 as may be shown by the fact that 80% times 125% is 
 unity, or by the fact that the reciprocal of the index 80 
 is the index 125. 
 
 Let us now consider an analogous problem involving 
 two commodities, with quantities and prices stated, as 
 follows : 
 
 FIRST YEAR 
 
 (q) (P) (v) 
 
 Commodity A, 10 lbs. @ $4 = $40 
 Commodity B, 3 bu. @ 7 = 21 
 
 Value index 61 
 
 SECOND YEAR 
 
 (q) (P) (v) 
 
 Commodity A, 12 lbs. @ $5 = $60 
 Commodity B, 2 bu. @ 6 = 12 
 
 Value index 72
 
 QUANTITY INDEXES AND THEIR USES 85 
 
 Instead of making use of average prices as standards 
 in computing quantities, we shall follow a common 
 usage and take the prices of the base year. Considering 
 the first year as the base, and its prices as the stand- 
 ards for finding quantity units (dollar's worths), we 
 have 61 as both the value and the quantity index. The 
 price index, V -f- Q, is therefore 100, as it should be in 
 the base year. In the second year V = 72 and Q = 62, 
 the latter result being obtained by summating the 
 quantities of the second year at the standard prices 
 of the preceding year. The price index for the second 
 year is therefore 72% -^-62%, or 116%, as compared 
 with 100% for the first year. 
 
 Reversing the computation, we now take the second 
 year as the base year, and its prices as the standard 
 prices. The price index for the second year will now 
 come to 100, since the value index is identical with the 
 quantity index. For the first year the value index will 
 be 61 as before; and the quantity index, obtained by 
 summating the quantities at the standard prices of the 
 second year, will be 68. The price index for the first 
 year is therefore 61% -^68%, or 90%. 
 
 Are our two results, obtained from different bases, 
 consistent with each other, as they were when only 
 one price index was used"? If so, the product of the 
 two indexes, 116% and 90%, should be unity. But 
 their product is actually 104%). The other test is that 
 the reciprocal of the index 90% should equal the index 
 116%. But actually the reciprocal of 90% is 111%. 
 
 It is obvious that the lack of consistency arises from 
 the failure to standardize quantity units by the use of 
 a constant price. If, now, we apply the method of
 
 86 INTRODUCTION TO ECONOMIC STATISTICS 
 
 standard quantities, we obtain the following results for 
 the two years, respectively : 
 
 P. = |=94 P. = r7=107 
 
 Considering the first year as a base, we obtain an in- 
 dex of 114 for the second year. If the second year is 
 taken as the base, a consistent result will necessarily 
 be obtained. It will be seen that the index for the 
 second year thus obtained is nearly midway between 
 the two indexes, 116 and 111, obtained before. 
 
 Fisher's Index. Thus the standard quantity method 
 may be used to avoid the inconsistency encountered 
 when different standard prices are used in measuring 
 quantities. But Professor Fisher has proposed an- 
 other solution which he regards as the theoretical ideal. 
 His method involves the finding of the geometrical 
 average of the two inconsistent results, as seen in the 
 two indexes, 116 and 111. This method will here give 
 a result of 114, which is equal to that obtained by 
 the method of standard quantities, though as a rule 
 the results are only approximately equal. The latter 
 method has a distinct advantage in that it will give a 
 consistent series of index numbers for a required inter- 
 val of time, and the series may be reduced to any de- 
 sired base. The weight of authority, however, supports 
 Professor Fisher's solution as being theoretically the 
 best. His formula, which is somewhat too complex for 
 ordinary use, is as follows : 
 
 2poqn Spoqo 
 in which the subscripts o and n indicate a base year 
 and a relative year, respectively. The development of
 
 QUANTITY INDEXES AND THEIR USES 87 
 
 the formula is sufficiently explained in the foregoing 
 discussion. The corresponding formula for quantities 
 (Qn) may be written by interchanging each p and q 
 in the price formula. 
 
 It will hardly be profitable to attempt to follow any 
 further the theoretical problems connected with the 
 compiling of price and quantity indexes. It must be 
 emphasized that at present the more significant prac- 
 tical problems relate to the securing of adequate and 
 reliable data rather than to the precision of the meth- 
 ods used in their elaboration. The choice of method, 
 as a rule, will not involve the danger of serious error ; 
 while work may be quite invalidated by deficiencies 
 and inaccuracies in the data. An understanding of the 
 theory of the subject will nevertheless be found to have 
 a practical value in the planning of work and in the 
 evaluation of results. 
 
 Quantity Theory Indexes. An interesting applica- 
 tion of price and production indexes to the measure- 
 ment of general business changes may be made on the 
 basis of the quantity theory of money. This theory 
 in its simplest form states that prices (P) change 
 directly as the circulating medium (M) and its rate of 
 circulation (R), and inversely as the number of phys- 
 ical units traded (N). This statement is expressed 
 algebraically as the following equation of exchange : 
 
 MR 
 ^~ N • 
 The equation may be elaborated by subdividing the 
 circulating medium into its various elements, as money 
 and credit, and dealing with each on the basis of its 
 own average rate of circulation. But as it is difficult to
 
 88 INTRODUCTION TO ECONOMIC STATISTICS 
 
 obtain accurate data bearing upon circulation rates, 
 such elaboration will not be attempted here. It should 
 also be noted concerning the quantity theory that there 
 has been much argument as to its validity. But the 
 arguments have not been concerned primarily with the 
 statistical aspects of the subject, to which little objec- 
 tion has been made. They have dealt, rather, with the 
 origin of changes in the equilibrium expressed by the 
 equation. 
 
 In order to make a certain provisional use of the 
 formula, further data must be adduced. Values for 
 the term M may be obtained from a study made by 
 Professor E. W. Kemmerer, entitled. High Prices 
 and Deflation, which gives estimates of the money 
 and bank credit in circulation during the war period. 
 Brought down to 1920, these estimates are as follows : 
 
 Circulation 
 (Millions of dollars) 
 Year Money Deposits 
 
 1913 3,390 12,678 
 
 1914 3,505 13,430 
 
 1915 3,682 14,411 
 
 1916 4,159 17,840 
 
 1917 4,914 21,273 
 
 1918 5,579 23,771 
 
 1919 5,793 27,928 
 
 1920 6,060 30,300 
 
 In combining money and deposits into a single index 
 of circulating medium, it is necessary to multiply de- 
 posits by two because they circulate, in the form of 
 checks, about twice as fast as money. The numbers
 
 QUANTITY INDEXES AND THEIR USES 89 
 
 obtained by addition may then be reduced to a 1913 
 base. For the value of iV, index numbers of physical 
 production will here be substituted. It is assumed that 
 this may be done because, as a rule, the volume of 
 trade parallels the volume of production. For the 
 value of P, the Bureau of Labor Statistics* index of 
 wholesale prices will be used. 
 
 An index of the rate of currency circulation may 
 now be readily obtained algebraically by the use of the 
 equation of exchange expressed in the form 
 
 PN 
 
 ^- "ir- 
 
 The resulting index will not, of course, register at all 
 accurately the percentage changes in the rate of cur- 
 rency circulation, owing to the fact that changes in the 
 rate of circulation of goods have been omitted by the 
 substitution of volume of production for volume of 
 trade. But it should show changes in the activity of 
 business by registering the increases or decreases of 
 monetary circulation relative to corresponding changes 
 in the circulation of goods. That is, it should indicate 
 the relative activity of the consumption market. 
 Speculative flurries, which increase the circulation of 
 both currency and securities, will therefore not regis- 
 ter at all. That the index does, in fact, show changes 
 in the activity of business from year to year with some 
 degree of certainty, is apparent from the figures. The 
 depression of 1914, the war activity of 1917, the tem- 
 porary slump of 1919, and the boom culminating in 
 1920 are all indicated. 
 
 The indexes entering into the equation of exchange 
 as just discussed are given below. Taken together,
 
 90 INTRODUCTION TO ECONOMIC STATISTICS 
 
 they present an epitome of the general movement of 
 business. Prices vary directly with purchasing power, 
 MR (demand), and inversely with production, N (sup- 
 ply). It should be observed that the values of R and N 
 here used are mere approximations. They are prob- 
 ably too high in 1914 and 1915, and too low in 1916. 
 
 Year 
 
 (P) 
 
 (M) 
 
 (E) 
 
 (N) 
 
 1913 
 
 100 
 
 100 
 
 100 
 
 100 
 
 1914 
 
 100 
 
 106 
 
 94 
 
 100 
 
 1915 
 
 101 
 
 113 
 
 97 
 
 109 
 
 1916 
 
 124 
 
 139 
 
 98 
 
 110 
 
 1917 
 
 176 
 
 165 
 
 123 
 
 115 
 
 1918 
 
 196 
 
 185 
 
 123 
 
 116 
 
 1919 
 
 212 
 
 214 
 
 106 
 
 107 
 
 1920 
 
 243 
 
 232 
 
 118 
 
 113 
 
 A complete statistical verification of the equation of 
 exchange would necessitate ampler data and more ex- 
 tended computations. The point of greatest difficulty 
 would be found to be the construction of an index of 
 the physical volume of trade (N). Theoretically this 
 should include all wealth and services traded in a given 
 period of time, measured in physical units of ''dollar's 
 worths" at average prices for the whole interval over 
 which the computation extends. Since MR is equal to 
 the value of the same quantities at current prices, the 
 equation of exchange reduces to the formula for find- 
 ing price index numbers by the standard quantities 
 method ; thus : 
 
 ° ~ N ~ Spmqn 
 
 Of course in actual work a process of sampling must 
 necessarily be employed, and various methods for ag-
 
 QUANTITY INDEXES AND THEIR USES 91 
 
 gregating quantities may be resorted to. The price 
 index thus obtained would theoretically vary somewhat 
 from the usual type of index, because the quantities in- 
 dicated in the formula refer to trade rather than to 
 production or consumption. Hence goods in which 
 there is unusually active speculation will in effect be 
 weighted heavily. 
 
 For discussions of the quantity theory and statis- 
 tical verifications of the equation of exchange the stu- 
 dent may profitably consult Kemmerer, Money and 
 Credit Instruments in their Relation to General Prices; 
 Fisher, The Purchasing Power of Money; and articles 
 published by King in the weekly service of the 
 Bankers' Statistics Corporation (1920). 
 
 Measurement of the National Income. We have thus 
 far considered production indexes principally with ref- 
 erence to their use in connection with price indexes. 
 But they are also of direct significance in that they 
 indicate changes in the total national income. By 
 national income is meant the sum of all consumption 
 values in economic goods, services, and property 
 usances, together with increases in stores of goods and 
 extensions of property. This conception differs slightly 
 from aggregate income as privately reckoned, inas- 
 much as the latter includes increases in capitalized 
 values, particularly of land. 
 
 While production indexes show changes in the na- 
 tional income, they serve merely as ratios unless sup- 
 plemented by estimates of the total income for one or 
 more years. Several such studies have been attempted, 
 but the most complete and authoritative study is one 
 recently made by the National Bureau of Economic
 
 92 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Research. This Bureau is a private organization 
 chartered in 1920 for the purpose of conducting statis- 
 tical investigations into subjects affecting the public 
 welfare. It is controlled by a board of nineteen di- 
 rectors representing various points of view and in- 
 terests. Its staff consists of Wesley C. Mitchell, Will- 
 ford I. King, Frederick R. Macauley, and Oswald W. 
 Knauth. This organization has recently issued a mon- 
 ograph containing estimates of the national income 
 for each of the years 1909-1918, inclusive. The mono- 
 graph is worthy of careful study, not only for its con- 
 clusions, but also as an excellent example of applied 
 statistical methods. The estimates of income were 
 made independently on two different bases; first, by 
 sources of production, and second by incomes re- 
 ceived. The two sets of results agree very closely, 
 the maximum difference being 6.9% in 1913. They 
 were averaged to give the final estimates, which were 
 reduced to terms of 1913 prices. As thus reduced, the 
 indexes were as follows, the income for 1913 being 
 34.4 billion dollars : 
 
 Year 
 
 Index 
 
 1909 
 
 88 
 
 1910 
 
 94 
 
 1911 
 
 92 
 
 1912 
 
 97 
 
 1913 
 
 100 
 
 1914 
 
 96 
 
 1915 
 
 102 
 
 1916 
 
 118 
 
 1917 
 
 119 
 
 1918 
 
 113 
 
 The Distribution of Income. 
 
 The National Bureau 
 
 of Economic Research has also made a study of the 
 individual distribution of income in 1918, based in
 
 QUANTITY INDEXES AND THEIR USES 93 
 
 large part on income tax returns. The total income 
 for that year was found to be about 58 billion dollars, 
 and the number of personal income recipients was 
 37.6 million, exclusive of men in active service in re- 
 spect to both items.^ The average income was $1,543 ; 
 the mode, $957; and the three quartiles were $833, 
 $1,140, and $1,574, respectively. The distribution is 
 summarized in the following derived tables : 
 
 FEEQUENCY DISTRIBUTION OF NATIONAL INCOME IN PER- 
 CENTAGES OF CIVILIAN INCOME RECIPIENTS, U. S., 1918 
 
 Income class 
 
 Per cent 
 
 Under $400 
 
 2.84 
 
 $ 400- 800 
 
 19.51 
 
 800-1200 
 
 32.16 
 
 1200-1600 
 
 21.53 
 
 1600-2000 
 
 9.88 
 
 2000-2400 
 
 4.64 
 
 2400-2800 
 
 2.63 
 
 2800-3200 
 
 1.61 
 
 3200-3600 
 
 1.07 
 
 3600-4000 
 
 .74 
 
 4000 & over 
 
 3.39 
 
 E PERCENTAGES OF 
 
 CIVILIAN INCOMl 
 
 3 AND OF NATIONAL 
 
 INCOME, U. S., 19 
 
 Recipients 
 
 Aggregate 
 
 of incomes 
 
 incomes 
 
 10 
 
 2.7 
 
 20 
 
 7.2 
 
 30 
 
 12.5 
 
 40 
 
 18.7 
 
 50 
 
 26.0 
 
 60 
 
 33.6 
 
 70 
 
 42.2 
 
 80 
 
 52.5 
 
 90 
 
 65.2 
 
 99 
 
 86.2 
 
 100 
 
 100.0 
 
 RECIPI- 
 
 ^When all are included, the average income becomes $1490.
 
 94 IXTEODUCTIOX TO ECOXOMIC STATISTICS 
 
 The first of these tables may he graphed as a fre- 
 quency curve, omittiug the last class of "$4,000 and 
 over," which if accurately represented should be ex- 
 tended to include incomes of several miUious.^ The 
 second may be graphed as a Loreuz curve, the first 
 column being represented on the base line, and the 
 vertical scale being drawn equal to the horizontal scale. 
 A diagonal drawn upward from left to right — the fine 
 of equal distribution — will serve as a basis of compari- 
 son. Or. the second table may be more simply graphed 
 by obtaining by subtraction the non-cumulative per- 
 centages of aggregate income, and representing them 
 as successive vertical blocks above a base hue on which 
 are measured the successive ten per cent groups of 
 income recipients. 
 
 Pareto's Law. If the percentages in the fre- 
 quency distribution are summated in reverse order, 
 and are plotted vertically against the corresponding 
 lower class limits on double-logarithmic paper (double 
 cycle on each scale), the so-called Pareto's law of in- 
 come distribution will be illustrated. This law states 
 
 ^ If the frequency distribution of incomes is graphed, as repre- 
 sented, on semi-logarithmic paper, with the number of incomes (income 
 recipients) plotted on the vertical arithmetic scale, and the magnitude 
 of the incomes plotted on the horizontal logarithmic scale, a somewhat 
 regular frequency curve is formed. This indicates a type of distribu- 
 tion which is normally symmetrical when the ratio departures from 
 the geometric mean, rather than the differences from the arithmetic 
 mean, are taken as the basis for measuring the dispersioiL In such a 
 distribution the median is therefore approximately the geometric mean 
 of the first and third quartiles. instead of the arithmetic mean. 
 
 S— 
 2— 
 1— 
 0— 
 
 I
 
 QUAXTITY INDEXES AXD THEIR USES 95 
 
 that the cm-ve of income distribution above the mode, 
 when logarithmically plotted, approximates a straight 
 line. Much the same result may also be obtained by 
 transferring the Lorenz cuiTe to double-logarithmic 
 paper ; or it may be more simply evidenced by plotting 
 the original frequency curve on similar paper. Pa- 
 reto's law merely caUs attention to the type of fre- 
 quency curve normally followed by income data. Thi^ 
 is a curve which when graphed in the ordinary manner 
 is seen to be strongly skewed to the right — somewhat 
 the same type as was suggested by the wage data 
 studied in the first two chapters. In foiTuulating the 
 law Pareto thought he had discovered a somewhat in- 
 flexible and permanent fact of economic relationships. 
 While he doubtless over-emphasized the inflexibility of 
 the law, yet he nevertheless pointed out an interesting 
 illustration of the strong tendency to statistical regu- 
 larity inherent in biological and social phenomena- 
 Income in Other Countries. The National Bureau 
 of Economic Eesearch has also made estimates of the 
 per capita income in several countries for the year 
 1914. These estimates are based upon prior studies 
 made by a well-known EngHsh authority. Sir Josiah 
 Stamp. The Bureau's data give the following ratios: 
 
 United States 
 
 100% 
 
 AustraUa 
 
 79 
 
 Tnited Kingdom 
 
 73 
 
 Canada 
 
 5S 
 
 France 
 
 55 
 
 Germanv 
 
 U 
 
 Italy 
 
 33 
 
 Austria-Hungary 
 
 30 
 
 Spain 
 
 16 
 
 Japan 
 
 9
 
 96 INTRODUCTION TO ECONOMIC STATISTICS 
 
 The Distribution of Property. A problem closely 
 related to the distribution of income is the distribution 
 of properiy ownership. For the United States the best 
 known investigation of the subject is one published by 
 Professor W. I. King in 1915. The total wealth for 
 1910 was estimated to have been about 200 billion dol- 
 lars, and its ownership was distributed in a form that 
 is suggested by the curve of income. The richest two 
 per cent of the families were thought to own over fifty 
 per cent of the property. But the data on which such 
 estimates rest involve many uncertainties, and the 
 meaning of the results is further obscured by lack of 
 knowledge of the rate at which fortunes are acquired 
 and dissipated. 
 
