Class J=<.iiAMA Book_^.Ri GopghtN? CfiEmiGtflT OfiPGSIK J JJ&^-3 RIVERSIDE TEXTBOOKS IN EDUCATION EDITED BY ELLWOOD P. CUBBERLEY PROFESSOR OF EDUCATION LELAND STANFORD JUNIOR UNIVERSITY DIVISION OF SECONDARY EDUCATION UNDER THE EDITORIAL DIRECTION OF ALEXANDER INGLIS ASSISTANT PROFESSOR OF EDUCATION HARVARD UNIVERSITY ll||||lMn||||M.l||||llM|||j||MM||j||lM,||,|M.MI|j|inM||j|ll.ll||j||ll.l||j| I|J|I ||||l"l||||MMM|||nMll|||lM.lll|lllM|||||llH,||||||M|,|j||,M|||j|||,lUi|||ltM||||llM||ljl|M.| -Jliinnlll 11I111..11II nlll illli lIliiMiillli.MiillliiMiillliiMiilllii.nilllnHiilLi.iilllin.iillliiHiilll illli llln I||imii|||ii..ii||||ii,ii|I|iimiii1--S STATISTICAL METHODS APPLIED TO EDUCATION A TEXTBOOK FOR STUDENTS OF EDUCATION IN THE QUANTITATIVE STUDY OF SCHOOL PROBLEMS BY '^ HAROLD O. RUGG ASSISTANT PROFESSOR OF EDUCATION THE UNIVERSITY OF CHICAGO HOUGHTON MIFFLIN COMPANY BOSTON NEW YORK CHICAGO M;.ll''!lll|ll'MI|l||MIM|||l."n||lU"ll||lll"U||inMU|j|imi,|jll..M||| tmi j|lH'.l|||ll'MI|||IMMl||llM.lll|||nni|j|||..|||||l.Mll|| l||||n.MI||llMMIl|| I--: ] !il„„lllll....llllM...llllU...lllll....llll lllhUllLMMlillnmllll.M.lLM,lllll,n,lllll,..,lllll,n,lllll.MMllln,.l.llll,M..lll I lllll....lllllm,.lll llli... COPYRIGHT, 1917, BY HAROLD O. RUGG ALL RIGHTS RESERVED rl-00 NOV -7 1917 CAMBRIDGE , MASSACHUSETTS U . S . A ©CLA477454 EDITOR'S INTRODUCTION Such a volume as the present number in this series of textbooks forms an interesting exhibit of the progress at present being made in the organization of instruction in the subject of education. Two decades ago there would have been almost no use for such a volume, as we had not then begun to make any accurate measures of the products of our educational efforts. Only the most general terms were then in use, while to-day the demand is for quantitative expres- sion in commonly used terms which students can under- stand. Especially within the past decade has there been a remarkable evolution of standards for educational work and quantitative units of measurement. To-day the educational investigator and the superintendent of instruction alike need to use refined tools in the measurement of educational results. To such, and to the students in our schools of edu- cation generally, the simple presentation of the mathematics underlying the accurate measurement and plotting of edu- cational results here presented should prove of large use- fulness. The author of this volume has stated the aims and pur- poses and plan of the work so well in his preface that little remains that an editor needs to say. The volume represents a very successful attempt to produce a book which will apply the mathematical theory of statistical work to educational problems, and as such it should find a hearty welcome from teachers of education in universities, colleges, and normal schools, educational investigators generally, and school oflSi- cers interested in making the best use of statistical data and displaying the results to their supporting public in the vi EDITOR'S INTRODUCTION most effective graphic form. The author has been particu- larly fortunate in the selection of what to include in the volume, and in the organization and presentation of what he has included. Ellwood p. Cubberley PREFACE During the past two decades a body of quantitative technique has developed in education which makes constant use of technical statistical methods. The school man, in trying to keep pace with the developing tools, has constantly demanded a complete exposition of them. At the same time he has made it very clear that the treatment which will appeal most pertinently to his needs must be couched in non-mathematical language. He has said frankly that his mathematical training has been limited to high-school alge- bra, and rather an ancient and, in some sense, obsolete al- gebra at that. He has told us that "graphs" are mysterious things to him; that equations of lines and formulae have no significance; that the use of "frequency distributions," "probability curves," "medians," "measures of variabil- ity," and "coefficients of correlation" can hardly be said to lend clearness to his thinking about his own school problems. Three courses are open to the writer who wishes to ac- quaint such persons with statistical methods of treating facts. First, he can say that the school man's lack of familiarity with college algebra, analytic geometry, the calculus, and least squares is his own lookout, and that it is impossible to write a "statistical methods" and to give statistical training without presupposing this particular kind of equipment. We have available now several books and many mono- graphs built on that basis which make use, more or less in detail, of the higher mathematics, but none of which are applied to educational problems. Second, he can give the student of education a manual of formulae and rule-of-thumb methods of computing the vari- viii PREFACE ous coefficients, without any explanation of the derivation of these constants, without an adequate exposition of how to discriminate the use of the different methods, and with- out making possible a complete and proper interpretation of the results of using the methods. To do this would commit the writer to the rather current theory that, for the educa- tionist, "statistics is arithmetic," and that his statistical equipment should include only the ability to compute the various coefficients and to follow rule-of-thumb methods of interpreting them {e.g., the rule that a coefficient of correla- tion of say .25 is "high," "low," or "what-not"). The few books and chapters of books which have so far applied sta- tistical methods to school problems have been very largely committed to this doctrine. Third, the writer in this field can assume that it is neces- sary to equip school men, generally, with a thorough-going knowledge of statistical methods; that in order for them to be discriminating in the use of the various methods in improving their school practice, this large background of knowledge must be developed; and that it is possible to ex- plain rather completely the reasons for and the significance of the principal statistical devices without expressing the explanation in technical mathematical language. This book has been written with a deep-rooted conviction that the third of these three courses is the proper one; with a complete recognition of the limitations in mathematical equipment of the "average" school administrator and teacher, which is the outcome of considerable classroom con- tact with this particular kind of student. It is based upon the knowledge, however, that it is possible to make clear the significance and proper use of the more important sta- tistical devices without expressing these in mathematical form. The necessary substitution of words for symbols in the PREFACE ix explanation of the derivation and common-sense significance of such devices has resulted in what, to the mathematically trained reader, will seem to be a "wordy" book. The pre- rogative of the "author's preface" leads the present writer to say frankly that in this book he has not been interested in writing for the mathematically equipped reader. At the same time, it is hoped that such a person can, indeed, get an initial view of statistical methods from the following chap- ters which he can use to advantage in a study of the second- ary and original works of Yule, Bowley, Elderton, Karl Pearson, and others who have constructed our statistical tools. The book throughout has been written in intimate contact with graduate classes in education. It is the direct out- growth of mimeographed notes written for seven of such classes, and elaborated and revised distinctly in terms of their specific needs and interests. Symbolic and word ex- planations have given way to graphic devices wherever necessary and possible. The many repetitional "back refer- ences," restatements of principles, reasons, etc., in succeed- ing chapters have been made with a full recognition of the possible inelegance in form, but with a firm conviction in the value of the resulting increase in clearness to the reader. Traditional usage in the form of textbook writing has been deliberately sacrificed to the one criterion of readableness. A very small group of students of education have made use, recently, of certain methods which have not been in- cluded in the discussion of this book. Outstanding among these is Yule's Partial Correlation, and Spearman's methods of "correcting" coefficients of correlation. To a very small group of educational psychologists these may seem unpar- donable omissions. However, neither set of methods could have been presented in the complete fashion necessary in the treatment of those topics without encroaching unduly upon X PREFACE the limited space of this textbook, already devoted to more important methods. Furthermore, it is doubtful if the for- mer of these methods will be used by more than a very small fraction of those working in educational research in our own generation. These persons should turn to Yule's complete original discussion. In regard to the latter of the two sets of methods, the writer is one of those who are still skeptical of the use of methods of "correcting" coefficients (the validity of which has not been established) which have been com- puted from material collected under conditions subject to such gross inaccuracies as are the conditions of educational research. It is fundamental to a clear comprehension of the writer's point of view to know that this book is based upon the doc- trine that statistical methods in themselves prove nothing, — that the methods selected for use in a particular situation must agree with the logic of that situation : in a word, that statistical methods are merely quantitative devices which we can use to refine our thinking about complex masses of data, and to refine our methods of expression. The example of Leonard P. Ay res in his discriminating use of statistical methods in school research, and his con- stant subordination of the exhibition of statistical form to clearness and simplicity of presentation, has been a potent factor in determining the writer's point of view, and has wrought a definite effect upon his practice. Harold O. Rugg. School of Education, University of Chicago, August 22, 1917. CONTENTS I. The Use of Statistical Methods in Education . 1 II. The Colijection of Educational Facts ... 28 III. The Tabulation of Educational Data ... 57 IV. Statistical Classification of Educational Data: The Frequency Distribution 74 V. The Method of Averages 97 "VT. The Measurement of Variability .... 149 VII. The Frequency Curve 181 VIII. Use of the Normal Frequency Curve in Education 207 IX. The Measurement of Relationship: Correlation 233 X. Use of Tabular and Graphic Methods in Report- ing School Facts 310 Selected and Annotated Bibliography . . . 361 Appendix . . . . 376 Index 405 LIST OF DIAGRAMS K Representing the rate of reading of a third grade . . ♦ . 6 2. Representing degree of comprehension of same children . . 7 3. Recording and computing device for determining class efficiency in arithmetic 8 4. Courtis's diagnostic curve for arithmetic ... . .10 5. Per cent of failures in each grade in three June promotions . .11 6. Per cent of failures in reading in each grade for two years . .12 7. Per cent of failures in arithmetic in each grade for two years . 13 8. Relative rank of Minneapolis for all school expenditures ... 14 9. Mean costs of high-school subjects 15 10. Difference between the various percentages of total expense and median percentages, for Washington, D.C 17 11. Scale of algebraic difficulty 18 12. Distribution of I.Q.'s of 905 unselected children 19 13. Hollerith tabulating card used in Oakland, California ... 69 14. Hollerith sorting machine opp. 70 15. Hollerith tabulating machine opp. 71 16. To illustrate use of "scale," "unit," "class-interval," and "fre- quency distribution" 78 17. Another form of the same . . . ". 79 18. To illustrate use of coordinate axes X and Y . . . . . 89 19. Frequency polygon representing integral measures .... 91 20. To illustrate plotting of frequency polygon for a grouped distribu- tion 92 21. To illustrate the plotting of a "column diagram" . . . . 94 22. Ideal curves to illustrate difference in variability in two distribu- tions, whose means are identical 98 23. Comparison of a plot of actual scores with smoothed curve . .102 24. Comparison of form of distribution of human traits with "normal probability" curve 105 25. To illustrate computation of the median . '. . . . .111 26. To illustrate the same Ill 27. To represent the use of "standard deviation," "mean deviation," and " quartile deviation " on normal and skewed curves . . 152 28. To illustrate the use of "standard deviation" and "probable error" as "unit distances on the scale " 153 29. To illustrate the computation of mean deviation by the short method 164 xiv LIST OF DIAGRAMS 30. Frequency polygon and column diagram to represent distribution of abilities of 303 college students in visual imagery . . . iSs 31. Comparison of "actual frequency " polygon with result of first and second "smoothings" 185 32. Distribution of 5714 marks given in plane geometry .... 187 33. Polygons representing various expansions 202 34. Graph of the line ?/ = 4x + 8 208 35. Distribution of measures in five groups under the normal curve . 217 36. Same, with different unit length and base line 218 37. "Normal" distribution of "errors" in averages 228 38. Distribution of correlated abilities in mathematics and languages 237 39. Same, plotted diflFerently 240 40. Same, plotted for 130 college students 241 41. Same, data plotted under assumption that all points are concen- trated at mean points of the class-intervals of the table . . 242 42. Data of Diagram 40, tabulated as in 41 243 [ 43. A Galton diagram for representing correlation graphically . . 246 , 44. Pairs of measurements plotted 248 45. To illustrate the first step in plotting a correlation table . . .261 46. To illustrate the computation of the correlation and the regression coefficients for the case of linear regression 264 47. A product-moment diagram 266 48. Relation between cost-per-student-recitation in English and the number of pupils instructed, in 148 Kansas high schools . . 277 49. Abstract from Table 43, to illustrate certain facts graphically . 280 50. Illustrating groups of measures set from type 295 51. To illustrate the computation of the "contingency coefficient" . 300 52. Comparison of board of education budget with that approved by the city council 319 53. Comparison of possible taxation for general purposes with that levied, for a series of years 321 54. Same for permanent improvements 324 55. Total city and school bonded indebtedness, for a series of years . 325 5Q. Rank of Cleveland in group of eighteen cities in expenditure for operation and maintenance of schools 329 57. Same in per capita expenditures for different municipal activities 330 58. Showing number of elementary teachers receiving various salaries 335 59. Showing training of teachers in Cleveland 337 60. Showing, for a series of years, degree to which the public schools of Grand Rapids are educating the children of school age in the city 340 61. Persistence of attendance at school in St. Louis 341 62. Showing the holding power of the schools 347 63. Per cent of children of each age and progress group in school at the close of the school year 348 LIST OF DIAGRAMS xv G4. Progress of ten typical pupils through the school system . . . 348 65. The environment of a minor during the principal periods of his growth 349 66. Distribution of pupils by nationalities in two elementary schools 349 67. How 915 children spent their spare time on two pleasant days in June 350 68. Average scores made in spelling in ninety-sk elementary schools 350 69. Some standards used in judging school buildings . . . .351 70. Ratio of glass area to floor area in Minneapolis schools. . . 352 71. Plan of educational organization in a small city ..... 353 72. Percentage distribution of non-administrative positions in office work in Cleveland 354 73. Nationality of workers in the building trades in Cleveland . .355 74. Illustrating spelling difficulties 356 75. Illustrating the importance of after-school activities . . . 357 76. Illustrating seating conditions in the school 357 77. Illustrating the school program 358 78. Illustrating promotions and failures 358 79. Showing the percentage of children having playgrounds of va- rious sizes 359 80. What the school records relating to medical examinations show . 360 LIST OF TABLES 1. Form A, grade record (H. A. Brown, 1916) 5 2. Correlation between first and second opposite tests . . . .20 3. Data on the careers of teachers 61 4. Present age of teachers 63 5. Relation of pedagogical and mental age 76 6. Relation of mental age and school marks 76 7. Advanced degrees held by members of normal school faculties . 82 8. Average salaries in North Central normal schools and colleges . 82 9. Class marks given to 123 high-school pupils in English . . 83 10. Table 9, rearranged under four different classifications . opp. 87 11. Number of factoring problems solved correctly by 137 pupils . 90 12. Comparison of approximate and true modes, pauperism data . lOl 13. Same, barometer data 101 14. Scores obtained by two groups of 11 pupils in factoring tests . 107 15. Same data, differently arranged 107 16. Distribution of marks in Latin given to 289 high-school pupils . 110 17: Cost for instruction in English in ten cities, illustrating long method of computing arithmetic mean 115 18. Cost per-pupil-recitation of English in 148 Kansas cities . . 116 19. Table 18 regrouped in class-intervals of two units each . . .118 20. Effect of grouping on size of arithmetic mean or median . . 119 21. Efficiency of 365 college students in tests for visual imagery, il- lustrating long method of calculating arithmetic mean . .120 22. Mean for Table 17, recalculated by short method .... 121 23. Table 21, recalculated by short method ...... 123 24. To illustrate the computation of quartile deviation, mean devia- tion, and standard deviation for the ungrouped series . . 157 25. To illustrate the computation of quartile deviation for the grouped series 158 26. The marks given 289 pupils in Latin, to illustrate computation of mean deviation by long method 161 27. To illustrate the computation of mean deviation with devia- tions stated in true values, but in units of class-intervals . . 162 28. To illustrate the computation of mean deviation by short method 163 29. Same, when true median falls below the assumed median . . 165 30. To illustrate computation of standard deviation by short method 171 31. Average percentile payments for general and municipal service . 176 32. Averages for spelling ability for 20,000 sixth-grade children . 226 33. School marks given a class in mathematics and modern languages 234 xviii LIST OF TABLES 34. Same, re-marked to show agreement 235 35. Showing relationships oi x on y 258 36. Showing relationships of y on x 258 37. To illustrate the second step in the tabulation of a correlation table 2G2 38. Columns corresponding to row 96-100 268 39. Columns corresponding to row 76-80 268 40. To illustrate computation of r without tabulation of the correla- tion table 275 41. Correlation between cost of instruction per-pupil-recitation, and the number of pupils taught by one teacher . . . ^ .281 42. Comparison of expenditures per pupil for various kinds of edu- cational service 290 43. Rank of measures in two series 295 44. Relative position of each pair of measures with reference to aver- age of both series 295 45. Relation between mental and pedagogical age 306 46. Date giving results of computing — — for each compartment of Table 45 307 47. Comparing the board of education budget and council allowance 318 48. Comparing possible school taxing capacity for a series of years, with probable actual tax levies 320 49. School bonded indebtedness, for a period of years .... 322 50. School bonded debt compared for a number of cities . . . 323 51. Total amounts of outstanding bonds maturing each year . . 326 52. Expenditure per inhabitant for schools compared .... 327 53. Expenditure per $1000 of wealth for schools compared . . .328 54. Distribution of current expenditures for schools, seventeen cities 332 55. Distribution of school officers and teachers in different grades of schools, for a period of years 333 56. Showing the distribution of teachers' ranks and salaries in St. Louis 334 57. Showing the general level of salaries in a city 336 I 58. Showing the years of teaching service of all teachers employed . 336 59. Showing the number of pupils per teacher, elementary grades . 337 60. Showing the number of pupils per teacher in different classes of schools 337 61. Number of pupils per teacher in nineteen American cities . . 338 62. Number of children of school census age 342 63. Showing total and average enrollment, and average attendance . 342 ,64. Showing ages of children in each grade . . . . • . 343 65. Showing years in school of children of each grade .... 344 66. Showing attendance in elementary schools for the year . . . 345 67. Showing promotions for the school year, of all kinds' . . .345 68. Cost data for nine fireproof elementary buildings in Boston . . 346 STATISTICAL METHODS APPLIED TO EDUCATION CHAPTER I THE USE OF STATISTICAL METHODS IN EDUCATION Problems and Methods in School Research Steps in the development of "scientific education." There are two groups of persons in the educational world who are directly interested in the application of statistical methods to school problems — the school administrator, and the teacher and educational psychologist. In corre- spondence to these two classes of interest, school problems may be said to be either administrative or pedagogical-ex- perimental in character. They arise either in connection with the attempt of the administrative agents of a school system to fit the "machinery of the system" to the needs and capacities of children, or to the attempt of the school man and the psychologist, working together, to determine more minutely the status of learning in the child. The school man's chief concern, then, is with these questions: First, how does the child learn .^^ Second, how may the course of study, methods of teaching, modes of classifying and pro- moting children, methods of organizing the school year, safe-guarding the health of school children, etc., be best adapted to the established facts of development and proc- esses of learning in children, and to the needs of their future life. The method of attacking the solution of such fundamental 2 STATISTICAL METHODS questions prior to our own generation was clearly tradi- tional and based on individual experience-. It was said by the representatives of the established sciences, and freely admitted by the pedagogues, that "education" was not a "science" ^ — that its method was not "scientific." By this was meant that school men did not make use of the fundamental steps in the scientific procedure of solving problems. Fundamental steps neglected. To be specific: (1) They did not systematically observe educational conditions, or collect necessary facts, recording their observations mi- nutely. More concretely this meant that they did not set about collecting the facts on the composition, training, certification, tenure, pay, and rating of the teaching staft'; the content of courses of study; the age-grade distribution of pupils, and their progress through the grades; the cor- responding measurement of instruction and the capacities of pupils; the extent of their elimination from and retardation in the public schools; the many facts concerning the central administration and organization of schools; school costs, school accounting, and the efiiciency of business manage- ment; operation of the plant and the handling of equipment and supplies, — the determination of each of which is neces- sary to the promotion of efficient school administration. Thus, the first step in the utilization of the scientific method — the collection of large numbers of facts — was not taken. (2) The indictment of our traditional pedagogy pointed out that students of "pedagogy" did not "measure" the results of school work, that no yardsticks were available by which the efficiency of school administration or school teaching could be evaluated; hence that little progress in the improvement of either one could come about. Nobody knew accurately to what extent boys and girls had mas- tered the elements of reading, writing, arithmetic, spelling. USE OF STATISTICAL METHODS 3 geography, history, and language. We simply knew that there was an accumulation of incapacity in particular grades of the public schools, relieved in part by rapid elim- ination of pupils from school. (3) In pedagogy, however, it was evident that since almost no collection of facts was made that no recourse was had to the development of mathematical or statistical methods of treating the data. Large quantities of data accumulat- ing in the biological and physical sciences had demanded and led to the development of sound methods of statistical treatment in those fields. Prior to fifteen years ago " peda- gogy," however, had made no use of the large body of sta- tistical technique that had been put together. (4) "Science" demands as the capstone of its procedure the utilization of a thoroughgoing experimental attack on the problem in question. Conditions must be "controlled" by the investigator, measurements must be made as mi- nutely as possible, records of results must be kept, and the data which have been collected must be systematically organized through the utilization of valid statistical methods. Again, prior to our generation, this experimental procedure had not been used in education. It is true that four dec- ades ago various German psychologists began the study of " learning " under isolated conditions, and with fairly refined laboratory technique planned a way for the transfer of their technique and certain of their grosser conclusions to class- room analysis of learning and teaching. This actual trans- fer, however, has come about in our own time. Lack of thorough collection of facts concerning educa- tional conditions, measurement of results, statistical treat- ment of the data, setting up of experimental methods of studying school practice, — these are the counts on which the older "pedagogy" was indicted. Recent developments^ The above statement of the ways 4 STATISTICAL METHODS in which pedagogy failed to utihze scientific method reveals specifically the steps in the development of " scientific edu- cation" during the past two decades, and leads naturally to an exposition of the use of statistical methods to school men. The school man has turned to exactly these steps of procedure in the attempt to determine the present status of school practice and to direct scientifically the course of its development. We have said above that school problems were either administrative or pedagogical-experimental in character. Our first task in taking up the study of " statistical methods applied to educational problems" is to recognize clearly the various school problems whose solution demands treat- ment by numerical methods. During the past fifteen years every phase of school administration and pedagogy has been subjected to quantitative methods of study arid ex- perimentation. Our educational literature abounds with *' factual " studies, our educational conventions are given over very largely to discussions of "measurement" and * 'standardization" of school processes. Outstanding at the present time, therefore, is the need for a clear, scientific, and complete statement of the statistical and graphic methods which the school man must call to his aid in this quantita- tive attack on educational problems. To get sharply before us a picture of the new emphasis, let us turn briefly to a few examples of the use of statistical methods in education. These have been so selected that the general field will be brought in review. I. Quotations from Recent Quantitative Literature 1. Measuring reading ability The checking-up of the results of school teaching by standardized tests is one of the most promising phases of USE OF STATISTICAL METHODS 5 the new movement. The tabulation and classification of the results of testing has led to the development of devices for recording school facts and for presenting the data. The need for tabular and statistical methods is well illustrated by the following quotation from Brown. ^ Tovm X School A Table 1. Grade Record Form A Date of Test, June 4, 1915 Grade III Pupil Name Age Rateof Devia- Compre- Devia- Reading Effi- ciency Deviation No. Yr. Mo. Reading tion hension tion 1 1.28 —1.79 75 +35 96.00 — 18.79 2 l.GO —1 47 38 — 2 60.80 — 53.99 3 1.85 —1.22 63 +23 116.55 + 1.76 4 2.07 —1.00 50 +10 103.50 — 11.29 5 2.15 — .92 40 86.00 — 28.7^ 6 2.33 — .74 70 +30 163.10 + 48.31 7 2.36 — .71 33 — 7 77.38 — 36.91 8 2.38 — .69 58 +18 +10 138.04 + 23.25 9 2.38 - .69 50 119.00 + 4.21 10 2.63 — .44 36 — 4 94.68 — 20.11 11 2.98 - .09 44 + 4 131.12 + 16.33 12 2.98 — .09 - 22 -18 65.56 - 49.23 13 3.00 - .07 17 —23 "51.00 — 63.79 14 3.15 + .08 22 —18 69.30 — 45.49 15 3.26 + .19 11 -29 35.86 — 78.93 16 3.28 -f- .21 33 — 7 108.24 — 6.55 17 3.32 + .25 55 +15 182.60 + 67.81 18 3.78 + .71 32 — 8 120.96 + 6.17 19 4.30 +1.23 32 — 8 137.60 + 22.81 20 4.60 +1.53 29 —11 133.40 + 18.61 21 5.83 +2.76 60 +20 349.80 +235.01 22 6.02 +2.05 14 —26 84.28 — 30.51 Average 3.07 0.90 40 15 114.79 40.39 Diagnosis of Class and Individual Needs In Table 1 are given, for purposes of illustration, the data from an actual third grade. This grade stood second among thirteen 1 Brown, H. A. The Measurement of the Ability to Read. Bulletin no. 1, Bureau of Research, New Hampshire Department of Public Instruction, Concord, N.H. 6 STATISTICAL METHODS third grades which were tested, and represents a somewhat satis- factory efficiency. Examination of the averages shows that the rate ^ of reading, the comprehension, and the reading abilitij of the X i • 5 /o II n 1} /I* fS lb n If 1$ 20 Zf 21 z3 2i Diagram 2. Curve representing Comprehension of the Saime Children as in Diagram 1 and in the Same Order The scale at the left shows the comprehension. (H. A. Brown, 191-). ciency, which is 114.79, is high. We find individual variations in comprehension and reading efficiency, but these are not nearly so great as in most of the classes thus far tested. In fact, it can be 8 STATISTICAL METHODS said that the class is in a rather satisfactory condition in this re- spect. While the rate of reading is high, there are, however, ten pupils whose rate is considerably below the average of the class. They should be given special quick percep- tion practice daily to bring their rate of reading up to a higher standard. There are ten pupils whose comprehension falls con- siderably below the average of the class, four of whom fall conspicuously low and can easily be identified in Diagram 2. They need special practice in rapid silent reading with special emphasis upon getting a maximum of content from what is read. The four who get the lowest marks in comprehension are seen on Fig. 3 to have a very low score for reading efficiency. We may now examine a number of indi- vidual cases. It is easy to see that Pupil No. 1 is deficient in the rate at which he can read. He gets a relatively large proportion of the content at his present rate of reading, but he reads so little in a unit of time that his efficiency is low. He should have practice to increase his speed, and if it is found that at a higher rate of reading his comprehension is p>oor, he should be given practice for the purpose of bringing about improvement along this line also. Pupil No. 2 has a dif- ficulty which is easy to diagnose. In the first place his rate of reading is not suffi- ciently rapid, but on quantity of reproduc- tion he stands high. His mark for quality, on the other hand, falls to zero. In other words, he gets a good many ideas in the rough but gets nothing accurately. We see in the case of this pupil one advantage of the method of scoring reading ability advo- cated in this bulletin. It enables us to find more correctly the exact location of defects in the reading ability of individual children. What this pupil gets is a mere smattering of the idea. His low mark for comprehension, together with his low Test Nc ).l Attempts ^ Rtgl,t3 1 FRQ. DEV FRQ. DEV. 24 23 22 21 20 19 18 17 / 16 o 15 1 14 1. 13 S 12 1 2. 11 o ^ 10 5 '^!i 9 5 ^•' 8 1 s 7 3 3 6 6 2. 5 7 ,f 1^ 4 M 4i 3 1 2 M- 1 3 'M Av.Wed 9.0 io Cor. .6 .(, Meil. 7.6 S.(. M.D. ^icCt^l^UD. -^ -sSZ V»t. i \' Diagram 3. Record- ing AND Computing Device for deter- mining Class Effi- ciency IN Formal Processes in Arith- metic Note the use of statistical methods. (S. A. Courtis, 1917.) USE OF STATISTICAL METHODS 9 rate of reading, gives him a low efficiency. He needs to work both for speed and for accuracy. Pupil No. 4 reads at a rate consider- ably below the average. He gets, in a rather rough way, a very large percentage of the ideas, but he is very inaccurate. Mr. Brown's material illustrates the use of averages, measures of variability and graphic methods for diagnosing weaknesses in school work. 2. Scientific supervision of arithmetic This type of statistical device may be supplemented by some of Mr. Courtis's recording devices in the improvement of teaching in arithmetic. Diagram 3 gives a simple chart for tabulating the number of pupils attempting various numbers of problems, the number of pupils working va- rious numbers of these correctly, the approximate median, (Ap. Med.) ; the correction (cor.) ; the true median (Med.) ; the mean deviation (M.D.) and the accuracy. . Diagram 4 presents Courtis's "Diagnostic Curve of Me- dian Development in Speed and Accuracy" in arithmetic, the use of which is explained in the following quotation : — In Diagram 4 are drawn curves for two school systems. Curve A is for a small village school in New Hampshire. Curve B again represents the scores made by the group of 29 school districts in Boston which have been tested every year since 1912. The work in school A is very poor. Grade four falls entirely out- side the diagram. Grades five and six in speed nearly equal the fourth- and fifth-grade standards, respectively, but in accuracy are way below the normal fourth-grade level. From the sixth grade on, the effect of school work is to emphasize accuracy, so that while the seventh and eighth grades approach more nearly the normal curve, i * Mr. Courtis's use of '^normal curve" in this connection should not be confused with the standard practice of reserving that term for the so-called "curve of error," the "normal probability curve," etc. There is great need for uniformity in practice in our statistical terminology. Such terms as "normal curve" have really become standard in our thinking and their specificness of meaning should not be clouded by multiplying terms. 