 REFERENCES 
 
 National Bureau of Economic Research, Income in the United 
 States, Volume I. 
 
 Day, Edmund E., **An Index of the Physical Volume of Pro- 
 duction," Review of Economic Statistics, September, 1920- 
 January, 1921. 
 
 Fisher, Irving, "The Best Form of Index Numbers" (and 
 discussion), Quarterly Publications of the American Statis- 
 tical Association, March, 1921, pp. 533-551. 
 
 Fisher, Irving, The Purchasing Power of Money. 
 
 Hoffman, F. L., "The Economic Progress of the United States 
 During the Last Seventy -five Years," Quarterly Publica- 
 tions of the American Statistical Association, December, 
 1914, pp. 294-318. For further data on the same topic, see 
 Appendix III. 
 
 Ingalls, W. R., "Labor the Holder of the Nation's Wealth and 
 Income," The Annalist, September 13, 20, and 27, 1920. 
 
 King, W. I., Wealth and Income of the People of the United 
 States. 
 
 Meeker, Royal, "On the Best Form of Index Numbers," 
 Quarterly Publications of the American Statistical Asso- 
 ciation, Sept., 1921, pp. 909-915. 
 
 Stewart, Walker W., "An Index Number of Production" 
 (and discussion), American Economic Review, March, 1921, 
 pp. 57-81.
 
 QUANTITY INDEXES AND THEIR USES 97 
 
 Walsh, C. M., The Measurement of General Exchange Value. 
 Working, Holbrook, ' ' What is to be the Future Price Level ? ' ' 
 The Annalist, June 27, 1921, p. 686. 
 
 EXERCISES 
 
 1. From Tables XIV and XIV- A (pages 76 and 77) find an 
 index of value production (a) for crops and (b) for min- 
 erals. Reduce each index to a 1913 base. 
 
 2. Divide each item in the index of value production of 
 minerals just obtained, by the corresponding item in 
 the index of physical production of minerals in the 
 United States. Explain the results. 
 
 3. Divide each item in the index of value production of 
 crops by the corresponding item in the index of physical 
 production of crops. Explain the results. 
 
 4. Plot on semi-logarithmic paper the index obtained in the 
 preceding exercise, together with the index of whole- 
 sale prices in the United States. Explain the divergence 
 of the two trends. 
 
 5. ''The Monthly Review," issued at the Federal Reserve 
 Bank of New York, gives the value of the ten leading 
 crops in the United States at average (standard) prices 
 for recent years as follows: 
 
 Value 
 (millions of ■ 
 Year dollars) 
 
 1910 5,873 
 
 1911 5,491 
 
 1912 6,549 
 
 1913 5,750 
 
 1914 6,397 
 
 1915 6,831 
 
 1916 5,907 
 
 1917 6,507 
 
 1918 6,418 
 
 1919 6,626 
 
 1920 7,284 
 
 1921 6,118 
 
 From these data derive an index of agricultural produc- 
 tion (base, 1913), and compare it graphically with the 
 corresponding index computed from three leading crops 
 (Table XV, page 81).
 
 98 INTRODUCTION TO ECONOMIC STATISTICS 
 
 6. From the data of crops and prices given below, find index 
 numbers of physical production, value production, and 
 prices, for the year 1914. Express each index in terms 
 of 1913 as a base. (It is assumed that the prices of 1913 
 represent average prices for a certain period.) 
 
 Wheat Corn Cotton 
 
 Year Bu. Price Bu. Price Bales Price 
 
 1913 760 $1.05 2450 $0.72 14 $61.00 
 
 1914 890 1.10 2670 .80 16 53.00 
 
 7. From the following data find indexes of the four terms 
 used in the equation of exchange (quantity theory) for 
 the year 1910, taking the year 1900 as the base. 
 
 Year: 1900 1910 
 
 Price index 80 100 
 
 Money in circulation 2.2 3 (billions) 
 
 Deposits subject to check.... 9.4 14.25 '' 
 Physical production 20 30 " 
 
 8. Plot together on semi-logarithmic paper the index of 
 wholesale prices in the United States for 1890 to 1920 
 inclusive, and a similar index (base, 1913) of per capita 
 circulation derived from the following data : 
 
 1890 $22.82 1905 $31.08 
 
 23.45 32.32 
 
 24.60 32.22 
 
 24.06 34.72 
 
 24.56 34.93 
 
 1895 23.24 1910 34.33 
 
 21.44 34.20 
 
 22.92 34.34 
 
 25.19 34.56 
 
 25.62 34.35 
 
 1900 26.93 1915 35.44 
 
 27.98 39.29 
 
 28.43 45.74 
 
 29.42 50.81 
 
 30.77 54.33 
 1920 57.04 
 
 9. The two indexes of physical production in the United 
 States (1913-1920) given below, are adapted (a) from 
 Day's and (b) from Stewart's comprehensive studies.
 
 QUANTITY INDEXES AND THEIR USES 99 
 
 Using data for prices and circulating medium cited in the 
 discussion of the quantity theory of money, derive two 
 indexes for R, 1913-1920, and compare them with the 
 corresponding index included in the data just men- 
 tioned. 
 
 Year (A) (B) 
 
 1913 
 
 100 
 
 100 
 
 1914 
 
 98 
 
 100 
 
 1915 
 
 105 
 
 111 
 
 1916 
 
 111 
 
 116 
 
 1917 
 
 114 
 
 123 
 
 1918 
 
 113 
 
 124 
 
 1919 
 
 106 
 
 119 
 
 1920 
 
 111 
 
 122
 
 CHAPTER V 
 TRENDS AND CYCLES 
 
 The Nature of Trends. The interpretation of a time 
 series of index numbers usually requires the determi- 
 nation of the trend. A trend is a derived series of in- 
 dex numbers following the general course of the given 
 items, but shortening or eliminating the fluctuations. 
 Its significance is best grasped by means of a graphic 
 representation. An inspection of such graphic work 
 will show that trends may vary from straight lines on 
 the one hand, to curves almost conforming to the 
 given items on the other. Whether a trend is dra^vn 
 as a straight or an irregular line depends in part on 
 the nature of the data and in part on the purpose to be 
 served. 
 
 The Free-hand Method. A good draftsman com- 
 monly determines trends merely by charting his data 
 and, without any preliminary computations, drawing 
 a straight or curved median line through them.^ Such 
 work may be done entirely free-hand, or by the use 
 of irregular curves and other drafting material, but in 
 either case it is classed as a free-hand method. For 
 many purposes this method will prove satisfactory, 
 and should be practiced by the student until it can be 
 used wdth facility. For even though more elaborate 
 
 * If a trend based upon {reometric means is desired, the data may be 
 plotted on semi-logarithmic paper. 
 
 100
 
 TRENDS AND CYCLES 
 
 101 
 
 methods are to be applied in actual work, practice in 
 the free-hand drawing of trends will be helpful. It 
 will develop an appreciation of the requirements of a 
 given problem, without which the best of mathematical 
 methods are likely to be misapplied. 
 
 An example of the free-hand method of drawing a 
 trend is given in the graph of the index of real wages, 
 
 IS70 
 
 mo 
 
 mo 
 
 im 
 
 1910 
 
 mo 
 
 FiQUBE 6. Trends and cycles. Upper line, index of real wages (hour 
 rates) in the United States (see Table X) and its trend; middle line, 
 index of per capita production (see Table XV) and its trend; lower 
 line, cycles of wholesale prices (see Figure 9). 
 
 Repriiited, with permission^ from Tlxe Quarterly Journal of tJie Univer- 
 sity of North Dakota, January, 1922. 
 
 in Figure 6. In this case a more precise method would 
 not be applicable because of the inaccuracies caused 
 by the substitution of wholesale prices for the cost of 
 living in the computation. As was previously stated, 
 there is reason to think that the marked rise in real 
 wages appearing during the decade 1890-1900 is an 
 exaggeration. The trend was therefore drawn so as to
 
 102 INTRODUCTION TO ECONOMIC STATISTICS 
 
 discount this rise. As a result it does not conform to 
 the usual rule that the deviations above and below it 
 should balance. 
 
 Method of Semi-Averages. When a straight-line 
 trend is to be drawn through a fairly long series, the 
 free-hand method may be improved upon by means of 
 a simple calculation. The average of the series may 
 be plotted on the middle ordinate, and the straight- 
 line trend drawn by inspection through this point. 
 The plus and minus deviations must then necessarily 
 balance. Or, better still, the series may be divided 
 into two equal parts and an average taken for each 
 part. These averages may be plotted on the middle 
 ordinate of each half series, respectively, and the trend 
 drawn through the two points. If the series consists 
 of an odd number of items, the middle item and unit 
 of time may be divided between the two parts. This 
 method of semi-averages is the one that was used in 
 drawing the trend of per capita production in Figure 
 6. The trend thus obtained is, in a long series, nearly 
 identical with the so-called line of least squares to be 
 discussed later. 
 
 The Moving Avera,ge. Among the trends that are 
 found by mathematical methods, perhaps the best 
 kno^vTi is the moving average. The process of com- 
 puting the moving average is simple in principle, but 
 it is usually tedious in practice even when calculating 
 machines are used. By this method the position of 
 the trend at any given period in the series is found 
 by averaging a certain number of items centering at 
 that period. Just how many items should be included 
 in the average must be determined by the nature of
 
 TRENDS AND CYCLES 103 
 
 the data. If the series shows a cyclic movement of 
 known length, then by taking the number of items 
 covering this length of time, the cycles will be 
 smoothed. If monthly data having a pronounced sea- 
 sonal swing are being studied through an interval of 
 several years, a twelve-month moving average is ap- 
 propriate.^ The method of finding the moving aver- 
 age is illustrated in the following table : 
 
 
 Per Capita 
 
 Moving A 
 
 -veragei 
 
 Year 
 
 Production 
 
 3-Year 
 
 5.Yea 
 
 1910 
 
 100 
 
 
 
 1911 
 
 95 
 
 101 
 
 
 1912 
 
 108 
 
 101 
 
 100 
 
 1913 
 
 100 
 
 102 
 
 101 
 
 1914 
 
 98 
 
 101 
 
 104 
 
 1915 
 
 106 
 
 103 
 
 104 
 
 1916 
 
 106 
 
 107 
 
 105 
 
 1917 
 
 109 
 
 108 
 
 106 
 
 1918 
 
 108 
 
 105 
 
 105 
 
 1919 
 
 99 
 
 103 
 
 
 1920 
 
 103 
 
 
 
 In explanation of the foregoing table it may be said 
 that the first number in the three-year moving average 
 (101) is obtained by averaging the first, second, and 
 third items (100, 95, and 108). The second number 
 (101) is obtained by averaging the second, third, and 
 fourth items (95, 108, and 100). The results are writ-' 
 ten to the nearest unit, and are placed opposite the 
 middle one of the three items averaged. In the same 
 way the succeeding averages are derived. The five- 
 year moving average is similarly computed, except 
 
 * The twelve-month moving average centers between the sixth and 
 seventh month in each computation. In order to make the trend thus 
 obtained fall on the same ordinates as the original items, it is neces- 
 sary to adjust it by deriving from it a two-month moving average. The 
 deviations of the original items may then be readily obtained.
 
 104 INTRODUCTION' TO ECONOMIC STATISTICS 
 
 that five items are averaged at a time. In practice 
 the work may be somewhat abridged, after the total 
 of the first group of items is found, by deriving the 
 next total from it. This may be done by adding to 
 the first total the difference between the next item 
 about to be included and the one about to be dropped. 
 
 WS^ 
 
 
 A /n^ 
 
 f06^ 
 
 
 A f^~\ 
 
 m^ 
 
 i 
 
 1 \ r'4/ \\ 
 
 /02^ 
 
 
 J\::^'iy/ \ / 
 
 
 7 
 
 'f\ v 
 
 93^ \ 
 
 / 
 
 ^ 
 
 96. ^ 
 
 4 
 
 
 9^. , 
 
 V 
 
 1 
 
 1 1 1 1 1 1 1 1 1 
 
 /P/0 
 
 
 /P/S /9Z0 
 
 Figure 7. Moving averages. A, Index of per capita production, 
 1910-1920. B, Three-year moving average of same. C, Five-year mov- 
 ing average of same. 
 
 Thus, the first five items above give 501 (average, 
 100). In obtaining the total of the second to the sixth 
 items, inclusive, the sixth item (106) will be added 
 and the first item (100) will be dropped; that is, a 
 balance of six will be added. The new total is there- 
 fore 501 + 6, or 507 (average 101). In the same way 
 the succeeding totals and averages may be derived.^ 
 
 ^ Mathematicians sometimes prefer a more complex form of the 
 moving average known as the progressive mean. This is similar to
 
 TRENDS AND CYCLES 105 
 
 The two moving averages just described, and the 
 index on which they are based, are plotted in Figure 
 7. The figure will serve to make clear the general rule 
 that the more inclusive the moving average, the 
 smoother the trend will be. A disadvantage of the 
 method will also be observed. The moving average 
 derived from a given series of items will always be 
 shorter than the series; and the more inclusive it is, 
 the shorter it will be. It is possible, however, to find 
 a tentative substitute for the lacking items in the trend 
 by repeating the extreme items in finding the aver- 
 ages. Thus, in the foregoing five-year moving average, 
 a trend item for 1911 might have been obtained as fol- 
 lows: 
 
 2 X 100 + 95 + 108 + 100 ^ ^^^ 
 
 5 
 
 and for 1910: 
 
 2 X 100 + 2 X 95 -^ 108 
 5 
 
 This method has the mathematical advantage of mak- 
 ing the sum of the trend items equal to the sum of 
 the data — a fact which may be found convenient to 
 use as a basis for checking the computations. It may 
 also be adapted to finding a current trend item for a 
 series of index numbers that is kept up to date. 
 
 The Line of Least Squares. If a straight-line trend 
 is at all adapted to a given series, the most satisfac- 
 tory mathematical trend to use is one known as the 
 
 the moving average, as LHustrat-ed and explained, except that weights 
 are used in taking the average. The weights are derived by the binomial 
 theorem, and are the frequencies of a theoretical curve of distribu- 
 tion. Thus in taking a five-year progressive mean, each group of five 
 terms is averaged by applying to the terms in succession the weights 
 1:4:6:4:1. Similarly, a seven year progressive mean would make use 
 of the weights 1:6:15:20:15:6:1 (cf. Slichter, Elementary Mathematical 
 Analysis, p. 194). 
 
 100
 
 106 INTRODUCTION TO ECONOMIC STATISTICS 
 
 line of least squares. The name is derived from the 
 fact that the line is so dra\\^l that the square of the 
 deviations of the data from it, as measured on the 
 ordinates, is always a minimum. The line of least 
 
 s. 
 
 
 
 
 o 
 
 /T ^U 
 
 4 . 
 
 
 
 
 /o • 
 
 J . 
 
 
 
 
 / • 
 
 2 - 
 
 
 
 
 / • 
 
 / . 
 
 
 
 
 / / 
 // 
 
 ^ AT' 
 
 
 
 
 
 r X 
 
 -/ . 
 
 
 
 // 
 
 > 
 
 -2 . 
 
 
 
 /o 
 
 
 -j . 
 
 / / 
 
 
 
 
 / 
 
 / 
 
 
 
 u/ 
 
 T'/ 
 
 ^O 
 
 
 
 -s . 
 
 
 
 
 y' ... 
 
 1 1 
 
 1 
 
 1 
 
 1 
 
 •^ 1 1 1 1 1 
 
 •S-4 
 
 -J 
 
 -£ 
 
 -/ 
 
 / a 5 4- s 
 
 FiouEE 8. Straight-line trend. TT', line of least squares for the 
 seven points indicated; UU', line of unit slope. 
 
 squares should be thoroughly understood, not only 
 because of its usefulness as a trend, but also because 
 it is the basis from which the principal method of com- 
 puting correlation is developed. In explaining it, the 
 following simple illustration will be taken :
 
 TRENDS AND CYCLES 107 
 
 Suppose that it is required to find a straight-line 
 trend for the index y (see next page), the average of 
 which is zero. The data are plotted upon coordinate 
 paper — the x and y scales having preferably the same 
 unit — as shown in Figure 8. The average of the data 
 is made to fall on the x-axis, and the middle item is 
 plotted on the y-axis. If we consider the vertical 
 distance of each index from the x-axis to represent a 
 force bearing upon that line, then the total moments 
 of these forces will be expressed by the sum of the 
 xy 's. This sum may be compared with the sum of the 
 moments of a line passing through the same ordinates, 
 and forming an angle of forty-five degrees with the 
 X and y axes at their point of intersection.^ Such a 
 line is said to have a unit slope; that is, it rises one 
 unit {y) for each unit {x) to the right. Its slope may 
 also be expressed by saying that the tangent of its 
 angle (UOX) is unity. The sum of its xy^s is, of 
 course, identical with the sum of its ic's squared. The 
 slope of the line of least squares is found by compar- 
 ing the moments of the data with those of the line 
 of unit slope, as expressed in the formula : 
 
 s = ^^^ 
 
 in which, 
 
 S = slope of the line of least squares, or tangent of 
 its angle 
 
 X ^ position of items relative to middle ordinate 
 
 y = items, as given 
 
 The data (y), their moments (xy), the moments 
 of the line of unit slope (x-), and the computation of 
 
 * The angle will vary, of course, if the x and y scales differ.
 
 108 INTRODUCTION TO ECONOMIC STATISTICS 
 
 the line of least squares, are shown in the following 
 table : 
 
 y X X* xy Trend 
 
 -3 
 
 -3 
 
 9 
 
 9 
 
 -4.5 
 
 -4 
 
 -2 
 
 4 
 
 8 
 
 -3 
 
 -2 
 
 -1 
 
 1 
 
 2 
 
 -1.5 
 
 -1 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1.5 
 
 5 
 
 2 
 
 4 
 
 10 
 
 3 
 
 4 
 
 3 
 
 9 
 
 12 
 
 4.5 
 
 = 
 
 
 28 
 
 S = 
 
 ) 42 
 = 1.5 
 
 
 The last column headed ** trend" gives the line of least 
 squares. This column is computed from the middle or- 
 dinate (0) by adding the slope (S = 1.5) once for each 
 successive ordinate in a positive direction, and sub- 
 tracting it once for each successive ordinate in a nega- 
 tive direction. 
 
 Data having an average of zero have here been 
 taken merely for the sake of simplicity; the same 
 process w^th but little modification may be applied to 
 any values. The dates or other numbers correspond- 
 ing to the items cannot be used, however, but must 
 be replaced by an x scale centering at the middle point 
 of the series. The average of the data is found, 
 and the trend is computed from this average by suc- 
 cessive additions of the slope in a positive direction, 
 and subtractions in a negative direction. The method 
 is illustrated by the use of the following data, which 
 parallel those used in the preceding illustration, except 
 that the average is 100. This increase in the values of
 
 TRENDS AND CYCLES 109 
 
 y disappears from 2xy because it affects equally both 
 the minus and the plus items. 
 
 Year 
 
 y 
 
 X 
 
 X2 
 
 xy 
 
 Trend 
 
 1900 
 
 97 
 
 -3 
 
 9 
 
 -291 
 
 95.5 
 
 1901 
 
 96 
 
 -2 
 
 4 
 
 -192 
 
 97 
 
 1902 
 
 98 
 
 -1 
 
 1 
 
 -98 
 
 98.5 
 
 1903 
 
 99 
 
 
 
 
 
 
 
 100 
 
 1904 
 
 101 
 
 1 
 
 1 
 
 101 
 
 101.5 
 
 1905 
 
 105 
 
 2 
 
 4 
 
 210 
 
 103 
 
 1906 
 
 104 
 
 3 
 
 9 
 
 312 
 
 104.5 
 
 7 
 
 )700 
 
 
 28 
 
 ) 42 
 
 
 (average) 
 
 A = 100 S = 1.5 
 
 The following details may be noted: (a) If there are 
 an even number of ordinates, the y-axis will lie mid- 
 way between the two middle ordinates, which are num- 
 bered as -0.5 and 0.5 respectively. The horizontal 
 positive scale will therefore read 0.5, 1.5, 2.5, etc., and 
 the negative scale will be the reverse, (b) It will some- 
 times be found that the value of ^ is negative. This 
 indicates a downward slope of the line of least squares, 
 (c) The position of the line of least squares is de- 
 scribed by designating the period coinciding with the 
 y axis as the point of origin, and by expressing the 
 value of y algebraically in terms of the average and 
 the slope. Thus in the above illustration the point of 
 origin is 1903, and the equation of the trend is y = 100 
 -f 1.5x.i 
 
 ^ It has been suggested that the method of least squares might be 
 applied to the finding of a price index {Quarterly Journal of Economics, 
 August, 1921, page 567). The expenditure for any given commodity 
 may be plotted on a coordinate chart as the value of y, and the num- 
 ber of units bought may be plotted as the value of x. The slant of a 
 line (tangent of the base angle) drawn from the intersection of the 
 axes to the point determined by the values of x and y, represents 
 the price. The average price of a number of commodities may be
 
 110 INTRODUCTION TO ECONOMIC STATISTICS 
 
 The Parabola Trend. A broad treatment of the sub- 
 ject of curve fitting would lead the student beyond the 
 range of ordinary statistical work. We shall not, 
 therefore, follow the subject farther, except to take up 
 
 /2o/ \ 
 
 m. 
 
 m^ 
 
 90. 
 
 io. 
 
 70. 
 
 60^ 
 /870 
 
 1 
 
 mo 
 
 mo 
 
 mo 
 
 /m 
 
 Figure 9. Index of wholesale prices in the United States (see Table 
 X) and trend. Indexes for 1870-1880 converted to a gold basis. 
 
 a simple method of adjusting a parabola to an index. 
 The method is one which is often used by engineers, 
 and has also recently come into use to some extent 
 among statisticians. 
 
 taken as the slant of the line of least squares determined by all the 
 coordinate pairs of x and y, and having the intersection of the axea 
 as the point of origin (y = Sx). In such a case, of course, all the 
 values of xy will be positive. To give definite comparisons at different 
 dates, this method would require the use of "dollar's worths" as 
 physical units. While the method is ingenious, it is of questionable 
 validity, since in effect it involves a weighting of the prices by the 
 square of the quantities in the process of finding the average. A 
 defense of tlic nietliod on the basis of the use of least squares in the 
 theory of errors docs not appear to be valid, since the theory of errors 
 would call for merely the arithmetic mean of the number of determina- 
 tiona.
 
 TRENDS AND CYCLES 111 
 
 The method of adjusting a parabola will be illus- 
 trated by applying it to the Bureau of Labor Statistics ' 
 wholesale price index for the years 1896 to 1915 inclu- 
 sive, as charted in Figure 9. This figure includes also 
 the same index for the years 1870-1895 (gold prices), 
 to which a line of least squares has been fitted. But it 
 may be easily seen that a similar trend would not be 
 suited to the succeeding index numbers. The some- 
 what regular curve of the latter portion of the index 
 indicates that a parabola of the second order would 
 be appropriate. 
 
 In fitting the parabola, the year 1895 has been taken 
 as the point of origin, though this year is not included 
 in the results. It is estimated by inspection of the 
 graph that the trend, if extended to 1895, would have 
 a value of 64 at that date. Two other points deter- 
 mining the trend may be similarly located, one at 
 about the middle of the series and one at the end. 
 These points have been taken as 88 for the year 1905 
 (x = 10), and 102 for the year 1915 (x = 20). We 
 have, then, these coordinate values of x and y: 
 
 If X r- 0, y = 64 
 
 If X = 10, y = 88 
 If X = 20, y = 102 
 
 The equation of a parabola of the second order is, 
 y = a -f- bx + cx^ 
 
 If the coordinate values of x and y as just stated are 
 substituted successively in this equation, the following 
 results will be obtained : 
 
 64 = a 
 
 88 = a + 10b + 100c 
 102 = a + 20b + 400c
 
 112 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Solving for the constants gives, 
 
 a = 64 
 
 b= 2.9 
 
 c= -.05 
 Substituting these values in the original equation gives 
 the equation of the required trend, 
 
 y = 64 + 2.9x -.OSx^ 
 from which the value of each item in the trend may be 
 found by substituting the coordinate value of x. 
 
 Since trends are used as a basis for measuring fluc- 
 tuations, the deviations of the data from the trend 
 are usually computed. This is done by subtracting 
 each item in the trend from the corresponding item 
 in the data. If the trend has been accurately con- 
 structed, the positive and negative deviations should 
 be practically equal ; that is, their sum should be zero. 
 Where the parabola has been used, a certain error will 
 probably be found to have resulted from the fact that 
 the original points determining the curve were located 
 merely by inspection. An adjustment (centering) to 
 remove this error may be made by finding the sum of 
 the deviations (2D), dividing it by the number of the 
 
 items (N), adding the result f-^j to each of the trend 
 
 items, and subtracting it from each of the deviations. 
 This is expressed in the equation of the trend simply 
 
 by adding the correction ( -^ j to the value of a. In 
 
 work in correlation, however, the correction may be 
 more easily made by another method, as will be evi- 
 dent later. When comparisons of the fluctuations in 
 different scries are to be made, either by graphing or
 
 TRENDS AND CYCLES 
 
 113 
 
 by the computation of a coefficient of correlation, it is 
 often necessary to find the standard deviation. For 
 graphic representation, the deviations are reduced to 
 multiples of the standard deviation, which serves as a 
 comparable unit. Table XVI shows the derivation of 
 
 TABLE XVI 
 
 DEEIVATION OF TREND AND CYCLES OF WHOLESALE PRICES, 
 
 BASED ON BUREAU OF LABOR STATISTICS INDEX, 
 
 UNITED STATES, 1896-1915 
 
 Equation of trend, y = 64 + 2.9x — .05x^ 
 Equation, corrected, y = 63.825 + 2.9x — ,05x» 
 
 Point of origin, 1895. 
 
 TEAR 
 
 PRICE 
 
 
 TREND 
 
 D 
 
 D 
 
 D» 
 
 CYCLES • 
 
 INDEi 
 
 X 
 
 ,y 
 
 CENTERED 
 
 D/a 
 
 1896 
 
 66 
 
 1 
 
 66.85 
 
 -.85 
 
 -.68 
 
 .4624 
 
 -.33 
 
 1897 
 
 67 
 
 2 
 
 69.60 
 
 -2.60 
 
 -2.42 
 
 5.8564 
 
 -1.17 
 
 1898 
 
 69 
 
 3 
 
 72.25 
 
 -3.25 
 
 -3.08 
 
 9.4864 
 
 -1.49 
 
 1899 
 
 74 
 
 4 
 
 74.80 
 
 -.80 
 
 -.62 
 
 .3844 
 
 -.30 
 
 1900 
 
 80 
 
 5 
 
 77.25 
 
 2.75 
 
 2.92 
 
 8.5264 
 
 1.41 
 
 1901 
 
 79 
 
 6 
 
 79.60 
 
 -.60 
 
 -.42 
 
 .1764 
 
 -.20 
 
 1902 
 
 85 
 
 7 
 
 81.85 
 
 3.15 
 
 3.32 
 
 11.0224 
 
 1.60 
 
 1903 
 
 85 
 
 8 
 
 84.00 
 
 1.00 
 
 1.18 
 
 1.3924 
 
 .57 
 
 1904 
 
 86 
 
 9 
 
 86.05 
 
 -.05 
 
 .12 
 
 .0144 
 
 .06 
 
 1905 
 
 85 
 
 10 
 
 88.00 
 
 -3.00 
 
 -2.82 
 
 7.9524 
 
 -1.36 
 
 1906 
 
 88 
 
 11 
 
 89.85 
 
 -1.85 
 
 -1.68 
 
 2.8224 
 
 -.81 
 
 1907 
 
 94 
 
 12 
 
 91.60 
 
 2.40 
 
 2.58 
 
 6.6564 
 
 1.25 
 
 1908 
 
 91 
 
 13 
 
 93.25 
 
 -2.25 
 
 -2.08 
 
 4.3264 
 
 -1.00 
 
 1909 
 
 97 
 
 14 
 
 94.80 
 
 2.20 
 
 2.38 
 
 5.6644 
 
 1.15 
 
 1910 
 
 99 
 
 15 
 
 96.25 
 
 2.75 
 
 2.92 
 
 8.5264 
 
 1.41 
 
 1911 
 
 95 
 
 16 
 
 97.60 
 
 -2.60 
 
 -2.42 
 
 5.8564 
 
 -1.17 
 
 1912 
 
 101 
 
 17 
 
 98.85 
 
 2.15 
 
 2.32 
 
 5.3824 
 
 1.12 
 
 1913 
 
 100 
 
 18 
 
 100.00 
 
 
 
 .18 
 
 .0324 
 
 .09 
 
 1914 
 
 100 
 
 19 
 
 101.05 
 
 -1.05 
 
 -.88 
 
 .7744 
 
 -.43 
 
 1915 
 
 101 
 
 20 
 
 102.00 
 
 -1.00 
 
 -.82 
 
 .6724 
 
 -.40 
 
 16.40 17.92 20)85.9880 
 -19.90 -17.92 
 
 20 )-3.50 
 K = - .175 
 
 0. a' = 4.2994 
 (J = 2.07 
 
 8.66 
 -8.66 
 
 ^ Any set of deviations taken from an average or trend, and intended 
 to measure cyclic movements, are commonly designated as cycles. They 
 need not necessarily be reduced to units of the standard deviation. In 
 some cases no complete cyclic movement may be discovered, but the 
 same designation may be used.
 
 114 INTRODUCTION TO ECONOMIC STATISTICS 
 
 the trend just discussed, together with the correction, 
 and the reduction of the deviations to multiples of the 
 standard deviation. Figures 10 and 11 show price 
 cj^cles as thus measured compared mth other cycles 
 similarly obtained. 
 
 The parabola may be used where compound curves 
 are required by adding the term dx^, and perhaps ex* 
 to the equation. For each term thus added, an addi- 
 
 1900 
 
 I I I 
 
 1905 
 
 
 I I I i I 
 
 I9IS 
 
 Figure 10. Cycles of wholesale prices (solid line) and commitments 
 to New York State prisons (broken line). 
 
 Eeprinted, with permission, from Tlie Quarterly Journal of the Uni- 
 versity of North Dakota, January, 1922. 
 
 tional point may be located by inspection, and the 
 trend may thus be more exactly fitted. But the work 
 of solving and applying such equations becomes very 
 laborious. With practice, however, the student will 
 find ways of abbreviating the process and modifying 
 it to suit his purposes. Often the terms of the equa- 
 tion may be estimated by simple experimentation. The 
 position of the point of origin may be varied to suit 
 given requirements. A compound curve shaped some-
 
 TRENDS AND CYCLES 
 
 115 
 
 what like an italic / may be obtained by using only odd 
 numbered powers of x in the equation, and taking the 
 point of origin near the middle of the original series. 
 "With a little ingenuity, sine curves and other trends 
 may be experimentally fitted.^ 
 
 Figure 11. Cycles of wholesale prices (solid line) and marriage 
 rate (broken line) in the United States. 
 
 Eeprinted, with permission, from The Quarterly Journal of the Uni- 
 versity of North Dakota, January, 1922. 
 
 Analyzing Business Barometers. A somewhat in- 
 tricate problem in trends is met when monthly data 
 are employed as indexes, or barometers, of business 
 conditions. Since such barometers are very generally 
 consulted as guides to business activities, their inter- 
 
 * When it appears that the trend of an index series increases or 
 decreases by approximately a fixed ratio, an exponential curve may 
 be readily fitted as follows. Find the logarithms of the data, plot 
 them, and construct a straight-line trend by means of the semi-aver- 
 ages, or by a line of least squares. Eead the items of the trend from 
 the chart, consider them to be logarithms, and find the corresponding 
 numbers. These numbers will be the items of the required trend. In 
 a long series the work may be abbreviated by grouping the original 
 items, aa by decades, and fitting the curve to the averages of these 
 groups.
 
 116 INTRODUCTION TO ECONOMIC STATISTICS 
 
 pretation is a matter of great practical importance. 
 The difficulty involved in their use lies in the com- 
 plexity of the influences playing upon them. For con- 
 venience of analysis these influences have been clas- 
 sified as (1) a seasonal variation usually due to the 
 dependence of industry upon weather conditions, illus- 
 trated by the rising of the interest rate mth the move- 
 ment of crops, (2) a cyclic movement covering an in- 
 terval of several years and marked by alternating in- 
 dustrial depression and activity, and (3) a secular 
 trend or gradual change due in most cases to the 
 growth movement, as seen in the increase in produc- 
 tion. In addition to influences which may be appro- 
 priately classed under one of these three headings, 
 there are others that must be looked upon as more or 
 less accidental interruptions, of which no exact account 
 can be taken. 
 
 Measuring Seasonal Variations. The most sat- 
 isfactory method of analyzing monthly business ba- 
 rometers is first to compute an index of seasonal va- 
 riations, and then to subtract it, month by month, 
 from an index of the data based upon the secular 
 trend. The result is an index of the cycles. As has 
 already been noted, the twelve-month moving average 
 is sometimes assumed to measure the data as distinct 
 from the seasonal variations. But such a method of 
 elimination takes into account the fluctuations of only 
 one year at a time. Other methods have therefore 
 been resorted to with the purpose of measuring the 
 seasonal swing more exactly on the basis of several 
 years. A simple method of this sort, which may be 
 considered valid for a period in which the cyclic in-
 
 TRENDS AND CYCLES 
 
 117 
 
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 118 INTRODUCTION TO ECONOMIC STATISTICS 
 
 fluences are moderate and well distributed along an 
 approximately straight-line trend, is illustrated in 
 Table XVII. 
 