10 STATISTICAL METHODS the increased accuracy is obtained at the expense of speed. The eighth-grade scores are lower than those of the seventh grade, and none of the scores reach the normal fifth-grade level. Curves of tliis character are evidences of lack of supervision, of poor, inef- fective teaching, and are far too common in country schools. Curve B, on the other hand, indicates good quality of work and steady progress. Note that the curve lies wholly above the normal Addition Diagnostic Curv« of Median Development in Speed and Aouuraey. Grades 4 to 8 inclusive June 191G. Scores 1 ^P""* Number of Esample8AtU.tai.wd |2 8 4 B 6 7 8 9 10 11 12 13 14 15 16 Accuracy l'"l'»"l .1 1 1 1 < 80 Any class whose position falls on this side oftl.e cupie Is high Id accuracy. Do not neglect speed. _6_ — ^ bB 1 - 4 5^^ -7^ ..^- £ er'^ f J / 1 1 1 A > I ■o .2 1 1 k^ rt 40- 00^ / .Any class whoso position falls on this side of the curro U low in accuracy. Work to increase accuracy. Positions to the rigl.t ( or lolt) of a grnde number in the median .urve indicate greater ( or le«a ) speed than tlie median speed for thalgmde. | Diagram 4. Courtis's Diagnostic Curve for Arithmetic (S. A. Courtis, 1917.) curve, and that each grade circle shows not only greater accuracy than normal, but greater speed as well. Note also that the largest growth occurs between the fifth and sixth grades, the second largest between the seventh and eighth grades. The curves of the schools doing the best work tend, in similar fashion, to approximate the 80% line in addition, although few attain to as high speed levels as those shown in curve B. S. Studies of failures in the public schools One of the most important types of administrative study that can be made of a school system is a study of the failures of its pupils in the different grades and different subjects. Somewhat recently these problems of non-promotion have USE OF STATISTICAL METHODS 11 been studied analytically, to the great benefit of the schools in question. A practical graphic device for revealing lack of adjustment between pupils and the work of specific Per cent 20 15 10 • .••'V ■^•r — \ V -J "" y / \ 191U 1915 1913 123V567S GRADES Diagram 5. Per cent of Failures in Each Grade FOR Three Successive June Promotions (C. H. Judd, Cleveland Survey Report, 1915.) grades or subjects is shown by samples from Mr. Judd's Report in the Cleveland Survey/ Diagrams 5, 6, and 7. 4-. Comparative method of analyzing city school costs In recent years many superintendents of schools have been adopting simple quantitative and graphic methods of ac- * Judd, C. H. Measuring the Work of the Public Schools. (Cleveland Survey Foundation Reports. 1916.) 12 STATISTICAL METHODS qualnting the public in their communities with school needs and school practice. Progressive among these has been Superintendent F. E. Spaulding, now of Cleveland. Dia- gram 8 illustrates his adaptation of the comparative method Per. cent 20 \ \ 15 \ - \ \ \ \ \ \ \ \ \ \ ^ \ \ 10 \ ♦ w \ \ \\ \\ \V 5 > ^;^'>. , x*» V* V % v^ ^;: ^^^ ^X- L ; 2 1 ^ \ ♦ ! 5 i > r s 1913 GRADES Diagram 6. Per cent of Failures in Reading in Each Grade for Two Successive Years (C. H. Judd, Cleveland Survey Report, 1915.) in studying the financial status of a school system. After a very detailed comparison of the expenditures of Minne- apolis for specific school activities, with those of twenty-four other cities, he sums up the situation in the following diagram: — USE OF STATISTICAL METHODS 13 5. Cost for high-school subjects The comparative method of studying school situations has led to the use of many statistical and graphic methods Per cent <^v ..'•••/ \ \ / 15 / \ »••** • / ^ / ~\ / ' y' '**'^^ 10 ( r. rY l! ^ ll \\ •» * 5 /; / 1 /. /^ /• / ; L 2 ; > » ^ I > ( > 1 1913 GRADES Diagram 7. Per cent of Failures in Each Grade FOR Two Successive Years (C. H. Judd, Cleveland Survey Report, 1915.) of presentation and interpretation. Mr. Babbitt's use of the middle 50 per cent (those between the two quartiles) to give a *'zone of safety," by indicating both the attain- ment and relative position of each city, school, or class in the group is shown in Diagram 9. 14 STATISTICAL METHODS 6. Use of ''ranking'* methods to determine relative standards in school efficiency Comparative "ranking" methods of studying school efficiency were used by Updegraff, in his Study of Expenses of City School Systems.^ In his discussion of method of treatment of data he says : — J. ' It has come to be generally accepted that the way in which to III 1 'o. 1 1 ^1 M 14 1 1 1 ll ^ 1 1 u 1 2 K 3 / \ 4 / \, K 5 / \ h 6 / \ \ 7 1 \ \ 8 \ \ 9 \ ^ 10 \ / \ 11 \ / \ 12 13 Media A \/ \ /\ \ 14 / \ \ 15 / \ \ \ 16 / \ \ 17 \ \ , \ 18 \ / \ / \ \ 19 \ / A/ \ 20 , / ~v— \ \ 21 \ / \ \ 22 \/ \ \ 23 V Only lli cilies repurteJ anj expenditure for tlie Promution of Health, and on that basjs. the rank' V \ 24 V \ 25 apoiis V « Diagram 8. Showing Relative Rank of Minneapolis for all School Expenditures Is Minneapolis spending too little, comparatively, for janitors' wages? The diagram shows Minneapolis was the thirteenth or median city for this item of expenditure. Do the supervisors receive too large a proportion of the total school expenditures? Minneapolis is twelfth compared with the other cities of her class. Has the average expenditure for five years been high for textbooks? Yes. It was the second in the list. But during part of the period high schools texts sold at cost to pupils were included under general maintenance. (F. E. Spaulding, 1916.) » Updegraff, H. Bulletin no. 5, U.S. Bureau of Education. (1912.) USE OF STATISTICAL METHODS 15 give the clearest and at the same time the most accurate measure of a series of numbers is to state the median of the series and the $90- Q.-70- 50- 40- Shop-work 93 Normal Training &2 Latin 71 English 51 Agriculture 48 20- Music 23 Diagram 9. Mean Costs of High-School Subjects " The variety of prices paid for the same quantity of instruction in the various subjects is shown. The subject of median cost stands at $62. The middle zone of variability shows a rcjM'/cfrom $55 to $70. For those that now stand above this zone of 'normal variability' it is possil)le that administrative readjustments are desirable for the purpose of bringing them down, and thus eliminating waste. For those below this normal range of variability classes need to be cut down in size, teachers better paid, or the teaching week shortened, so as to bring them at least nearer the range of normality. In other words, just as it is possible to de- termine standard costs for each of the various subjects separately, out of the practical situa- tions where those subjects are taught, so it may be possible to determine flexibility standards of cost for the entire situation applicable to the entire range of subjects. Whether or not this can be profitable can be known only after such standards have been derived for high schools of hnmogeneous classes, and involving large numbers. After the matter has been tried out its worth can be known." (F. Babbitt, 1916.) limits of the middle 50 per cent. In time past the arithmetical mean or average has been used for this purpose, and it still has its value. Nevertheless its disadvantages, especially that of the undue weight 16 STATISTICAL METHODS exercised by a number which is very large or very small as com- pared with the others in the series, are causing the increased use of the median wherever practicable. The second feature of the general method of treatment is the ''ranking'' of the various amounts in each column by groups. The "rank'' of an item is its place in the series, as arranged for the determination of the median and the middle 50 per cent, as just described, the item lowest in value being given rank 1, the next to the lowest rank 2, and so on. In other words, the "ranks" are the result of the process of the numbering of the series, which neces- sarily precedes the determination of the median and the middle 50 per cent. No element of comparative worth is attached to the numbers given. In some items, as in fuel, it is creditable to a city to have a low number; in others, a high number. The purpose for the insertion of the columns entitled "rank" in the tables is merely to facilitate the comparison of items. As an illustration of his use of the method, we may quote : — Comparison of distribution of expenses in one city with distribu- tion of expenses in other cities of the same group. This may be done in a cursory manner by extending the process just indicated to all items, and forming a rough judgment as to the items in which the city is low or high as compared with the group as a whole. The more accurate method consists in computing the differences between the percentages of the various classes of expenses for the city and the corresponding medians, and arranging the excesses and deficiencies in separate lists. As those items that vary most from the medians are of greatest importance, and as variation from the median to the extent of the limits of the middle 50 per cent may be regarded as normal, the computation of differences m cases wherein the city's percentage is within the limits of the middle 50 per cent may be for all practical purposes neglected. The following diagram (10), presents the result of such a computation for the city of Washing- ton. 7. Use of the ''normal curve'* in designing school tests In attempting to improve the marking of pupils and the planning of school tests, much recourse is had now to the USE OF STATISTICAL METHODS 17 normal probability curve. One example can be given from the writer's discussion of standardized tests in algebra.^ A more complete quotation from this study is given in Chapter VIH. This briefer one will serve to illustrate the method : — DEFICIENCIES 7.0 e^O 5jO 4.0 8.0 2j0 'i° . . . <^ ■ EXCESSES « r i*. GENERA). CONTROL J-EVENINQ SCHOOLS I MISCELLANEOUS j EXPENSES Diagram 10. Differences between the Various Percentages of Total Expenses that Lie outside the Limits of the Middle Fifty Per cent, and the Median Percentages for the Same Items, FOR Washington, D.C. (H. Updegraff, 1912.) Let Diagram 11 represent the distribution of algebraic abilities in the pupils represented by our 27 school systems. The base line then represents a "scale of algebraic difficulty" ranging, let us say, from nearly ability to nearly perfect or 100 per cent ability. . . . Taking as our unit of measurement on the base line, sigma, <'', or the "standard deviation" of the distribution (indicated graphically on Diagram 11), and laying it off 2.5 times each way from the mid- point of the curve, gives us 5 divisions (which may be conveniently divided into 10 divisions, corresponding "practically" to our public-school marking system). In doing this we are arbitrary to the extent of neglecting only 0.62 of 1 per cent of our pupils at each end of the base line. If this 0.62 of 1 per cent is thrown into the middle of the curve where the individuals are more closely grouped, it is a negligible factor. Calling the point 2.5 x sigma * School Review, February and March, 1917. 18 STATISTICAL METHODS from the mid-point 0, and setting the successive points 10, 20, 30, etc., to 100, we now have a practical working "scale of algebraic difficulty" over the successive points of which the corresponding percentages of our pupils may be indicated. Doing this, we see in So '^' /w/i^ 9^ /V^/.^g />y ^care^ 'O ^O 5o ^ >^5 €0 TO 60 90 7\ 90 too Diagram 11. Scale of Algebraic Difficulty Distances on the base line represent, to scale, relative difficulty of problems. Area under the curve represents total number of pupils that were tested for ability to translate verbal problems. and 100 points set arbitrarily at I.SXa from the mean. Mean is set arbitrarily at 50. Area of the curve between and any point on base line represents percentage of pupils who failed the problem placed at that point. (H. O. Rugg, 1917.) Diagram 11 the proportions of our group of pupils that correspond to various degrees of difficultj^ on the base line. Thus a problem which is failed by 96.6 per cent of the group falls at the point marked 85; that failed by 84.8 percent is scored 70, etc., througliout the list. To enable us to mark in an accurate way, a table has been com- puted in which the base line has been divided into 500 parts. USE OF STATISTICAL METHODS 19 8. Distribution of general intelligence in school pupils The study of the distribution of general intelUgence in pupils in our public schools is likewise making use of quanti- tative methods. Terman ^ points out the symmetry of the plotted results of testing the intelligence of 905 school chil- dren, as follows : — The I Q's were then grouped in ranges of 10. In the middle group were thrown those from 96 to 105; the ascending groups in- 66^ 66-76 76-85 86-95 96-105 106-U5 116-125 120-135 136 -Itf iSfS £9jS BJ5* SOa< 88.99^ 2SM 9.0J$ 2.3« J55i DiAQBAM 12. DiBTBIBUnON OF I Q's OP 905 UnSELECTED CHHiDREN, 5-14 Years of Age . (L. M. Terman, 1916.) eluding in order the I Q's from 106 to 115, 116 to 125, etc.; cor- respondingly with the descending groups. Figure 12 shows the dis- tribution found by this grouping for the 905 children of ages 5 to 14 combined. The subjects above 14 are not included in this curve because they are left-overs and not representative of their ages. The distribution for the ages combined is seen to be remarkably symmetrical. The symmetry for the separate ages was hardly less marked, considering that only 80 to 120 children were tested at each age. In fact, the range, including the middle 50 per cent of I Q's, was found practically constant from 5 to 14 years. The tendency is for the middle 50 per cent to fall (approximately) between 93 and 108. 1 Terman, L. M. The Measurement oj Intelligence, p. 66. (Houghton Mif- flin Co., 1916.) 20 STATISTICAL METHODS 9. Correlation between mental tests A quotation from Freeman's ^ discussion of methods of testing in the laboratory shows the following use of corre- lation: — Table 2. Correlation between First and Second Opposites Tests X y Individ- Score Score dif. of dvff. of ual in I in II scores in I from average scores in II from average x"^ y^ xy 1 15 10 -4 -3 16 9 +12 2 15.5 10 -3.5 -3 12.25 9 +10.5 3 IG 6 -3 -7 9 49 +21 4 17.5 10 -1.5 -3 2.25 9 + 4.5 6 17.5 11 -1.5 -2 2.25 4 + 3.0 6 17.5 18.5 -1.5 +5.5 2.25 30.25 - 8.25 7 18.5 11 - .5 -2 .25 4 + 1 8 19.5 13 + .5 .25 9 20.5 10 +1.5 -3 2.25 9 - 4.5 10 20.5 13 +1.5 2.25 11 20.5 20 +1.5 +7 2.25 49 +10.5 12 22 17.5 +3 +4.5 9 20.25 +13.5 13 23.5 16 +4.5 +3 20.25 9 + 13.5 14 24 18 +5 +5 25 25 +25 Average . 19 13 105.5 226.5 101.75 'S.x- y 101.75 101.75 V105.5X226.'5 " 154.6 65.8 V2 xa • 2 1/2 sum of the produc ts of x and y square root of (the sum of x2 X the sum of y^) Table 2 illustrates a form of procedure which is necessary, in many cases, to obtain a reliable calculation of correlation, that is, the determination first of the reliability of the measures secured in each test by itself. This is secured by finding the correlation 1 Freeman, F. N. Experimental Education, pp. 177-79. (Houghton Mifflin Company. 1916.; USE OF STATISTICAL METHODS 21 between the two performances in the same test, using, where the nature of the test demands it, different subject-matter in the two performances. If this correlation is not fairly high — above ,60 — the degree of correlation between this test and others is of little significance, since the scores are not accurate measures of the ability in question. A formula has been developed by Spearman to correct a coefficient of correlation when it is reduced by lack of precision in the results in the individual tests, but the reliability of this formula is doubtful, and it is far better to perfect the methods of giving the test until their results are consistent. In the case before us two series of opposites were used with the same persons. The correlation between them appears from the table to be satis- factory (r = 65.8), though it might well be higher. Use of quantitative methods. The foregoing quotations offer but a crude and inadequate picture of the extent to which students of education are making use of quantita- tive methods in attempting to solve their administrative and pedagogical problems. They merely serve to illustrate the principal statistical and graphical methods Vv^hich will be taken up in the succeeding chapters. It is felt, however, that there is a need for a more complete organization of the "field of educational research" than as yet has been made. Many quantitative studies have appeared in each of the various phases of scientific education. The student is baf- fled by a maze of scattered material. To aid him in organ- izing his thinking, by cataloguing the various educational problems and the methods by which school men are trying to solve them, Plate I is included in Chapter X. On this plate the writer has attempted to give definite reference to all the studies that are of any importance to school men. The chart is so built as to indicate two important charac- teristics; it states: (1) who has studied each of the various problems; and (2) by what methods these persons have attempted to solve these problems. The key number at- tached to each name refers to the position in the complete 22 STATISTICAL METHODS bibliography given at the end of Chapter X. ^ It will be noted that no attempt has been made to include the studies in the field of educational psychology. To give the student the key to this field, selected references containing complete bibliographies are given. II. The More Important Groups of School Problems In summarizing the discussion of this chapter let us bring in review a brief statement of each of the more important groups of school problems. To enumerate them we find : — 1. Administrative problems Study of the curriculum. There have been concerted at- tempts to establish minimum essentials in the course of study in our schools, — question-blanks have been sent out covering various phases of the content of the curriculum; textbooks have been analyzed in a tabular way; judgments of specialists have been secured concerning the proper organization of subject-matter; industrial, economic, and social conditions in various types of communities have been studied with a view to adapting school practice to them. Facts about the teaching staff. By means of question- blank methods and personal investigation of state school laws, city school-board by-laws, manuals, rules, and records, and Federal, state, and city school reports, — quantitative facts have been collected about the teacher : who she is, what home environment she came from, how much training and experience she has had; facts about her appointment, cer- tification, salary, progress in the teaching profession, and her classroom efficiency. Problems centering about the pupil. Personal study of individual systems, supplemented by the question-blank, has been used by private and public agencies to ascertain USE OF STATISTICAL METHODS 23 the status of pupils in our schools; in what way they are distributed through the elementary and secondary grades, according to relative ages; non-promotions and rates of progress through the grades; how pupils are eliminated from school; administrative devices ("promotion systems" or " plans ") for adapting the machinery of the school system to the capacities, needs, and interests of the child; method of "marking" the pupils' achievement. Status of school finance. Recently school costs and busi- ness management have been studied in this same quanti- tative way. Originally by question-blank, but mainly by individual investigation of school laws, charters, and rec- ords, specialists are establishing the legal basis of school finance, the status of city and state school revenues and expenditures, unit costs for education, methods of raising and apportioning school funds, and the efficiency of the business management of our city schools. Measurement of school and teaching efficiency. During the last seven years the school world has at last turned to the construction and use of tests and "scales" to measure the results of teaching. Accompanying the attempt to study the content of the curriculum, to clarify and make definite the aims and outcomes of teaching, there has developed a most promising and important movement of educational measurement. In answer to the critics of the "older peda- gogy" the newer and more scientific "educationist" is de- vising and using tests to measure the results of teaching in practically all of the "skill" or "formal" subjects. There are now available six handwriting "scales," of varying de- grees of usefulness to classroom work; as many standard- ized reading tests; many discussions of measuring spell- ing "ability"; a fairly large and definite body of results in testing arithmetical abilities, some extensive work in the field of algebra tests, — with little or nothing done in the 24 STATISTICAL METHODS remaining subjects. Accompanying this material, we now have a growing body of critical data on the validity of such tests. Furthermore, during the past five years more than fifty American school systems have been "surveyed" by groups of outside specialists — men who came into the systems in question and collected, by detailed personal investigation from the ofl&cers, teachers, and records of the system itself, sufficient facts to adequately typify the practice of educa- tion in that city. " School measurement " has seen its most thorough-going development in this school-survey move- ment. Problems of central organization and administration. Even the board of education in American cities has been subjected to the same type of quantitative study. Its pres- ent status as to size, qualifications for membership, tenure, compensation, and methods of selecting board members; their functions, powers, and duties, and the way they carry on their business, have been numerically determined by both question-blank analysis and by personal study of the charters, by-laws, rules, and records of city school systems. Miscellaneous educational activities. In the same fashion, various miscellaneous educational activities have been can- vassed in a tabular way, — problems of school hygiene, medi- cal inspection, rearrangements of the school year, etc. All of the above types of problems are administrative in character. In each of them we have noted the recurrence of the fundamental initial methods of statistical inquiry, — the collection of educational data by either (1) question- blank; or (2) some method of personal investigation. These will be discussed definitely in Chapter II. In addition to these outstanding administrative "prob- lems," we must bring into our perspective of the " field of educational research" a statement of the more important USE OF STATISTICAL METHODS 25 experimental problems of learning and teaching. For our purposes a brief enumeration of the principal types of study will have to suffice. 2. Pedagogical-Experimental problems Problems of "learning" were first studied in a controlled way under isolated conditions in the psychological labo- ratory three decades ago. The names of the leaders of va- rious German schools, Ebbinghaus, Meumann, Kraepelin, Lay, etc., are linked up, literally, with scores of specific quantitative studies of isolated learning. These may be listed under the following points : — Studies of the " practice " or ** learning " curve. Data were collected and interpreted on the improvement of sub- jects in doing a particular mental act {e.g., memorizing se- ries of nonsense syllables) ; facts were collected on the rate of improvement, the amount of improvement, the limit of improvement, the mental qualities that conditioned im- provement, changes in the rate and the permanence of im- provement. Each of the studies involved the use of many quantitative methods. During the past fifteen years these studies have come out rapidly from American laboratories, and gradually have been extended to include specific types of mental work done in the class room.^ Formal discipline. Since James suggested the use of quan- titative methods in studying the possibilities in formal dis- cipline in 1890, thirty-odd reports have been made on the influence of training in one field of mental activity on per- formance in another field of mental activity. The old tra- ditional a priori method of controversial discussion has given way to an experimental and statistical attempt to es- ^ For fairly complete bibliographies on the "Practice Curve," "Mental Fatigue," "Mental Work," and "Mental Types," see Thorndike's Educor tional Psychology, vol. ii, entitled The Psychology of Learning. 26 STATISTICAL METHODS tablish scientifically the status of the possibility of ''trans- ference of training." ^ Mental work and fatigue. In the same way the condition- ing factors of "mental fatigue" and "mental work" have been tested under controlled experimental conditions, and a fairly large body of data collected. General intelligence and mental inheritance. A very voluminous literature is already available giving the results of the application of experimental and statistical technique to this group of problems. Similarly, many studies have been reported on problems of mental inheritance, carrying over the same statistical methods from the field of biologi- cal investigation. 2 These, then, are the administrative and experimental problems which the school . man of to-day is trying to solve. During the past ten years he has turned decidedly to quan- titative methods in studying school practice. Each phase of school work is being subjected to "counting" methods of study. School discussions are becoming thoroughly factual. III. Steps in Educational Research In revealing the problems of school research we have pointed out the outstanding methods of collecting educational data. At this point the student should have in mind at least a rough persj)ective of the general steps in the com- plete procedure of working out a statistical problem. In 1 For a complete summary of all published literature see the present writer's Experimental Determination of Mental Discipline in School Studies. (Warwick & York, Baltimore, Md., 1916.) 2 Complete bibliography on these fields of study can be found in Thorn- dike (referred to above); Whipple, G. M., A Manual of Physical and Mental Tests (2 volumes, Warwick & York, Baltimore, Md., 1916); and Stern, W., Psychological Methods of Testing Intelligence (Warwick & York, 1916); or in Meumann, E., Psychology of Learning (1916). USE OF STATISTICAL METHODS 27 bringing to a close this introductory discussion we should now connect this first step in school research with the re- maining steps. To merely enumerate them at this point, a complete statistical analysis of a carefully-defined educa- tional problem would necessitate the following steps : — A. Necessary steps. 1. The careful definition of the problem. 2. The collection of educational data. 3. The original tabulation or arrangement of data. 4. The systematic classification of data (in frequency distributions) . 5. The summarization or condensation of data. Two general methods: 1. analytic; 2. graphic. B. Analytic methods. These are classified as: — 1. The method of "averages" — representing the typical condition or "central tendency." 2. The method of "variability," representing the extent to which data vary around the average. 3. The method of relationship between various sets of data. 4. The method of reliability — establishing the amount of dependence that one may place on the statistical results of his investigation. C. Graphic methods or the reporting of school facts. The use of various types of frequency curves, dia- grams, charts, etc.; the application of "type" fre- quency curves {e.g.^ the normal probability curve) to educational data. These steps and methods will be taken up and explained and illustrated in the chapters which follow. CHAPTER II THE COLLECTION OF EDUCATIONAL FACTS If a superintendent of schools or an "interested citizen" wished to collect facts on any of the types of problems men- tioned in Chapter I, he would have access to four principal sources of original data. These may be stated, in tabular form, as follows : — I. General Sources of Original Educational Data These general sources may be enumerated under the fol- lowing main headings : — A. State school laws and city board of education charters. ■ B. Published official reports. I. Federal reports, generally published annually. a. Annual reports of the United States Bureau of the Census. b. Annual reports of the United States Bureau of Edu- cation. c. Annual reports of the United States Bureau of Labor Statistics. n. State reports. a. Reports of state superintendents of public instruction (or equivalent oflficer), or state boards of education, in each of the States. b. Reports of other state departments: e.g., Indiana Bureau of Statistics; state census reports; etc. III. Publications of city school systems. a. Manuals, by-laws, and rules and regulations of city boards of education. b. Periodic "proceedings" or "minutes" of meetings: 1. Of city boards of education. COLLECTION OF EDUCATIONAL FACTS 29 2. Of permanent and special committees of boards of education. (Former are publislied in medium- sized and larger systems; latter are not.) c. Annual reports of city boards of education. d. Special bulletins, issued either by the superintendent or by some other school official, or, in a few cities, by the bureau in charge of school research. C. Types of school research by private agencies that may con- tain "original" data. I. School survey reports. Published reports are now available for forty to fifty cities, and eight States, few of which, however, contain "original" data. Material mostly of "summarized" and comparative type. XL Published reports of studies made by educational foun- dations {e.g., Russell Sage Foundation, Division of Edu- cation; Carnegie Foundation for the Advancement of Teaching; General Education Board). III. Published reports of studies made by individuals, con- taining, in rare cases, "original" data. D. The original records of: I. Federal and state bureaus or departments. II. City school systems. These, then, are the sources ^ which are now available for the collection of facts about educational practice and con- ditions. It will be of some value to describe briefly each of ^ Each student of school research should also secure, each year, a bulle- tin issued by the United States Bureau of Education, entitled Educational Directory (for 1915-16, 1916-17, etc.). This pubHcation contains complete lists of the names of officers of (1) United States Bureau of Education; C2) state school systems; (3) state library commissions; (4) superintendents of schools in cities and towns of 2500 population and over;- (5) associate and assistant superintendents in larger cities; (6) county superintendents; and (7) officers of miscellaneous institutions; e.g., schools of pedagogy, normal schools, colleges, and universities, schools for blind and deaf, feeble-minded, etc., schools of art and of industry, parochial schools, directors of museums, library schools, church educational boards and societies; state, national, and international educational and other learned and civic organizations. 