 The Method of Averages. The problem stated and 
 solved in the table is the finding of the seasonal varia- 
 tions in the interest rate on the somewhat slender basis 
 
 la^Q 
 
 I8 60 
 
 18 80 
 
 I900 
 
 1920 
 
 Figure 12. Average interest rate on commercial paper in the United 
 States (lower line) compared with index of wholesale prices (upper 
 line). 
 
 Adapted from Monthly Review, New York Federal Reserve Bank. 
 
 of the five years 1909 to 1913, inclusive. The monthly 
 rates as given are first averaged both by columns and 
 rows in order to obtain the annual averages and the 
 month averages. The process may therefore be called 
 the method of averages. 
 
 If the interest rate had maintained approximately 
 the same level from year to year during the interval
 
 TRENDS AND CYCLES 119 
 
 studied, nothing more would be needed than to reduce 
 the month averages, or types, to a percentage of their 
 own annual average. The results would form an index 
 of the seasonal variations. But if there is a secular 
 movement, it must be canceled from such an index. 
 If an interval of half or three-quarters of a century 
 is taken into account, a general downward trend of the 
 interest rate may be discovered, as may be seen in 
 Figure 12. But the five years here studied happen to 
 be an exception. By means of the line of least squares, 
 a positive slope of 0.019 monthly is revealed. By ap- 
 plying this slope to the average of the month types, 
 an annual trend is constructed, having its point of 
 origin midway between the June and July items. It 
 will be noted that one-half the slope must be added to 
 the average to obtain the July trend item, and the same 
 amount subtracted to obtain the June item. In ob- 
 taining the remaining trend items, the slope is applied 
 as previously explained. 
 
 Confusion may perhaps here arise from the fact 
 that the slope was computed from the five annual aver- 
 ages, but was applied to the construction of a trend 
 with which to compare the monthly data. But it should 
 be obvious that in this case a slope obtained from the 
 month averages would be materially affected by the 
 seasonal swing. This we wish to retain, while the secu- 
 lar trend we wish to cancel. Of course we could find 
 the secular trend from the monthly data taken consecu- 
 tively as sixty items, but such a method would be un- 
 necessarily laborious. Hence we find it from the five 
 annual averages, and apply it to the construction of 
 a line that will serve to cancel the secular trend from
 
 120 INTRODUCTION TO ECONOMIC STATISTICS 
 
 the month types. This cancellation is accomplished by 
 dividing the month averages, item by item, by the 
 trend. The quotients should be centered, if necessary, 
 by reducing them to percentages of their common aver- 
 age. The result is an index of seasonal variations, 
 from which the percentage deviations, month by month, 
 may be directly stated.^ 
 
 Applying the Seasonal Index. The method of 
 applying the seasonal index has already been sug- 
 gested, and may be described as follows. A secular 
 trend for the whole period under consideration is con- 
 structed — in this case the line of least squares already 
 found may be extended — and the data are reduced 
 month by month to percentages of the trend. From 
 each month's item as thus found the seasonal index 
 for the same month is subtracted. The remainders 
 are assumed to measure the cycles, and may be plotted 
 as deviations above and below a horizontal axis. The 
 seasonal index may, with caution, be applied to other 
 years than those from which it is derived; of course, 
 the greater the number of normal years from which 
 it is derived, the safer such an extension of its use 
 to comparable years becomes. 
 
 The Link-relative Method. A complex but more ac- 
 curate method for measuring seasonal variations has 
 
 ^ Another variation of the method of averages — one that is per- 
 haps theoretically preferable, but which involves more extended cal- 
 culations — may be briefly described as follows: A twelve-month 
 moving average of the monthly data tlirough a given scries of years 
 is first ^ound. This is adjusted to make it conform to the ordinatea 
 of the original series by deriving from it a two-month moving average. 
 There is then obtained, for each month^ the ratio of the original monthly 
 item to the corresponding adjusted moving average. The median of 
 the ratios so obtained for the Januaries is taken as the index of sea- 
 sonal variation for January; and index numbers for the other months 
 are similarly obtained. The twelve results are then centered, if neces-
 
 TRENDS AND CYCLES 121 
 
 been developed by Professor Persons, and applied to 
 the analyses appearing in the early numbers of the 
 Review of Economic Statistics. In a somewhat sim- 
 plified form, this method is illustrated in Table 
 XVII- A, which is based on the data of Table XVII. 
 Briefly stated, the method consists in finding what are 
 called '' link-relatives"; that is, the percentage which 
 the index of each month is of the preceding month. 
 The median link-relatives are then selected from each 
 month's series, and are tabulated as the month types.^ 
 Beginning with December as a base (100%), the types 
 are multiplied consecutively, producing an index series 
 from January to December for a typical year. If the 
 final December item fails to come out to 100%, in con- 
 formity with the base in the preceding December, a 
 secular trend is evidently disturbing the index. The 
 discrepancy, if moderate, may be removed by distribut- 
 ing it throughout the year ; that is, by subtracting one- 
 twelfth of it from the January index, two-twelfths 
 from the February index, and so on through the year.^ 
 The results are designated an adjusted index. The ad- 
 justed items are next centered by reducing them to 
 a base of the average of the series. In making the 
 computations the figures were carried to one more 
 place than is shown in the table. 
 
 The superiority of this method lies in the fact that 
 in taking the link-relatives, and in selecting their 
 medians as the month types, the effects of the cyclic 
 
 sary, by reducing them to percentages of their common average (Cf. 
 Jordan, Biisiness Forecasting, p. 212). 
 
 * To get the best results, the median should be based on a larger 
 number of years than are here taken. 
 
 'A more exact method is to apportion the discrepancy geometrically; 
 that is, to divide the January index by the twelfth root of the final 
 December index (written as a decimal), the February index by the 
 square of this root, and so on.
 
 122 INTRODUCTION TO ECONOMIC STATISTICS 
 
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 < 
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 « 
 
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 TRENDS AND CYCLES 123 
 
 movements and of chance influences are minimized. 
 The secular trend is satisfactorily eliminated by the 
 process of adjusting. The index thus obtained should 
 be accurate for the years just preceding the establish- 
 ment of the Federal Reserve System. Since that time 
 seasonal changes have lessened.^ 
 
 Business Cycles. The Review of Economic Statis- 
 tics, in the analyses just referred to, has made exhaus- 
 tive studies of the cyclic movements of the commonly 
 used business barometers. After measuring the in- 
 fluence of seasonal variations by a process similar to 
 that just described, it determined the cycles on the 
 basis of lines of least squares. It then combined 
 twelve of the principal barometric indexes into three 
 composite indexes which are taken as measurements of 
 speculative activity, business activity, and banking 
 strain, respectively. A chart of these three composite 
 indexes for the years 1903 to 1913, inclusive, is here 
 reproduced (Figure 13). The chart shows very clearly 
 the general stages of the business cycle, from the pre- 
 dominance of the speculative activity which marks the 
 awakening from a period of depression, through the 
 period of intensified production, and into the period 
 of banking strain which heralds another depression. 
 The various barometric series used in constructing the 
 figure are indicated in the explanation accompanying 
 the title. It should be added that while New York bank 
 loans point to the speculative aspects of the cycle, those 
 outside New York conform more closely to business 
 
 *For the most exhaustive study of seasonal variations in the interest 
 rate, the student should consult "Seasonal Variations in the Relative 
 Demand for Money and Capital in the Fnited States" (National 
 Monetary Commission, 1910), by E. W. Kemmerer, See also the 
 Monthly Review, Federal Reserve Bank of New York, Feb. 1, 1922.
 
 124 INTRODUCTION TO ECONOMIC STATISTICS 
 
 activity. For practical purposes measurements of the 
 general aspects of the cycle need to be supplemented 
 by indexes showing the position of particular indus- 
 tries, as indicated by relative production and stocks of 
 goods on hand. This need is now being met in part by 
 the Federal Reserve Bank of New York, and by other 
 agencies. 
 
 Several phenomena more psychological than eco- 
 nomic in nature show a tendency to fluctuate more or 
 
 Figure 13. The index of general business conditions, 1903-14. 
 
 A, Speculation: New York Bank clearings, average price of indus- 
 trial stocks, average price of railroad stocks, and average price of 
 railroad bonds. 
 
 B, Business: Bank clearings of the United States outside New York 
 City, Bradstrect's index of wholesale commodity prices, United States 
 Bureau of Labor Statistics' index of wholesale commodity prices, and 
 pig-iron production. 
 
 C, Banking: Interest rates on 60-90 day and on 4-6 months com- 
 mercial paper in New York City, loans and deposits of New York City 
 clearing house banks (both inverted). 
 
 Reproduced from Beview of Economic Statistics, by permission of 
 the editors. 
 
 less closely with the business cycle. Professor A. H. 
 Hansen has recently showTi statistically that since 
 1898 strikes increased with prosperity, though before 
 that date, when the trend of prices was downward, 
 they increased with depression (cf. American Eco- 
 nomic Review, Dec, 1921, pp. 617-621). Mr. Roger 
 Babson has traced a connection between church growth 
 and the business cycle; religious activity being ap- 
 parently intensified during a period of depression.
 
 TRENDS AND CYCLES 125 
 
 Unemployment, failures, suicides, and crime generally, 
 also increase during a period of depression. On the 
 other hand, immigration, the marriage rate, and ex- 
 travagance, tend to increase with a period of prosper- 
 ity (cf. Figures 10 and 11, pages 114 and 115). 
 
 As an example of the application of statistical meth- 
 ods to the practical problem of forecasting the cyclic 
 movements of business, the ''Annalist Barometer and 
 Business Index Line" may be cited. This index is 
 published in graphic form, together with a brief ex- 
 planation, each week in the Annalist, a well-known 
 financial paper of New York. It is derived from the 
 data elaborated by the Review of Economic Statis- 
 tics, already briefly described. The index is the recip- 
 rocal of a weighted average of the deviations from 
 normal of commodity prices, interest rates, pig iron 
 production. New York bank clearings, and bank clear- 
 ings outside of New York. Since these series of data, 
 when directly combined, measure the later phases of 
 the business cycle, their decline precedes and fore- 
 casts the rise in stocks marking the beginning of the 
 next cycle; and their rise similarly forecasts a decline 
 in stocks. By taking the reciprocal of the deviations 
 of the combined series, the forecast is made direct 
 instead of inverse. By a comparison of the forecast- 
 ing index and the movement of stocks in former years, 
 the decisiveness of change in the former necessary to 
 constitute a forecast of the latter has been determined. 
 A detailed account of the construction and use of the 
 index will be found in the Annalist of March 28 and 
 of October 24, 1921. 
 
 The student who desires to inquire more intensively
 
 126 INTRODUCTION TO ECONOMIC STATISTICS 
 
 into the statistics of the business cycle should consult 
 for himself the data published in the Review of Eco- 
 nomic Statistics already mentioned. In addition he 
 should become famihar wdth Wesley C. Mitchell's 
 standard work on Business Cycles, and with a 
 more recent work by D. F. Jordan on Business Fore- 
 casting. In connection with the underlying causes 
 of the cycle, reference should be made to the interest- 
 ing but very technical works of H. L. Moore (cf. Eco- 
 nomic Cycles, and articles in the Quarterly Journal 
 of Economics, February, August, and November, 
 1921). Professor Moore discovers some relation to 
 exist between a weather cycle of heavier and lighter 
 rainfall and the business cycle, the average duration of 
 each cycle being about eight years. The moist years 
 bring as a rule larger crops, with some tendency to a 
 lowering of the general price level, followed during 
 the drier years by a rise in the price level. This rela- 
 tion seems to be clearer for English prices than for 
 American. The weather cycle is shown to be synchro- 
 nous in several countries, and to be correlated with a 
 cycle of barometric pressure, which in turn may have 
 astronomical causes. But while the subject is very 
 interesting and valuable theoretically, the correlations 
 disclosed are too irregular to be of great practical 
 value. 
 
 REFERENCES 
 
 Babson, Roper W., Business Barometers. 
 
 Davies, G. R., "Social Aspects of the Business Cycle," Qimr- 
 
 ierly Journal of the University of North Dakota, January, 
 
 1922. 
 Hurlin, Ralph G., "The Long-Time Trend of Prices in the 
 
 United States," The Annalist, July 4, 1921.
 
 TRENDS AND CYCLES 127 
 
 Jordan, D. F., Business Forecasting. 
 
 Kemmerer, E. W., High Prices and Deflation. 
 
 Mitchell, Wesley C., Business Cycles. 
 
 Moore, Henry L., Economic Cycles: Their Law and Cause. 
 
 Peddle, John B., The Construction of Graphical Charts, 
 Chapter VI. 
 
 Persons, W. W., "Construction of a Business Barometer 
 Based upon Annual Data," American Economic Review, 
 December, 1916, pp. 739-769. 
 
 Piatt, Andrew A., National Monetary Commission. 
 
 Tingley, Richard H., "Another Yardstick of Banking Condi- 
 tions," The Annalist, November 28, 1921, p. 511. 
 
 EXERCISES 
 
 1. Plot the data for production and price of wheat, 1870- 
 1920 (Tables XIV and XIV-A, pp. 76 and 77), and 
 draw a free-hand trend for each series. 
 
 2. As in Exercise 1, construct free-hand trends for the pro- 
 duction and price of com. 
 
 3. Plot the index of physical production of crops, 1870- 
 1920 (Table XV, p. 81), and draw a straight-line trend 
 by inspection. 
 
 4. Apply the method of semi-averages to the construction 
 of a straight-line trend for the data of the preceding 
 exercise. 
 
 5. Compute a five-year moving average of per capita pro- 
 duction, 1890-1918. Plot both the trend and the data 
 from which it is derived on 17"x22" cross-section paper, 
 and measure graphically the deviations from the trend. 
 
 6. Plot on a horizontal axis the deviations obtained in 
 the preceding exercise. Find the average deviation, and 
 indicate this on the graph for both the plus and the minus 
 deviations. 
 
 7. Compute and plot a straight-line trend (line of least 
 squares) for the following price index. 
 
 Year Prices (3 articles) 
 
 1897 85 
 
 1898 70 
 
 1899 90 
 
 1900 130 
 
 1901 125
 
 128 INTRODUCTION TO ECONOMIC STATISTICS 
 
 8. Find straight-line trends (lines of least squares) for 
 the production and price of pig iron as given below, 
 taking each year separately. 
 
 PEODUCTION AND PRICE OF PIG IRON (Iron Age) 
 (000 omitted from production) 
 
 Month 
 
 1909 
 
 1910 
 
 1911 
 
 1912 
 
 1913 
 
 
 Tons 
 
 $ 
 
 Tons 
 
 $ 
 
 Tons 
 
 $ 
 
 Tons 
 
 $ 
 
 Tons $ 
 
 Jan. . 
 
 1,797 
 
 16.25 
 
 2,608 
 
 17.25 
 
 1,759 
 
 14.25 
 
 2,057 
 
 13.25 
 
 2,795 16.95 
 
 Feb.. 
 
 1,707 
 
 16.13 
 
 2,397 
 
 17.06 
 
 1,794 
 
 14.25 
 
 2,100 
 
 13.31 
 
 2,586 16.69 
 
 March 
 
 1,832 
 
 15.05 
 
 2,617 
 
 16.30 
 
 2,188 
 
 14.25 
 
 2,405 
 
 13.50 
 
 2,763 16.31 
 
 April . 
 
 1,738 
 
 14.25 
 
 2,483 
 
 15.37 
 
 2,065 
 
 14.25 
 
 2,375 
 
 13.75 
 
 2,752 15.65 
 
 May.. 
 
 1,883 
 
 14.50 
 
 2,390 
 
 15.00 
 
 1,893 
 
 13.95 
 
 2,512 
 
 14.15 
 
 2,822 14.94 
 
 June. 
 
 1,930 
 
 14.70 
 
 2,265 
 
 14.85 
 
 1,787 
 
 13.44 
 
 2,440 
 
 14.25 
 
 2,628 14.06 
 
 July.. 
 
 2,103 
 
 15.75 
 
 2,148 
 
 14.75 
 
 1,793 
 
 13.25 
 
 2,410 
 
 14.70 
 
 2,560 13.75 
 
 Aug.. 
 
 2,248 
 
 16.38 
 
 2,106 
 
 14.31 
 
 1,926 
 
 13.45 
 
 2,512 
 
 15.06 
 
 2,543 14.06 
 
 Sept- . 
 
 2,385 
 
 17.35 
 
 2,056 
 
 14.25 
 
 1,977 
 
 13.31 
 
 2,463 
 
 15.87 
 
 2,505 14.25 
 
 Oct.. 
 
 2,599 
 
 17.88 
 
 2,093 
 
 14.25 
 
 2,102 
 
 13.25 
 
 2,689 
 
 16.80 
 
 2,546 14.35 
 
 Nov.. 
 
 2,547 
 
 17.75 
 
 1,909 
 
 14.25 
 
 1,999 
 
 13.20 
 
 2,630 
 
 17.25 
 
 2,233 13.87 
 
 Dec. . 
 
 2,635 
 
 17.45 
 
 1,777 
 
 14.25 
 
 2,043 
 
 13.19 
 
 2,782 
 
 17.25 
 
 1,983 13.95 
 
 Total 
 
 25,410 
 
 16.12 
 
 26,855 
 
 15.16 
 
 23,329 
 
 13.67 
 
 29,383 
 
 14.93 
 
 30,722 14.90 
 
 (Reprinted, with permission, from Babson's Desk Sheet.) 
 
 9. Find the average index of per capita physical pro- 
 duction in the United States (page 81) for each dec- 
 ade from 1870 to 1919. Using the resulting five aver- 
 ages, construct a line of least squares. Plot the original 
 data and the trend thus found. 
 