30 STATISTICAL IV^ETHODS the types of data that can be secured from each source, naming the kind of facts to be found, and characterizing the relative vahdity of each. A. School Laws and City School Charters At the present time a codification of the state school laws (a summary of all legislation affecting the conduct of schools in each State) is issued by the Department of Edu- cation (or of Public Instruction) in nearly every State. Those desiring to collect detailed facts on the legal status of any aspect of school administration should turn to these sources. Various compilations of state legal provisions, and decisions of state and federal courts on school matters, have been made under the direction of the United States Bureau of Education. Bulletin no. 47 (1915), Digest of State Laws Relating to Public Education, in Force January i, 1915, is a rather extensive compilation of the actual legal basis of American school administration. In addition to this, the United States Bureau of Education has issued each year a compilation of legislative and judicial decisions on educa- tion for the current year. Of all these sources, the codi- fications of the state school laws themselves are the only ones containing the detailed legislation. City school-board charters are found in various published sources, sometimes published and bound with certain issues of the annual report of the board of education; more com- monly published and bound with the rules and by-laws of the board. Thus they are quite generally reprinted only on dates of revision. Students who desire to study the legal basis of any as- pect of city or state school administration should turn to the original statement of the law itself, found in one of these sources. - COLLECTION OF EDUCATIONAL FACTS 31 B. Published Official Reports 1. Federal reports Educational statistics have been published annually by three federal agencies : the Bureau of the Census, the Bureau of Education, and the Bureau of Labor Statistics. Let us characterize each of these briefly. (a) Educational statistics in reports of the United States Bureau of the Census. Prior to 1912 this bureau published completely analyzed data on public-school finance. The most immediate sources were found in an annual bulletin called Financial Statistics of Cities, and covered all Ameri- can cities of 30,000 population and over. The published facts included complete descriptions of methods of securing the data, of the accounting terminology used by school statisticians, detailed statistics of receipts and disburse- ments, property valuations and municipal indebtedness for all city departments including schools, classified in such a way as to permit intelligent study of school costs. These data, as reported to and including the year 1911, were collected by agents of the bureau by personal tabula- tion from the records of the school systems in question. Data on school cost, to be comparable, must be classified on a uniform basis. Prior to 1911 it was a very evident fact there was no semblance of uniformity in the accoimting methods of different city school systems. Hutchinson in 1914 reported that he visited thirty-eight cities trying to secure comparable data on school costs, and found the sum- marized statistics worked out on so many different bases that it was impossible to make comparative statements about the cost of different kinds of school service and school activities from these summary statements. The agents of the Bureau of the Census, therefore, rendered a distinct 32 STATISTICAL METHODS service in classifying, in detailed fashion, and on pertinent educational bases, various educational financial data. The best assumption the student can make about the validity of original administrative data on school costs is that those in the reports of the Bureau of the Census are approximately accurate. The relative validity of these data and those in the reports of the United States Bureau of Education will be discussed below. In addition to the purely educational statistics that can be found in the Financial Statistics of Cities, in detailed form through 1911, and in general summary form since 1911, the Bureau of the Census published many reports containing municipal, economic, population, and industrial statistics. Various special bulletins can be secured by ad- dressing the Director of the Census, Department of Com- merce and Labor, Washington, D.C. (b) Annual Reports of the United States Bureau of Education. The Commissioner of Education publishes each year an annual report, in two volumes. Volume 1 contains descriptive summaries of educational movements, past and present. Volume 2 reports detailed statistics of all phases of public and higher education in this country, for cities and towns of 2500 population and over. These include all facts on school finance analyzed in very detailed fashion, facts on the distribution, grades, experience, training, age, sex, and pay of teachers; facts on attendance, enrollment of pupils in public and higher special schools, etc. In addition to these the bureau also publishes, intermittently, compila- tions of original statistics covering particular aspects of school administration, as, for example, salaries paid to va- rious grades of teachers, together with number of teachers receiving these salaries; salary schedules, etc., for all cities above 2500 population, etc. Validity of data in reports of the Bureau of Education. COLLECTION OF EDUCATIONAL FACTS 33 These data have always been secured by question-blank methods; almost never by personal investigation of the records of the school systems by agents of the bureau. They are collected annually on a detailed blank form, the business and statistical clerks of the various systems filling in the required data. The result of the use of this method has been that the statistics have been very unreliable (for com- parative purposes), both absolutely and relatively. Prior to the year 1911 they were distinctly so, due to the fact that there was almost no uniformity in city school account- ing methods, and there was comparatively little agitation (at least prior to 1905) for getting cities to use uniform sys- tems of records and reports. During the years 1905 to 1910, a growing demand for improvement in these conditions led to the cooperation of the United States Bureau of Educa- tion, the National Education Association, and the newly formed National Association of School Accounting Officers (1910) in an attempt to standardize accounting and sta- tistical methods in city schools. A joint committee of these agencies recommended the adoption of a " Standard Form," for recording and reporting all types of school sta- tistics. The Bureau of Education adopted this form in 1911 for its annual collection of data, and a decided improvement has taken place in the character and validity of the school statistics during the past five years. It is estimated that fully five hundred American city systems are now classifying their records in accordance with this form. It is true, how- ever, that many cities, particularly some of our larger cities, having school officers of initiative and originality, have been slow to change their school accounting systems to accord with the standard scheme. Even to-day some of these, although laboriously retabulating their statistics for the Commissioner's Report each year, use their own inde- pendent system of accounting. 34 STATISTICAL METHODS Thus, it is believed that, since 1911, the educational sta- tistics of the Bureau of Education have increased steadily in reliability for "comparative ranking purposes," although still collected by question-blank methods. It is to be re- gretted that, with the use of the standard form by the Bu- reau of Education, the Bureau of the Census stopped making its detailed classification of educational statistics in 1911, reporting since that time only very general summaries of school receipts, expenditures, indebtedness, etc. In making the study of the Public School Costs and Busi- ness Management in St. Louis (1916), the writer attempted to establish the validity of the statistics of the Bureau of Edu- cation for purposes of comparing various cities by arranging them in "rank" or "serial" order in their various financing activities. It was assumed that the financial statistics of the Bureau of the Census to and including the year 1911 were approximately correct. It was found that the Bureau of Education in the same year, 1911, published the same type of financial statistics, thus providing an opportunity for direct comparison of the absolute figures compiled by two agencies on identical school activities. Tables in the com- plete survey report give, as obtained from each source, the total expenditures and differences in amount spent for each of a list of cities, for nine different kinds of school service — special supervision, principalships, instruction, supplies, etc. Tables computed and stated in the survey report give the per pupil cost for each of these nine kinds of service, to- gether with the rank of each of the cities in the group for each item. It is clear from inspection of those tables that we have to discuss the validity of the data as collected from these two sources strictly in terms of the use we are going to make of them. First : if we merely are going to rank cities in terms of per pupil cost, then the statements made in the survey report are valid, namely : — COLLECTION OF EDUCATIONAL FACTS 35 With few exceptions the tables show a very satisfactory agree- ment in position, the costs for supervision and principalships being the ones for which less agreement would be expected than for any other activities. The conclusions that we form from one set of rec- ords will not be unlike those formed from the other set of records. Especially is this true in the case of the one city in which we are interested, St. Louis. We may summarize its position in all the tables as follows: — Salaries of Textbooks Supervisors Principals Teachers Repairs Janitors Bureau of Cen- sus 4 4 7 6 11 9 8 6 4 3 8 Bureau of Edu- cation 8 The largest displacement in the ranking for St. Louis is two places. As a result of the tabulation and ranking it is believed that the interpretations made on the financial situation in St. Louis, from cost tables computed from the Annual Report of the United States Bureau of Education, 1915, will be valid. Especially is this true since 1912 was the first year in which the bureau collected statis- tics on the standard form, and much improvement has come about since in the completeness and accuracy with which city systems report their school facts. The most frequent use that school men want to make of educational statistics is of this very '* comparative" and "ranking" type. One point should be noted, however. These cities are the largest cities in the country, and have the most thoroughly equipped accounting and statistical staffs, supervised by specialists in this field. The experience and investigations of the writer lead to the belief: — (1) That considerable reliance may be placed on the com- parability of the classification of educational statistics for groups of medium-sized cities (15,000 to 40,000 for example). These cities are following the "Standard Form," even more 36 STATISTICAL METHODS closely than are the larger cities. The comparative financial statistics of the Bureau of Education for twenty-one cities in Indiana, Illinois and Wisconsin (between the sizes of 15,000 and 25; 000, and within 150 miles of Chicago) have been checked with care. The results show a fair agreement be- tween the records as compiled by the bureau and by other agencies. The methods have been checked personally in three of these cities, and show that considerable reliance may be placed on the absolute expenditures reported, as well as on the "position" of each city in the group. (2) In the study of larger cities, however, it was found that, if we wish to deal with the absolute statistics of cost, attendance, teaching staff, etc., we must make decided men- tal reservations in our acceptance of the Bureau of Edu- cation figures. In the first place, there are occasionally very large differences in reported figures due to incorrect classifica- tion (for example, expenditures for supervisors and principals in certain cities). In the second place, differences of 10 to 20 per cent are relatively common in these tables. The present study, however, can merely warn the student of the large inaccuracies in the absolute figures reported by certain cities to the Bureau of Education. (c) Annual reports of the United States Bureau of Labor Statistics. If the school man desires statistics on the occu- pational situation, distribution of workers as to grade, trade, salaries paid, etc., he can find such data in annual reports and bulletins of this bureau, by addressing the director. COLLECTION OF EDUCATIONAL FACTS 37 2. State school reports The superintendent of public instruction, or the depart- ment of education in each of the states, now issues either biennial or annual reports on educational activities in the state. A very considerable body of original statistical ma- terial may be found in these. It is fairly common, for ex- ample, to classify the statistics by counties, instead of enu- merating them for cities and towns. On the whole, it is rarely that one can find detailed data on city schools in state school reports. Furthermore, it is uncommon to find data detailed enough on town and rural schools to permit of comparative studies of educational practice in specific communities. The reports are filled up with narrative reports from county and other school ofiicers, from various special and higher institu- tions controlled by the state, state courses of study, reports on county institutes, digests of school laws and legal deci- sions, state examination questions, and other types of de- scriptive material. They all give certain detailed financial and attendance statistics on the "common schools," ar- ranged by counties. In exceptional cases, good comparative data can be obtained. For example, the state report for Missouri contains a detailed financial analysis for several hundred towns and cities in the state. It is possible to use the data in making a comparative cost study for particular communities, grouped in various ways. 3. Publication of city school systems (a) Manuals, by-laws, rules, and regulations. All of our larger cities, and many of the smaller ones, print annually handbooks or "manuals" giving miscellaneous data con- cerning the administration of the city schools. They may include certain fiscal data for various city departments, and sometimes the "charter" under which the board oper- ates; the "by-laws" enacted by the board to govern its con- 38 STATISTICAL METHODS duct and to create a complete working organization for the schools of the city, to endow and state specifically for each of the officers his powers and duties, and to state the "rules" governing the schools. They also contain, very probably, the districting of the city system, rules governing: (1) pupils; (2) grading, salary schedules, eligibility, appointment, pro- motion, etc., of teachers; (3) operation of departments out- side the educational department. (b) "Proceedings" or "minutes" of meetings of city boards and their committees. These are now very generally printed for the larger cities, monthly, semi-monthly, or weekly. They very often are found to duplicate the fiscal facts printed annually in the school report; they often con- tain the detailed itemization of school facts that properly ought only to be typewritten and filed in the boards' offices (e.g.y financial itemization of all vouchers paid, regardless of amounts; lists of names of pupils graduating from various schools, etc.) (c) Annual school reports. It is a fairly common practice now for cities of 30,000 to 50,000 population, and above, to print an annual school report. During the past ten years dis- tinct changes have come about in the types of original data that these contain. The tendency toward standardization, uniformity, and a clearer classification of School facts is evidenced by the better organization of data. To a student desirous of making a comparative study of school conditions (say of cost, elimination and retardation, non-promotion, teaching staff, or what-not), the statement should be made that even now, with all the improvements which have been made in recent years, comparable statistics on these or other phases of school practice are not to be obtained from the annual school reports of our cities. This is true even for the very largest cities, with their well-organized accounting staffs. COLLECTION OF EDUCATIONAL FACTS 39 The above, in the main, comprise the larger sources in which students of educational administrative problems may find original data. In rare cases one can discover original detailed statistics in studies made by individual students, either working as officers of a city bureau of school research, or in some educational institution or "foundation." In summing up this brief discussion of the sources and validity of original data, the writer would urge the direct collection of statistics and data from the records and persons in the school systems in question. Question-blanks sent out by individuals rarely have resulted in sound comparative conclusions that benefit school practice. The tendency at the present time is for question-blanks to receive a decreasing amount of respectful attention from a much overburdened school world. When economically possible, personal collec- tion gives much more valid results. It leads to: (1) a more consistently uniform original record; (2) a complete original record {i.e., no data are suppressed); (3) thoroughly com- parable bases of interpretation; (4) a more consistent inter- pretation of the facts as expressed in original and summary tables. Studies which demand recourse to state school laws, charters, rules, and other official state and city documents rest, of course, upon a perfectly valid basis. II. Methods of collecting Educational Data The source and validity of the various types of educa- tional statistics having been discussed, we now turn to the methods by which data are collected. The analysis of these methods, as given in Chapter I, made many references to the two most important methods: (1) use of the question- blank; (2) personal investigation. We shall next take up the detailed discussion of these two general methods, turning to the question-blank method first. 40 STATISTICAL METHODS A, The question-blank method of collecting educational data - • Plate I shows the very great use that school men have made of the question-blank in studying their problems. There is hardly a phase of school administration that has not been subjected to that type of analysis. Present practice and conditions as to the content of the course of study have been established in arithmetic by Jessup and Coffman, and by Van Houten; in algebra by Denny and Mensenkamp; in spelling by Pry or; in handwriting by Freeman; in the high-school subjects by Koos, etc.^ The present status of the teaching staff is tabulated from the " Standard Form " replies each year by the United States Bureau of Educa- tion. It has been studied by the use of the same method by Coffman, Thorndike, Coffman and Jessup, Ruediger, Manny, and Boice; by committees of the National Educa- tion Association and other organizations. The question- blank method has given facts on the age-grade distribu- tion of pupils, — collected by Thorndike, by Strayer (both through the agency and authority of the United States Bureau of Education), and by Ayres, working through the Russell Sage Foundation. Data on promotion plans have been collected at various times by the United States Bureau of Education. The study of current practices in school fi- nance by the question-blank method, by Strayer, and by Elliott, although not leading to basic comparable results themselves, has stimulated the standardization of school financial methods very much. In the same way the status of certain phases of central administration has been de- termined by the work, for example, of Shapleigh, working for the Public Educational Association of Buffalo, in such studies as his determination of the effect of commission form * For details of speci6c references in this chapter see bibhography at end of Chapter X. COLLECTION OF EDUCATIONAL FACTS [; 41 of government on city school administration — forty-eight cities — and the present status of janitorial service in city school systems. Enough has been said here, therefore, to indicate the frequent use that has been made of the question-blank in school research. As indicated above, use of this method by persons working in no official capacity, or by an organiza- tion of the government, has done little more than stimu- late discussion of present practice and the need for greater standardization. Even the Federal Bureau of Education has had no real " extractive power " in its search for school facts, and we have already indicated the large inaccuracies in its original records. However, under various conditions we shall be forced to make some use of the " questionary " in our attempt to determine the status of current practice. For that reason it will be pertinent to give here a discussion of its design and use. 1. The design of the question-blank Principal types of question-blanks. Question-blanks for the collection of educational data can be distinguished into three classes, in terms of the source and reliability of the facts for which they ask: (1) those asking for facts in the personal information of the reporter; (2) those asking for facts to be found in school records; and (3) those asking for introspective or retrospective analysis, judgments of spe- cialists, etc. I. Question-blanks asking for facts in the personal in- formation of the reporter. In the statistical studies in edu- cation many examples may be found of this type. They in- clude questions relating, for example, to the age and sex of the teachers, number of years of training in particular types of institutions, number of years of experience in various grades of public-school work, salary received dur- 42 STATISTICAL METHODS ing various stages of the teacher's career, certificates held, etc. Such questions all relate to the personal history of the person reporting the facts. It is probable that more reliance may be placed on such types of fact than on any other col- lected by the question-blank method. They do not involve the labor, on the part of the reporter, of going to the records of class, school, or system to get the data, with the con- sequent chance for error in transcription and of decrease in number of returns caused by the inability of the reporter to take the time necessary to make the search. A second sort of data obtained from the personal in- formation of the reporter pertains to facts concerning par- ticular phases of school practice. For example, a question- blank sent to teachers of English in high schools contained questions such as the following : — 1. Do you have a special teacher in oral composition? Yes ; No 2. Do you use a text in oral composition? Yes ; No If so, what? 3. Have you a printed course of study in oral composition.'^ Yes ; No 4. Do you have a course in public speaking? Yes ; No 5. Is the work in oral composition given in connection with public speaking? Yes ; No 6. Do oral lessons precede work in written composition? Yes ; No 7. Who selects the topics in oral composition? The student ; teacher Inquiries conducted for the purposes of getting facts con- cerning the content of a particular course of study, names of textbooks used, methods of grading pupils, etc., all fall within this class. Providing questions have been clearly asked and cannot be misinterpreted, data of this type should be very reliable. Question-blanks demanding information in the COLLECTION OF EDUCATIONAL FACTS 4S immediate possession of the reporter ought to result in a very large percentage of returns to the investigator. If the blank is clearly written, well planned, short, and definite, it should result in a return of two thirds to three fourths of the blanks sent out. • 2. Question-blanks asking for facts to be found in school records. In this group we include the collection of facts, concerning, for example, the age-and-grade distribution of pupils in schools; various inquiries of specialists which demand detailed copying of records {e.g., on the problem of retardation and elimination), total expenditures for various types of school activities, administration, instruction, oper- ation, maintenance, etc.; distribution of teachers' time to various subjects; statements from payrolls, class enroll- ment records; total and unit costs, etc. With this type of inquiry nothing but intimate acquaintance with the aims, and full recognition of the importance of the investigation, will cause the reporter to take the time to give comparable and complete data which will lead to the improvement of school practice. 3. Question-blanks asking for intrbspective and retro- spective analyses, judgments of specialists, etc. In this group are found various types of psychological question- blanks; e.g., those from inquiries aimed at determining the status of methods of study. For example, a recent inquiry of this sort quotes liberally from an article on How My Brain Functions, and asks the reporter to check his own mental processes against those of the author, and tell him the result. The following excerpt illustrates this type : — Question: Often he "thinks of nothing." In this state he ex- periences euphoria, a feeling similar to that of the con- valescing patient, who prefers to lie absolutely quiet. He experiences together with this intellectual lethargy a physical inertia. While in this condition he per- 44 STATISTICAL METHODS suades himself "to postpone until to-morrow what he should do to-day." Answer: Do you note a similar phenomenon in your own ex- perience? Please state wherein your experience differs from that of Beaunis. The early stages of the child-study movement were quite given over to the '* questionnaire" method, masses of judg- ments being accumulated concerning child life, mental and moral activity, and growth. Such studies involved most extreme types of " judgment " questions, and as such are the farthest removed from a purely factual basis. It is no doubt true that the compilation of data concern- ing particular phases of school practice. by the question- blank method will be necessary for some time to come. Since governmental agencies, such as the United States Bureau of Education and the various state departments of educa- tion, have no "real extractive power " as yet, it is clear that individuals must do the work. It is also clear, as will be shown later, that few question-blank inquiries have resulted in establishing beyond a doubt the status of the particular question they were designed to study. This is largely due to the fact of hasty and incomplete planning of inquiry blanks, and lack of recognition of the many issues and difficulties arising in the carrying on of the problem. For this reason it appears worth while to discuss, somewhat in detail, the necessary steps in the carrying on of an in- quiry of this sort. 2. Essential steps in school research by the question-blank method First step : acquaint yourself with the literature covering the field of your problem. Your first duty is to know what others have contributed to the solution of such problems. Read carefully, take notes, and make many comments on COLLICCTION OF EDUCATIONAL FACTS 45 every study made in your field. Many studies have contrib- uted little because of this very lack of acquaintance with what other workers have done. In this way needless dupli- cation will be avoided, needed repetition will be secured, and the mistakes and the excellencies of others' research will be utilized to advantage. Our great need is to have the vital gaps in our knowledge filled in. The careful study of the literature of a specific field of work will lead to the selection of the exact problem upon which research is most urgently needed. Second step : specific definition of the problem. The suc- cess of your investigation depends upon the clearness with which you recognize the exact problem at hand, — espe- cially its educational implications. Write out a very specific and detailed statement of it. Visualize the carrying on of the study from the first step to the last. Ask yourself at every turn — what has this to do with school practice? What kind of facts shall be collected to throw light on this point .f^ Does this point really belong in this inquiry? Plan in a rough way the tables to be made up as a result of sending out the ques- tion-blanks. In a word — project yourself through the en- tire investigation in order to be able to start with a per- fectly clear idea of what you are to study. It is probable that a most specific definition of your problem can come only after you have read the literature on the subject, and after you have actually worked through, at least in a preliminary way, the design of the question-blank itself. Third step : exact delimitation of the extent of the inquiry. Your study of the literature and your attempts to define your problem should lead to an exact determination of the points to be covered and the questions to be asked in your study. Plan the number and kinds of questions to be asked in the light of a careful estimate of the labor of tabulation and summary of results. Decide the number of replies 46 STATISTICAL METHODS needed to establish definitely the status of your problem. In doing this, count on a return of from one third to three fourths of the blanks sent out, depending on the length of the blank, the possibility and ease of giving the information on the part of the reporter, and the clearness with which the pertinency of the investigation to the needs of school prac- tice is recognized by those to whom blanks are sent. In de- ciding the number of blanks to be collected, make use of methods of determining the minimum number of cases, such as are described in Chapter VIII. Secure enough cases to satisfy statistical '* criteria of reliability,'* and no more than are necessary to secure ac- ceptance of the results of your inquiry by the persons to whom you will present them. One study known to the writer involved the collection of 30,000 blanks, the original tabulation alone of the returns from which would have taken at least 700 hours of clerical labor. Careful study showed that the same conclusions could be derived from the tabulation of one fourth as many cases, with the reliability of the investigation established at every point. Further- more, the delimitation of the extent of the study calls for careful weighing of the relative value of having a small number of questions and a large number of replies, or of having a large number of questions with a small number of replies. Fourth step : design of the questions on the blank. Noth- ing is more important to the success of the study than the careful placing and wording of the questions. The most detailed analysis should be made of each one. Ask yourself concerning each one: Is this question so worded that the reporter cannot misinterpret it.^^ Has every term been clearly defined, so that the returns from different reporters will be exactly comparable? Is the question ambiguous ? Can it be answered by Yes, No, a phrase, a number, or a COLLECTION OF EDUCATIONAL FACTS 47 check mark? Has the person who will answer this question the information desired? Is there sufficient space allowed for the most complete answer desired? Will the questions lead to specific quantitative statements? Are they factual? Have I eliminated all confusion that might arise because "factual" and "judgment" questions have been put to- gether? A fundamental point to be kept in mind in this connection is: Can the replies to this question be tabulated so that the data can be definitely summarized and inter- preted? Still better, can the data called for at this point be more completely secured by tabulation on the question sheet itself ? Such points will be illustrated thoroughly in the next section. Fifth step : design of the originai tabulation forms. Chap- ter III will take up in detail the tabulation of educational data. It should be pointed out here that an absolutely es- sential step in the design of a sound question-blank is the preparation of the forms upon which the original tabulation of the data is to be made. This means the planning of the specific headings of the tables to be compiled, and will re- quire definite decisions concerning the arrangement of questions and the probable types of returns to be secured. Preparation of the tables will lead to a clear-cut, logical arrangement of questions, so put together as to facilitate a clear presentation and discussion in the report. A little time spent at this stage of the work will aid much in the later organization of the completed discussion. Sixth step: preliminary collection of data on tentative question-blanks. Having decided on the wording and arrangement of the questions, collect some data for pre- liminary analysis of your blank and tabulation forms. Have your blank mimeographed, making say 20 to 30 copies, and ask members of the group to whom will be sent the final in- quiry to fill in the questions. Tabulate these returns on your 48 STATISTICAL METHODS forms, and note the diflficulties of tabmlation and errors in interpretation of the questions. Only in this way can your blank or your forms be made thoroughly usable. Careful study of the returns will enable you to revise both the word- ing and the arrangement of the blank. Mimeograph it again and try it on another group, tabulating the returns. Re- vise once more and prepare the final copy for the printer. In selecting a group to fill in the preliminary blank, take the persons entirely at random {e.g., arrange them alpha- betically and take every nth one). This will enable you to foretell from the returns, roughly, the proportion of the entire number of eases that you can expect to receive and will aid you in deciding how many blanks to send out. Seventh step: preparation of printed blank. If the in- vestigation is at all extensive the blank should be printed. Practical criteria of handling and filing returns should con- trol the selection of the material to be used. If financially expedient and practically possible, use a light-weight card instead of paper. If this is done use standard sizes, either 3 by 5 inches, 5 by 8 inches, or 8| by 11 inches. This will facilitate filing the returns later. These, then, are the necessary steps in the design of a sound question-blank : — know the literature concerning the problem; define the problem specifically; limit the extent of the inquiry carefully; scrutinize minutely each question included on the blank; design the forms upon which the original tabulation is to be made; organize the tentative question-blank, and try it on 20 to 30 persons; tabulate the returns, revise the blank, try it on another group, and tabulate again; print the final copy on standard- sized material, using cards where possible. COLLECTION OF EDUCATIONAL FACTS 49 3. Guiding 'principles concerning the content and form of the question-blank There are many important points concerning the selec- tion of questions and the form of the blank that need to be commented upon before leaving this question. 