 10. Find the average index of production of iron and copper 
 by five year periods from 1870 to 1914. Plot these aver- 
 ages, and fit to them a parabola of the second degree. 
 
 11. Compute and plot the cycles of the interest rate, 1909- 
 1913, using the data and index of seasonal variations 
 presented in Table XVII, page 117. 
 
 12. Using the data given below, find an index of seasonal 
 variations in exports of merchandise (a) by the method 
 of averages, and (b) by the link-relative method.
 
 TRENDS AND CYCLES 129 
 
 EXPORTS OF MERCHANDISE, UNITED STATES, 1909-1913 
 (In Millions of Dollars) 
 
 1909 1910 1911 1912 1913 
 
 January 157 144 197 202 227 
 
 February 126 125 176 199 194 
 
 March 139 144 162 205 187 
 
 April 125 133 158 179 200 
 
 May 123 131 153 175 195 
 
 June 117 128 142 138 163 
 
 July 109 115 128 149 161 
 
 August 110 135 144 168 188 
 
 September 154 169 196 200 218 
 
 October 201 208 210 255 272 
 
 November 194 207 202 278 246 
 
 December 172 229 225 250 233 
 
 (December, 1908, 170) 
 
 13. The following table, adapted from the Yearbook of the 
 Department of Agriculture, 1918, gives the farm price 
 of wheat in the United States (cents per bushel) on the 
 first day of each month for the years 1909 to 1913 
 inclusive. Using the method of link relatives, derive an 
 index of seasonal variations. 
 
 1909 1910 1911 1912 1913 
 
 January 1 . . . . 93.5 103.4 88.6 88.0 76.2 
 
 February 1 ... 95.2 105.0 89.8 90.4 79.9 
 
 March 1 103.9 105.1 85.4 90.7 80.6 
 
 April 1 107.0 104.5 83.8 92.5 79.1 
 
 May 1 115.9 99.9 84.6 99.7 80.9 
 
 June 1 123.5 97.6 86.3 102.8 82.7 
 
 July 1 120.8 95.3 84.3 99.0 81.4 
 
 August 1 107.1 98.9 82.7 89.7 77.1 
 
 September 1 . . 95.2 95.8 84.8 85.8 77.1 
 
 October 1 94.6 93.7 88.4 83.4 77.9 
 
 November 1 ... 99.9 90.5 91.5 83.8 77.0 
 
 December 1 . . . 98.6 88.3 87.4 76.0 79.9 
 
 (December 1, 1908, 92.2) 
 
 14. Using the method of averages, derive an index of sea- 
 sonal variations from the following table of farm prices 
 of wheat in the United States (cents per bushel) for the 
 years 1909 to 1918, inclusive.
 
 130 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Yearly- 
 
 
 Monthly 
 
 
 averages 
 
 ; 
 
 averages 
 
 ; 
 
 1909 
 
 101.3 
 
 Jan. 1 
 
 109.4 
 
 1910 
 
 96.5 
 
 Feb. 1 
 
 115.2 
 
 1911 
 
 86.9 
 
 March 1 
 
 115.2 
 
 1912 
 
 87.4 
 
 April 1 
 
 116.4 
 
 1913 
 
 78.4 
 
 May 1 
 
 125.6 
 
 1914 
 
 88.4 
 
 June 1 
 
 126.0 
 
 1915 
 
 105.2 
 
 July 1 
 
 117.7 
 
 1916 
 
 125.9 
 
 Aug. 1 
 
 117.9 
 
 1917 
 
 200.8 
 
 Sept. 1 
 
 117.4 
 
 1918 
 
 204.3 
 
 Oct. 1 
 
 116.5 
 
 
 
 Nov. 1 
 
 119.7 
 
 
 
 Dec. 1 
 
 118.6 
 
 15. Compute and plot the cycles in the price of wheat, 1909- 
 1913, as measured from a line of least squares. Use the 
 data of Exercise 13. 
 
 16. From financial journals and other sources obtain monthly 
 or weekly quotations for recent and current dates illus- 
 trating barometric subjects such as are mentioned on 
 pages 66 and 124. Plot the data and construct trends. 
 On the basis of these barometers and such other informa- 
 tion as may be available, make a forecast of business 
 conditions for the immediate future, allowing for seasonal 
 variations.
 
 CHAPTER VI 
 
 CORRELATION 
 
 Correlation Defined. A study of the cycles of busi- 
 ness barometers leads to the problem of classifying 
 and measuring the relationships among them. These 
 relationships may be discovered in various forms and 
 degrees. For example, the cycles of building permits 
 and pig iron production will be found to move some- 
 what closely together. Then again, stock prices and 
 commodity prices form similar waves, though the lat- 
 ter usually follow a few months behind. On the other 
 hand, commodity prices and business failures show 
 opposite movements — when one is up the other is down. 
 All such relationships between two sets of data are 
 known as correlations. When the two sets of cycles 
 agree, the correlation is called positive ; when they dis- 
 agree, it is negative. When the cycles are not quite 
 coincident in point of time, the one which follows is 
 said to show a **lag" of a given interval.^ Correla- 
 
 * The term ' ' lag ' ' is also sometimes used to designate a smaller 
 degree of variation occurring in one series than in another comparable 
 to it. Thus the lag of retail prices behind wholesale prices ia largely 
 a matter of degree, and only slightly a matter of time. But it is the 
 time element only that enters into the calculation of correlation. In 
 allowing for the lag, the series coming later in time is considered aa 
 moved back by the length of the lag, and the corresponding items are 
 then compared. When the length of the lag is difficult to determine, 
 estimates must be made, and the correlation computed on the basia of 
 each estimate. The lag resulting in the most marked correlation is 
 assumed to be the correct one. In determining the lag it is often 
 necessary to take into account the causal relation existing between the 
 two series under consideration, as in a case where the assumption of a 
 lag for one series results in a positive correlation, while the transfer 
 of the lag to the other series results in a negative correlation. 
 
 131
 
 132 INTRODUCTION TO ECONOMIC STATISTICS 
 
 tion carries the idea of a fundamental relationship: 
 either one phenomenon acts or reacts upon the other, 
 or both are due to common causes. The principle is 
 not limited to time series. Comparison might be made, 
 for example, between the advertising and the rate of 
 earnings of given business firms. But in any case 
 the principle would be the same as before, and the 
 methods used would be practically identical. 
 
 The Graphic Method. A fairly good study of cor- 
 relation in economic phenomena can often be made 
 mthout any more elaborate methods than those al- 
 ready described in isolating and plotting the cycles. 
 Two time series, reduced to standard deviation cycles 
 and plotted on equal horizontal scales may be very well 
 compared by superimposing one on the other. To fa. 
 cilitate comparison, one may be drawn on a trans- 
 parent medium, such as tracing cloth ; or a mimeoscope 
 may be used. By shifting the superimposed cycles 
 back and forth, the lag may be fairly accurately deter- 
 mined. The correlation may be described as posi- 
 tive or negative, high, moderate or low, and the lag 
 and its consistency may be stated. This is the method 
 adopted by the Review of Economic Statistics in its 
 study of the correlations existing among 24 business 
 barometric series for the years 1903-1914. 
 
 Method of Concurrent Deviations. It is often desir- 
 able, however, to measure the degree of correlation in 
 precise mathematical terms. This is particularly true 
 when correlated data are being; used in support of a 
 given theory. In order to obtain a precise result, 
 mathematical methods developed originally for use in 
 biometrics have been borrowed and adapted.
 
 CORRELATION 133 
 
 As an introduction to the mathematical methods of 
 measuring correlation, we may take up a simple for- 
 mula which is well adapted to the comparison of short- 
 time fluctuations. The formula is the expression of 
 what is called the method of concurrent deviations. It 
 may be illustrated by applying it to a comparison be- 
 tween the short-time fluctuations of real wages and per 
 capita production in the United States (cf. pp. 51, 53, 
 and 81). The fluctuations may be most readily deter- 
 mined by reference to a graph of each series (cf. Fig. 
 6, p. 101). If at any given year the line makes an in- 
 verted angle, like a caret (A), the fluctuation is reg- 
 istered on the index as positive ( + ). If the angle 
 is V-shaped, it is registered as negative (-). If no 
 angle is formed, the year is indicated as neutral (0). 
 In some cases it may not be possible to determine from 
 the graph whether the angle is neutral, or slightly posi- 
 tive or negative; in which case resort may be had to 
 the data. 
 
 After the deviations of both series have all been 
 registered, they are compared across, item by item. 
 If in a given year both indexes show a positive fluctua- 
 tion, one agreement is counted. If positive and nega- 
 tive meet, one disagreement is counted. If one or both 
 of the fluctuations of a given year are neutral, one- 
 half is added to both the agreements and disagree- 
 ments. When this summation is complete, the larger 
 of the two totals thus obtained is designated as the 
 number of concurrent deviations, denoted in the for- 
 mula by the letter C. The sign of the coefficient to be 
 obtained by the use of the formula is determined by 
 the nature of the concurrences. If they are agree-
 
 134 INTRODUCTION TO ECONOMIC STATISTICS 
 
 ments, the sign is positive ; if disagreements, the sign 
 is negative. The formula for correlation (B) as thus 
 measured is : 
 
 In the case of the wage and production indexes just 
 mentioned, the number of disagreements, or concur- 
 rences, totals to 33>4 and the number of comparisons is 
 49. The formula therefore becomes: 
 
 = — .61 
 
 The derivation of the formula is of little importance, 
 as it is patterned empirically on the one next to be 
 described. The significance of the coefficient will be- 
 come evident in the same connection. 
 
 The Pearson Method. We come now to the so-called 
 Pearson **r", the most satisfactory method to apply 
 to straight line correlations: that is, to those which 
 when graphed show an approximation to a straight 
 line rather than a curve. This statement does not refer 
 to the trends of the two series taken separately, but 
 only to the trend formed by the two sets of cycles 
 plotted as x and y, respectively. In explaining the 
 method, it is most convenient to begin with two sets 
 of cycles, or deviations, already reduced to units con- 
 sisting of their respective standard deviations. The 
 following table gives two such series, and the process 
 of finding their correlation. The two cycles are 
 graphed as coordinates in Figure 14, page 136. 
 
 Since the units used in both cases are standard devia- 
 tions, the spread on the two axes, as measured by the
 
 CORRELATION 135 
 
 CORRELATION OF PRICES (X) AND EMPLOYMENT (Y) 
 JANUARY, 1920, TO JANUARY, 1921 
 Cycles, in Units of Standard Deviation 
 
 Totals 
 
 X 
 
 y 
 
 X^ 
 
 xy 
 
 .75 
 
 .59 
 
 .5625 
 
 .4425 
 
 .94 
 
 .85 
 
 .8836 
 
 .7990 
 
 .91 
 
 .51 
 
 .8281 
 
 .4641 
 
 .88 
 
 1.10 
 
 .7744 
 
 .9680 
 
 .89 
 
 .68 
 
 .7921 
 
 .6052 
 
 .57 
 
 .51 
 
 .3249 
 
 .2907 
 
 .37 
 
 .34 
 
 .1369 
 
 .1258 
 
 .18 
 
 .42 
 
 .0324 
 
 .0756 
 
 -.14 
 
 -.09 
 
 .0196 
 
 .0126 
 
 -.53 
 
 -.34 
 
 .2809 
 
 .1802 
 
 -.98 
 
 -.59 
 
 .9604 
 
 .5782 
 
 -1.74 
 
 -1.27 
 
 3.0276 
 
 2.2098 
 
 -2.10 
 
 -2.71 
 
 
 4.4100 
 
 5.6910 
 
 
 
 13.0334 
 
 12.4427 
 
 
 12.4427 
 
 
 
 r = 
 
 1 o nnoA — 
 
 .96 
 
 
 13.0334 
 
 cycles squared, must necessarily be equal. If every 
 deviation in one series concurs with an equal deviation 
 in the other series, the points when plotted will neces- 
 sarily fall on a diagonal sloping upward from left to 
 right at 45°. If positive deviations concur with nega- 
 tive, the points will lie in a diagonal sloping downward 
 from left to right at 45°. In the first case a line of least 
 squares drawn through the points will necessarily have 
 a slope of -f 1, and in the second case of — 1. These 
 are obviously the largest results, both positive, and 
 negative, that could be obtained from two such correl- 
 ative series. A neutral result of zero would be ob- 
 tained if the points as plotted fall in haphazard posi- 
 tions about the two axes. The slope of the line of 
 least squares (the tangent of its angle \vith the X-axis)
 
 136 INTRODUCTION TO ECONOMIC STATISTICS 
 
 is therefore taken as the measure of correlation. Its 
 basic formula is : 
 
 2 X y 
 ^""2x2 
 
 In ordinary work it is, of course, necessary first to 
 find a trend for each series, if the cycles are to be meas- 
 ured. If the deviations are taken from the average of 
 each series, the general direction and form of the two 
 lines will be contrasted. This is equivalent to assum- 
 
 ^<r 
 
 /<r 
 
 O 
 
 -/a- 
 
 -2<T 
 
 X' 
 
 / 
 
 At 
 A/ 
 
 • 
 
 y 
 
 -^<r -/<r 
 
 /<r ^<r 
 
 FiOTjRi; 14. Correlation of price cycles (i) and employment cycles (y), 
 expressed in units of the standard deviation of each series, respectively. 
 Line of least squares (solid line), and line of unit slope (broken line).
 
 CORRELATION 137 
 
 ing a horizontal trend as a basis of measurement. In 
 certain cases this comparison may be desired. But 
 usually it is advisable to measure either the two trends 
 by eliminating the cycles or to measure the cycles as 
 taken from the trend. The latter is the usual pro- 
 cedure, since interest generally lies in a comparison 
 of the cycles. 
 
 The computing of a coeflficient of correlation from 
 data which require the finding of trends is illustrated 
 in Table XVIII. The work is for the most part self- 
 explanatory, since the trends are found by processes 
 already explained. Instead, however, of using the 
 standard deviations as the units in which to express 
 the cycles, the original units are employed throughout. 
 The reduction to standard deviation units is in effect 
 obtained by inserting a^ and fTg in the denominator of 
 the formula for r. But a substitute must be found for 
 2x-, which in the formula as just used was also in terms 
 of the standard deviation. The required substitute is 
 N. This may be seen by recalling that in the standard 
 
 deviation series /l/ -^ == a = 1. Hence ^x^ must 
 
 equal N. The formula for the line of least squares as 
 applied to the previous problem in correlation may 
 therefore be transformed into the Pearson correlation 
 formula, thus : 
 
 „ (both X and y being expressed in 
 
 r = -y-^ units of their respective standard 
 deviations.) 
 __ Sxy (in which x and'i/ are expressed in 
 Njidg terms of the original units.) 
 _ ^xy (an alternate form obtained by sub- 
 V2x22y2 stitution.)
 
 138 INTRODUCTION TO ECONOMIC STATISTICS 
 
 « 
 
 
 PM 
 
 
 H 
 
 
 O 
 
 
 < 
 
 .^ 
 
 K 
 
 * 
 
 W 
 
 m 
 
 ;> 
 
 
 < 
 
 P3 
 
 
 rr 
 
 OCQ 
 
 ^ 
 
 -f^ 
 
 <1S 
 
 
 <1/ 
 
 ^ 
 
 )-l 
 
 o 
 
 "4H 
 
 o 
 
 1— ( 
 
 
 
 
 
 
 Oh-) 
 
 t-H 
 
 
 p4 a 
 
 
 o 
 
 f^ 
 
 <*H 
 
 O 
 
 T) 
 
 
 O 
 
 ^ 
 
 
 r-) 
 
 en 
 
 M 
 
 rt 
 
 ^ 
 
 a 
 
 
 
 O 
 
 t-. 
 
 WQ 
 
 A. 
 
 ^-^ 
 
 !^ 
 
 
 P 
 
 
 O 
 
 o 
 
 M 
 
 E-i 
 
 -< 
 
 W 
 
 o 
 o 
 
 00 o ■<*< CO i-i o 00 <D kn fo cci 
 
 CC 50 CO to O CO uo lO to lo lo" 
 
 in 00 CO tJ^ i-H rH tJH i-H CO ITS 
 
 ino'cocoeoocJO'-^ci-^' 
 
 I 1 I I > 
 
 COCMrHOr-ICNCOTtlkn 
 
 oooooooooco 
 i-t t-- CO t-- ,-( in 1-H (M L-- a> CI 
 
 b-^ CQ in CO 00 in CO in oo ti< •>*' 
 
 rHr-li— li-(Csll-Hi— li-liHrHi— i 
 
 <M CO -^^ CO t-; 00 O i-H O] CO in 
 
 oo' ci o >-I oj co' in co' t-' oo' oi 
 
 i-(i-I(M(M(M(MCJ(M(MIM(M 
 
 ooO(©ao cccDi— looo 
 
 ocooJoinoin-^iHoo'id 
 
 OJcocoinoQ (M»nt~i— iin 
 
 I I I I 1 "^M 
 
 ^ fe 
 
 o in o CO 00 C5 00 CO t^ t~ o 
 00 CO CO in in in in t-^ co' oi "-H 
 
 ,— II— I<M(M(Mt-IC4Cv>OJ(MCO 
 
 CO -^ in CO t^ 00 ci o t-i ci CO 
 
 OoOOOOOi-HrHi-Hi-t 
 d Ol 05 Ci Ol C3 05 O". O; <»5 Oi 
 
 o CD 
 
 in o 
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 02 
 
 O © © A 
 
 C3 rH en o 
 
 to '^ 3 ^ 
 
 J^ aj tx CI 
 
 •2?fl 
 
 - 9 !3 la 
 
 ,Xi « 3 o 
 
 ff 3 0^ £ ■ 
 »; TO J3 
 
 •n " 
 
 g 
 
 en 
 Cj (S 
 
 
 
 ;M 
 
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 Pi m 
 
 .a .2 3 t^ 
 
 11 gat 
 a| o £-§ 
 
 cj TO 'n'*^ g 
 
 •a o " 0) 5? 
 