1. ** Factual " questions. A fundamental principle for the selection of questions is that they must be as "factual" as possible: Thus, questions should involve a minimum of "judgment," discrimination, or "deferred memory" on the part of the reporter. For example, in this question, asked in an inquiry on the economic condition of the members of the general teaching staff of the country : — Check the item that would most nearly represent the parental annual income when you began teaching : — $250 or less, $250 to $500, etc. The answer demanded memory of a situation many years past, in addition to the calculation of various items entering into the answer. The data obtained must be of very ques- tionable value. 2. Difficulties with ** general " questions. Education question-blanks have abounded in "general" questions. One type is the sort that nearly always can be answered **Yes," while at the same time it is nearly impossible to reply more specifically. For example, in a recent state survey of commercial education we find such questions as: — Do you have difficulty in obtaining clerical help? Do you find pupils, 14 to 18 years of age, who come from ele- mentary and high schools, deficient in general education? Another type of the " general " question is that which leads to unanalyzed and practically unanalyzable statements. It tends to hide up the specific facts out of which it might be possible to construct a valid general statement. To illustrate. 50 STATISTICAL METHODS we quote a question on the cost of teacher's education, asked in a recent survey : — Estimated cost of your education beyond the high school, in- cluding specific items, as cost of tuition, books, board and room, etc ; and estimated cost of time as measured by the amount that you could have earned at productive employ- ment during this period of training; Total The answers to "general" questions seldom can be tabu- lated and definitely interpreted. It is a safe rule to follow that data which do not lend themselves to tabulation and statistical treatment are of negligible value to the investi- gator. That is, answers should be definite and susceptible of tabular classification and this should be a controlling criterion in the planning of questions. Such questions as those below, taken from a "study" of the course of study, hardly render themselves subject to that kind of treat- ment. 1. In what way (if at all) is your teaching of the following sub- jects determined by the peculiar needs and opportunities of the '^ local community or district served by the school: — Agriculture Manual training Arithmetic Geography etc. 2. What in general is the attitude of the parents toward "home work" in school studies 3. What is the attitude of your community toward: — (a) Taking pupils on excursions to study neighboring in- dustries, etc Many of these "general" questions demand of the re- porter a type of discriminative judgment that but few people possess, and those only specialists trained to that particular thing. To illustrate : — COLLECTION OF EDUCATIONAL FACTS 51 What difference in training do you notice between public high- school commercial graduates and graduates of the common private business colleges? 3. Ambiguity of statement. Many difficulties in tabulation and interpretation arise from ambiguity of statement of the question. For example, the following question, on which thousands of replies had been collected from elementary public-school teachers, had to be eliminated from the study because of the confusion in interpretation of the word "school" by many of those reporting. Total number of pupils in the entire school Number of teachers in the school, including superintendent or principal if he teaches Number of pupils in the high-school department In the grades The returns indicated that a large proportion of the re- porters had interpreted "school" to mean "school system." Many teachers from the same system reported on the same conditions, thus permitting a check. Of course, the question should never have been asked of teachers at all, but of the administrative officer. 4. Information difficult to obtain. Apropos of asking for facts in the immediate personal information of the reporter, it will be recognized that we must not ask for general facts that the reporter cannot give without considerable search on his part. For example, we find on a question-blank con- cerning the distribution of workers in certain occupations, sent to the superintendent of schools of the city in question, the following : — 1. Number of children in your community between 14 and 16 years of age at work or idle What are they doing if at work Number of families of this whole out-of -school group to whom this income of the youth is necessary 52 STATISTICAL METHODS 2. State the number of workers (in this pursuit), male and female, with different ages; the number 14 years old, , 15, ,16 , etc., up to 80. The number of years of schooling of each of these workers by age and sex , etc. The impossibility of the reporter in question filling in these data is evident. In this connection it is clear that we should not ask for data which cannot be given accurately and in detail by the reporter, when at the same time the detailed information is available in printed records. For example, this question, on a blank directed to each of the teachers in various systems, should not have been asked : — State the population of the village, town, city, or district in which you teach 5. Other t3rpes of information. If you desire to compute percentages, plan the questions concerning the number of items so that percentages can be worked from the collected data. For example, desiring to know the proportion of brothers and sisters who lived to adulthood, one investiga- tor asked : — How many brothers and sisters lived to early adulthood or longer? He omitted to ask for the total number of brothers and sisters. In studying problems involving many stages of growth or progress one must be careful to include all the possible stages or grades. A detailed question of this type is: — You attended country district school years, village school years, city graded school years, one- teacher high school years, larger high school years, private academy years, normal school years, military school years, college or university years, graduate school years, etc. COLLECTION OF EDUCATIONAL FACTS 53 -4. Rules governing the form of question-blanks In concluding the discussion on the design of question- blanks there are certain rules of form that well may be set down: — 1. State the questions specifically. Beware of general headings or word or phrase captions. Use complete sentences or phrases long enough to convey your exact meaning to your most careless reporter. Discount the ability of your reporter to discriminate and interpret what is meant. Define and, if necessary, redefine each term which is in any respect technical, or which possibly can be misinterpreted by the least intelligent reporter in your group. 2. Plan the introductory or explanatory paragraph so clearly and completely that it will acquaint the reporter fully with what you are doing and enlist his interest in your problem. Be careful to show the pertinency of your inquiry to the improvement of his conditions or at least to the improvement of school practice in some particular. If you cannot do this your investigation is of doubtful value, and cooperation will not be given you. 3. Questions of arranging the form of the sheet are very important. Striking defects of nearly all question-blanks are (a) lack of clear organization of questions; {h) lack of suffi- cient space for answers; (c) lack of tabular schemes by which the reporter can give numerical data. 4. If the tabulation forms are designed in advance, and the complete plan of the report is sketched, the questions will be systematically organized on the sheet with this view. Arrange them in the order in which you wish to tabulate and to discuss the points of your report. Logical organiza- tion at such early stages of the procedure will enhance the clarity of your later discussion. 5. Plan sufficient space for the longest possible answer to 54 STATISTICAL METHODS the question. This can be done effectively by insisting on a preHminary filhng-in of the blanks. If you do this you will be almost sure to redesign the blanks in order to give longer spaces. It is rare that question-blanks are well planned in this particular. 6. When asking for continuous numerical data, provide a tabulation form on the question-blank upon which the re- porter can fill in the data. Plan this tabulation form very carefully, so as to prevent errors in interpretation and in subsequent tabulation. To illustrate, this point, a portion of a question-blank on junior high-school costs is quoted here- with: — V. Omitting all names, will you give the individual yearly salaries that were paid junior high-school principals, teachers, supervisors of special subjects, and principals' clerks during the school year 1914-15. To make it easier for you, the salaries are arranged in groups in one column. In the opposite column (marked "No. re- ceiving"), will you place the number who received the salary stated? Example. If four women teachers and one man teacher re- ceive annual salaries between $800 and $825 respectively, enter them thus : — Annual salaries Men Women 800-825 1 4 COLLECTION OF EDUCATIONAL FACTS 55 Principals' Number receiving each salary given Teachers' salaries Number receiving each salary given salaries Men Women Men Women 1000-1099 500- 549 1100-1199 ■550- 599 1200-1299 600- 649 ^'3(30-1399 650- 699 1400-1499 700- 749 1500-1599 750- 799 lGOO-1699 800- 849 1700-1799 850- 899 1800-1899 900- 949 1900-1999 950- 999 2000-2099 1000-1049 2100-2199 1050-1099 2200-2299 1100-1149 2300-2399 1150-1199 2400-2499 1200-1249 2500-2599 1250-1299 2G00-2699 1300-1349 2700-2799 1350-131)9 ■ 2800-2899 1400-1449 2900-2999 1450-1500 3000-3500 Principals' clerks or other admin- istrative clerks Number receiving each salary given Supervisors of special subjects'^ Number receiving each salary given ■ Men Women Men Women 250-299 700- 799 800- 899 300-349 350-399 900- 999 400-449 1000-1099 450-499 1100-1199 500-549 1200-1299 550-599 1300-1399 600-649 1400-1499 650-699 1500-1599 700-750 1600-1700 For example, supervisor of art, music, etc. 56 STATISTICAL METHODS B. Methods of Personal Investigation The foregoing pages have set before us essential principles and methods to govern our practice in the collection of edu- cational facts by the use of question-blanks. It is undoubted that the more important contributions to the improvement of school practice will come through personal collection of facts and actual contact with the school situation itself. We should next bring in review ways and means of utilizing such methods. Since these make such complete use of tabu- lar analysis we will take them up in connection with the tabulation of educational data, in the next chapter. CHAPTER III THE TABULATION OF EDUCATIONAL DATA As students of education have turned to quantitative methods of solving their problems, the use of " questionary " methods of collecting facts has rapidly given way to inten- sive personal investigation. Plate I shows semi-graphically the extent to which such methods have been used to estab- lish the status of various types of problems. A brief summary of them may be given at this point. I. Methods of Personal Investigation of Educational Problems A. Statistical compijation from printed material. Under this heading we have: — 1, Tabular analysis of provisions of public school laws and .^xcity charters, to establish the legal status of various ad- ministrative problems. For example: current practices in the various states concerning the certification of teachers; constitutional and state-school-code provisions for the ad- - \ -ministration of education in rural and city districts; the com- position, methods of selecting, powers, tenure of office, and O compensation of boards of education; methods of raising and apportioning school funds. 2. Tabular analysis of rules, by-laws, and manuals of city boards of education. By this method we determine the status of the appointment, pay, and tenure of teachers and other employees; the powers and duties of committees of the board and of its officers, and the basis of carrying on the instructional and business activities of the system. 58 STATISTICAL METHODS 3. Tabular analysis of textbooks and printed courses of study, to determine the present status of the content of the course of study and the relative efficiency of tlie order of presentation of topics in various subjects of study. Jf. Tabular analysis of the data given in federal, state, and city official reports. The types of data, vaHdity of each, and problems which can be treated from these have been taken up in the previous chapters. 5. Tabular analysis of the data found in the records of city school systems. Only by personal tabulation of facts from such records can we expect to make real progress in making known the facts on present school practice in this country. The school " survey movement " in this connection is lending great impetus to the work. Detailed comparative analysis of groups of cities is establishing definitely : — • facts on the teaching staff; age-grade, elimination, and re- tardation facts on the pupil; facts on the marking system; detailed and systematic compilation of facts on revenue, ex- penditures, and unit costs; and facts on the central admin- istration and business management of public schools. 6. Tabular analysis of facts in experimental and statistical descriptive literature. B. Tabular analysis of results of experimentation. Under this heading we have : — 7. Tabular analysis of the ''abilities'' or ''achievements'* of pupils, through the design of "mental" or "educational" tests. 8. Tabular analysis of facts from experiments in " learn- ing," "mental discipline," etc. 9. Tabular analysis of facts concerning the efficiency of teaching secured through the personal observation of teaching, with or without the aid of "efficiency score-cards," or schedules of "qualities of merit in teaching." TABULATION OF EDUCATIONAL DATA 59 Systematic tabulation of facts. It will be noted that the collection of data in the "scientific" study of education is either: (1) straight statistical compilation of facts, from vari- ous principal sources; or (2) dependent upon the preliminary setting up of auxiliary devices for measurement (stand- ard tests, score cards, etc.), and the conducting of experi- mentation. Either procedure necessitates the same funda- mental auxiliary method: the systematic tabulation of facts. Experience has shown that the thoroughness and insight displayed in planning and carrying through the original tabulation is an important factor in determining the success of the investigation. We have seen already the necessity for planning the scheme of tabulation in detail at the time of designing the question -blank. The two steps in the general research thus must be carried on together — the efficiency with which one is done contributing to the success of the other. Although it is recognized that the planning of tabula- tion forms is a task, the detailed execution of which must be carried on so as to fit each particular problem, there are cer- tain general guiding principles which, if discussed here, may save the student or investigator much wasted time and effort. Original and secondary tabulations. We speak of tabula- tion in general as "original" tabulation and as "secondary" tabulation. By original tabulation we shall mean the prep- aration of detailed tables on which are compiled the original data. By secondary tabulation we shall mean the prepara- tion of tables which summarize the original data, and which permit comparisons of "groups" by means of "averages," measures of "variability," and measures of "relationship." The discussion of this chapter relates to the original tabu- lation of educational data. The complete treatment of second- ary tabulation is included in Chapters IV to IX inclusive. 60 STATISTICAL METHODS II. The Original Tabulation of Educational Data i. hand tabulation There are two important phases to the work of tabulation. The first has to do with the selection of the general scheme of tabulation, while the second deals with the method of tabulation. We first face the question : What general scheme shall be used in compiling the original data — ruled cards, large ruled sheets, or ruled blank books? Two criteria control the selection of the general scheme: (1) How many separate points are to be covered by the inquiry and how many cases are to be tabulated? (2) Which method of tabulation is to be used, the '* writing method" or the "checking method"? Since the selection of the general scheme depends so completely on the adopted method of tabulation, that will be discussed next. 1. The method of tabulation The writing method vs, the checking method. In com- piling the original data of the inquiry, whether from ques- lion-blank returns or from original records, the investigator can adopt one of two procedures. He can write out the de- tailed data covering each point of his inquiry in the fashion indicated by Table 3. The data in the table are quoted from the illustrative tables in a study covering the social conditions and careers of more than five thousand teachers. It will be noted that the original data, compiled from the question-blanks, are written out in detail on this sheet. The only abbreviation of the data occurs in such questions as that covering "parental income," in which each of the ■pWOQ MMXHMMM^MM UOlflSOJ oioe«oeoeoect-io-* uoiiipuoo ffjimvjj C^rHrHTHCOiHi-Hi-lrHrH SJ,SfS2S pUD «DiO'*o©ooeoioeor-( uoi^vdnooo s^juaxDj ■o, io^iHiaeoeoioeou'5 gevnOuvj puwofvpi Ht^WWWWW^WO gdvnduvi 'ivui9;vj ft;,tgwHWWW^WO n^aipti i-iiHb> 1001(08 uaiox •*00|0-*^0-<*00 jooi{08 jvmy ilOOOCq'rHOOOt- 98v s^JLOuuidoq OOOO^OOOOt-OTJD muovi jiod flivpg {2Sg|l§SSS12? siipiovi fo j,3qmn^ oooooocsooo 9moom 8jU3J,vj iri(NOC5'*Tt<«)oooiH 96v S?5g5S^^S^SS iaqmnu pnptaipuj SSSSiSSSSSS 62 STATISTICAL METHODS various intervals $250 or less, $250-$500, $500-$750, etc., is given a code number, and these numbers are tabulated, 5, 2, 9, 9, 4, etc. This is done, however, merely to save the time of writing the complete record for each teacher, and does not contribute at all to the more rapid summarization of the data later. In fact, in having to apply the code number to each case the tabulator is very likely handicapped in the rapidity and accuracy with which he compiles the data. It should be noted carefully that as a result of such a detailed original tabulation the original records are transcribed in full at a very considerable expense, but that no summarization has been done and none is possible on this table. The compu- tation of averages and measures of variability and relation- ship for various comparable groups cannot be done without complete re tabulation of the data. This brings us to a fun- damental principle of tabulation: the original tabulation should lead at once to group " totals^' and to the rapid computa- tion of the necessary statistical measures, averages, measures of variability, etc. It is clear that the "writing method" of tabulation does not do this, and that for extensive inves- tigations it is not an economical method. The checking method. This brings up the checking method of tabulation, and we can illustrate its use by representing the same data given in Table 3. Its first dis- tinctive feature is found in the form of the "heading" pre- pared for the tabulation. Now, instead of using a general blanket heading "age," for example, or " number of months " ("for which present contract is drawn") etc., we prepare a scheme of column headings, one column of the table being left for each possible reply, or perhaps for the smallest range covered by such replies. To illustrate, the column headed "age" in Table 3, now becomes a series of columns as in Table 4. TABULATION OF EDUCATIONAL DATA 63 Table 4. Present Age of Teachers Teacher's NumbLir 7i 1 1 1 GO 1 1^ I <^o ^ i? 1 ■5- § § 1 51 52 53 54 55 56 57 58 59 60 X X X X X X X X X X Totals 1 1 2 2 1 1 1 1 1 The records of "age" from Table 3 are retabulated by "checking" the appropriate column for each teacher. A second distinctive feature now stands out, — this method of tabulation at once permits grouping of data, and the im- mediate and easy compilation of totals, and of "averages" and measures of "variability." The student should be cau- tioned to classify his records carefully at the start so as to per- mit the tabulation of the data on a perfectly uniform group of individuals on one sheet or page. For example, the data given in Tables 3 and 4 should refer to teachers who are teaching under the same conditions, or who are from other standpoints perfectly comparable with each other. If this is done, as far as possible, the labor of retabulation in the subsequent statistical treatment of the data will be cut to a minimum. To adopt the checking method on such points as "Age," in which many columns are needed, raises the question " How large shall the interval be made, — 1 year, 3 years, 5 years, or what?" Chapter IV discusses the statistical classification of data in great detail, and this question can best be answered for the reader by suggesting the reading 64 STATISTICAL METHODS of that chapter, with the subsequent rereading of this dis- cussion. In that treatment a complete discussion of the size of the interval, its position, and best methods of marking limits, etc., are given. To use the checking method, therefore, we must plan, at the start, a series of column headings sufficiently detailed to cover the range of possible replies on each point. The thought will arise immediately in the mind of the reader: *'But the preparation of column headings is expensive, both of material and of the time of the tabulator." The first point is admittedly of not sufficient weight to demand consider- ation. The second is important, however. As the result of the detailed experience of the writer with both the writing and the checking methods, it can be said that the latter is by far the more economical in the long run. To offset the utilization of time in preparing column headings we have three distinct savings: (1) that due to checking answers instead of writing them out in detail; (2) that due to the possibility of totaling the data in each column rapidly and accurately; (3) the fact that averages and other statistical meas- ures can be computed for the various groups of data from the original record. To these we should add that the checking method gives a more accurate perspective of the returns, per- mits better preliminary planning of the treatment of the data, and leads to a more adequate interpretation of results. Schemes for tabulation. We said above that the selec- tion of the scheme of tabulation depended not only on the method of tabulation, but also upon the number of points to be covered by the inquiry and the number of cases to be collected. In deciding on the scheme of tabulation we have a choice of the use of : (1) the ruled card; (2) the large ruled sheet; and (3) the ruled blankbook. I. Use of the ruled card. It will be evident that the ruled card (regulation sizes, 4 by 6, 5 by 8, 8| by 11) is adapted TABULATION OF EDUCATIONAL DATA 65 to only the most restricted investigations, — those covering a comparatively small number of separate points and in which but few cases (perhaps 25 to 50) are to be collected. It has the advantage of facilitating manipulation and filing of the data. Such a scheme is excellently adapted to those compilations of data in which a single question may be put on a card, rulings being adapted in such a way as to give the data from each case on this particular point. This scheme is well adapted to the collection of data on various phases of city school administration by buildings, by kinds of schools, or by kinds of activities. 2. Use of the large ruled sheet. This is adapted to some- what more extensive investigations, — those covering per- haps 50 to 100 points and as many cases. To the tabu- lation, for example, of the content of courses of study, or of the study of the content of textbooks, the large ruled sheet (19 by 24 inches is a standard size and easily manip- ulated) is well fitted. Its chief advantage lies in the clear perspective which it gives of all of the data covering a par- ticular group of items or cases. It also permits easy second- ary tabulation. It is used in large city systems, in many phases of the office tabulation of records; for example, in the standardizing of school supplies, both as to kind and amount, tabulation of *' building'* records, tabulation of bids, etc. 3. Use of the ruled blankbook. Nearly all educational investigations are extensive enough in number of points covered and in number of cases collected to demand tabu- lation of the original records in ruled blankbooks. A good rule is to use a book of standard size (say 8 by 10 inches) with cross-sectional ruling (to facilitate the non-uniform rul- ings which will be needed for data of the particular inquiry at hand) and including perhaps 60 to 100 pages. Thirty to forty cases can be tabulated on the length of the page. If 66 STATISTICAL METHODS the checking method of tabulation is being used, the column headings should be arranged in the order of questions on the question-blank (if it is a question-blank inquiry), and the edge of the pages should be "cut-back" sufficiently to permit the use of the original list of names or numbers, writ- ten on the first page of the record. In this way, the entire record of an individual appears on the same line of the tab- ulation even though it may cover many pages in length. If the questions on the blank have been numbered consecu- tively as they should, these numbers could be used as column headings. There is almost no type of extensive investiga- tion to which the ruled book is not well adapted, and in general it should have wide usage. II. THE MECHANICAL TABULATION OF EDUCATIONAL STATISTICS A recent development in statistical work. The discussion thus far has dealt with problems of school research which have implied the use of hand tabulation. For the tabulation and manipulation of the detailed educational and business records of a school system, the experience of school men is proving that electrical mechanical tabulation is both more economical and more efficient. Within the past few years. New York, Philadelphia, Rochester, Oakland (California), and other cities have adopted such methods and have proven their superiority to hand methods. To get the meth- ods clearly before us, together with the consensus of prac- tical judgment on their availability, quotations from recent discussions of the matter will be given. Four methods. The following statement, by the Audi- tor of the Board of Education, New York City,^ indicates four distinct methods of preparing statistical data : — 1 Cook, H. R. M. "The Standardization of School Accounting and of School Statistics "; in American School Board Journal, June and July, 1913. TABULATION OF EDUCATIONAL DATA 67 1. By means of the electrical tabulating and sorting machine and electrical battery adding machine, and by the use of perforated cards. 2. By means of cards of uniform size, on which are printed the statistical classifications, while the figures or amounts are inserted by hand. The margin of the card may be perforated by hand. This last process permits of a limited range of information being assembled. It also affords a means of assembling quickly all cards which relate to one or more items of classification. When assembling statistics from these cards, the use of the adding machine is ad- visable. 3. By the use of a columnar collateral ledger, exhibiting the various statistical classifications under which may be recorded the salient feature of the expenditure as shown by the voucher or by the voucher register. 4. By so planning the books of accounts as to include analysis columns in which should be entered, synchronously with the passage of a voucher, that particular statistical classification to which the expenditure may be applicable. The first method is suitable either for large or for moderate-sized school systems, in fact, it may be profitably used anywhere, ex- cept in the case of the small rural organizations. In any city or town where the population exceeds 20,000 inhabitants, the installa- tion of a statistical plant of this kind would be advantageous. Not only is it possible to make a complete distribution of school expen- ditures, but school facts of important character, both educational and physical, may be recorded with great speed, accuracy, and minuteness. A uniformly printed card, a few square inches in size, is susceptible of use for the purpose of recording thousands of facts of most varied nature. No matter how the cards may be fed through the machine, the sorting machine automatically separates each fact. The widest imaginable range of statistical information can be produced by the adoption of the first method. The system involves the compilation of a "code" in which each statistical point of information or fact is assigned to a number or combination of numbers. An illustration of the form of card used will be found among the diagrams. The cost of rental and operation of this type of statistical outfit in a small or moderate sized school system would be about the same as the salary of a clerk. In a large system it might reach the cost of two such clerks. The second method is suitable for a system of any size and is 68 STATISTICAL METHODS very elastic, but it lacks the speed and wide range of the first- described method. It was actually and successfully employed for some years in one of the largest school systems in the worW. It was only displaced because of the superiority of the first-described method, because the rental of a machine is cheaper than clerk hire. The cost of stationery is about the same. In a small school system the total cost would probably trend the other way, but not sufficiently far to make up for the extra efficiency and wide range of the mechanical device. An illustration suggestive of a suitable form of card will be found among the diagrams accompanying this treatise. The third method represents a purely hand-made system, and is intended to operate in parallel with the regular books of account. The volume of the expenditures in the fund accounting will neces- sarily equal the volume of the statistical accounting between given points. This method permits of the preparation of data sufficient for the purposes of the standard blanks of the United States Bureau of Education, but it does not afford any very wide range of in- formation which it might be desirable to collect for local purposes. The fourth method is a modification of the third just-described method. It is suitable for school systems of a size which are re- quired to present information for the purposes of the " abridged " standard blank adopted by the United States Bureau of Education. All of the foregoing methods are practical. They have been tried and found to work successfully. They will furnish results within their limits and scope. The Oakland, California, method. That the utilization of "mechanical tabulation" is not confined to the largest sys- tems, but that it is efficient and economical in any city in which Tabulating Service Bureaus have been established, is shown by a recent report of Mr. Wilford E. Talbert, Director of Reference and Research, Oakland, California. After discussing the way in which the statistical reports of teachers, principals, and other employees are compiled by time-saving methods he says : — Transferring reports to Hollerith cards. As soon as the teachers' reports are received in the Superintendent's office, the information they contain is punched onto Hollerith cards by a clerk who, be- DCPT. OF REFERE. NCE & RESEARCH E iOARD OF EDUCATIO N, Oakland, Cal.' suonoui-ad o ' »- CM CO 1 ■* in to ! 1^ CO O) sin.oJd'o.adS O T- CM CO 1 ^ lO (O 1 r- 00 a> O ^ CM CO ; •* LO to 1 r- 03 a Absent on Accoun ' of Illness o T— CM CO rj- in iO I r~- 00 m o T- CM CO "* LO CO r- 00 a> o T- CM CO ^ lO B}t?9S 4UB0BA o T-. CM CO ^ in r- OD CO I H z s It a ^ o T- CM CO ">* in Oi SPBJO o ■r- CM CO Tt- m ® r^ CO o> \- o' ^ • •— CM P5 "* in CO r*- 03 o> Z 5 111 .2^ • ^ CM CO •* in CO r- 00 <5> epBJD • »— CM CO "* in CO r^ "i Cli • ^ CM CO ■* in CO r- CO Oi III • ,- CM CO ■* in CO r- 00 a z 1 • • T— CM CM CO CO in in CO CO 1^ 00 00 Gi apBJO • T- CM CO >* in CO h- 03 o' o> Q • ^ CM CO >* in CO r- 00 o < -J < • T- CM CO "* in CO r- 00 o> ^ • r- CM CO 't in CO r- 00 Oi M • T- CM CO «* in CO r- 00 o 9 ^ CM CO ^ in CO r- 00 O) si 9 r- CM CO v* in CO r- 00 o> o ^ • CO ■<4' in CO r^ 00 a> og o • CM CO ^h in CO r^ 00 <3> • T- CM CO ■* in CO r- 00 Oi • ,- CM CO ■* in CO r^ 03 a> • ^ CM CO "* in CO . r- 00 OD to ^ o ,. CM ^ •<* in CO r- 00 o> ^i o T- CM CO 9 in CO r- 00 O) q| o ,- CM ro rf in CO r- 9 o> • T- CM CO ■t in CO ! r- 00 <3> • ,- CM CO ■* in CO ! r- CO Oi 1 o ,- CM CO "* in CO ! "^ 00 9 .teqnmi^ o ! '- CM CO -* • CD r- 00 o> ^qoBajG o 1 ^ • CO rf in CO 1 r- CO CJ> asqmnjsr o • j: CM CO CO "* ■^ in in CO CO 1 ^~ 1 r- CO 00 <3> Oi jaqiutiM looqog o o ; i CM CM CO in U5 CO CO i ^ 1 r- 00 00 ay muojM o CM CO ;® in CD 1 f^ CO a> looqos A o •ul jCM I < a ^co ^93 CO 1 "^ !