 ® ^ t -(^ *i 
 "^ eg ° f..^ 
 § fl OJ.S ® 
 
 1^2 +3 
 
 .9 
 
 
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 C4 vu 
 
 -^a 
 
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 tc a p,S
 
 CORRELATION 
 
 139 
 
 2 3 u 
 s a ^ 
 
 o Oi o Tt< t- 05 cvi ■* CO iq •* 
 
 f O C<i T-H rH CO * l' «? f f 
 
 * CO I* * i> r ' f iH r f 
 
 t~ «o CO ai <M 
 
 N 00 «D t-; i-j c> 00 cci iq ■* iq 
 
 l' N N CO CO t-- * r-J CO r-H rH 
 
 COTj<lO«Dt^00050lHCC]CO 
 
 OOOOOOOT-HrHrH'-i 
 
 (N_ iq 
 
 CO CQ 
 
 oO 
 
 O 05 
 00 (M 
 
 in 
 
 1 
 
 ■<}< 
 
 1 
 
 1 
 
 o 
 
 1 
 
 
 <M 
 
 
 r-( 
 
 1 
 
 
 in 
 
 r*. 
 
 
 X 
 
 >n 
 
 Oi 
 
 CM 
 
 CO 
 
 
 CO 
 
 
 
 X 
 
 1 
 
 
 
 
 i 
 
 tH 
 
 fH 
 1 
 
 s 
 
 to 
 
 
 1 
 
 jz; 
 
 co 
 'i 
 
 1^
 
 140 INTRODUCTION TO ECONOMIC STATISTICS 
 
 The Probable Error. It will be seen that the first 
 part of Table XVIII records merely the finding of the 
 trends (lines of least squares) for the two series. The 
 figures have not been carried out beyond approximate- 
 ly one per cent accuracy. The production and price 
 cycles shown in the latter part of the table are obtained 
 by the usual method of subtracting the trend items 
 
 'Z<T, 
 
 Figure 15. Variations in production and price of pig iron, United 
 States, 1908. Standard deviation units, measured from the line of 
 least squares (r = .68). 
 
 from the corresponding items in the original series. 
 The standard deviation of each of the resulting series 
 of cycles and the summation of the xy 's are next found. 
 The Pearson formula is then applied, and the result, 
 r = .45, is obtained. To this result is appended what 
 is known as the ''probable error" (P.E.), the word 
 ** error" being here used merely in the sense of di- 
 vergence, as in the theory of least squares. The for- 
 mula for the probable error expresses the quartile
 
 CORRELATION 141 
 
 deviation from the coefficient of .45 which would nor- 
 mally appear by the operation of the laws of chance. 
 The probable error therefore gives some idea of the 
 range over which the value of r has an even chance 
 of deviating, and may be used in estimating the sig- 
 nificance of the correlation. On the basis of experience 
 it is assumed that if the value of r is as low as thirty, 
 or if the probable error is as high as one-third of r, 
 
 Feb. Mar. Apr. May Jan. Jul. Aiy. Sep. Oct Nw. Dec 
 
 Figure 16. Seasonal variations in the visible supply and the price 
 of wheat, United States, 1909-1913 (r = — .87, prices preceding one 
 month). For data, see Exercise 8, page 150. 
 
 correlation is barely indicated. If, however, r is as 
 high as fifty, and if the probable error is not more 
 than one-fifth of r, correlation is clearly indicated. 
 Between these limits a correlation may be regarded 
 as more or less tentatively indicated. 
 
 The Relation of Output to Price. The result 
 obtained in Table XVIII indicates that in the iron in- 
 dustry production is directly adjusted to meet demand 
 as reflected in the price. When the price is high, pro-
 
 142 INTRODUCTION TO ECONOMIC STATISTICS 
 
 duction therefore is high, and vice versa. This relation 
 is seen to be very close in certain years if monthly 
 data are used (cf. Figure 15). The case is doubtless 
 t}T)ical of a large part of manufacturing, particularly 
 when the process is relatively short. But in agricul- 
 ture, where the maturing of the output is determined 
 by the seasons, contemporaneous movements of out- 
 put and prices are negatively related. This fact is 
 shown indirectly by Figure 16. About 75 7^ of the 
 world 's wheat crop is harvested in the months of June, 
 July, and August, with a consequent rapid depression 
 
 Figure 17. Comparison of corrected figures for index of physical 
 production of crops and index of crop prices. The long tine movements, 
 or secular trends, have been eliminated, and the two series have been 
 expressed in comparable units. 
 
 Reproduced from the Eeview of Economic Statistics, by permission 
 of the editors. 
 
 of the price. Extremely large crops are normally fol- 
 lowed for several months by unusually low prices, and 
 vice versa. This fact is depicted in Figure 17, where 
 agricultural x^roduction is plotted against the subse- 
 quent price, rather than the average price for the 
 year. A comparison of the cycles of crops and of 
 manufactures suggests that the former exert some pos-
 
 CORRELATION 
 
 143 
 
 
 
 
 
 
 -* CO 
 
 tH 
 
 00 
 
 
 
 
 
 
 lO 1-1 
 
 H^ O^ H^ 
 
 
 
 
 
 
 i-l (M 
 
 C2 CO iH 
 
 
 
 g 
 
 
 <» to 00 o o O C-) 
 I— 1 CO (M I— 1 1— 1 I— 1 
 
 rH 
 
 oi 
 
 II II I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r)< oj '^ o o o <r> 
 
 tH 
 
 tH 
 
 
 
 
 
 I— 1 t— 1 r-l iH 1 
 
 Oi 
 
 to 
 
 m CO 
 
 (M 
 
 
 
 
 1 
 
 
 '^ 
 
 OO CO o 
 
 CO •* 
 
 
 
 
 
 ' ^ 
 
 
 •O iH o 
 
 iH CO 
 
 
 
 
 
 II II 
 
 o ^- • 
 
 CO f 0"J tH 
 
 rH CO 
 iH 
 
 
 
 
 
 
 
 
 
 MW 
 
 II II'PII 
 
 " t^ II ^ r> 
 
 111! 
 
 
 b 
 
 
 t-H ■* l^ O t^ O CO 
 
 (M 
 
 
 
 
 1-1 1-1 >-l 
 
 UO 
 
 n 
 
 
 
 
 
 II 
 
 ^ .|. W 
 
 to t. 
 
 
 
 
 
 ^ 
 
 
 
 1-1 
 
 t^ 
 
 1-1 
 
 iH 
 
 
 t- 
 
 05 
 
 
 
 
 «£> 
 
 iH 
 
 iH 
 
 
 to 
 
 CO 
 
 CO 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 T-( tH iH rH 
 
 ^ 
 
 
 o 
 
 o 
 
 
 (M 
 
 
 
 
 
 C<1 
 
 o 
 
 iH 
 
 
 M 
 
 ^ 
 
 '^ 
 
 iH (M (M iH 
 
 CO 
 
 
 Tj( 
 
 CO 
 
 
 o 
 
 (M 
 
 
 
 
 
 (M 
 
 ^ 
 
 
 S 
 
 ^ 
 
 CO 
 
 rH iH 
 
 W 
 
 
 to 
 
 OO 
 
 
 es 
 
 <M 
 
 
 
 
 
 
 iH 
 
 
 P 
 
 
 
 
 
 
 
 
 
 b 
 
 J^ 
 
 oq 
 
 iH iH 
 
 (M 
 
 
 Tt* 
 
 GO 
 
 
 o 
 
 <5 
 
 W 
 
 
 
 
 
 
 
 
 CO 
 
 i-t 
 
 (M iH 
 
 CO 
 
 
 CO 
 
 CO 
 
 
 « 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 o 
 
 tH CO <M (M 1-1 
 
 05 
 
 
 o 
 
 o 
 
 
 <) 
 
 CO 
 
 
 
 
 
 
 
 
 1 
 
 ;?? 
 
 iH 
 
 rH N 
 
 CO 
 
 
 CO 
 
 CO 
 
 
 in 
 
 CO 
 
 1 
 
 
 
 
 1 
 
 
 
 9 
 
 ^tH 
 
 (M 
 
 iH Tjf rH 
 
 to 
 
 
 CO 
 
 TjH 
 
 
 3 
 
 s^ 
 
 1 
 
 
 
 
 iH 
 
 CI 
 
 
 o 
 
 CO 
 
 
 
 
 
 1 
 
 
 
 a< 
 
 
 CO 
 
 rH 'J* CM tH 
 
 00 
 
 
 •<* 
 
 oa 
 
 
 S 
 
 ^ 
 
 1 
 
 
 
 
 O] 
 
 t^ 
 
 
 CO 
 
 1-1 
 o 
 
 
 
 
 
 
 1 
 
 
 
 
 1 
 
 tH cq 
 
 CO 
 
 
 tH 
 1 
 
 00 
 
 
 o 
 
 
 
 
 
 
 
 
 
 CO 
 
 
 ■? 
 
 <M 
 
 w 
 
 
 o 
 
 iH 
 
 1 
 
 o 
 
 
 
 J^ 
 
 CO 
 
 iH .-( 
 
 (M 
 
 
 CO 
 
 CCl 
 
 
 
 1 
 
 
 
 
 iH 
 
 t^ 
 
 
 
 Tt< 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 II 
 
 
 II 
 
 II 
 
 
 
 o 
 
 
 T}H CO CO iH O 1-1 <M 
 
 ^ 
 
 
 
 
 
 
 
 
 O lO O LO C lO O 
 
 
 
 
 
 
 
 O 00 OO t- I- to to 
 
 
 
 
 
 En 
 
 trai 
 
 ice examination 
 
 gra 
 
 des — weight* 
 
 d averages. I 
 
 1^ 
 
 X 
 
 M 
 
 - I 
 t> I 
 
 i
 
 144 INTRODUCTION TO ECONOMIC STATISTICS 
 
 itive influence upon the latter within a period of a year 
 or two, but the relation is not very regular.^ 
 
 Correlation from Frequency Tables. A somewhat 
 diflScult application of the Pearson method of meas- 
 uring correlation is encountered when the two series 
 which are to be compared are each compiled in fre- 
 quency tables. The case is illustrated in Table XIX, 
 where average entrance examination grades and aver- 
 age scholarship grades for the four college years are 
 compared. The entrance examination data are ex- 
 pressed in per cents, tabulated to the nearest multiple 
 of five. The scholarship grades were expressed pri- 
 marily in six groups, ranking downward in order from 
 the first to the sixth. The averaging of such groups 
 for four years gave results carried out to fourths of a 
 group, as shown in the table. For convenience of cal- 
 culation the values of both scales are converted into 
 unit intervals measuring the deviations, the new scales 
 centering at a zero set opposite the values (70 and 
 314) which are assumed as the averages. The number 
 of frequencies for the combined series is written at 
 the appropriate coordinate points in the body of the 
 table, while the frequencies for each scale taken inde- 
 pendently are written to the right and below (column 
 F and row F). 
 
 The initial steps in the computation will be readily 
 understood by reference to the short-cut method of 
 finding the standard deviation. By this method the 
 standard deviations for both series, respectively, are 
 determined. The finding of 2xy is a somewhat long 
 
 * H. L. Mooro, in Economic Cycles, finds a positive correlation be- 
 tween an eight-year crop cycle and general prices, allowing a four- 
 yeaj lag to the latter.
 
 CORRELATION 145 
 
 process, since each of the frequencies at the coordinate 
 points in the body of the table must be taken into ac- 
 count. Each of these frequencies is multiplied by its 
 two coordinate values, and the products are totaled. 
 To illustrate, the first three columns of frequencies 
 give the following results: 
 
 F 
 
 X 
 
 y 
 
 Fxy 
 
 1 
 
 -6 
 
 
 
 
 
 1 
 
 -6 
 
 -2 
 
 12 
 
 2 
 
 -5 
 
 -1 
 
 10 
 
 1 
 
 ^ 
 
 1 
 
 -4 
 
 2 
 
 -4 
 
 -1 
 
 8 
 
 By continuing this computation, a total result of 148, 
 as the value of 2xy, is obtained. 
 
 Since the two inserted scales measuring the devia- 
 tions are not centered with precision at the two axes 
 of the table, as determined by the averages of the two 
 series, respectively, the plus and minus deviations from 
 the assumed averages will not exactly balance. Hence 
 a correction must be made in the 2xy, just as in the 
 two standard deviations. The corrections (Ki and 
 K2) applied to the finding of the standard deviations 
 are, of course, merely 2FD -^ N. It may be shown 
 that Sxy will be increased by the product of the two 
 corrections, for every item included. The corrected 
 summation of the moments about the coordinate axes 
 is therefore expressed : 
 
 2xy - NK.Ks 
 In other respects the formula is as previously used. 
 For convenience, however, it is written in the revised 
 form shown at the foot of the table. ^ Applying the 
 
 * In correlations where the deviations to be contrasted are intended 
 to be taken from the average of each series, the coefficient may be
 
 146 INTRODUCTION TO ECONOMIC STATISTICS 
 
 formula to the data in question, the value of r is found 
 to be .62 ± .06, a well-marked correlation. 
 
 TABLE XX 
 
 COERELATION OF RANKING OF STATES IN MANUFACTUR- 
 ING AND IN LITERACY, U. S., 1860 
 
 STATE 
 
 Alabama 
 
 Arkansas 
 
 Connecticut 
 
 Delaware 
 
 Florida 
 
 Georgia 
 
 Illinois 
 
 Indiana 
 
 Iowa 
 
 Kentucky 
 
 Louisiana 
 
 Maine 
 
 Maryland 
 
 Massachusetts. . . 
 
 Michigan 
 
 Mississippi 
 
 Missouri 
 
 New Hampshire. 
 
 New Jersey 
 
 New York 
 
 JSTorth Carolina. . 
 
 Ohio 
 
 Pennsylvania. . . . 
 Rhode Island. . . 
 South Carolina. . 
 
 Tennessee 
 
 Vermont 
 
 Virginia 
 
 Wisconsin 
 
 lUNK IN 
 MANUFAC- 
 TURES 
 (CAPITAL 
 PER SQUARE 
 MILE) 
 
 24 
 29 
 
 3 
 
 8 
 
 28 
 
 23 
 
 16 
 
 14 
 
 26 
 
 15 
 
 25 
 
 12 
 
 9 
 
 2 
 
 17 
 
 27 
 
 19 
 
 7 
 
 4 
 
 6 
 
 22 
 
 10 
 
 5 
 
 1 
 
 21 
 
 18 
 
 11 
 
 13 
 
 20 
 
 RANK IN 
 
 LITERACY 
 
 OF NATIVE 
 
 WHITES 
 
 23 
 24 
 
 2 
 21 
 22 
 27 
 14 
 17 
 13 
 25 
 18 
 
 5 
 15 
 
 1 
 
 9 
 16 
 19 
 
 6 
 11 
 
 7 
 29 
 12 
 10 
 
 8 
 20 
 28 
 
 3 
 26 
 
 4 
 
 1 
 5 
 1 
 
 13 
 6 
 4 
 2 
 3 
 
 13 
 
 10 
 7 
 7 
 6 
 1 
 8 
 
 11 
 
 1 
 7 
 1 
 7 
 2 
 5 
 7 
 1 
 
 10 
 8 
 
 13 
 
 16 
 
 D" 
 
 1 
 
 25 
 
 1 
 
 169 
 
 36 
 
 16 
 
 4 
 
 9 
 
 169 
 
 100 
 
 49 
 
 49 
 
 36 
 
 1 
 
 64 
 
 121 
 
 
 
 1 
 
 49 
 
 1 
 
 49 
 
 4 
 
 25 
 
 49 
 
 1 
 
 100 
 
 64 
 
 169 
 
 256 
 
 r = 1 - 
 
 62D» 
 
 P.E. 
 
 .10 
 
 NCN^'-l) 
 
 =: 1 - 
 
 6X1618 
 
 29x840 
 
 .60 
 
 1618 
 
 found directly from the original items. This is done by the use of 
 the formula given in Table XIX. An average of zero is assumed for 
 both series, and the items are treated as positive deviations from this 
 average. The standard deviations are found by the modified formula 
 explained on page 41.
 
 CORRELATION 147 
 
 The Method of Rank-differences. One further modi- 
 fication of the Pearson method of correlation, known 
 as the method of rank-differences, may be noted. This 
 method has the advantage of simphcity, and is espe- 
 cially applicable to comparisons which are made on the 
 basis of approximate measurements only. In Table 
 XX this method is illustrated by applying it to a com- 
 parison of the ranking of twenty-nine states in 1860 
 for manufacturing and literacy. The rankings as here 
 shown are based upon the census of 1860. In arrang- 
 ing such rankings, ties may sometimes occur. In such 
 a case the average rank of the tied items is applied to 
 each of the items. Thus if the second and third items 
 happen to be equal, each is ranked 2i/^ ; if the second, 
 third, and fourth are equal, each is ranked 3. When 
 the rankings have been tabulated, as shown, the dif- 
 ference between the two ranks for each state is found. 
 These differences are then squared, and the squares 
 totaled. The formula, as given at the foot of the table, 
 is an adaptation from the one last discussed. Apply- 
 ing the formula, we find that a correlation of .60 
 exists between the two series. This comparison is 
 an illustration of a number of interesting relationships 
 which may be shown to exist between the economic 
 and the social environment. 
 