< 1 00 a> 70 STATISTICAL METHODS cause she specializes on this sort of work, is at least as apt to detect errors as any but the most careful principals. After the cards are punched they are all checked for accuracy by reading back the data to another clerk holding the original reports. On the Hollerith cards (Diagram 13 reproduces an Oakland card), error has been carefully guarded against by color of cards, by clipping of corners, and by code numbers. Also an attempt has been made to foresee every possible kind of information that might ever be wanted from the reports for the given period. By the use of code numbers for sorting fields, this becomes very simple under the Hollerith system. In fact, someone has called the Hollerith cards "canned information," and, like canned goods, they are always on hand, they are compact, and their contents is al- ways readily available. For example, it is possible in a few min- utes on the sorting machine to take from the entire year's reports, the cards for special classes, for any given teacher, for any desired grade, for any teacher's register in the city (even though that regis- ter itself may have been burned), and all the data on these cards can be quickly tabulated in any desired way, even though none of this information is tabulated from month to month. The work of tahidating results. As soon as all reports are received and all data have been transferred to the Hollerith cards, the lo-tter are called for in the morning by the Tabulating Service Co., and the following four reports are returned by the same evening: — 1. Attendance and absence by schools and kinds of schools. (188 sums.) 2. Total enrollment by schools, by kinds of schools, and by de- partments of each school. (141 sums.) 3. Distribution of enrollment in non-departmental, and in de- partmental classes of the elementary schools, showing the number of classes of each size from the smallest to the largest, and giving the location by schools of all classes which are either exceptionally large or exceptionally small. (790 signifi- cant figures reported last month.) These reports are all typed and arranged in such shape that this office can readily write in the names of schools and such aver- ages, etc., as it is necessary to compute on the calculating machine. Even the typing is mechanically checked for error, so that we have absolutely reliable and unchangeable data as a basis for further computations. Diagram 14. Hollerith Sorting Machine for CLASSIFYING SCHOOL STATISTICS TABULATION OF EDUCATIONAL DATA 7X Use of the plan at Rochester, New York. The statistical bureau of the school system of Rochester, New York, uses mechanical tabulation methods. Mr. J. S. Mullan, Sec- retary of the Board of Education, discusses the method in part as follows: ^ Up to the present time, the analysis of school expenditures and the development of school statistics have been restricted because of cost and the time element, — that is, whether the information would be worth what it would cost, and whether it could be compiled in time for administrative and legislative use. With the adaptation of machinery to statistical purposes, we are entering upon a new era of statistical possibilities. With mechanical tabulation, cost and the time element are being reduced to the minimum. In fact, statistical analyses, heretofore prohibitive and practically impos- sible, are now being compiled, used, and demanded. With mechanical tabulation, the bookkeeping division becomes a machine shop. The machinery consists of card-punching ma- chines operated by hand (for individual cards and cards in gangs), a card-sorting machine [pictured in Diagram 14] operated by elec- tricity, and a tabulating machine [pictured in Diagram 15], also operated electrically. The cards used in connection with the ma- chines are somewhat larger than regular index cards. Upon the cards are printed what are technically known as "fields," each field representing an item of information. The field consists of vertical lines of varying distances apart, in which appear numerals, each field containing one or more perpendicular rows of numerals according to the requirements of each of the fields. The card which has been adopted for use in the accounting division of the Rochester school system shows the year and month; voucher number; vendor; school building; day, night, continuation or normal school ; function ; sub-function; educational subject; character of expenditure; quan- tity; unit of measure; commodity; class and number; price; amount; fund; and whether contract, open-market order, pay-roll, or mis- cellaneous expenditure. Expressed in a numerical code, the information is punched on the cards by the operator striking keys which perforate the cards with ^ "Mechanical Tabulation of School Financial Statistics"; in Proceed- ings of the Fifth Annual Meeting of the Nnfional Association of School Ac- counting Officers, p. 43, 72 STATISTICAL METHODS small holes. Any data appearing on the requisition, invoice, pay- roll, or voucher can thus be transferred to the cards, after which the cards are ready for sorting and tabulation. It can be seen that once the cards are punched and checked with the original docu- ment, the period of detail checking is over. All the data punched on the cards are elemental. The total of the cards is the sum of the elements. Once punched and checked, the cards go to the sorting machine, where by electrical contact through the holes in the cards they are sorted into any pre-determined group; thence they go to the tabulating machine, where in the same way they are tabulated by groups and in total, the totals when obtained being entered on a prearranged form. The sorting and tabulating may be repeated until all the fields on the cards have been covered, the final totals of the various sortings being the automatic check. Furthermore, the punching of the cards and their tabulation are accomplished in a comparatively short period of time, so that any group result or com- bination of results is expeditiously produced, and at a minimum of cost. Compare the possibilities of this procedure with distribution by hand posting, including the factor of possible clerical error, the diffi- culty of attempting to carry on more than one analysis at one and the same time, i.e., functional service, amounts of compensation, quantities and prices of commodities, repairs, interest, refunds, bond payments, etc., and the confusion of thought in handling such a conglomerate, — and we begin to appreciate the possibilities of mechanical tabulation and its superlative advantages. III. Secondary Tabulation The future chapters. In Chapter II the initial steps in the study of an educational problem were shown to be the careful definition of the problem and the collection and origi- nal tabulation of the educational data. These were to be followed by the systematic classification of the data in the frequency distribution, and its summarization by means of various analytic and graphic methods. The discussion of the checking method pointed out that systematic planning of column headings for the original tables really amounted to TABULATION OF EDUCATIONAL DATA 73 the statistical classification of the facts. The principles and methods controlling this work are treated in detail in Chap- ter IV. The succeeding chapters, V to IX inclusive, take up the remaining steps in the statistical treatment of facts. In a fashion, they may all be called Secondary Tabulation. Chapter V presents the various methods of typifying data by " averages.*' Chapter VI shows how the data may be represented somewhat more completely by measures of "variability." Chapter VII discusses the methods of graphic representation of educational facts, and their connec- tion with ideal frequency curves. Chapter VIII shows the application of such type curves to practical educational problems. In Chapter IX will be given a complete discus- sion of ways and means of determining the possibility and degree of relationship that exists between various aspects of school work. CHAPTER IV STATISTICAL CLASSIFICATION OF EDUCATIONAL DATA: THE FREQUENCY DISTRIBUTION I. Introductory Statistics of attributes and variables. The study of the quantitative problems with which we deal in education re- Veals two principal statistical methods of treating the meas- urement of human traits: (1) the method of "attributes"; and (2) the method of *' variables." The measurement of human traits may vary in refinement all the way from the mere counting of the presence or absence of a trait (treated by the method of attributes) to the rather minute quanti- tative measurement of the trait (better treated by the method of variables). The grouping of individuals accord- ing to the presence or absence of a trait may ^e illustrated by : the counting of the number of pupils in a class that have passed or not passed; the number that are of normal men- tality, or are mentally deficient; the number that have light hair or dark hair, are tall or short, blind or seeing, sane or insane, and so on. The methods by which we would treat statistics collected in this way have been denoted by Yule, *'THE STATISTICS OF ATTRIBUTES," and they are to be thought of as somewhat distinct from the methods of treating statistics collected by more refined methods of measurement. These latter, which are known as the " STA- TISTICS OF VARIABLES," imply that the specific mag- nitude of the trait has been measured with reference to a scale made up of known units. In general the statistics gathered in educational research are those of measurable traits, i.e., the statistics of variables. For example, we can STATISTICAL CLASSIFICATION OF DATA 75 measure, in a fairly refined way, the ability of pupils in arithmetic, or in algebra; the mental age of children; the cost of teaching various subjects of study; the retardation of pupils in the public school, and so on. We should have clearly in mind therefore that the statistical methods with which we treat one kind of statistics — those of attributes — may be different from those w ith which we treat the other kind — those of variables. The term variable as used in this book may be taken to mean a varying quantity or human trait, — for example, arithmetic ability, teaching skill, the height of men, etc. Thus, these traits are subject to statisti- cal study by either mere enumeration or counting methods, or may be subject to fairly accurate measurement. II. CLASSinCATION OF STATISTICAL DaTA Grouping of data into classes. Whatever may be the method -by which, or the degree of refinement with which data are colle^ed, when we turn to their organization so that we may interpret the situations that they represent, we face the problem of " grouping." Clear thiijking about large num- bers of facts necessitates the condensation and organization of the data in systematic form. That is, we are forced to group our data in ** classes," and the statistical treatment of the data depends upon the determination of these "classes." A statistical CLASS, whether of attributes or of variables, may be illustrated by Tables 5 and 6.^ They picture the re- lation that exists, for example (Table 5) between the peda- gogical standing and the mental standing of school children. To do this, the pedagogical ages of the children in question are grouped in three classes, — "retarded," "normal," and "advanced," — ^and the mental ages according to whether they are "retarded," "at level" {i.e., normal) and "ad- 1 From Stem's Psychological Methods of Testing Intelligence, pp. 59 and 61. 76 STATISTICAL METHODS vanced." Corresponding to this same classification of mental age, Table 6 "groups" the pupils in three classes according to whether their school marks were poor, satisfactory, or good. Thus the 14 pupils '* retarded "in both pedagogical age and mental age form a "class"; 16 that were "normal" in pedagogical age and "retarded" in mental age form an- other "class." Or, turning to the "total" columns, in the entire group of 101 there are found: a class of 24 pupils re- tarded, 65 pupils normal, and 12 pupils advanced. Because of the fact that refined quantitative methods were not em- ployed in classifying the records we call these data, "STA- TISTICS OF ATTRIBUTES." Table 5. Relation of Pedagogical and Mental Age* Pedagogical Age Retarded Normal . . Advanced. Total Mental Age Retarded At level Advanced 14 16 9 33 5 1 16 7 30 47 24 Total 24 65 12 101 * Binet. Table 6. Relation of Mental Age and School Marks f School Marks Mental Age Tntnl Retarded At level Advanced Poor 29 26 17 79 13 21 31 46 Satisfactory 126 Good 44 Total o5 109 52 216 t Bobertag. STATISTICAL CLASSIFICATION OF DATA 77 Suppose, however, that the standing of the 216 pupils represented in Table 7, instead of being grouped as poor, satisfactory, and good, had been given in terms of numerical marks on a 100 per cent scale, say, — 87, 82, 54, 76, 91, etc. It will be clear that the grouping of these data now neces- sitates setting definite numerical limits to the classes in which the various measures (individual marks), are going to fall. Now, instead of being called poor, satisfactory, good, the marks will be found to fall within some definite interval of the scale, 85.0 to 89.99; 80.0 to 84.99; 50.0 to 54.99; 75.0 to 79.99; 90.0 to 94.99, etc. Our data thus illustrate again the STATISTICS OF VARIABLES, and point out the differ- ences between the method of treating such measures and the ATTRIBUTES represented in Stern's tables. Distribution on scales. This discussion of the grouping of measures has made use of several important concepts, which must be clearly grasped by the student. Fundamental to the practice of measurement are the concepts of SCALE and UNIT. We shall think always of mental and social measurements as distributed over a "scale" — i.e.y a linear distance or a difference in numerical magnitude which will represent or stand for the magnitude of the measures in question. For example, the ability of a group of children in hand- writing may vary in magnitude, let us say, from 40 to 75, when measured on a total scale of "handwriting merit," such as is given in the Scale for Measuring Handwriting, devised by Dr. L. P. Ayres, from 20 to 90. Scholastic abilities are measured, very generally, by the percentile marks of teachers, which are taken to represent the relative position of pupils on a one hundred per cent scale. The per-pupil costs of teaching the various high-school sub- jects may be pictured clearly as distributed over a "cost- scale." The scale may be pictured in numerical or graphic STATISTICAL CLASSIFICATION OF DATA 79 terms. Let us illustrate these points by graphic illustra- tions. The student will be aided in grasping the reasons for certain steps in statistical computation if he will always C/c?3^'/nfer\/o/, 3ca/^, Fre*QC/enc^^ A/o. of f^OJO//>S 20- Z 9. 9 3 7 ZO' 39' 9 9 30 - - a/ ^O -^^-^^ ^O — - 4J ^o-S9 99 50 - - (>S 6o ^6 9 99 60 - - 39 70 ^ — To - 7 J, 99 fd do - 9o 9d Diagram 17. To illustrate Use of "Scale," "Unit," "Class- Interval," AND "Frequency Distribution" supplement his numerical thinking about the "scale" with a graphic picture of it. For example, Diagrams 16 and 17 give a numerical representation of the handwriting scores of 198 pupils grouped in various CLASS INTERVALS along 80 STATISTICAL METHODS a SCALE, whose RANGE (the distance from the smallest measure to the largest measure) extends from 20 per cent to 90 per cent. III. Classes and Class Limits Manifold classification. In distinction from the rough grouping of attributes illustrated above, this numerical classification of measures is called '* manifold-classification." The student should be cautioned that the classes should be clearly marked off from each other by definite numerical limits, 50.0-54.99; 55.0-59.99; or 47.5-52.49; 52.5-57.49, etc., if the class-interval is to contain, for example, five units. There are three different ways in which the limits may be set to the intervals on the scale: The first method of setting class limits is to give the limits themselves, as: 5.0-9.99; 10.0-14.99; 15.0-19.99, etc., as is given in Diagram 17. The student, in beginning the tabulation of frequency distributions, is advised to make use of this definite method, clearly distinguishing the position of the intervals. Especially is it a helpful devise in increasing the accuracy of tabulation. The use of the method 5-10; 10-15; 15-20, etc., leads to many errors in tabulation. The routine statistical work should be safeguarded at every pos- sible step. Clear marking off of class-intervals will tend to reduce errors in this particular. The second method of setting class limits is to express the interval in terms of the mid-value of the class-interval; for example: 7.5; 12.5; 17.5. From the standpoint of accuracy in tabulating the frequencies this is a very poor method, and leads to many errors in tabulation. The third method is to state the interval in words in the form, " 5 and less than 10 "; '* 10 and less than 15 "; " 15 and less than 20," etc. As cautioned above, the use of the same STATISTICAL CLASSIFICATION OF DATA 81 numbers in expressing the numerical limits of class-intervals, 10, 15, 20, etc., and the complication of the word-heading leads to error. It should be clear that, at least for the novice in statistical work, class-intervals should be defined very carefully. Students consistently make more mistakes in the routine tabulation of measures than in the computation of means, measures of variability, etc., after the data have been arranged. IV. The Frequency Distribution: The Steps in ITS Construction Arranging a frequency distribution. The grouping or clas- sifying of measures consists, therefore, (1) in noting the length of the range, i.e., the distance between the largest and the smallest measures; (2) in deciding on the number of class-intervals (or, the size of a class-interval) into which you are to divide the total range of the measures; (3) set- ting the position of the class-intervals (i.e., determining the specific class-limits) ; and (4) tabulating the FREQUENCY of occurrence of the measures in each of the class-intervals. The result of such grouping of measuries is called a "FRE- QUENCY DISTRIBUTION," and is made up of two columns of figures, first a serial list of the " CLASS-INTER- VALS," arranged preferably with the smaller measures at the lower end of the scale; second, a column of '* frequencies," which gives the number of measures tabulated in each class interval. Tables 7 and 8^ give illustrations of the fre- quency distribution as it is used in the study of educational problems, and which make use of the method of defining class-limits very carefully. ^ Judd, C. H., and Parker, S. C, Problems Involved in Standardizing State Normal Schools, pp. 17, 18, 19. Bulletin no. 12, U.S. Bureau of Edu- cation. (19160 82 STATISTICAL METHODS Table 7. Advanced Degrees held by Members of Normal School Faculties Percentage of faculty Colleges and universities Normal schools Ph.D* Master* Ph.D.-\ Master f to 9 2 11 16 13 6 10 2 3 1 1 1 2 7 8 11 11 15 6 22 8 2 3 10 to 19 6 20 to 29 5 30 to 39 7 40 to 49 6 50 to 59 4 60 to 69 70 to 79 1 80 to 89 90 tolOO Total 63 63 32 32 * Nine not reporting. t Three not reporting. Table 8. Average Salaries in North Central Colleges and Normal Schools Salaries Universi- ties and colleges Normal schools Salaries Universi- ties and colleges Normal schools $900 to $999 1000 to 1099 1100 to 1199 1200 to 1299 1300 to 1399 1400 to 1499 1500 to 1599 3 4 8 6 7 6 1 1 2 1 5 3 $1600 to $1699 1700 to 1799 1800 to 1899 1900 to 1999 2000 to 2099 2100 and over No information 9 9 2 5 1 2 10 3 3 5 1 ... 3 7 Total 34 13 Total 38 22 The first step in constructing the frequency distribution. To make clear the construction of the frequency distribution STATISTICAL CLASSIFICATION OF DATA 83 let us work through a problem with the following illustrative data. Table 9 gives the "original measures," — in this case, the marks given to 123 pupils in English. Running down each of the columns we note that the lowest mark given was 20; the highest, .95- Thus the range is 75. In the treat- ment of these data our aim is to classify them in such a way that, for example, an "average," computed for the data in the classified or condensed form, will be very closely the same as the " true average," which would be computed from the entire list of the original measures themselves. Table 9. Class Marks given to 123 High-School Pupils in English 80 57 45 74 95 80 73 87 59 80 57 52 75 75 63 75 84 50 77 76 63 90 79 80 58 71 60 85 76 76 72 73 56 75 84 80 87 85 69 85 40 66 78 79 73 86 88 75 80 79 80 60 87 80 78 82 52 75 67 80 77 80 66 74 73 79 60 66 57 74 76 70 55 87 87 72 73 68 87 81 60 75 35 73 15 67 78 86 73 79 40 82 55 65 80 86 79 6.5 73 5Q 71 73 80 67 78 62 79 79 81 77 82 78 93 78. 70 72 79 45 81 75 20 80 30 The second step in constructing the frequency distribu- tion. This is : deciding on the number of class-intervals into which the range shall be divided; i.e., how many units on the scale shall be included in one class-interval. Two ques- tions have to be answered : — (1) How large may the class-interval be made and still give reasonably small errors in the computation of "aver- ages," etc. The larger we make the interval — that is, the more greatly measures are condensed — the more do we cut down our labor of arithmetic computation. In an ex- tensive investigation, which includes many frequency dis- tributions made up from data that show similar characteris- 84 STATISTICAL METHODS tics as to variation, it may be feasible to take the time to group the data in several different frequency -distributions, computing, say, some average value for each. If the student does so he will note that as he makes the size of class-inter- val smaller there will be an "optimum size" beyond which further reduction will not give an increase of accuracy of the average. In most educational investigations, however, an em- pirical rule can be given to guide the student in his work. In general, when the units of the scale covered by the range are as few as 10, 15, or even 20, nothing is to be gained by group- ing the data in fewer classes, and we may let each unit repre- sent a class-interval. For example, in the problem given in Table 18 there are 12 different unit costs of teaching English, the frequency of the occurrence of each of which is given for 148 Kansas cities. The true mean may be rapidly computed without grouping. On the other hand, the 123 class marks given in the foregoing problem cover a range of 75 units, and obviously must be grouped. A practical rule is to condense to not more than 20 intervals, and to choose a size that gives ease of tabulation. In this case class-intervals of 5 units convert a range of 75 units into 15 class- intervals, a good working number. (2) In what ways are the measures concentrated around certain average values? For example, are most of the marks in the illustrative example grouped in the middle of the scale,, with about the same number of measures on each side (that is, do they form a fairly ''symmetrical" distribution), or are they widely scattered over the scale, each value occurring only a few times .^ This question can be answered roughly by careful inspection of the lists of original measures. If such inspection leads to the conclusion that the measures are fairly well concentrated, or are symmetrically dispersed over the scale, the particular method of grouping will not cause a fluctuation in the value of the "average" or the STATISTICAL CLASSIFICATION OF DATA 85 "measure of variability" that is computed from the fre- quency distribution. Fundamental assumption underlying grouping. There is one fundamental assumption that we make in all "group- ing" of measures in a frequency distribution, namely, that all the values in any class-interval are concentrated at the mid-point of the interval, and may be represented by the value of this mid-point. For example, if the data of Table 9 were grouped in class-intervals of 5 per cent, as in 73.0- 77.99; 78.0-82.99; etc., then the values of 74, 73, 75, 76, 77 all fall within the interval 73.0-77.99, and for all practical purposes are each assumed to be equal to the mid-value, 75.5. It will be clear that, with very unsymmetrical dis- tributions, the assumption is untenable as large errors of computation come about. For example in the cost prob- lem on page 116, grouping the original distribution in class intervals of 2 makes at the low end of the range a very ma- terial error in the first interval, one city actually having a per-pupil recitation cost of one cent, and 26 cities a cost of two cents, the "grouping" causing us to assume that 27 cities each have a cost of 1.5 cents. The error in computing the "average," due to this assumption, is partly compen- sated for, however, by the next interval in which are com- bined 46 measures at three cents, and 26 measures at four cents, offsetting in part the "skewing" of the average toward the low end of the scale. In some educational investiga- tions the data are either so scattered, or are concentrated unsymmetrically, — more heavily at one end of the scale, — that it is necessary to be cautious about grouping. It should be pointed out that with most educational measurements the data are concentrated fairly near the middle of the scale, and tend to be fairly symmetrical. This is a fortunate con- dition, and makes relatively easy for the student the problem of grouping his data. In general he may accept it as a rule, 86 STATISTICAL METHODS for guiding the preparation of frequency distributions, that he should get a working number of class-intervals, say from 10 to 20, but at the same time should make the interval as small as is necessary to reveal any particularly predominant points on the scale. Summary as to class-intervals. Summing up the fore- going statements on the question of deciding the number of class-intervals, we see that the class-interval must be made as large as is possible, and at the same time give relatively slight error in computation from the frequency distribution; that the intervals should not exceed approximately 20 in number, or, in general, be less than 10; that the grouping can be done much more completely if the measures are con- centrated fairly near the middle of the range, and are dis- tributed in a somewhat symmetrical manner on both sides of this general point of concentration; that in all grouping we make the very important assumption that all measures in a class-interval are grouped at the mid-point of the inter- val, and are equal to it in value, and that this assumption is pertinent to the determination of the size of the interval, the larger and more unsymmetrical the distribution of measures in the interval the greater the error made in making the assumption. Third step in constructing the frequency distribution — ■ determining the position of the class-intervals. In dividing up the range into class-intervals we are forced to decide at what digits to set the numerical limits of the intervals, — 50.0, 55.0, or 4^.5, 52.5, 57.5, or 53, 58, 63, etc. Two criteria control this decision: First, the interval should be set at such points on the scale as will lead to the greatest ease and accuracy of tabulation. The experience of the writer and his students leads to the belief that to satisfy this criterion, intervals should not only start and stop with digits, but should make use of the basic tens system wherever pes- a o 2 § GV r) n :: § « :S 1^ .§ r^ •?; S? 1— 1 0. HH 2 g J* & .« Q O rd u ''^ lil s o o m ri tC S !^ o O H ';3 CM e^ «o CS( u to c^ 5 H >> w C 0) 3 «H o S^^ 1 - ^- ^ ^ 13 •S t«o r-j ° c 1 (3 > 1^ s^ i II ^ .£3 1 ^^ > ^ ^ 1 1 ^ \ O "O '^ a •f ^■ V :1 V 'S" U § irS I :.»o lO »o lO ta o »c >o S 1* ^'05 00 t- CO o th CO (M 'S .>3 ' ^ "' ^ 1 J_ <-l Ss 5 , "? OS 05 OS OS OS OS ^ > o Oi OS OS OS OS OS OS B o OS OS OS OS OS OS OS a r-' 00 ^7 '© XO o CO o STATISTICAL CLASSIFICATION OF DATA 87 sible. Thus the measures given in Table 10, Classifications I and II, make use of this method, — 50.0-54.99; 55.0-59.99; 60.0-64.99, etc. The second criterion has to do with the later manipulation of the measures in the frequency dis- tributions, — such as is required in the working of the weighted arithmetic mean. Such computation requires the multiplication of the frequencies by the mid-points of the class-intervals. To cut down the arithmetic labor involved in this process would seem to demand that the mid-point be an integer, — for example, 55, 60, 65, 70, etc. Classifications I, II, III, and IV of the data in Table 10 illustrate the differ- ences in the computation with the mid-points at integral and decimal points. The later discussion of the computation of averages and variability shows, however, that the actual multiplication may all be reduced to mental processes (by the use of short methods). For this reason the second criterion should not hold in deciding on the position of class-intervals. It is the WTiter's judgment that accuracy and rapidity of tabulation should guide the student, and cause him to use that classi- fication of limits for his intervals that bring about the most rapid and most accurate tabulation. It is recommended that for distributions covering a large portion of the percentile range, intervals of 5 be used, and that their limits be set at 20.0, 25.0, 30.0, 35.0, etc. V. The Graphic Representation of Educational Data Importance of graphic representation. The fundamental aim of all statistical organization of educational data is to secure clear interpretation of the situation represented by the data. The numerical classification of large numbers of facts in the frequency distribution is certainly the first im- 88 STATISTICAL MliTHODS portant step in condensing the original measures so that the mind can deal clearly with them. It will be shown in the next two chapters that there are two major numerical methods of further condensing the material, — the method of *' averages," and the method of ** variability." Each of the methods condenses the facts of the frequency distribution into a single number, and aids materially in the interpre- tation of the data. But thorough use can be made of such measures only by the most experienced manipulator of statistical methods. The student needs still more concrete methods of representing facts. Probably the greatest aid to sound interpretation of statistical data will come from the graphic representation of the facts in question. At this point, then, it will be well to take up a brief discussion of the plotting of frequency distributions. Representing a frequency distribution. There are two principal methods of representing a frequency distribution by a graph: (1) that which gives the FREQUENCY POLY- GON; and (2) that which gives the HISTOGRAM or COLUMN DIAGRAM. The two methods are illustrated by Diagrams 20 and 21, which graphically represent the data of Table 10, in three different classifications. In both types the horizontal base line represents the scale silohg which the class-intervals of the frequency distribution are laid oft\ The class-intervals are laid off on this scale by making use of the largest ** unit " that the width of the paper will permit. The vertical lines represent the number of measures found to fall in a particular class-interval or at a particular point on the scale. General directions for plotting. All graphing is done on two basic lines or axes. Using our established notation we may call these OX and OY. Keeping the accepted alge- braic methods of graphing we shall lay off all units on the horizontal scale from left to right, and all units on the STATISTICAL CLASSIFICATION OF DATA 89 vertical scale from bottom to top. Doing this, as in Dia- gram 18, the steps in making the frequency polygon are these : — 1. Note the numerical amount of the range of the fre- quency distribution. 2. Lay off the units of the frequency distribution on the base line OX. Make the units as large as possible and yet get all of the distribution on one piece of paper. Obviously the selection of units must be left to the judgment of the draftsman. Mark clearly the limits of the class-intervals on the base line. 3. At the mid-point of each class-interval draw a vertical line, the length of which rejpre- sentSy to any selected scale, the number of measures that have been found to fall within that class-interval. If your data are definite integral records, varying by units of one each, such as the number of prob- lems solved by large numbers of pupils in arithmetic, — say, 10 solving 5, 14 solving 6, 72 solving 7, 158 solving 8, 49 solving 9, 10 solving 10, etc., — then draw the vertical lines representing the number of individuals at these definite unit points, 5, 6, 7, 8, 9, 10, etc. No grouping of records is done, and no assumption is made that the measures are concen- trated at the mid-point of the class-interval. Size of unit. The selection of the size of the "unit" in laying off the number of measures on the vertical lines is arbitrary. Two principles of construction should control it however: (Ij The units should be made large enough that the whole distribution may be pictured on one graph — the Diagram 18. To illustrate Use of Coordinate Axes X AND Y All measures plotted on OX are called "a; "; all on OF, "m." 90 STATISTICAL METHODS size of the paper chosen will determine this point. (2) The units must be large enough to make very clear the charac- teristic features of the distribution. This means that the horizontal and vertical scales shall be so taken that the polygon is sufficiently "steep" to indicate distinct changes in the distribution of the data. Especially is this true of graphs which picture rates of increase, in which case we should avoid using a small scale, which will result in a very "flat" polygon. The student should be directed to indicate very clearly on the graph: (1) the limits of the class-intervals; (2) the distribution of the units along the vertical axis OY. The frequency polygon. Diagrams 19, 20, and 21 illus- trate the plotting of the frequency-distribution for two kinds of records: (1) ungrouped measures expressed in integral units; (2) measures grouped in class-intervals. The measures for the first illustration are arranged in the frequency dis- tribution, shown in Table 11. Such a table is then plotted as is shown in Diagram 19. Table 11. Number of Factoring Problems solved cor- rectly BY 137 Pupils in First-Year Algebra No. of problems No. of pupils 13 1 12 3 11 8 10 14 9 29 8 35 7 21 6 16 5 5 4 3 3 1 2 1 137 ' Diagram 20 illustrates the plotting of measures which have been grouped in the frequency polygon. The student STATISTICAL CLASSIFICATION OF DATA 91 should be reminded again of the fundamental assumption underlying this method, namely: the values of all meas- ures in the class-interval are assumed to be equal to the mid-value of the interval, and in 'plotting are actually con- centrated at this mid- value. An important corollary to the Y / \ / \ X / \ 20 / / \ / \ \ 10 / \ \ ■^ \ "x^ 5 6 7 8 9 Number of Problems solved 10 12 13 Diagram 19. Frequency Polygon representing Integral Measures Based on Table 11, showing the number of factoring problems solved correctly by ' 137 pupils in first-year algebra. above statement, then, is this : the total number of measures in the frequency distribution is equal, to scale, to the total length of all the vertical distances laid off above the mid- points of the class-intervals. The histogram, or column diagram. Thus the procedure stated above for the plotting of frequency polygons repre- sents the frequency distributions by the length of vertical lines erected at the mid-points of class-intervals. Another specific method of graphically representing the distribution of measures over a scale is to assume that the measures may be represented by the area of rectangles, constructed with f\ s ^ 40 Distrib 1 class int 1 ution of 123 marks 1 i! \ ervals of 10 units e ! .K 1! ii \ / II ji \ 250 ^ / ll j \ 3 J . 1! \ 15 v - \ . y y 1 \ < ii \ 20 30 40 50 60 70 80 90 100 30— /!\ 20I 15-'o ,0^ 1 1 1 1 1 Distribution of 123 marks 1 '!j \ inc lass nter rals ( >f 6i mits each L !J \ s. / 11 nil \ log r !^ /"- -^ y 1 ll" \ V ^ y 1 lljl \J .^ 20 25 30 35 25- 40 45 50 55 65 70 75 85 90 96 100 i Distribution of 123 marks in class intervals of 3 units each ^^v^ \/ ^^ 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 Diagram 20. To illustrate the Plotting op the "Frequency Polygon" for a Grouped Distribution STATISTICAL CLASSIFICATION OF DATA 93 the base of the rectangle equal to the length of the class- interval, and the altitude equal (to the chosen scale) to the number of measures in that class-interval. (Diagram 21.) It will be clear that such 'plotting of measures makes the definite assumption that measures are distributed uniformly throughout the interval. This assumption is to be contrasted with the one made in the case of the frequency polygon, — namely, that all measures in the class-interval are concen- trated at the mid-point of the interval. Plotting of this kind has to do with a definite and gen- erally fairly small number of measurements which have been made, in educational research, with rather rough measuring instruments. With the development of very refined meth- ods of measuring, and the collection of a large number of measures, the measures would be found to vary from each other by very small amounts. Furthermore, human measure- ments, when compiled in large numbers, point to the fact that the numbers of measures at consecutive points on the scale are closely the same. That is, as we increase the accuracy of measurement and the number of observations, the mid-points of our class-intervals move more and more closely together. Furthermore, the tops of the ordinates erected at these points tend to form a continuous curve, instead of a polygon of broken lines. This curve we speak of then, as a FREQUENCY CURVE, and the total area he- tween the curve and the base line represents the total number of measures. This is important for the student to hold in mind in connection with the later graphic treatment of measures. It will be evident that the area under the frequency poly- gon represents very inadequately the number of measures in the distribution, in those cases in which the number is small and the range is relatively large. In such cases it is suggested that the column diagram be drawn, as typifying more clearly, by its area, the true status of the measures. il 45 40 35 30 25 2C Ij li ft Distribution of ,123 marks ■ 1 1 % ^ Class in ervals of 10 units < ,acb 1 il S !l ii 10 II ii C ^1 1 ■c < ■^ ^:l^ 21 20 30 40 50 60 70 8 90 100 2£ 2( 16 10 1 1 "I Oh o 2 . Cla Is of ach. ii ss m ;erv£ 5 ur its e 1 II il !JL 2 1 i? 1 |! i-s ' "^ "*! 1 "^ 20 25 3( 35 40 45 50 55 60 65 70 75 80 85 90 95 100 'iJ — 1 20 p, ll^ 1 fs ?. :ia 3S nt >rv als of 3 mi ts 2ac h. i — — 10 1 -^ i\ 5 y; _ — B II -^1 — _ s 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97100 Diagram 21. To illustrate the Plotting of a "Column Diagram' STATISTICAL CLASSIFICATION OF DATA 95 ILLUSTRATIVE PROBLEMS* 1. Tabulate the following series of measures in 3 frequency distribu- tions, using class-intervals of 5, 10, and 15 units each respectively. DiSTEIBUTION OF NuMBER OF PUPILS TaUGHT BY OnE TeACHER IN Science in 125 Small Cities 147 66 70 61 126 63 85 54 92 73 96 44 53 45 95 87 48 98 76 75 52 50 115 36' 78 51 58 52 77 45 37 62 53 60 38 109 40 41 75 90 93 57 94 64 52 44 84 97 94 50 46 85 71 46 73 67 77 47 54 47 93 102 60 54 152 151 108 81 80 86 50 117 78 62 74 55 72 86 93 143 78 92 41 91 87 82 89 52 145 76 79 50 35 76 56 105 48 88 61 40 121 71 39 132 88 101 70 95 91 71 64 50 107 91 74 59 77 63 62 72 82 111 83 58 * These illustrative problems are quoted from Rugg, H. O., Illustrative Problems in Edu- cational Statistics, published by the author to accompany this text. (University of Chicago, y 1917.) 2. Tabulate each of the following series of measures in a frequency distri- bution. Use your own judgment concerning the best size and position of dass-interval. Average Annual Cost per Pupil for Stationery in 122 St. Louis Schools, 1910-11 and 1914-15t Series I (1910-11) Series II (1914r-15) 2.78 .61 .29 .80 .53 1.80 .50 .38 .61 .46 .18 .88 .64 .54 .59 .49 1.20 .58 .37 .44 .55 .50 1.61 .41 .58 .51 .53 1.38 \38 .43 .51 .57 .45 1.15 .74 .39 .52 .51 1.58 .54 .41 .37 .45 .15 1.50 .71 .53 .60 .47 1.61 .58 .45 .51 .53 .26 1.59 .66 .72 .70 .58 1.78 .46 .39 .40 .47 .20 .43 .54 .66 .41 .43 .39 .40 .41 .56 .47 .28 .48 .49 .50 .52 .51 .41 .44 .45 .56 .47 .26 .54 .61 .66 .53 .50 .45 .33 .52 .48 .58 .55 .50 .45 .67 .57 .53 .52 .45 .51 .43 .49 .16 .59 .62 .46 .45 .47 .62 .48 .35 .56 .36 .33 .50 .48 .38 .43 .45 .46 .68 .54 .38 .57 .50 .5Q .56 .58 .62 .25 .54 .39 .58 .37 .41 .57 .32 .64 .91 .32 .47 .53 .44 .26 .54 .62 .71 .38 .44 .33 .55 .35 .57 .59 .48 .50 .50 .56 .43 .39 .40 .58 .43 .62 .48 .71 .51 .60 .53 .18 .43 .52 .31 .36 .44 .34 .65 .72 .49 .14 .51 .40 .34 .59 .49 .56 .63 .57 1.04 .24 .53 .57 .52 .42 .72 .44 .42 .58 .57 .71 .41 .55 .58 .68 .57 .30 .56 .50 .59 .51 .28 .31 .43 .43 .84 .81 .34 .44 .45 .47 .62 .65 .50 1.66 t Data from Annual Reports, Board of Education, St. Louis, Missouri, 1914-15. 96 STATISTICAL METHODS 3. Plot a frequency polygon for each of the three distributions of problem No. 1. Select such a scale on X and Y that you can plot the three graphs one above the other on one cross-section sheet. Place the graphs so that corresponding points on the scales of the three distributions will fall on the same vertical line. 4. For the data given in each series in problem No. 2, plot a column dia- gram. Select such a scale on X and Y that you can plot the three graphs one above the other on one cross-section sheet. Place the graphs so that corresponding points on the scales of the three distributions will fall on the same vertical line. CHAPTER V THE METHOD OF AVERAGES The First Method or describing a Frequency Distribution 1, General statement of methods of describing a frequency- distribution Measures of condensation and organization. Having discussed the method of organizing material in the form of a frequency distribution, we are now prepared to take up the consideration of methods of statistically treating the distri- bution. It seems clear that the organization of the material in serial class arrangement, as in a frequency distribution, is but a preliminary step to the definite quantitative treatment of the material itself. The frequency distribution with its accompanying diagrams may represent adequately the status of the numerical data. It does not, however, enable definite comparison of its central tendency {e.g., the "aver- age") with that of the typical status (the "average") of other distributions. To make these comparisons, to be able to portray the typical numerical situation concisely and completely, we need measures of condensation and organiza- tion. There are three principal methods of typifying fre- quency distributions that will aid in comparison. Central tendency and variability. The first method is the method of averages or of "central tendency "; i.e., the method that shows how distributions differ in position, as shown by the size of the measure around which the measures largely cluster. The second method is, method of varia- bility; i.e.y the method that indicates the way in which 98 STATISTICAL METHODS eon , Med /'on and A/o(/e . the separate measures of two distributions, "spread," or "fluctuate," around the "average." It is clearly not suffi- cient to be able to compare the status of two distributions by stating their average value. The average value may be and often is a deceptive value for use in comparative work. In fact, any one statistical measure will probably be an inadequate means of fully describing any group of data. This is clearly shown by the frequency curves drawn in Di- agram 22, in which the average value of the two distribu- tions is identical, but in which one distribution is twice variable as the as Diagram 22. Ideal Curves drawn to illus- j trate Difference in Variability in Two Distributions, whose Means are identical other. In this illus- trative case it w^ould be quite incorrect to infer from the identity of the average standing of the two classes that the distributions of abilities are equal. In this case at least we need a measure which will enable us to compare the variation of ability in the two classes about the average ability of each. The method of relationship. The third method of treat- ing frequency distributions is the method of relationship. For example, we need to know how one type of ability is related to another type of ability; how one type of activity "cor- responds" to, or "correlates" with, another; what effect one type of learning has on another type; how ability in mathe- matics is related to ability in languao'es, etc., etc. THE METHOD OF AVERAGES 99 Throughout the discussion of the use of averages and meiisures of variabiHty we should have in mind the fact that for an adequate comprehension of the status of a group of numerical data one needs to study and interpret the whole distribution. Averages, and variability measures, and meas- ures of relationship are but means of representing central tendency by statements of the most probable value of the measure in question. For example, the arithmetic mean (the commonly used " average ") maj^ be said to be the " most 'probable value of a series of measures.'^ The value of the corre- lation coefficient, "r" (.17, .33, .42, or what-not) may be said to be only a statement of the most probable value of the degree of relationship which exists between the two traits in question. If it were said that the coefficient of correlation, "r " for the relationship that exists between scholastic ability in mathematics and that in languages, is .70, we should be able to use this value of .70 as a statement of the probability that as ^'ability in language'' increases, so does '^ability in mathematics'' tend to increase. That is, that pupils high in mathematics tend to be high in languages. It is desired to emphasize this precaution against the wholesale acceptance of statistical devices, and to point out the need for a thorough tabulation of the original data in such complete fashion that detailed study and interpretation may be made of the raw material. With this brief preliminary statement we shall turn at once to the treatment of the problem of " averages." 2. Discussion of averages used in educational research Averages describe frequency distributions by pointing out central tendencies. The attempt to describe a classi- fication of educational data by a single number must be an arbitrary process. The student thrown on his own resources and forced to invent a way of typifying a distribution would doubtless hit upon one of the several accepted ways 100 STATISTICAL METHODS of doing that. Suppose that he had plotted a frequency polygon from his data, as in Diagram 19, and had raised the questions — What is the most evident "central tendency" of these data? What is their most characteristic or typical feature? It is evident that the most outstanding charac- teristic is shown by the high point or peak existing in the frequency polygon. Since the height of the vertical ordi- nate in each case represents the number of measures, this high point means that a larger number of pupils solved eight problems correctly than any other particular number of problems. The corresponding pojnt on the scale, then, may be called : — I. The Mode First method of pointing out central tendencies. The mode is simply that value on the scale which occurs most fre- quently. It is clear that we have here a rough device for indicating the typical tendency of a mass of data. The value that occurs the most frequently obviously points out cen- tral tendencies, provided a large enough number of meas- ures is included in the frequency distribution to make it a representative or "random" sample of the total group. We shall clear up the question of "sampling" in a later chapter, but may point out now that the number of meas- ures in a distribution is large enough to form a "random sample" when it reaches such a number that the addition of another similar group of measures will not cause a fluctua- tion in the magnitude of the "average" that is computed from it. Under such a condition, the mode roughly typifies the distribution in question. The student should recognize two distinct problems arising in connection with the use of the mode in interpreting his data. (1) It may be used only as an approximate "inspec- THE METHOD OF AVERAGES 101 tion'* average. Inspection of the frequency polygon re- veals the modal value (or modal values if there should prove to be more than one distinct peak in the polygon). This specific modal value, which is the mid-value of the class- interval that contains the largest number of measures, depends — within a certain range over the scale — upon the size and position of class-intervals. To a considerable extent, with distributions of limited numbers of cases, and with distributions decidedly unsymmetrical in shape, this crude inspection mode is an unstable average. On the whole w^e should caution against using it for any purpose except as a very rough aid in the preliminary inspection of the fre- quency distribution, as an aid in " characterizing the type,'* — picking out summit points and central tendencies in the frequency curve. (2) The term "mode" should be technically reserved for the "theoretical mode" (introduced by Professor Karl Pearson in 1902), which is thoroughly mathematical in its origin. It was noted above that the mid-value of the class- interval containing the largest frequency depends upon the selection of the size and position of class-intervals. It should be emphasized that the larger part of our statistical work in school research is done on a distinctly limited number of measures, and with unrefined measuring instruments. If we will postulate the increase of the number of measures to a number relatively large, and an increasing refinement of the measurement itself, then the frequency polygon (or column diagram), which we draw to represent our actual recorded data, may be said to approach continually a "continuous" or "ideal" frequency curve as a limit. That is, the smooth frequency curve (for example. Diagram 23, from the data of Table 11) rei)resents the ideal situation, — the law that would be obtained by refined measurement of a very large number of cases. 102 STATISTICAL METHODS Practically, however, we cannot attain to sufficiently re- fined measurement of an infinitely large number of cases, and we have to content ourselves with a theoretical fitting 35 o / \ «» -Q 'l^ / \ 30 OS \ 1 fc P -Q V a a jy' ^ 1 ^ bo 3 A \ ' ^V ^0 \ 25 £? ^i;;- n \\ 03 II 1 1/ I \ •^ ^ ? cs-e \\ ^ c ^■« / \\ 1=! Aid 11 I \ 1 e C C y^ // \ 2 2 a / J \ » 1 S / \\ ! 1 / \\ I // \ / / \ / 1 \\ ^ ;/ \ ^ 5 6 7 8 9 10 Number of Problems solved 11 12 13 Diagram 23. Comparison of Plot of Actual Scores of 137 Pupils in Solution of Algebra Problems with "Smoothed Curve," representing a Probable Arrange- ment OF A Very Large Number of Cases of some frequency curve, whose equation is known, to the actual measurements. This takes the student at once to the advanced theory of curve-fitting, the thorough under- standing of which implies a considerable amount of mathe- matical training. Thus, the discussion of the calculation of THE METHOD OF AVERAGES 103 the "true mode" is clearly beyond the scope of the present work.^ Pearson's empirical rule for calculating mode. Fortu- natelj^ most of our distributions in educational research are but "moderately skewed," — that is, the measures are largely concentrated somewhere near the central portion of the range. For such distributions (for example Diagram 23, from data on Table 12) Pearson has given us an em- pirical rule for quickly calculating an approximation to this mode, which will very closely approach the true mode. It depends, however, on the previous computation of the arithmetic mean and the median. (These averages will be taken up next in this discussion.) This may be expressed as: — The mode = Mean — 3 (mean— median). That is, with moderately unsymmetrical distributions the median, mean, and mode stand in such a relation that the median is always about one third of the distance from the mean towards the mode. Applying this to our illustra- tion in Diagram 20 we find, mean = 72.6; median = 75.64; difference between them = 3.04. Therefore the approxi- mate mode is 81.72. To give an estimate of the closeness with which the mode calculated by the use of this empirical relation approaches the "true" mode we give on page 104 two tables from Yule. II. The Median Second method of pointing out central tendencies. If the commonest measure or value on the scale is the most evident ^ Complete directions are given in the complete bibliography in the Appendix concerning methods of finding the mathematical literature covering the theory of curve Ctting. 104 STATISTICAL METHODS Table 12. Comparison of the Approximate and True Modes in the Case of Five Distributions of Pauperism (Percentages of the Population in Receipt of Relief) IN the Unions of England and Wales * Year Mean Median Approximate mode True mode 1850 6.508 5.195 5.451 3.676 3.289 6.261 5.000 5.380 3.523 3.195 5.767 4.610 5.238 3.217 3.007 5.815 1860. 4.657 1870 5.038 1881 3.240 1891 2.987 * Yule, Jour. Boy. Stat. Soc, vol. ux, p. 122. (1896.) Table 13. Comparison of the Approximate and True Modes in the Case of Five Distributions of the Height of the Barometer for Daily Observations at the Sta- tions NAMED t Station Mean Median Approximate mode True mode Southampton . . Londonderry.. . Carmarthen . . . Glasgow Dundee 29.981 29.891 29.952 29.886 29.870 30.000 29.915 29.974 29.906 29.890 30.038 29.963 30.018 29.946 29.930 30.039 29.960 30.013 29.967 29.951 t Diatributiona given by Karl Pearson and Alice Lee, Phil. Trans., A, vol. cxc, p. 423. (1897.) method of pointing out central tendencies in a distribution, the second is plain: find some pertinent middle value. Such a value is the median ^ defined rigorously as that point on the scale of the frequency distribution, on each side of which one half of the measures falls. It will be helpful to the student to do his thinking strictly in terms of the linear scale which represents the frequency distribution. The completeness with which the student refers to a scale all of his work with frequency distributions will be determined largely by the THE METHOD OF AVERAGES 105 number of cases involved, and the distribution of their respective values. For example, we face two distinct prob- lems in averaging. Continuous series of measures. First — we have to do with two distinctly different kinds of measures in educational research: continuous series of measures, and discontinuous series of measures. A continuous series of measures is one in Distribution of Intelligence Quotients of 905 unselected children, 5 to 14 years of age (After Terman, p. 66) "Normal" Frequency Distribution 126 M Distribution of heights of 12 year old Boys (After Whipple, p. SO) 60 62 64 66 68 70 72 74 76 78 Stature in inches Frequency distribution of Stature for 8585 Adult Males born in the British Isles (After Yule, p. 89) Diagram 24. Comparison of Form of Distribution of Human Traits with "Normal Probability" Curve 106 STATISTICAL METHODS which the quantities are subject to any degree of division. For example, the arithmetical ability of a class of boys, as shown by the scores made on tests or by their class marks; their heights, weights and other anthropometrical measure- ments; in fact nearly all anthropometrical and social attri- butes such as we meet in educational research. We shall comment in detail in a later chapter on the form of the dis- tribution of such human traits, but we may point out here, in order to illustrate the point of the discussion, that most human measurements have been found to conform roughly to some such smooth cm-ve as is given in Diagram 24, Fig. 2. The base line of this curve represents, in each case, the status of the trait in question, for a very large number of persons. The method of testing such ability results in inte- gral scores, it is true, but in each case these integral scores represent the mid-values of various class-intervals on the scale. For example, Tables 14 and 15 give the scores obtained by two groups of eleven pupils in a test for ability in factoring. Each of these scores, 24, 23, 22, etc., means that the pupil had solved 24, or 23, or 22 problems, and was working on the next. That is, we let the integral score 22, for example, represent a distance in the scale, say the distance (or class- interval-) from 21.5 to 22.5, or from 22.0 to 22.99. We spoke of it, in the previous chapter, as the mid-value of the class- interval. Thus we see that such measures form continuous series, — that if we re'fine our methods of testing we will get scores of 22.1, 22.2, etc., instead of 22, 23, 24. Discontinuous series of measures. On the other hand, although most of our measurements are of the foregoing type, we do meet discontinuous series in our study of educational problems. For example, all our records of attendance contain gaps, — whether by classes, schools, or grades; the salary schedules of teachers contain distinct gaps, — we advance THE METHOD OF AVERAGES 107 Scores obtained by Two Groups of Eleven Pupils IN A Test for Ability in Factoring Table 14. Group I Table 15. Group II Number of problems right Scale Number of pupils 24 23 22 21 20 19 18 17 IG 15 14 ■I I T 1 1 1 1 Total 13 1 11 Total •• 12 With 12 measures, median = 18.5 With 11 " " =19. Number of problems right Scale Number of pupils 24 1 23 1 22 1 21 20 1 19 18 1 17 16 1 1 15 14 13 12 11 10 9 1 8 7 6 1 5 __ Total •• 11 108 STATISTICAL METHODS teachers by jumps of $25.00, or $50.00, or $100.00, etc. It should be clear, however, that for purposes of pointing out central tendencies these measures may best be distributed along a scale and grouped, each group thus representing a distance on the "salary scale." The second definite problem that has to be clear to the stu- dent who wishes to grasp sound methods of "averaging'* takes account, first, of the differences in proper methods to use in the case of small numbers of measures, as opposed to large numbers, and, second, of the shape of the frequency distribution. This latter point takes account of the degree to which the measures are concentrated at different points on the scale, — whether near the middle or at the extreme ends. The two points must, however, be discussed together. With small numbers of measures (perhaps 10 to 20 or 30), and anything but a very symmetrical distribution over a fairly short range, the wisdom of using any average to typify the measures is questionable. Rather than do this, the whole distribution should be presented and discussed in detail. Furthermore, the form of the distribution or the way in which the measures are concentrated at particular points on the scale may render any single measure decidedly ficti- tious. For example, suppose that the distribution showed a large proportion of measures largely concentrated at the very end of the range, but with decided numbers scattered throughout the entire range. The attempt to find some one typical measure to point out the central tendency of these measures must result in a partially fictitious statement of affairs. On the other hand, with the distribution of algebra scores shown in Diagram 19, a middle value, say 8, in Table 11, typifies the group very well. So, in turning to the discussion of the finding of the median, a central-most value, we should take up the discussion with a full recognition of the limitations of such single measures in typifying distri- butions of certain kinds. THE METHOD OF AVERAGES 109 We said that the mode was an '* inspectional average." In the same way, the median is a counting average. Its de/ termination includes two steps: (1) the arrangement of the measures in serial or rank order, placing the largest one first and the smallest one last or vice- versa; (2) the counting in of the measures from one end to determine the point on each side of which half of the measures fall. The specific computation of the median depends upon whether the meas- ures are arranged in a simple series or in a grouped fre- quency distribution. Computation of the median (A) With the measures in a simple series. By a simple series we mean a distribution of values on a scale, each of which values occurs once. Thus Tables 14 and 15 give simple series of an odd number of measures, 11. In Table 14 it is clear that the middle-most measure, the sixth (19) is the median of the series, regardless of whether we define the median carefully as the point on the scale on each side of which there is an equal number of cases, or as the middle measure. Many people have been defining the median as the middle-most measijre in the series. Ob- viously if we add a twelfth measure (say one case of 13 prob- lems in Table 14) we now have no middle measure. We are forced to assume that the median is the value half-way between 18 and 19, or 18.5. This latter way of defining the median assumes that the median is the (N + l)th measure 2 in the series (e.cr., 11 + 1 ^ . \ ttt i n i r> = otn measure;. We snail define it throughout this discussion as a point on the scale on each side of which N /9> measures are found to fall. Table 14 therefore offers no very real difficulty in typi- fying the distribution, — the measures are uniformly dis- 110 STATISTICAL METHODS trihuted by units of one. In Table 15, however, the 11 meas- ures are scattered over a wider range. Now the middle measure is 10, the median under the A^ + 1 definition. ~^ Adding a measure, say 21, makes our total 12, with no middle measure but a hypothetical median at 13, half way between 10 and 16. It should, of course, be stressed that with 11 measures, any "average" is a questionable measure of central tendency. (B) With the measures grouped in a frequency distribu- tion. As the number of measures becomes larger (30 or 40, perhaps, and upward) we are forced to group our measures in a frequency-distribution. For purposes of computing the median we now make an important assumption: the meas- ures in any class-interval are distributed uniformly through- out the interval, but may be represented by the value of the mid-point. The computation may now be illustrated by the distribution in Table 16. Table 16. Distribution of Marks IN Latiis 289 High-School Pupils Class-interval No. of pupils 95.0-100.00 22 90.0- 94.99 68 85.0- 89.99 51 80 0- 84.99 28 75.0- 79.99 47 70.0- 74.99 33 65.0- 69.99 21 60.0- 64.99 9 55.0- 59.99 6 50.0- 54.99 2 45.0- 49.99 1 40.0- 44.99 . 1 N = = 289 Half the measures, i.e., iV/2 = 144.5. Therefore we wish a point on the scale on each side of which there are 144.5 THE METHOD OF AVERAGES 111 measures. Counting down from the top the three class-inter- vals 95.0-100.0, 90.0-94.99, and 85.0-89.99 contain 141 measures. That is, 141 measures have values greater than 85.0. In the class-interval 80.0-84.99, there are 28 measures, assumed to be distributed uniformly throughout the interval. Diagrams 25 and 26 show graphically the method of finding e2 ^8 s/ 26 I / i6i> 95 SO T J44S 80 75 '70 ^3 f^opi/s ^/ /=^op//^ P^Up/73 Q65urr?- /nbc/T^c/ un/- ■65 A4A/r form/y f^roag/?. *'^-^ cot /nferira/^ -60 .55 -50 ■40- /^JRi/f^/Js 9S 90 ^ , ■> r7c/= 3^.37 /Vc/=60t -75 ^8 - 6^.37 /20 rn&eysOrG^s to h^riS' Diagrams 25 and To ILLUSTRATE COMPUTATION OF THE MeDIAN the median point on the scale. It is found to fall at a point in the interval, 3.5/28ths of the distance from 85.0 to 80.0. In numerical terms, then, the median is: 85.0 — 3.5/28 X 5 = 85.0 - 0.63 = 84.37. The same result is obtained working up from the bottom of the scale. Thus, class-intervals, 40.0-44.99 to 75.0-79.99 inclusive (or from 40.0 to 80.0) contain 120 measures. We 112 STATISTICAL METHODS wish the pKoint on the scale on each side of which there are 144.5 cases. Therefore we need to go up into the class-in- terval 80.0-84.99, 24.5/28ths of the entire distance in the intervals. In units on the scale this means 24.5/28 X 5 added to 80.0, which is the value of the lower limit of the scale, = 84.37 as before. It will be noted that, to define the median as the point on the scale on each side of which there are N /9> measures, makes it possible to compute the median from either end of the scale and secure a constant value. This calls attention to the fact that the definition of the median as the {N + l/2)th measure leads to inconsistent results. For example, in the computation of the following simple problem: — Class-interval Frequency f 20.0-24.99 ^sh" 15.0-19.99 10.0-14.99 29 5.0- 9.99 -|.6 0.0- 4.99 Total 95 Working from the 20-25 class-interval downward the me- dian equals : — 48-40 15.0- X 5 = 15 -1.379 = 13.621 29 Working upwards from the 0-5 class, the median is : — 48- 26 10.0 -f X5 = 10 + 3.793 = 13.793 29 Thus, computing the median from one end of the distribu- tion gives 13.621; from the other, 13.793. The method used should give the same result, regardless of the direction of computation. It is suggested here that the student should always check his work by counting in from both ends of the THE METHOD OF AVERAGES 113 distribution. Using the method of computing the median, adopted here, the work checks up as follows : (a) Working from 20-25 : median equals the point on the scale on each side of which there are A^/2 or 47.5 measures. Therefore 47 5 — 40 Md=15- -^— X 5 = 15 - 1.30 = 13.70 29 (6) Working upward from the bottom of the distribution : — Md=10 + ^^ X5 = 13.70 29 Summary of steps in computing median. In concluding the discussion of the median let us summarize the steps in its computation for the frequency distribution. First : compute N/2 measures. Second : Beginning at either end of the distribution, say the lower end, count the number of measures included in all class-intervals to the interval that contains the median. N Third : From — measures subtract the total number below the 2 interval (obtained in step 2). This number of measures is the number that is needed to be included from the next interval to bring the computation to the median point on the scale. Fourth : Divide this remainder by the number of measures in this interval (containing the median). This is the proportion of the total measures in the interval that are needed to bring the com- putation to the median point. Fifth : Multiply this ratio by the number of units in a class- interval. The product is the number of units on the scale that need to be added to the value of the lower limit of the class-interval to give the median. Sixth : Add this number to the value of the lower limit of the class-interval. This is the median point on the scale. Ease of computation and checking will be facilitated by expressing the value of the lower and upper limits of class- intervals as whole numbers, 80.0, 85.0, 90.0, etc., instead of 79.99, 84.99, 89.99, etc. 114 STATISTICAL METHODS This whole process can be duplicated from the upper end of the scale by subtracting instead of by adding. III. The Arithmetic Me.