 Conclusion. The purpose of this chapter will have 
 been served if the student has gained a knowledge of 
 the simpler methods commonly employed in measuring 
 correlation. The full theory of the subject is very 
 complex, and is hardly within the scope of an introduc- 
 tory course. A caution must be sounded, however, 
 against an undiscriminating application of the meth-
 
 148 INTRODUCTION TO ECONOMIC STATISTICS 
 
 ods here explained. In particular, conclusions stating 
 causal relationships should never be based on mathe- 
 matical processes alone. The data, their methods of 
 collection, and the concrete realities they are assumed 
 to measure, must all be subjected to careful scrutiny. 
 The same caution may indeed very properly be ex- 
 tended to the whole field of statistical methods. These 
 methods should prove to be valuable tools in the inter- 
 pretation of physical, biological, and social phenomena, 
 but they may be a source of positive error if their use 
 is not directed by an adequate comprehension of the 
 field of knowledge in which they are employed. 
 
 REFERENCES 
 
 Bowley, Arthur L., Elements of Statistics (4th Edition), Part 
 II, Chapters VI-IX. 
 
 Jevons, W. Stanley, The Principles of Science. 
 
 King, W. I., "The Correlation of Historic Economic Vari- 
 ables," Quarterly Publications of the American Statistical 
 Association, December, 1917, pp. 847-853. 
 
 Persons, W. W., "The Correlation of Economic Statistics," 
 Quarterly Publications of the American Statistical Asso- 
 ciation, December, 1910, pp. 287-322. 
 
 Secrist, Horace, Readings and Problems in Statistical Meth- 
 ods, Chapter X. 
 
 West, Carl S., Introduction to Mathematical Statistics. 
 
 Yule, G. U., An Introduction to the Theory of Statistics, 
 Chapters IX-XII. 
 
 EXERCISES 
 
 1. On separate sheets of cross-section paper having the same 
 horizontal scale, and! with the vertical scales so ad- 
 justed as to bring the deviations as nearly as possible 
 to the same measured average, plot the cycles of pro- 
 duction and price as obtained in exercises 1 and 2 of 
 the preceding chapter. Similarly plot the cycles in the 
 interest rate, measuring them from Figure 12, and the
 
 CORRELATION 149 
 
 price cycles as shown in Figure 6 (pp. 118 and 101). 
 Copy these cycles on tracing paper. Describe the correla- 
 tion of production and price of the two crops (allowing a 
 lag of one year for prices), and of the interest and price 
 cycles. 
 
 2. By the method of concurrent deviations, measure the 
 following correlations (Tables X, XI, and XV, pp. 51, 53, 
 and 81) : (a) Wholesale prices and per capita produc- 
 tion (both concurrently, and allowing a lag of one year 
 for prices), and (b) Wholesale prices and real wages. 
 
 3. Using the table given in exercise 8 of the preceding chap- 
 ter, and the lines of least squares there obtained, measure 
 by the Pearson "r" the correlation of production and 
 price of iron for each year there studied. 
 
 4. From the data on page 53, find the correlation between 
 wages and the cost of living for the years 1913-1920 
 inclusive (Pearson "r"). Measure the deviations from 
 the average of each series, respectively; that is, assume 
 a horizontal trend, 
 
 5. Reduce the deviations obtained in the preceding exer- 
 cise to units of the standard deviation of each series, 
 respectively, and plot as coordinates the two sets of 
 deviations thus obtained. Compute the line of least 
 squares for the points so plotted, and show that the 
 slope of this line is identical with the coefficient of cor- 
 relation. 
 
 6. Correlate the following indexes (Pearson "r") taking 
 the deviations from the average, without finding a trend. 
 
 Explain the significance of the result. 
 
 Year Prices Unemployment 
 
 1912 110 70 
 
 1913 100 120 
 
 1914 90 140 
 
 1915 90 100 
 
 1916 110 70 
 
 7. Find the Pearson coefficient of correlation for the in- 
 
 dexes of normal seasonal variations of merchandise ex- 
 ports from the United States and the price of sterling 
 exchange at New York, as follows:
 
 150 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Month Exports Sterling 
 
 January 110 100 
 
 February 95 108 
 
 March 99 109 
 
 April 90 115 
 
 May 87 116 
 
 June 80 120 
 
 July 78 119 
 
 August 85 106 
 
 September 98 74 
 
 October 125 70 
 
 November 123 80 
 
 December 130 83 
 
 (a) Find the Pearson coefficient of correlation measur- 
 ing the relationship between the following indexes of 
 
 seasonal variation in the visible supply and the price of 
 wheat in the United States, based on the years 1909-1913. 
 
 (b) Find the coefficient, as before, but assuming that 
 prices tend to anticipate the supply by about a month. 
 
 Visible Price 
 
 Month supply (first of mo.) 
 
 January 139 97 
 
 February 130 100 
 
 March 122 101 
 
 April 112 101 
 
 May 89 103 
 
 June 69 106 
 
 July 52 104 
 
 August 60 100 
 
 September 77 97 
 
 October 99 97 
 
 November 118 98 
 
 December 133 96 
 
 The following correlation table presents entrance groups 
 (vertical scale) and scholarship groups (horizontal 
 scale) for a certain class of students. Find the coef- 
 ficient of correlation (Pearson "r") and the probable 
 error.
 
 CORRELATION 
 
 151 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 
 
 
 2 
 
 1 
 
 1 
 1 
 
 1 
 
 2 
 1 
 
 
 
 
 1 
 
 
 1 
 
 4 
 
 2 
 
 
 3 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 2 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 3 
 
 2 
 
 
 1 
 
 
 2 
 
 
 
 
 10. The following correlation table classifies to the nearest 
 twenty-five per cent sixty-nine important commodities 
 according to their price indexes (base, 1913) in May, 
 1920, and May, 1921 {Monthly Labor Review, Aug., 
 1921, pp. 84-85). The correlation measures approxi- 
 mately the evenness of the price changes occurring 
 between the two dates. Find Pearson's "r." 
 
 400 
 ^ 375 
 2 350 
 .325 
 ^ 300 
 § 275 
 g-250 
 
 ^ 200 
 M 175 
 flj 150 
 •i 125 
 Ph 100 
 75 
 
 Price Indexes, May, 1920 
 
 oicoiooicoiooiooiooinoiooiooinoiooiooio 
 oocimt^ocMiot^ooainii^oiMkot^oocimi^-oiMiot^ooa 
 
 1 
 
 113 
 
 1 2 
 
 1 1 
 
 1248859444545111 
 
 1 69 
 
 11. The following table shows the ranking of states in 
 (a) Noted men bom in state, per 1000 population in 
 1880, (b) Population per square mile in 1890, and 
 (c) Per cent of urban population in 1890. By the 
 method of rank-differences measure the correlations 
 existing among these three series. 
 
 (a) 
 
 Alabama 23 
 
 Arkansas 29 
 
 (b) 
 
 (c) 
 
 24 
 
 25.5 
 
 28 
 
 28
 
 152 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Connecticut 4 
 
 Delaware 8 
 
 Florida 28 
 
 Georgia 26 
 
 Illinois 16 
 
 Indiana 15 
 
 Iowa 19 
 
 Kentucky 18 
 
 Louisiana 25 
 
 Maine 5 
 
 JMaryland 10 
 
 Massachusetts 2 
 
 Michigan 17 
 
 Mississippi 27 
 
 Missouri 21 
 
 New Hampshire 3 
 
 New Jersey 11 
 
 New York 7 
 
 North Carolina 22 
 
 Ohio 9 
 
 Pennsylvania 12 
 
 Rhode Island 6 
 
 South Carolina 20 
 
 Tennessee 24 
 
 Vermont 1 
 
 Virginia 13 
 
 Wisconsin 14: 
 
 4 
 
 3 
 
 9 
 
 11 
 
 29 
 
 20 
 
 22 
 
 23 
 
 10 
 
 10 
 
 11 
 
 17 
 
 20 
 
 19 
 
 12 
 
 21 
 
 26 
 
 18 
 
 27 
 
 9 
 
 7 
 
 8 
 
 2 
 
 2 
 
 18.5 
 
 14 
 
 25 
 
 29 
 
 16 
 
 16 
 
 14 
 
 6 
 
 3 
 
 5 
 
 5 
 
 4 
 
 21 
 
 27 
 
 8 
 
 12 
 
 6 
 
 7 
 
 1 
 
 1 
 
 17 
 
 25.5 
 
 13 
 
 24 
 
 18.5 
 
 13 
 
 15 
 
 22 
 
 23 
 
 15
 
 APPENDIX I 
 
 LABORATORY MATERIAL AND REFERENCES 
 
 Equipment for Graphic Work. While the larger part of 
 statistical work may be done without any systematic train- 
 ing in mechanical drawing, yet some degree of skill in this 
 field is necessary if graphic representations are to be satis- 
 factorily prepared. The necessary degree of skill may readily 
 be acquired. The student should provide himself with a 
 drawing board, celluloid triangles and irregular curves, a 
 ruler with decimal subdivisions of the inch, a ruling pen, 
 some round-pointed and fine pens for lettering, India ink, 
 several styles of cross-section paper, and a loose-leaf note- 
 book. If he is unfamiliar with the use of drafting materials, 
 he should read the introductory directions given in an ele- 
 mentary treatise on mechanical drawing. 
 
 In addition to the material just mentioned, some of the 
 more complicated apparatus used in an engineer's drafting 
 room will be found useful. This equipment may include a 
 drafting machine, a pantagraph, a line-spacer, a map-meas- 
 urer, and a planimeter. A blue-print outfit is also very use- 
 ful, indeed is almost a necessity, unless some improved copy- 
 ing device like the photostat is available for use. For ele- 
 mentary work a simple S^xlO'' photography printing frame, 
 and corresponding blue-print paper, will be found quite sat- 
 isfactory and inexpensive. 
 
 Lettering. It is not difficult to learn to draw freehand the 
 italic letters used by draftsmen. Directions for such work will 
 be found in "Lettering for Draftsmen, Engineers, and Stu- 
 dents," by Charles W. Reinhardt (D. Van Nostrand Com- 
 pany, New York), or in other books on the same subject. 
 
 Other Material. There are several aids to statistical work 
 that will materially lighten the drudgery incidental to long 
 mathematical processes. The most common of these is the 
 slide rule. A ten-inch rule, giving squares and cubes, will 
 be found sufficient for the greater part of the work involving 
 multiplication, division, and powers or roots. The slide rule 
 
 153
 
 154 INTRODUCTION TO ECONOMIC STATISTICS 
 
 is not difficult to use, and should be mastered by every stu- 
 dent of statistics. Besides being an inexpensive and portable 
 device for mathematical operations, it will be found useful 
 in the drawing of logarithmic or ratio graphs, which are now 
 coming into general use. If, however, it is necessary to ob- 
 tain products or quotients accurate to four or five significant 
 figures, a large cylindrical slide rule may be used, such as the 
 Thatcher, though this is less convenient and considerably 
 more expensive. For powers, roots, and reciprocals, elaborate 
 printed tables are obtainable. If possible, an adding machine 
 (listing) should be available for occasional use, such as the 
 Dalton, the Burroughs, or the Federal. While such a machine 
 is a convenience, a calculating machine is an absolute neces- 
 sity if very extensive work is to be attempted; and it is well 
 for the student to become acquainted with its operation. Sev- 
 eral successful models are now on the market, among which 
 may be mentioned the Burroughs, the Comptometer, the Mon- 
 roe, and the Marchant. The last two are dial machines, par- 
 ticularly adapted for subtractions and divisions. For certain 
 kinds of statistical work tabulating machines (Hollerith and 
 Powers types) are required, but these machines are so com- 
 plex and expensive that they can hardly be made available 
 except in the larger laboratories. 
 
 Recording. Laboratory exercises and other statistical work 
 should be recorded fully, and should be put in clear and neat 
 form. Every graph should be accurately labeled, and the 
 units used in each scale should be indicated. When the scales 
 of a graph do not begin at the zero point, the initial coor- 
 dinates should not be drawn more heavily than the others, 
 since they are likely to be looked upon as base lines if so 
 drawn. In so far as is practicable, the tables of data from 
 which a graph is drawn should accompany the figure, and 
 the source should be noted. Graphing and lettering should 
 be done in pencil first; the pencil draft may then be com- 
 pleted wnth India ink, and the pencil lines erased with a soft 
 rubber. Errors in calculation should not be tolerated. All 
 mathematical operations should be performed twice, or some 
 other reliable method of checking should be adopted. Data 
 copied from an original source should always be carefully 
 verified. Tables and mathematical processes may most con- 
 veniently be recorded on cross-section paper having one-fifth 
 or one-sixth inch spacing. In any given study the conclu- 
 sions should be brought out clearly, and their significance ex- 
 plained. 
 
 Various Types of Graphs. Most of the types of graphs in
 
 APPENDIX I 155 
 
 common use have been illustrated in the preceding pages. 
 In addition, mention may be made of certain elementary- 
 types. One of these is the bar diagram, in which bars of 
 uniform width, and proportional in length to given magni- 
 tudes, are used. They are drawn horizontally, except in time 
 series. When they are subdivided, the parts are distinguished 
 by various kinds of cross-hatching and shading. Another is 
 the "pie diagram," in which a circle is subdivided by radii. 
 This diagram is particularly adapted to the representation of 
 percentage subdivisions, such as the relative expenditures for 
 certain classes of goods in a family budget. Another type is 
 the polar chart, designed for graphing seasonal data. Draw- 
 ings of similar surfaces and solids are sometimes used in 
 the representation of given magnitudes. It should be re- 
 membered that, geometrically, magnitudes compared by the 
 use of similar surfaces vary as the square of the dimensions; 
 and by the use of similar solids, as the cube of the dimensions. 
 Sometimes in such drawings it is explicitly stated that the 
 ratio is represented by one dimension only, as when the 
 military forces of different countries are set forth by means 
 of drawings of soldiers whose heights are proportional to the 
 size of the armies. Such drawings are not scientific, how- 
 ever, and are justified only in material of a very popular 
 nature. In general, the use of similar surfaces and solids 
 in the representation of magnitudes is to be avoided. A more 
 complex type of graph is the statistical map. This may be 
 drawn in so many different ways that a general description is 
 impossible. The student having occasion to use it should con- 
 sult the excellent examples contained in the Statistical Atlas 
 of the United States. 
 
 Sources, References, and Tables. Brief summaries are 
 given below of the principal sources of statistical material, 
 and the textbook references and statistical tables which are 
 most likely to be of use in connection with an introductory 
 course. 
 
 SOURCES OF STATISTICAL DATA 
 
 Aldrich Report {Senate Report No: 1394) 
 
 Annalist 
 
 Bradstreet's 
 
 Commercial and Financial Chronicle 
 
 Dun's Review 
 
 Federal Reserve Bidletin 
 
 Financial Review {year hook)
 
 156 INTRODUCTION TO ECONOMIC STATISTICS 
 
 Monthly Labour Review 
 
 Monthly Review (Federal Reserve Bank of New York) 
 Monthly Summary of Foreign Commerce of the United States 
 Review of Economic Statisitics [Harvard) 
 Statesman's Yearbook 
 
 Statistical Abstract of the United States (yearbook) 
 Statistical services: Bahs&n's, Banker's Statistical Corpora- 
 tion, Brookmire's, and Prentice-Hall. 
 Survey of Current Business. 
 United States Census 
 
 Weather, Crops, and Markets (U. S. Dept. of Agriculture) 
 World Almanac 
 Yearbook of the Department of Agriculture 
 
 TEXTBOOKS, TABLES, AND GENERAL 
 REFERENCES 
 
 American Econmnic Review (Bi-monthly) 
 Bailey and Cummings, Statistics 
 Barker, E. H., Computing Tables and Formulas 
 Barlow's Tables 
 
 Bowley, A. L., Elements of Statistics 
 Brinton, W. C., Graphic Methods for Presenting Facts 
 Copeland, M. T., Business Statistics 
 Davenport, C. B., Statistical Methods 
 Jordan, D. F., Business Forecasting 
 Journal of Political Economy (Monthly) 
 King, W. I., Elements of Statistical Method 
 Marshall, Wm. C, Graphical Methods 
 Quarterly Journal of Economics 
 
 Quarterly Publications of the American Statistical Associa- 
 tion 
 Secrist, H., An Introduction to Statistical Methods 
 Secrist, H., Readings and Problems in Statistical Methods 
 West, C. S., Introduction to Mathematical Statistics 
 Whipple, G. C, Vital Statistics
 
 
 
 APPENDIX II 
 
 
 
 
 TABLE OF POWERS AND ROOTS 
 
 
 
 
 
 Square 
 
 Cube 
 
 No. 
 