\n Third method of pointing out central tendencies. We des- ignated our first method of pointing out central tenden- cies, the mode, as a rough *' inspectional average"; our second method, the median, as a *' counting average." It is clear that either of these methods take account but indi- rectly of the VALUES of the measures in the distribution. That is, the mode is determined by the number of measures that happen to be concentrated most largely at a certain point, — that is, the mode is a "position" average. Simi- larly, the median takes account of the actual VALUES of the measures only in their serial or rank order arrangement. The remaining steps in the computation of the median rec- ognize each measure equally with all other measures. For example, in Table 10, and Diagram 20, the extreme meas- ures of the distributions have equal weight with all other intermediate measures in the middle part of the range. We have definite need, however, for a measure of central tendency which will take account not only of the position of each of the measures, but also of their actual numerical value. Such a measure is the arithmetic mean, which is very gen- erally called the "average," or "arithmetic average." It should be pointed out here that the term "average" should be regarded as a class term which will include all of the vari- ous measures of central tendency that we are discussing in this chapter, and not as applying specifically to any one of them. / THE METHOD OF AVERAGES 115 Definition and computation of the arithmetic mean The arithmetic mean may be defined as the sum of the values of all the measures in the distribution, divided by the number of measures. Throughout this book we shall let M represent the arithmetic mean of the distribution, m repre- sent the value of any measure, and A^ the number of cases. Thus the formula for the arithmetic mean becomes M: We must next make clear the distinctions which arise in connection with the problem of averaging by the arithme- tic mean, — namely, the computation of the simple and weighted arithmetic means, considered in connection with the distribution of measures, first, in the simple (ungrouped) series, and second, in the grouped frequency distribution. I. The computation of the arithmetic mean with the measures reported at their true values ; i.e., in ungrouped or simple series. This may be illustrated by introducing the following table: — Table 17. Annual Cost per Pupil for Instruction in Eng- lish IN 10 CiriES, with Long Method of computing the Arithmetic Mean of the Simple Series City Cost Frequency / Frequency X measure A $46 46 B 42 42 C 57 57 D 71 71 E 51 51 F 61 61 G 50 50 H 22 22 I 31 31 J 21 21 10)452(45.2 = simple arithmetic mean 116 STATISTICAL METHODS It will be noted that the data of Table 17, although forming a simple series (each measm-e occurring at its actual value) also represent a simple frequency distribution, the frequency of each value being one. The arithmetic mean that is computed from such a series is a simple arithmetic mean. Table 18 also presents an illustration of the simple series, — that is, no grouping of measures has been done, and each measure appears at its true value. In this case. Table 18. Cost per Pupil-Recitation of teaching English IN 148 Kansas Cities* Cod per student-recitation in Number of school The measure X corresponding frequency f.m. cents. " The Measure " m systc ms. ' 'Frequency " / 12 1 12 11 1 11 10 1 10 9 5 45 8 3 24 7 6 42 6 9 54 • 6 23 115 4 26 104 3 46 138 2 26 52 1 1 1 Total N = 148 148)608 4.11 = the true weighted arith- metic mean of the above distribution "^ One case marked " above 12 " omitted in all computations on these data. Monroe, W. S., Cost of Instruction in 148 Kansas High Schools. Bulletin no. 2, Bureau of Educa- tional Measurements and Standards, Kansas State Normal School, Emporia, Kansas. however, we are dealing with a WEIGHTED frequency dis- tribution, because the frequency of occurrence of measures of any particular value is, in many cases, greater than one, — 46 at 3 cents, 26 at 4 cents, etc. In this weighted frequency THE METHOD OF AVERAGES 117 distribution there has been no approximation, however, for each " class" in the distribution is a single unit, — 46 cities actually paid 3 cents a pupil-recitation, 26 cities 4 cents, 23 cities 5 cents, etc. The weighted arithmetic mean. The mean that is com- puted here is called a weighted arithmetic mean, because cer- tain values occur more frequently than others. In this case, however, the student should note that it is a true mean, just as the simple mean computed from the simple fre- quency distribution in Table 17 is a true mean. Further- more, theoretically there is no difference in the principle underlying the computation of the simple mean and the weighted mean. In both, the value of each measure is mul- tiplied by the frequency of occurrence of that measure, the products are added, and the sum is divided by the number of measures. The expression for the weighted arithmetic mean now becomes : where m represents the numerical value of any measure, / the corresponding frequency of occurrence and N the total number of measures. In the actual computation of the simple mean we merely add the values of the separate measures, and divide by the number of measures. In Table 17 each of the measures has been reported as having a frequency of 1, merely to make clear that there is no theo- retical difference between the simple and weighted mean, and that, with the former as well as with the latter case, in which the data consist of ungrouped measures, the com- putation results in a true mean. The computation of the weighted mean by multiplying the actual value of each measure by the corresponding frequency involves a large amount of numerical labor. It will be shown later that this 118 STATISTICAL METHODS labor may be very materially cut down by a short method of computation, when dealing with either the simple or the weighted arithmetic mean. 2. The computation of the arithmetic mean with the measures grouped in the frequency distribution. In the previous discussions we have noted that in grouping meas ures in class-intervals we make two fundamental assump- tions, — (l) that the measures are distributed uniformly throughout the interval, and (2) that for computation pur- poses they are all numerically represented by the value of the mid-point of the class-interval. For the measures in Table 18, the effect of this is illustrated by the following grouping of the measures : — Table 19. The 148 Measures of Table 18, grouped in Class-Intervals of 2 Units each Cost per student-recitation in cents. "The measures " The class-interval Mid-point of the interval Number of cities "Frequency "f The measures X their corre- sponding frequencies f'm 1- 2 3- 4 5- 6 7- 8 9-10 11-12 1.5 3.5 5.5 7.5 9.5 11.5 27 72 32 9 6 2 40.5 252.0 176.0 67.5 57.0 23.0 148 148)616.0 4.16 = approximate weighted arithmetic mean of the distribution We note that grouping the classes of the original distribu- tion in this fashion changes the "average*' cost per pupil recitation by five cents, a difference from the true mean of about one per cent. With a more symmetrical or *' skewed '* distribution we would have found that grouping even THE METHOD OF AVERAGES 119 two consecutive intervals together would have affected the mean more considerably. With distributions that are fairly symmetrical, however, we see that the "grouping" of class-intervals changes the mean but slightly. The deci- sion as to grouping of data, size of class-interval, etc., must depend on the data in hand. In each, there is no need of further grouping. In the case of data like those given in Table 21, showing the distribution of the percentile effi- ciency of 365 students in a test for visual imagery, it would be a waste of time to compute the mean of the entire ungrouped distribution. Since we have a range of 100 per cent, it will be convenient to divide the distribution into 20 class-intervals of five per cent each. The frequency distri- bution is then as given herewith in Table 21. Table 20 illustrates the effect of grouping on the size of the arithmetic mean and median, by giving results for the true mean computed from the 123 original measures of Table 9, p. 83, ungrouped, and the approximate mean for grouping these same measures in class-intervals of three units, five units, and ten units respectively. These results may be tabulated as follows : Table 20. Effect of Grouping on the Size of the Arithmetic Mean or Median True mean or median (i.e., data in original Mean or median with data grouped in class- intervals of units of 1 each) 3 units 5 units 10 units Arithmetic: Mean 72.17 Median 72.6 75.64 73.11 76.05 72.89 75.10 The frequency distributions and polygons representing these data, as given in Table 9, and Diagrams 20 and 21, 120 STATISTICAL METHODS are seen to be but moderately skewed. For such a type of distribution it is clear that grouping the measures changes the *' average," either mean or median, relatively little. Table 21 gives the detailed computation of the arith- metic mean of the achievement of 365 college students in tests for visual imagery, by the traditional or LONG method. This method groups all measures in a class-interval at the mid-point. This example makes it clear that the computation of a mean by this method is unwieldy. Short methods of computation are therefore desirable. Table 21. Efficiency of 365 College Students in Tests for Visual Imagery {The long method of computing the weighted arithmetic mean) Class-intervals Frequency f Value of mid-point m fm 95.0-100.0 ■ 8 97.5 780.0 90.0- 94.99 2 92.5 185.0 85.0- 89.99 9 87.5 787.5 80.0- 84.09 8 82.5 660.0 75.0- 79.99 24 77.5 1860.0 70.0- 74.99 16 72.5 1160.0 65.0- 69.99 33 67.5 2227.5 60.0- 64.99 11 62.5 687.5 55.0- 59.99 35 57.5 2012.5 50.0- 54.99 18 52.5 945.0 45.0- 49.99 59 47.5 2802.5 40.0- 44.99 20 42.5 850.0 35.0- 39.99 56 37.5 2100.0 30.0- 34.99 20 32.5 650.0 25.0- 29.99 18 27.5 495.0 20.0- 24.99 6 22.5 135.0 15.0- 19.99 12 17.5 210.0 10.0- 14.99 4 12.5 50.0 5.0- 9.99 4 7.5 30.0 0.0- 4.99 2 2.5 5.0 365 365) 18,632.5 51.05 = weighted arithmetic mean THE METHOD OF AVERAGES 121 3. A short method of computing the arithmetic mean. The short method to be presented to the student shortens the labor of multipUcation by making three conditions: (1) that we treat the class-interval as a unit of 1 on the scale, in- stead of as an aggregation of many units; (2) that we assume the value of any measure or class-interval as the estimated mean, or as the one which contains the estimated mean; (3) that we compute the difference between the true mean and the estimated mean, rather than compute the true mean it- self. Let us illustrate it first by application to the simple series of cost data reported in Table 17. Table 22. Mean for Table 17, recalculated by Short Method Deviation from estimated mean in actual units Cost Estimated mean + - 51 5 42 - 4 57 11 71 25 46 46 61 15 50 4 22 -24 31 -15 21 -25 60 -68 Total = 10 measures 60 Sd=_"8 — Sd „ — =.8 n M = Estimated mean H n = 46+ (-.8) = 45.2, as by the long method 122 STATISTICAL METHODS The practicableness of the use of the short method in saving time in a simple series is doubtful. This simple illus- tration is included here to make clear the principle under- lying the use of the method in the case of the frequency dis- tribution. With grouped series it has practical value as a labor-saving device. Let the student note clearly, however, before turning to the more complicated illustration given below, that the short method merely estimates the mean, and then adds a correction (c) which is the arithmetic mean of the deviations from this estimated mean, c = —rr and the •^ N formula for the mean becomes : — M = estimated mean + correction M = estimated mean + —X number of units in the interval in which d is the deviation of any class-interval from the interval containing the estimated mean. Obviously the method will hold true regardless of the frequency of occur- rence of the measures. We pointed out in the previous sections that there is no theoretical difference between the simple and weighted mean. It will be noted that the multi- plication can now be done mentally. In Table 23 we apply the method to the data of Table 21. The value of the whole method lies in the fact that it is a time- and labor-saving device. We estimate the "as- sumed mean," and compute mentally the correction that has to be made to the assumed mean. The example in Table 23 illustrates the use of the short method. The entire distribution of 365 cases is first grouped in 20 class-intervals, each interval having a range of five per cent. This step results in the frequencies 8, 2, 9, 8, 24, etc. These are then totaled, giving 365. Instead of next multiplying the mid-point of each interval (a three-place THE METHOD OF AVERAGES 123 Table 23. Efficiency of 365 College Students in Visual Imagery Class-intervals Frequency f Deviation from the assumed mean interval d Frequency X deviation fd 95 0-100 0. 8 2 9 8 24 16 33 11 35 18 59 20 56 20 18 6 12 4 4 2 10 9 8 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -7 -8 -9 80 90.0—94.99 18 85.0- 89.99 72 80.0- 84.99 5Q 75.0- 79.99 144 70.0- 74.99 80 65.0— 69.99 132 60.0- 64.99 33 55.0- 59.99 70 50.0- 54.99 45.0- 49.99 40.0- 44.99 35.0- 39.99 30.0- 34.99 18 703 703 - 20 -444 -112 259 - 60 25.0- 29.99 - 72 20.0- 24.99 15.0- 19.99 - 30 - 72 10.0- 14.99 5.0- 9.99.../ 0.0- 4.99 - 28 - 32 - 18 365 444 259 divided by 365 = .71; .71 X 5 = 3.55, the correction to be added to assumed mean to get the true mean. The true mean = the as- sumed mean + correction. Assumed mean = 47. 50 Correction = 3 . 55 True mean = 51 . 05 number) by the corresponding frequency, we estimate the class-interval, 45.0 to 49.99, the mid-point of which most closely approximates the position of the true mean. This can be determined by inspecting the frequency dis- tribution, countir g up from one end until half the cases are included. Pr ctice in this "scanning" of the distribu- tion will give skilj in closely approximating the true mean. 124 STATISTICAL METHODS The labor involved in mental multiplication will be further reduced by taking the assumed mean at the portion of the distribution at which the measures are most heavily con- centrated. Use of class-intervals with the short method. The next step consists of tabulating the number of units distant that the mid-point of each class-interval is from the mid-point of the interval containing the assumed mean, 47.5. These dis- tances are called deviations *'(i," and in the example are: interval 50.0-54.99 is 1 unit above, or larger than, 45.0- 49.99, therefore its deviation is +1; d for 55.0-59.99 is + 2; for 40.0-44.99 is - 1 ; for 35.0-39.99 is - 2, etc. Thus, the whole short method merely treats the class-intervals as units of 1 instead of 5 or whatever they may be. With the traditional method of finding the weighted mean we would next multiply the mid-point of each class- interval by its frequency. Instead, we now multiply the *^ deviation'' of each class-interval from the assumed mean by the frequency of the class, 8 X 10, 2X9, 9X8, etc., giving the column headed/^. Two points must now be kept in mind, — first, the fd's occupy the same place in the computation by the short method that the fm's do in the traditional method; second, that having assumed a mean, all of the deviations above the mean will be positive, and all below the mean will be negative. If we were dealing with the true mean of the distribution, the sum of the positive deviations should be equal to the sum of the negative deviations. Since only in rare cases does the true mean fall at the mid-point, it will always be necessary to ADD to the estimated mean a "correction" (denoted "c"). Just as we find the sum of the fm's in the long method, so do we find the ALGEBRAIC S UM of the fd's. That is, we total the positive deviations and the negative deviations, and subtract the smaller from the larger. This difference is then the toU , amount that the THE METHOD OF AVERAGES 125 mid-points of all the class-intervals deviate from the assumed mean. However, we wish the average amount of the devia- tions of the mid-points of the class-intervals from the as- sumed mean. This will be the correction " c'." The average amount is found by taking the arithmetic mean of the sum of the deviations, — in the example given this amounts to dividing the difference of the positive and negative devia- tions, 259, by the total number of cases, N = 365. This gives a correction c' = 4- .71, which means that the assumed mean is smaller than the true mean by .71 of the range of the class-interval. Note that to find the true mean we do NO T add this value of the correction to the assumed mean, for this value has been computed on the basis of the class- interval of 1, instead of 5. Therefore the true correction is .71 X 5 = 3.55, which, if added to the assumed mean, 47.5, will give the true mean, 51.05. The accuracy with which the mean worked by this method checks the mean worked by the longer method, depends merely upon the number of decimal places to which the arithmetic work is carried by the two methods. In Table 5 the problem dis- cussed above is worked by the long method to permit a com- parison of the two methods. Summary of steps in the computation of the arithmetic mean by the short method. In conclusion, let us summarize the method as follows : — 1. Group the original measures in a frequency distribution. 2. Total the frequencies. 3. Estimate the interval that contains the mean. The value of the mid-point of this interval is the value of the estimated mean. 4. Treating each class-interval as a unit, record the number of units that the mid-point of each class-interval deviates from the estimated mean, indicating as positive all intervals whose mid-value is greater and as negative all those whose mid-value is smaller than the estimated mean. These distances will nearly always be less than 10, because most distributions 12C STATISTICAL METHODS will contain less than 20 intervals, and the estimated mean will be taken approximately in the middle of the distribution. A safe rule is to take the estimated mean in the heavily concen- trated portion of the distribution. In this way the mental multiplication will involve smaller numbers. 5. Multiply each deviation (d) by its corresponding frequency (/) taking account of signs. 6. Find the algebraic sum of the positive and negative deviations. 7. Divide this sum by (N) the number of measures. This gives the correction (c'), which is the arithmetic mean of the devia- tions from the estimated mean, in units of class-intervals. 8. Multiply c' by the number of units in an interval, giving c. 9. x\dd c to the estimated mean to get the true mean. For most of the "averaging" problems of school re- search the three methods discussed in the foregoing pages suffice. A detailed analysis will be given later of the specific use of various methods of averaging. There are two prob- lems involving the computation of averages, when time rates or rates of increase are in question, that have to be treated by special averaging methocjs. We shall turn to these next. IV. The Harmonic Mean The averaging of time rates. At the present time the measuring movement in education consists largely in the establishment of "norms of attainment" in the various school subjects, and for various levels of scholastic develop- ment. We have many " grade norms " in handwriting, spell- ing, reading, and arithmetic in the elementary school, and for algebra in the secondary school. The "norm" is taken to be the "average" performance (expressed as so many words written in one minute, or read in one minute, etc.) of large groups of pupils found at the different years of school life who are actively taking work in the various studies. "Average" performance has quite universally been taken to be the arithmetic mean of the performances of the THE METHOD OF AVERAGES 127 individual pupils. Generally this has been true irrespective of the conditions of work involved in the testing. In fact, it seems quite clear that there is no general recognition of the fact that there is an issue involved in the averaging of time rates. ^ We shall therefore call attention to certain points in the use of statistical averages which may have been overlooked by workers in educational research. It is desired to establish the following points : — 1. That there are two distinctly different methods of averaging time rates : (a) Averaging by the arithmetic mean of the rates; (b) Averaging by the harmonic mean of the rates; 2. That with given material average performances computed by the two methods will not be comparable; 3. That these two methods imply two different units of compu- tation, *'the unit of work" and the "unit of time"; 4. That a method of averaging must be selected appropriate to the unit of computation which is being used, — with the unit of work we must use the harmonic mean of the rates (the arithmetic mean of the absolute times) ; with the unit of time we must use the arithmetic means of rates. To get the problem clearly before us let us use the follow- ing simple illustration : — Suppose a group of five boys to have been tested for speed of solving the algebra problems used in the writer's Test 1, Series A, by assigning a definite amount of time (2 minutes) and noting the amount of work done. Let us ex- press the results in two ways : (l) express the efficiency as "the number of problems worked correctly in one minute "; (2) express the efficiency as the " number of seconds required to solve one problem" (assuming the problems to be uniform in difficulty, on which basis the test was designed). The 1 This issue was first pointed out to the writer by Dr. L. P. Ayres. The writer later met the problem in his own work and is alone respon- sible for the present method of treatment. 128 STATISTICAL METHODS first method expresses performance as a rate, or in terms of a unit of time; the second method expresses performance in terms of a unit of work (the amount of time required to do a unit of work). Let us now find the "average" per- formance of the results of the testing, by computing the arithmetic mean (the method commonly used) of the in- dividual records in the two series. The computation is as follows: — Number of problems solved per minute Number of seconds required to solve ov£ problem 12 10 8 6 4 5 6 7.5 10 15 5)40(8 problems solved on the average per minute 8)60(7.5 sec. required to solve one problem 5)43.5(8.7 sec. required to solve , one problem, or an average rate of 6.897 problems per minute Formula of the harmonic mean. The question arises, why is not the time required to solve one problem as obtained by one arithmetic mean the same as the time required when obtained by the arithmetic mean in the other series .f* It is noted that the rate as determined by the two methods differs as much as fifteen per cent. The answer to the question is: The two series are not comparable until reduced to the same base. The base required is : What part of a minute is required to solve one problem? In the second series this is the base used (i.e., the number of seconds required to solve one prob- lem). Each member of the first series needs to be reduced to that base. In other words, the reciprocal of each measure should be obtained instead of the rates themselves, and these should be averaged by the arithmetic mean. This amounts to finding the harmonic mean of the series of rates. We may define the harmonic mean as follows: it is the reciprocal of the arithmetic mean of the reciprocals of the in- THE METHOD OF AVERAGES 129 dividual measures of the series. following formula : — • It may be expressed by the H N^ my where N = number of cases, and m represents any individual measure. It should be stressed that the harmonic mean of the rates is the same thing as the arithmetic mean of the corresponding time. The work now checks up as follows : — Number of problems solved Reciprocal of number of problems Number of seconds required per minute solved per minute to solve one problem 12 .08333 5 10 . 10000 6 8 . 12500 7.5 6 . 16667 10 4 . 25000 15 5)40(8 5).72500(.1450 5)43.5(8.7 sec. 8)60(7.5 sec. re- 1 required to solve one quired to solve one = 6.897 problems problem. Rate = 6.897 problem, accord- can be solved in one problems solved per ing to the arithmetic minute = rate. minute, according to the mean of the rates 6.897)60(8.7 sec. re- arithmetic means of the quired to solve one prob- absolute times lem according to the har- monic mean of the rates It has been recognized that the harmonic mean of a series of rates will always be less than the arithmetic mean. This simple problem shows that it will be less by as much as fifteen per cent with distributions of large variability. Natur- ally the two means approach each other in value as the variability decreases. It is clear that there are two distinctly different ways of approaching the problem of establishing standards of attain- ment in various mental or physical abilities. They are plainly to be distinguished on a basis of the unit involved, the unit of 130 STATISTICAL METHODS work, or the unit of time. To repeat them here, they are: (1) the unit of work: How much time is required to do a' unit of work? (2) the unit of time: How much work is done in a unit of time? It will be agreed that in order to get com- parable average measures we must use the same method of averaging individual records in the two series. Proper method of averaging with each of the different units. Granted that there are two distinctly different methods of averaging time rates {i.e., two different units of computation), and that results computed by the arith- metic mean on the basis of one unit are not comparable with those computed on the other unit, the question arises : Which method of averaging should be used; (1) with the unit of work; (2) with the unit of time? It must be recognized at the start that the taking of an average to represent or typify large numbers of measures is in a sense an arbitrary process. It is merely an attempt to select one numerical index (out of several possible ones) which shall represent adequately the status of the entire group. To state our problem clearly let us turn to the stock problem of the men rowing a boat at different rates. We may then adapt the conclusion of the matter to our own problem of educational measurement. First, the unit of work: Assume A and B each are to row one mile (or work one problem) and the time is to be taken. A rows the mile in 7.5 minutes, i.e., he rows the mile at the rate of 8 miles an hour. B rows the mile in 5 minutes, i.e., he rows 1 mile at the rate of 12 miles an hour. Together they row 2 miles in 12.5 minutes, or 1 mile in 6.25 minutes, or at the average recific use of the geometric mean. We may define this mean as the nth root of the product of the separate measures in the series, — that is, Mg =V (^1^'2^3 . . . . Xn). There has been practically no use made of the geometric mean in educational research, in spite of the fact that with problems of averaging rates of increase the average to use is the geometric mean. For example: — Suppose an individual's performance, as shown by testing, had improved fifty per cent in ten practice periods, say in ten weeks. What is the average weekly rate of improve- ment? Is it five per cent, as shown by the quotient of the total improvement divided by the number of weeks? On the contrary it is found by taking the 10th root of 1.50 and sub- tracting the initial efficiency, i.e., X/Y^ ~~ 1» which gives us 4.1 per cent. In other words a weekly improvement of 4.1 per cent will increase the efficiency by 50 per cent in ten weeks. It is clear that we cannot take the arithmetic mean of such geometrical increase as the above. To do so in this case would give a total improvement of 63 per cent, instead of 50 per cent. The geometric mean, practically adapted only to the THE METHOD OF AVERAGES 133 solution of short series, can easily be computed by the aid of logarithms. Thus the logarithm of the geometric mean of a series of measures is the arithmetic mean of the loga- rithms. The expression would read thus: — - - . S log X log Mq = ^ . Steps in the computation of the geometric mean. The steps in the computation of a geometric mean are therefore as follows : — 1. Find the logarithm of each of the measures. 2. Find the arithmetic mean of the series of logarithms. 3. Find the number corresponding to the arithmetic mean of the logarithms; — this is the geometric mean of the original series of measures. Another illustration of the use of the geometric mean is given below: — 1. Suppose a group of boys to have gained skill in practicing shooting, 90 per cent in 3 months. What has been the average 90 gain each month? Not — = 30 per cent but o \/Y9d-1.0=lM - 1 = 24 per cent. That is, at the end of the first month their gain in efficiency is 24 per cent + 100 per cent =124 per cent of their initial efficiency, which was 100 per cent. At the end of the second month they have gained 24 per cent of 124 per cent = 29.76 per cent, which added to 124 per cent gives an efficiency of 153.76 per cent. The third month they gain 24 per cent of 153.76 per cent = 36.24 per cent, and their final efficiency is 190 per cent of their initial efficiency, a gain of 90 per cent as stated above. Thus we have described and discussed the computation of five specific averages, which are available for use by students of educational research: — 134 STATISTICAL METHODS 1. The mode (approximate or true), a "position" average. 2. The median, — a "counting" average. 3. The arithmetic mean, — an arithmetic average based on the value of each measure. 4. The harmonic mean, — an average for use in averaging time rates. 5. The geometric mean, — an average with particular uses in averaging rates of increase. Of these, by far the greater use is made of the median and arithmetic mean. It is necessary next to establish the proper function, limitations, and specific use of each of these meth- ods of averaging. VI. Statistical Fallacies and Mooted Points in Averaging As students of education have turned to the use of quanti- tative methods, many fallacies have been evident in their manipulation of statistical devices. These are found in con- nection with methods of averaging, of measuring dispersion, of measuring correlation, and of determining the rehabihty of measures. We shall discuss each of these pitfalls in the use of statistical methods as we take up each phase of the work. We must point out here, then, certain typical fallacies in averaging, and proceed to the thorough analysis of the proper use of averages. Fallacies of averaging have been of two kinds : (1) those in which the wrong average has been used (e.g., the arithmetic mean instead of the harmonic mean); (2) those in which an incorrect use has been made of an average, — that particular average being tlie proper one to use in the given problem (e.g., the use of the simple instead of the weighted mean). A. Use of wrong average. In this type of mistake in aver- aging we find : — 1. Averaging time rates by the arithmetic mean. We have THE METHOD OF AVERAGES 135 already shown that rates of achievement, obtained by the "unit of work" method of testing, may not properly be averaged by finding the arithmetic mean of the rates them- selves, but rather by finding the arithmetic mean of the corresponding "times" (i.e., by the harmonic means of the rates). 2. Averaging rates of increase by the arithmetic mean. In the last section we pointed out the difficulty of defining the average of a series of percentage increases by taking their arithmetic mean. We found rather that the nth root of the product of the measures {i.e., their geometric mean) might better be taken to represent the average status of the measures. B. Incorrect use of an average. Of this second general class of incorrect uses we find the use of the simple arithmetic mean for the weighted arithmetic mean. To this may be added the error of assuming that the measures of a distri- bution are distributed uniformly throughout the range of the distribution. The most evident of such mistakes found recently is : — 1. Taking the arithmetic mean of the extremes of a distribu- tion as the ^^averajge'* of the measures in the distribution. This is one of the most patent of fallacies in averaging. It is illus- trated clearly by the following data. A State commission appointed to survey the State's higher educational institu- tions, collected data on the occupancy of classrooms in three state institutions by this method, — viz: "The maximum occupancy of any room (the maximum number of students regularly in the room at any period of the week) plus the minimum occupancy, divided by 2, equals the average occu- pancy." Furthermore, to obtain the occupancy ratio for any building or group of buildings, they obtained the " aver- age " for each room and took the simple arithmetic mean of these averages. This report gives no complete data upon 136 STATISTICAL METHODS which to compare the actual occupancy with these fictitious figures. They are obviously fictitious, however, and are based on two unsound assumptions: first that the distribu- tion of classroom occupancies is uniform, and second, that the frequency of occurrence of each size of classroom is constant. A recent report calling attention to this fallacy says: "Room 32 of a similar building in another state had a minimum occupancy of 1, and a maximum of 56." The actual occupancy was then stated as follows: — Ckuia Size of class Number of times room is used by each class f fm A B C D E F G 56 23 19 6 5 4 1 3 2 3 6 4 2 3 168 46 57 36 20 8 3 Total 114 23 338 Average occupancy = 23)338(14.7 By the method of taking the arithmetic mean of the extremes the occupancy is, — - — = 28.5. In brief, such an error in the use of averages is really caused by using the simple mean instead of the weighted mean, in that it assumes that the frequency of use of each classroom is the same. It also mistakenly assumes that the sizes of class are dis- tributed uniformly over the entire range. This obviously is not true in school practice. Which mean to use. The question as to which arithmetic mean to use (simple or weighted) in averaging educational data is one of great importance at the present time. The THE METHOD OF AVERAGES 137 answer can be given the student only in terms of the relation between the nature of the data at hand and the purpose of interpretation. Use that method of averaging which will give the truest picture of the central tendencies evident in your data. In the foregoing example the actual occupancy of classrooms is clearly typified better by the weighted mean than by the simple mean of the two extreme measures of the distribution. Educational conclusions based on such a method as the latter must necessarily hamper the progress of scientific education. Let us give some concrete illustra- tions of the use of the simple and weighted mean. The most frequent demand for "averages" is in connec- tion with the attempt to measure various aspects of school efficiency. Our figures are stated, for example, in terms of the achievement of pupils determined by testing; unit (aver- age) costs of various school activities; average age, experi- ence, training or salary of teachers; average amount of time devoted to this, that or the other subject of study, etc. Measurement of classes, schools, and systems of schools gives distributions of data that are to be expressed in terms of central tendency. Shall we express this by weighting every class, school, or system with the number of pupils in each, number of teachers, number of rooms, etc., or by taking the simple arithmetic mean of the records of classes, schools, buildings, or systems? This amounts to asking in the case of the achievement of pupils, — what is the basic unit in our data — the pupil or the class? the ability of the pupil re- gardless of training, or the specific type of training to which he has been subjected? Take the case of testing pupils' efficiency in algebra, as the writer has done it in 50 school systems. The number of al- gebra pupils per school varied from 30 to 100; the average achievement varied among schools by very large amounts on any one test. Shall the score of the school of 100 pupils be 138 STATISTICAL METHODS weighted 100, and the score of the school of 30 pupils, 30? Or, shall they each be regarded as of equal weight with all the others, and the simple mean be computed? The answer must be made in terms of the basic unit — the unit clearly is the class, not the pupil. We are testing the result of the pupil's training in algebra, his skill in doing a specific thing he has been trained to do. We are testing the results of a score of types of training, and these are the basic units. Contrast this situation with the determination of average height of school boys, the average age of teachers, etc. Here the basic unit is very clearly the individual boy or teacher, not the class into which he or she may be grouped, and the re- cords of classes, schools, groups, etc., should be weighted by the number of individuals. Another commonly occurring problem nowadays is the school-cost problem. We meet a series of heating costs com- puted say, for 20 school systems, by buildings, in units of, "per cubic foot," "per classroom," or "per pupil in average daily attendance." In such a problem we should first classify buildings in groups in terms of like heating conditions, — similar heating apparatus, like number of rooms, etc. If this is impossible then unit heating costs for city systems clearly should be computed with the basic unit taken to be the classroom, cubical contents, or number of pupils, and not the building. Homogeneity of data. Still another very important prob- lem of averaging is raised in connection with the question of "homogeneity of data." It is fundamental to the sound treatment of numerical data that we include in any one statistical group only individuals who have been subject to the same conditioning factors. For example, the attempt to com- pute the "average" salary of all teachers in a school system cannot possibly result in a clear statement of "average" sal- ary which will definitely be comparable to that computed for THE METHOD OF AVERAGES 1S9 another system. The "average" in this case is computed from a distinctly non-homogeneous group of persons, — elementary teachers, secondary teachers, elementary princi- pals, secondary principals, supervisors of grades and special subjects, assistant superintendents, superintendent, and other special administrative officers. To secure comparable measures we clearly must average separately for each statis- tical group, making sure that each is made up of persons whose salary status is determined by the same set of causes. The student should guard constantly against the fallacy of computing averages from non-homogeneous data. He will meet series of data, continually, in which he has included items that are caused by conditions qualitatively different, and which should be eliminated from the gi'oup. For example, suppose that we have tested classes of pupils in arithmetic. In a class of 20 there are three who do not attempt any problems of the test. Should we sum the scores of the class, and divide by 20 or by 17.