 Square 
 
 Cube 
 
 Root 
 
 Root 
 
 1 
 
 1 
 
 1 
 
 1.000 
 
 1.000 
 
 2 
 
 4 
 
 8 
 
 1.414 
 
 1.259 
 
 3 
 
 9 
 
 27 
 
 1.732 
 
 1.442 
 
 4 
 
 16 
 
 64 
 
 2.000 
 
 1.587 
 
 6 
 
 25 
 
 125 
 
 2.236 
 
 1.709 
 
 6 
 
 36 
 
 216 
 
 2.449 
 
 1.817 
 
 7 
 
 49 
 
 343 
 
 2.645 
 
 1.912 
 
 8 
 
 64 
 
 512 
 
 2.828 
 
 2.000 
 
 9 
 
 81 
 
 729 
 
 3.000 
 
 2.080 
 
 10 
 
 100 
 
 1,000 
 
 3.162 
 
 2.154 
 
 11 
 
 121 
 
 1,331 
 
 3.316 
 
 2.223 
 
 12 
 
 144 
 
 1,728 
 
 3.464 
 
 2.289 
 
 13 
 
 169 
 
 2,197 
 
 3.605 
 
 2.351 
 
 14 
 
 196 
 
 2,744 
 
 3.741 
 
 2.410 
 
 15 
 
 225 
 
 3,375 
 
 3.872 
 
 2.466 
 
 16 
 
 256 
 
 4,096 
 
 4.000 
 
 2.519 
 
 17 
 
 289 
 
 4,913 
 
 4.123 
 
 2.571 
 
 18 
 
 324 
 
 5,832 
 
 4.242 
 
 2.620 
 
 19 
 
 361 
 
 6,859 
 
 4.358 
 
 2.668 
 
 20 
 
 400 
 
 8,000 
 
 4.472 
 
 2.714 
 
 21 
 
 441 
 
 9,261 
 
 4.582 
 
 2.758 
 
 22 
 
 484 
 
 10,648 
 
 4.690 
 
 2.802 
 
 23 
 
 529 
 
 12,167 
 
 4.795 
 
 2.843 
 
 24 
 
 576 
 
 13,824 
 
 4.898 
 
 2.884 
 
 25 
 
 625 
 
 15,625 
 
 5.000 
 
 2.924 
 
 26 
 
 676 
 
 17,576 
 
 5.099 
 
 2.962 
 
 27 
 
 729 
 
 19,683 
 
 5.196 
 
 3.000 
 
 28 
 
 784 
 
 21,952 
 
 5.291 
 
 3.036 
 
 29 
 
 841 
 
 24,389 
 
 5.385 
 
 3.072 
 
 30 
 
 900 
 
 27,000 
 
 5.477 
 
 3.107 
 
 31 
 
 961 
 
 29,791 
 
 5.567 
 
 3.141 
 
 32 
 
 1,024 
 
 32,768 
 
 5.656 
 
 3.174 
 
 33 
 
 1,089 
 
 35,937 
 
 5.744 
 
 3.207 
 
 34 
 
 1,156 
 
 39,304 
 
 5.830 
 
 3.239 
 
 36 
 
 1,225 
 
 42,875 
 157 
 
 5.916 
 
 3.271
 
 158 INTRODUCTION TO ECONOMIC STATISTICS 
 
 
 
 
 Square 
 
 Cube 
 
 No. 
 
 Square 
 
 Cube 
 
 Root 
 
 Root 
 
 36 
 
 1,296 
 
 46,656 
 
 6.000 
 
 3.301 
 
 37 
 
 1,369 
 
 50,653 
 
 6.082 
 
 3.332 
 
 38 
 
 1,444 
 
 54,872 
 
 6.164 
 
 3.361 
 
 39 
 
 1,521 
 
 59,319 
 
 6.244 
 
 3.391 
 
 40 
 
 1,600 
 
 64,000 
 
 6.324 
 
 3.419 
 
 41 
 
 1,681 
 
 68,921 
 
 6.403 
 
 3.448 
 
 42 
 
 1,764 
 
 74,088 
 
 6.480 
 
 3.476 
 
 43 
 
 1,849 
 
 79,507 
 
 6.557 
 
 3.503 
 
 44 
 
 1,936 
 
 85,184 
 
 6.633 
 
 3.530 
 
 45 
 
 2,025 
 
 91,125 
 
 6.708 
 
 3.556 
 
 46 
 
 2,116 
 
 97,336 
 
 6.782 
 
 3.583 
 
 47 
 
 2,209 
 
 103,823 
 
 6.855 
 
 3.608 
 
 48 
 
 2,304 
 
 110,592 
 
 6.928 
 
 3.634 
 
 49 
 
 2,401 
 
 117,649 
 
 7.000 
 
 3.659 
 
 50 
 
 2,500 
 
 125,000 
 
 7.071 
 
 3.684 
 
 51 
 
 2,601 
 
 132,651 
 
 7.141 
 
 3.708 
 
 52 
 
 2,704 
 
 140,608 
 
 7.211 
 
 3.732 
 
 53 
 
 2,809 
 
 148,877 
 
 7.280 
 
 3.756 
 
 54 
 
 2,916 
 
 157,464 
 
 7.348 
 
 3.779 
 
 55 
 
 3,025 
 
 166,375 
 
 7.416 
 
 3.802 
 
 56 
 
 3,136 
 
 175,616 
 
 7.483 
 
 3.825 
 
 57 
 
 3,249 
 
 185,193 
 
 7.549 
 
 3.848 
 
 58 
 
 3,364 
 
 195,112 
 
 7.615 
 
 3.870 
 
 59 
 
 3,481 
 
 205,379 
 
 7.681 
 
 3.892 
 
 60 
 
 3,600 
 
 216,000 
 
 7.745 
 
 3.914 
 
 61 
 
 3,721 
 
 226,981 
 
 7.810 
 
 3.936 
 
 62 
 
 3,844 
 
 238,328 
 
 7.874 
 
 3.957 
 
 63 
 
 3,969 
 
 250,047 
 
 7.937 
 
 3.979 
 
 64 
 
 4,096 
 
 262,144 
 
 8.000 
 
 4.000 
 
 65 
 
 4,225 
 
 274,625 
 
 8.062 
 
 4.020 
 
 66 
 
 4,356 
 
 287,496 
 
 8.124 
 
 4.041 
 
 67 
 
 4,489 
 
 300,763 
 
 8.185 
 
 4.061 
 
 68 
 
 4,624 
 
 314,432 
 
 8.246 
 
 4.081 
 
 69 
 
 4,761 
 
 328,509 
 
 8.306 
 
 4.101 
 
 70 
 
 4,900 
 
 343,000 
 
 8.366 
 
 4.121 
 
 71 
 
 5,041 
 
 357.911 
 
 8.426 
 
 4.140 
 
 72; 
 
 5,184 
 
 373,248 
 
 8.485 
 
 4.160
 
 APPENDIX II 159 
 
 
 
 
 Square 
 
 Cube 
 
 No. 
 
 Square 
 
 Cube 
 
 Root 
 
 Root 
 
 73 
 
 5,329 
 
 389,017 
 
 8.544 
 
 4.179 
 
 74 
 
 5,476 
 
 405,224 
 
 8.602 
 
 4.198 
 
 75 
 
 5,625 
 
 421,875 
 
 8.660 
 
 4.217 
 
 76 
 
 5,776 
 
 438,976 
 
 8.717 
 
 4.235 
 
 77 
 
 5,929 
 
 456,533 
 
 8.774 
 
 4.254 
 
 78 
 
 6,084 
 
 474,552 
 
 8.831 
 
 4.272 
 
 79 
 
 6,241 
 
 493,039 
 
 8.888 
 
 4.290 
 
 80 
 
 6,400 
 
 512,000 
 
 8.944 
 
 4.308 
 
 81 
 
 6,561 
 
 531,441 
 
 9.000 
 
 4.326 
 
 82 
 
 6,724 
 
 551,368 
 
 9.055 
 
 4.344 
 
 83 
 
 6,889 
 
 571,787 
 
 9.110 
 
 4.362 
 
 84 
 
 7,056 
 
 592,704 
 
 9.165 
 
 4.379 
 
 85 
 
 7,225 
 
 614,125 
 
 9.219 
 
 4.396 
 
 86 
 
 7,396 
 
 636,056 
 
 9.273 
 
 4.414 
 
 87 
 
 7,569 
 
 658,503 
 
 9.327 
 
 4.431 
 
 88 
 
 7,744 
 
 681,472 
 
 9.380 
 
 4.447 
 
 89 
 
 7,921 
 
 704,969 
 
 9.433 
 
 4.464 
 
 90 
 
 8,100 
 
 729,000 
 
 9.486 
 
 4.481 
 
 91 
 
 8,281 
 
 753,571 
 
 9.539 
 
 4.497 
 
 92 
 
 8,464 
 
 778,688 
 
 9.591 
 
 4.514 
 
 93 
 
 8,649 
 
 804,357 
 
 9.643 
 
 4.530 
 
 94 
 
 8,836 
 
 830,584 
 
 9.695 
 
 4.546 
 
 95 
 
 9,025 
 
 857,375 
 
 9.746 
 
 4.562 
 
 96 
 
 9,216 
 
 884,736 
 
 9.797 
 
 4.578 
 
 97 
 
 9,409 
 
 912,673 
 
 9.848 
 
 4.594 
 
 98 
 
 9,604 
 
 941,192 
 
 9.899 
 
 4.610 
 
 99 
 
 9,801 
 
 970,299 
 
 9.949 
 
 4.626 
 
 100 
 
 10,000 
 
 1,000,000 
 
 10.000 
 
 4.641 
 
 Note: In the above table the last two columns are correct to three 
 decimal places, without allowance for decimals dropped.
 
 York 
 
 1 r89 
 
 2 »21 
 
 3 [39 
 
 5 193 
 
 6 k02 
 
 7 .91 
 
 8 Ms 
 
 9 m 
 
 10 111 
 
 11 54 
 
 12 90 
 
 13f75 
 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 
 23 , 
 
 24 57 
 
 1850 
 
 2,995,772 
 
 23,191,876 
 
 7.74 
 
 7,135,780,000 
 
 307.69 
 
 63,452,774 
 
 2.74 
 
 63,452,774 
 
 3,7:2,393 
 
 0.16 
 
 31,981,739 
 
 1,866,100 
 
 147,395,456 
 
 131,366,526 
 
 278,761,982 
 12.02 
 
 1840 
 
 1,793,299 
 
 17,069,453 
 
 9 . o2 
 
 3,573,344 
 0.21 
 
 3,573,344 
 
 174,698 
 
 0.01 
 
 1,675,483 
 
 1,726,703 
 
 79,336,916 
 
 106,968,572 
 
 186,305,488 
 
 10.91 
 
 1830 
 
 1,793,299 
 
 12,866,020 
 
 7.17 
 
 48,565,407 
 
 3.77 
 
 48,565,406 
 
 1,912,575 
 
 0.15 
 
 643,105 
 
 2,495,400 
 
 26,344,295 
 
 61,000,000 
 
 87,344,295 
 
 6.79 
 
 1800 
 
 843,246 
 
 5,308,483 
 
 0.30 
 
 1 
 2 
 3 
 
 4 
 
 5 
 
 82,976,294 6 
 
 15.63 7 
 
 82,976,294 8 
 
 3,402,601 9 
 
 0.64 10 
 
 317,760 11 
 
 224,296 12 
 
 16,000,000 \^ 
 
 15 
 
 16 
 
 17 
 
 18 
 
 10,600,000 19 
 
 26,500,000 20 
 
 5.00 21 
 
 22 
 
 23 
 
 24 
 
 25
 
 A Picture of the Progress of the United States During 120 Years of National Life 
 
 CompUed (roni umdal Sources by O. F. Austin. StatisUclan or The NaUooal City Bank of New York 
 in. 1910 1900 189U 188U 1870 1800 
 
 
 
 
 K i^ 
 
 
 
 
 rcipts, not pr^loualf I 
 
 
 d )lBS.e80.01T. ropMtlrelr. 
 
 b 
 
 o
 
 INDEX 
 
 Aggregate method, 56 
 
 Aldrich Eeport, 4 
 
 American Economic Review, 53, 
 
 96, 124, 127 
 American Statistical Association, 
 
 96-148 
 Annalist, The, 97 
 Annalist Barometer, 125 
 Arithmetic mean, 20 
 Average, 20 
 Average deviation, 36 
 
 B 
 
 Babson, Roger W., 76, 124, 126 
 
 Babson 's index, 65 
 
 Bailey, W. B., 19 
 
 Bowley, Arthur L., 19, 44, 67, 148 
 
 Bradstreet's index, 64, 124 
 
 Bureau of Labor Statistics, 51, 56, 
 
 63, 113, 124 
 Barnett, George E., 66 
 Brinton, W. C, 44 
 Business barometers, 115 
 Business cycles, 123 
 Business statistics, 3 
 
 Circulation, 98 
 Concurrent deviations, 132 
 Continuous series, 5 
 Correlation, 131 
 Cost of living, 53 
 Corn, 76, 77 
 Cummings, John, 19 
 Cycles, 101, 113, 137 
 
 Day, E. E., 19, 96 
 Derived table, 13 
 Discrete series, 5 
 Dun's index, 64 
 
 E 
 Exports, 129, 150 
 
 F 
 
 Falkner, R. P., 3 
 
 Farm wages, 70 
 
 Federal Reserve Bank of New 
 
 York, 124 
 Federal Reserve Board, 64 
 Federal Reserve Bulletin, 65 
 Field, J. A., 44 
 Fisher, Irving, 44, 67, 96 
 Fisher's index, 86 
 Food prices, 56 
 Foreign exchange, 68 
 Free-hand method, 100 
 Frequency curve, 9 
 Frequency polygon, 16 
 
 G 
 
 Graphic work, 153 
 
 Hansen, A. H., 124 
 Howard, Stanley E., 49, 67 
 Hoffmann, F. L., 96 
 Hurlin, Ralph G., 126 
 
 Income distribution, 92 
 
 Indexes, 47, 62, 65 
 
 Index of physical production, 75 
 
 Index of quantity, 74 
 
 Index of value, 74 
 
 Ingalls, W. R., 96 
 
 Interest rates, 118 
 
 Interpolation, 30 
 
 Jevons, Stanley W., 148 
 Jordon, D. F., 121, 126, 127 
 
 161
 
 162 
 
 INDEX 
 
 Kemmerer, E. "W., 88, 123, 127 
 King, W. I., 44, 96, 98 
 Knauth, Oswald W., 92 
 Koren, John, 19 
 
 Lettering, 153 
 
 Line of least squares, 105, 135 
 Link-relative method, 120 
 Lorenz curve, 40, 44 
 
 M 
 
 Machine tabulation, 18 
 
 Macauley, Frederick E., 92 
 
 Marshall, Wm. C, 44 
 
 Median, the, 27 
 
 Meeker, Royal, 67, 96 
 
 Method of rank-differences, 147 
 
 Method of averages, 118 
 
 Method of semi-averages, 102 
 
 Method of standard quantities, 83 
 
 Mitchell, Wesley C, 67, 92, 127 
 
 Mode, 21 
 
 Monthly Labor Review, 29, 70 
 
 Monthly Review, 118, 123 
 
 Moore, H. L., 126, 127, 144 
 
 Moving average, 102 
 
 N 
 
 National Bureau of Economic Re- 
 search, 91, 96 
 
 National income, 91 
 
 National Industrial Conference 
 Board, 54 
 
 Normal curve, 10 
 
 
 
 Ogive, 34, 36 
 
 Primary table, 5 
 Probable error, 30, 140 
 Property ownership, 96 
 Proportional expenditure method. 
 
 57 
 Pareto's law, 94 
 
 Q 
 
 Quantity indexes, 74 
 Quantity theory, 87 
 Quartile deviation, 29 
 Quartile dispersion, 32 
 Quarterly Journal of Economics, 
 109, 126 
 
 Rank-differences, 147 
 
 Ratio paper, 33 
 
 Real wages, 48, 51, 53 
 
 Rectangular histogram, 25 
 
 Reference list, 156 
 
 Review of Economic Statistics, 96, 
 
 121, 123, 124, 126 
 Royal Statistical Society, 67 
 Rugg, H. O., 19 
 
 S 
 
 Salaries, 16 
 
 Salaries in Universities, 28 
 Schedules, 4 
 Seasonal index, 120 
 Seasonal variations, 116 
 Secrist, Horace, 19, 44, 67, 148 
 Semi-logarithmic paper, 33 
 Skewness, 43 
 Slichter, C. S., 105 
 Sources of data, 155 
 Standard price, 78 
 Statistical units, 17 
 Stewart, Walter W., 67, 96 
 
 Parabola trend, 110 
 
 Pearson method, 134 
 
 Persons, W. W., 121, 127, 148 
 
 Peddle, John B., 127 
 
 Pig iron, 76, 77, 128 
 
 Physical production in U. S., 80, 
 
 81, 98 
 Pratt, Andrew A., 127 
 Price indexes, theory of, 82 
 Prices, 51, 55, 60, 63, 65, 77, 113, 
 
 128, 142 » » > 
 
 Tabulation methods, 18 
 Tally sheet, 11 
 Tingley, Richard H., 127 
 Trends, nature of, 100 
 Types of graphs, 154 
 Types compared, 27 
 
 Value indexes, 74 
 
 Visible supply of wheat, 141, 150
 
 INDEX 163 
 
 W Weighted average, 20 
 
 Weights, cost of living, 71 
 Wages and salaries, 16 West, Carl S., 148 
 
 Wage indexes, 49, 51, 53, 70 Wheat, 76, 77, 129, 141, 150 
 
 Wage roll, 7 Whipple, G. C, 44 
 
 Walsh, C. M., 97 Working, Holbrook, 97
 
 r^ov 
 
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