^ The arithmetic mean will be distinctly different in the two cases, and our interpretation of comparisons correspondingly so. Such cases must be decided by reference to the question of " homo- geneity of data." If the class is under our immediate con- trol it will be possible to tell if these three pupils in iniel- lectual capacity, previous training and physical condition on the day of the test are qualitatively different from the other 17 members of the class, who solved problems varying in num- ber from 3 to 18. Comparison of the scores in various tests in the same subject will also help us to decide. If they prove to be so, they should be eliminated from the group and the average computed from the records of the 17. Another illustration from the field of school costs will make the point clearer. A recent study on the relationship between the cost of instruction and the number of pupils taught by one teacher gives the data reproduced in Dia- 140 STATISTICAL METHODS gram 47, Chapter IX. It will be noted that the table in- cludes all groups of data on the number of pupils taught by a teacher, from 25 to above 170. Careful examination of the table will show, however, that the investigator has two distihct groups of conditions included in his study. It is evident that for the range from 25 pupils to 80 pupils taught by one teacher there is a very high degree of rela- tionship, i.e., that as the number of pupils increases the cost decreases in a definite way. This relationship miay be ex- pressed by a coefficient of correlation of — .84. From 80 pupils throughout the rest of the table it is evident that, as the number of pupils increases there is no decrease or increase in cost, and the coefficient is practically 0. The investigator has thrown the two distinctly different groups together, and computed relationship for a non-homogen- eous group. His coefficient of — .47 and his averages and measures of variability are largely fictitious for that reason, and conclusions based on them are of questionable value. Averaging ** samples." Before leaving this introductory discussion of the uses of particular averages we should refer briefly to the effect of averaging inadequate "samples'* of our total mass of data. In educational research we are constantly forced to form conclusions from a relatively small group of data. How large, for example, should a group be, or how many times should a test be given to permit gen- eral conclusions to be drawn concerning similar individ- uals in the mass or similar testing work.^ In other words, how many cases must we have to give us an "average," typical of a very large number of similar individuals? To illustrate the point : suppose that we wish to determine the spelling ability of 20,000 pupils in a city system, represent- ing the achievement in part by some measure of "average" attainment. It is not expedient to test all of them. How many pupils shall we test to get a "random sample".? A THE METHOD OF AVERAGES 141 common sense way to define such a sample is this: A sample of any total population is "random" when numerical coeffi- cients, for example averages, computed from any number of samples similarly selected and of similar size will be approximately constant. (The more technical phases of "sampling" in statistics will be discussed later.) Functions and limitations of particular averages. We have thus introduced the subject of the functions and limitations of averages by a concrete exposition of particular difficulties that the student will meet in pointing out central tendencies in his data. It should be recalled here that these difficulties are of two types: (1) those which may involve the taking of the distinctly wrong average; (2) those which involve the application of any average to non-homogeneous data, or to an inadequate sample, or to an improper determination of the basic unit. The second point has been discussed com- pletely enough to lead to a thorough presentation of the former point. Therefore, we turn next to the question of the properties of each of the G.ve averages, their proper functions, their limitations, and the specific purpose for which each should be used. Enough has been said to make it clear that the process of averaging is one of selecting the best single quantity to characterize the central tendency of a distribution; that any average that is used must have certain properties which will show it to be a good representation of type. Summary of essential properties of a valid average.^ It will aid the discussion to list here the essential character- istics of representative averages : — 1. If it is to be completely representative of the entire distribu- tion, it must be contributed to by all the measures of the dis- tribution. ^ The writer has been aided in making a complete summary of these properties by Yule's discussion, Introduction to the Theory of Statistics, chapter on averages. 142 STATISTICAL METHODS 2. It should be purely quantitative, — defined by the numerical data alone, and should not involve the judgment of the ob- server. 3. It should be so constructed as to be relatively simple in com- putation. 4. It should be stable; that is, it should be of such a nature that representative samples taken from the total population will give a fairly constant average value. All other factors being equal, that average which gives the smallest fluctuation in value as we take different samples from the total group, is the best average to use. 5. An average must not be much displaced by slight changes in the arrangement of the frequency distribution. References to the discussion of the arrangement of data in the frequency distribution will make clear the importance of getting an aver- age that will be fairly stable, regardless of the size or position of class-interval that is selected. Furthermore, the average must be as little as possible affected by errors in observation. 6. Since the purpose of averaging is to point out clearly central tendencies to the reader, the average which is selected should be of such simple and definite nature that the lay reader will grasp easily its typifying significance. In this characteristic the geometric and harmonic means show themselves to be poor averages, the arithmetic, median, and mode being much more easily understood. Complete success in using an average must depend on the student and the reader being able to think clearly in terms of the average. 7. From the standpoint of mathematical treatment, in the refined use of averages, it is important that an average be susceptible of algebraic manipulation. For example, it has been repeatedly pointed out that it should be possible to express an average obtained from the combination of two or more samples of the same data in terms of the averages of each of the samples. VII. Use of the Different Measures of Central Tendency It will now be possible to come to some agreement con- cerning the proper use of averages by checking each against the foregoing list of essential properties. THE METHOD OF AVERAGES 143 Function of the mode as a measure of tjrpe. Taking up our })roperties in order, these conclusions seem evident: 1. The mode is not contributed to by all the measures. On the contrary it may be determined by a relatively small propor- tion of the total number of measures, concentrated in one class. 2. It is quantitative in the sense that it is defined by the fre- quency of the largest class. 3. The empirical mode is an inspection average, and thus is the easiest of all the averages to determine. Furthermore, it may be determined without any detailed knowledge of the extremes of the distribution except that the frequency of measures there is small. On the other hand, the theoretical mode is the most difficult to compute of any of the averages, depending on the most advanced theory of "curve-fitting." 4. It is more unstable than the arithmetic mean or median in its fluctuations, due to the taking of different samples from a given group of data. 5. In any but closely symmetrical distributions it is relatively unstable in the way in which it depends very closely on the method of grouping of the class-intervals. The manner in which it fluctuates is illustrated by Diagrams 20 and 21, as we change the size and position of the class-interval. For fairly refined work it is evident that the mode is too unstable for ef- fective use. 6. The mode has the advantage of being the most easily com- prehended of any of the averages. It is the ''newspaper aver- age"; the average of the man on the street, and for the lay reader has a clearer meaning than most of the other averages. Here it finds its principal function in describmg skewed distri- butions of many class-intervals, with distinct concentration of measures in certain class-intervals. Furthermore, it serves a good purpose in the graphic representation of measures, be- ing marked by distinct peaks in the frequency polygon. 7. It is clear that the empirical mode (the only one in which the student of education is interested) has no mathematical signifi- cance, and is not susceptible of algebraic treatment as is the arithmetic mean. In resume, it should be clear that the empirical mode is only a rough inspection average; that it may be indicated 144 STATISTICAL METHODS to the reader as one means of pointing out central tenden- cies; but that its capacity for representing the central ten- dency is very limited. Dependence on it beyond preliminary inspection of a distribution is not to be recommended, except in very symmetrical distributions. The geometric mean as a measure of central tendency. With the exception of problems involving the averaging of rates of increase, the student of educational research will have comparatively little need for using the geometric mean. Its computation is rather laborious; it is not readily com- prehended by the lay reader (not having come into popular use) ; and its mathematical properties are abstract, although valuable in certain forms of problem work. The principal function of the geometric mean is found in treating data which involve rates of increase, and which thus take the form of geometric series. For example, in problems in averaging increases in population, attendance in school, growth in the teaching staff, budget, etc., aver- age status can be more consistently defined by means of the geometric mean. A second valuable property of the geometric mean is found in connection with the discussion of index numbers or ratios. It may be said that the mathematical properties of the geometric mean establish the superiority of that mean over that of the arithmetic mean or the median, in aver- aging such index-numbers. Use of the harmonic mean in measuring central tendency. In connection with the discussion of the harmonic mean on pages 126-131, its specific function as an average of progress rates was pointed out. This valuable property of the harmonic mean should be kept in mind, and brought into use in all problems of that nature. The median as a measure of central tendency. With refer- ence to the median, the following conclusions seem evident: THE METHOD OF AVERAGES 145 1. The median is contributed to by all the measures of the se- ries, the magnitude of each, however, being taken account of only indirectly. That is, the median is an average de- pending on the serial order of values and on the actual nu- merical value only as it determines this serial arrangement. 2. It is .quantitative, being defined at least indirectly by the values of the measures. 3. It has the great advantage that it is the most easily com- puted of all the numerical averages; it is a "counting aver- age," depending for its determination on (a) the serial ar- rangement of the measures (with the use of the frequency distribution this is a necessary step of the computation of the arithmetic mean also) ; (6) the counting in of half the meas- ures to reach the median point on the scale. 4. Fluctuations in the size of the median may be larger with the taking of small samples. At the same time the median may give a more stable average from small samples, due to fluc- tuation in the size of extreme values. In this particular it should be pointed out that the median is affected less by the extremes of the distribution — that is by unusually large or small measures — than is the arithmetic mean, which takes full account of these values. The student must decide care- fully, in connection with his specific distribution, whether the "average" should or should not be contributed to by unu- sually large or small values. If they are regarded as important the arithmetic mean is the best representative of central ten- dency; if not, then the median is the better measure of type. Again, the location of the median depends only partially on a small group of measures; in this, it differs distinctly from the mode. However if it happens that the measures in a distribution are largely concentrated in a few intervals, it may result that the median (falling at a point on the scale at which many measures are concentrated) will be very in- definite. 5. With the types of distribution commonly met in educational problems, the median is but little subject to fluctuation with rearrangement in the size and position of class-intervals. Reference to Diagrams 20 and 21 shows the relatively stable position of the median in the distribution of fairly large num- bers, with a form not more than moderately skewed. 6. The median must rank high in the ease with which its mean- 146 STATISTICAL METHODS ing may be grasped by the lay reader. Partly for this reason, it is being adopted rapidly by students of education. 7. The median does not lend itself to algebraic treatment. (a) The median of component parts of a distribution cannot be expressed in terms of the median of whole distribution ; this is true because the distribution depends on the form of the component distributions, and not on their medians alone. (6) No theorems can be expressed for the median values of measurements subject to error. Use of the arithmetic mean as a measure of central ten- dency. Applying the criteria of the essentials of a valid average to the arithmetic mean we find that it outranks all the others as a sound measure of central tendency. It conforms to all the stated properties for a desirable mean as listed above. It is definitely and numerically de- fined; is based on all the measures; is popularly known and commonly used, hence will always be readily grasped by the lay reader; is very easily calculated (in this it ranks high as a mean, e.g., the short method of computing the mean is also a necessary step in the determination of the standard deviation and of the correlation coefficient) ; the. aggregate and the number of cases are sufficient to enable the compu- tation of the mean, i.e., the specific individuals do not need to be treated; and in adaptation to algebraical treatment it has a great advantage over the other means. For exam- ple, important properties of this mean are : — 1. The algebraic sum of the deviations from the arithmetic mean equals 0; 2. The average of a series may readily be expressed in terms of the means of component parts of the series. From this it can be deduced that the approximate value of a mean in a fre- quency distribution is the same whether we assume that all the values in any class are identical with the mid- value of the class-interval, or that the mean of all the values in the class is identical with the mid-value of the class-interval; 3. The mean of all the sums and differences of corresponding measures in the two series (of equal number of measures) is equal to the sum or difference of the means of the two series. THE METHOD OF AVERAGES 147 The arithmetic mean is also characterized by the fact that the sum of the squares of the deviations of measures from the mean is a minimum. The arithmetic mean has properties of fundamental importance in the field of math- ematical statistics, especially in connection with the theory of errors and the theory of probabihty (e.g., the arithmetic mean can be shown to be the most probable value of a series of measures). Accidental errors of observation tend to neu- tralize each other around the arithmetic mean. The error of the average is considerably smaller than the error of a single measure, and the accuracy of the arithmetic mean varies directly with the square root of the number of the measures. The median and mode have no similar properties. ILLUSTRATIVE PROBLEMS* 1. Find the arithmetic mean and the median of each of the following dis- tributions. In the computation of the mean use the short method. 1. Achievement of 5th Grade 2. Distribution of Monthly Pupils in Spelling 25 Words FROM Column L of Ayres's "Scale for Measuring Spell- Salary paid to Teachers OP English in 149 Kansas High Schools ING Ability No. words Fre- spelled quency correctly (/) 25 7 24 5 23 11 22 14 21 21 20 13 19 8 18 7 17 5 16 3 15 5 14 3 13 N = 2 Monthly salary (/) $120.0-124.99 3 115.0-119.99 110.0-114.99 1 105.0-109.99 1 100.0-104.99 2 95.0- 99.99 90.0- 94.99 3 85.0- 89.99 10 80.0- 84.99 30 75.0- 79.99 36 70.0- 74.99 31 65.0- 69.99 20 60.0- 64.99 8 55.0- 59.99 1 50.0- 54.99 N = 1 * These illustrative problems are quoted from Rugg, H. O., Illustrative Problems in Edu- cational Statistics, published by the author to accompany this text. (University of Chicago, 1917.) 148 STATISTICAL METHODS 3. Achievement op Pupils in SOLVING Problems of Test 3, "Standardized Tests in 1st Year Algebra " No. prob- lems solved correctly / 21 3 20 5 19 3 18 11 17 16 16 21 15 29 14 20 13 17 12 10 11 5 10 3 9 7 8 3 7 N = 2 4. Distribution of Monthly Salary paid to Teachers OF Science in 147 Kansas High Schools* Monthly salary (/) $135.0-140.00 1 130.0-134.99 3 125.0-129.99 4 120.0-124.99 4 115.0-119.99 2 110.0-114.99 10 105.0-109.99 7 100.0-104.99 26 95.0- 99.99 8 90.0- 94.99 16 85.0- 89.99 22 80.0- 84.99 15 75.0- 79.99 15 70.0- 74.99 5 65.0- 69.99 4 60.0- 64.99 N = 2 * Data from Monroe, W. S., Cost of Instruction in Kansas High Schools. CHAPTER VI THE MEASUREMENT OF VARIABILITY Second Method of describing a Frequency Distribution Need for measures of variability. It has been pointed out in the last chapter that the average of a distribution cannot possibly completely represent the measures of the distribu- tion. At best, it is but a partial measure of type, arbi- trarily selected to represent central tendency. We have indicated with what relative degree of success the different averages do this. Frequently the student of education will have to compare two distributions in which the average is closely the same, but in which the FORM of the distribu- tion is very different. This calls attention to the need of interpreting our data only after careful examination of both the entire distribution and the frequency polygon plotted from it. For example. Diagram 22, Chapter V, is drawn to repre- sent the achievement of two classes. The average, as shown by the arithmetic mean or the median, is the same in both distributions. If one should compare the two distributions on the basis of the average achievement alone his interpre- tation concerning the outcomes of teaching in the two classes would most certainly be wrong. This is evident by a study of the characteristic differences in the two fre- quency distributions: (1) the "RANGE" in the one case is nearly as long as in the other; (2) on the other hand, the measures in one distribution are very much more concentrated near the middle of one group than of the other. One 150 STATISTICAL METHODS could say, for example, that the "middle half'* was dis- tributed over a portion of the scale not much more than half as large in one case as in the other. This certainly means that the teaching has in one case served to develop a rather compact group, that is, teaching emphasis has been so distributed that differences in achievement have been largely smoothed out. In the other case the teaching has resulted in a widely scattered group, certainly calling for reclassification of pupils in connection with any further learning in that subject. In pointing out the characteristic differences between such distributions we make clear the kind of measure that is needed with which to supplement the use of an average. We need some measure which will indicate the degree to which the measures are concentrated around the average, or — to express it another way — a measure which will point out concretely the degree to which the measures vary away from the average. That is, we need measures of variability or dispersion. Variability a distance on a scale. We found that a measure of central tendency, such as an average, is always expressed as " position," — as a point on the scale. We now find that with symmetrical distributions, a measure of variability is always expressed as that distance on the scale, which includes a particular proportion of the measures in the distribution. Although educational distributions are not perfectly sym- metrical, it will be a helpful device for pointing out the de- gree of concentration or lack of concentration of the meas- ures to say: ** approximately such cl proportion of measures is included between such unit distances on the scale. '^ We have already emphasized the importance of the term "unit" and " scale." The student now will find that his measure of va- riability is nothing but a unit distance on the scale. Of the different unit distances that we have for measuring v^tria- MEASUREMENT OF VARIABILITY 151 bility, each includes a certain proportion of the measures under the frequency curve. Four measures of absolute variability. For example, the four measures of absolute variability that we use, and the approximate proportion of the measures included within their limits, when laid off on the scale, are: — 1. The range: includes all of the measures in the distribu- tion. 2. The quartile-deviation or median-deviation: when laid off on each side of the average: includes only roughly half of the measures. 3. The standard deviation: when laid off on each side of the average : includes approximately the middle two thirds of the distribution. 4. The mean deviation: when laid off on each side of the average: includes approximately the middle half of the measures. Unit distances with normal and skewed distributions. We should emphasize the fact here, that with distributions that are not perfectly symmetrical (e.g., Diagram 27), we are able to state the proportion of measures included by these unit distances on the scale only very approximately. With a sym- metrical distribution, for example, the "probability curve'' shown in Diagram 28, we are able to state the proportion of measures exactly. Diagrams 27 and 28 illustrate this dis- tinction, and, although we shall take them up more thor- oughly in Chapter VIII, we. may point out the important features here. On these diagrams are indicated graphically and literally the chief characteristics of these measures of variability. It will be evident to the reader that with the perfectly symmetrical distribution. Diagram 28, any unit distance may be laid off from the average (in this case arithmetic mean, median, and mode coincide) either way and include 4' ^.^ t kg S 114! > P V II 2 ^ EH H < M >.^ W - w o H O MEASUREMENT OF VARIABILITY 153 the same proportion of cases. Thus, it will be shown in Chapter VII that between the curve, the base line, and ordi- nates erected at a unit distance from the mean called the Probable Error (P.E.)y 25 per cent of the measures is in- cluded, or 50 per cent between P.E. and the curve and the base line. In this case it is clear that on the " Normal Curve'' the quartile deviation (defined as half the distance Se/n^ee-o o one/ 2 /9^^^/./3 % one/ 3 /?£ =a '^7 BS ^ / >=i£. 2/^^. Diagram 28. To illustrate the Use of ''Standard Deviation," = 51 10.0- 14.99 3 5.0- 9.99 3 0.0- 4.99 2 , iV = 300 iV/4=75 1. Divide A^ by 4. 300/4 = 75 measures. 2. For Q3, there are 61 measures above 70.0. We need 75-61, or 14 measures from the 26 in class-interval 65.0-69.99. 3. Therefore Qa = 70.0 - 14/26 X 5 = 70.0 - 2.692 = 67.308. 4. Similarly for Qi; since there are 51 measures in the intervals 0-4.99 to 30.0-34.99 inclusive, we need 75-51, or 24 measures from the 44 in class-interval, 35.0-39.99. 5. Therefore Qx = 35.0 -|- 24/44 X 5 = 35.0 + 2.727 = 37.727. 6. Therefore since Q = Qz- Qu we have 67.308 - 37.727 = 2 29.581 = 14.791. MEASUREMENT OF VARIABILITY 159 Properties of the quartile deviation. On account of the simple meaning of the quartile deviation, it is a good meas- / ure of variability to use in presenting facts for the lay reader. s\ It further has the advantage of being the most easily com- puted of any of the measures of variability. In brief, Q is the inspectional or approximate measure to use in expressing variability, in the treatment of any data in which numerical precision is not necessary, or where the theory does not imply algebraic treatment. There are many opportunities to-day, in educational research, to use the quartile deviation as a measure of variability. 3. The Mean Deviation What the mean deviation is. We pointed out that, strictly speaking, the quartile deviation is not a measure of deviation from a particular average. Expressed in another way this means that the quartile deviation takes but indirect account of the form of the frequency distribution, — of the relation between the values of particular measures and the frequency of their occurrence. There are two measures of variability that do this, however: the mean deviation, and the standard deviation. They differ only in the fact that in the former case simple deviations are averaged without regard to sign, and in the latter case the deviations are averaged after each has been squared, with the necessary subsequent step of extract- ing the square root. The mean deviation of a series of measures is the arith- metic mean of their deviations from a selected average (median or arithmetic mean) the deviations being summed without re- gard to sign. In the computation of an average deviation, the taking account of signs would result in a fictitious measure of deviation, the difference between positive and negative de- viations being always very small, and equal to when the deviations are taken from the arithmetic mean. Therefore, 160 STATISTICAL METHODS to average simple deviations we are forced to disregard signs. From the practical standpoint, the deviations may be taken from either the arithmetic mean or from the median. The computation in columns (3) and (4), of Table 24, show that in the case of that simple series, fairly uniformly distributed as it is, the mean deviation computed from either average is the same to the second decimal place, 9.04. This will be true also with the data grouped in the symmetrical distri- bution, and with those not more than moderately unsym- metrical in form. From the theoretical standpoint, however, it is proper to take the deviations from the median, for that is the point on the scale about which the mean deviation is the least. Because it is much simpler of computation, and because of this mathematical relation, the recommendation is made that the student adopt the practice of computing deviations from the median. Computation of the mean deviation : (a) data ungrouped. Let us list the steps in the computation when the data are ungrouped. Each step is illustrated by the data of Series II, Table 24: — 1. Compute the median: 80.5. 2. Compute the deviation of each measure from this value, 15.5, 14.5, 13.5, etc., in column 3. 3. Sum these deviations: 235.0. 4. Find the arithmetic mean of the deviations by dividing the sum, 235, by the number of measures, 26, giving the mean deviation, 9.04. Column 4 gives corresponding deviations from the arithmetic mean, 79.97, a total of 235.06, and the same value for the mean deviation, = 9.04. Computation of the mean deviation : (6) data grouped in the frequency distribution. The computation still may fol- low the steps given above, which may be called "the long method." The work given in Table 26 illustrates this method. MEASUREMENT OF VARIABILITY IGl Table 26. Disthibution of Marks given to 289 High-School Pupils in Latin. To illustrate Computation of Mean Deviation by Long Method Mid- Class-interval j)oint of class-in- terval m Frequennj f Deviation d fd 95.0-100 97.5 22 13.12 288.64 90.0- 94.99 92.5 68 8.12 552.16 85.0- 89.99 87.5 51 3.12 159.12 80.0- 84.99 82.5 28 -1.88 52.64 75.0- 79.99 77.5 47 ■ -6.88 323.36 70.0- 74.99 72.5 33 11.88 392.04 65.0- 69.99 67.5 21 16.88 354.48 60.0- 64.99 62.5 9 21.88 196.92 55.0- 59.99 57.5 6 26.88 161.28 50.0- 54.99 5^.5 2 31.88 63.76 45.0- 49.99 47.5 1 36.88 36.88 40.0- 44.99 42.5 1 41.88 41.88 iV = 289 2623.16 N 2623 16 ^=144.5 M.D.= Z^ 2 289 True median = 84. 38 = 9.08 ' To make use of this long method necessitates a large amount of computation. The arithmetic labor may be cut down very materially by using the principle employed in the short method of computing the arithmetic mean. To do this here would involve these fundamental steps : — 1 . Compute the total deviations about an assumed median. This can easily be done by taking the assumed median at the mid-point of the class-interval which contains the true median. 2. Correct these total deviations about this assumed me- dian by an amount equal to the difference between the devia- tions about the assumed median and the total deviations about the true median. 162 STATISTICAL METHODS It will be shown below that the sum of the deviations about any assumed median must always be less than the sum of the deviations about the true median. Hence, whatever be the relative position of the assumed and true medians, the correction of the deviations around the assumed median to the true median must always be added. Let us contrast in Table 27, the computation by the short method with the correction applied in this way, and with the reduction to class-intervals of one unit each, but first, with the deviations stated in their true value, 0.62, 1.62, 2.62, etc., instead of 1, 2, 3, etc. We can then go, second, a step farther and compute the deviations in terms of units of 1, 2, 3, etc., and correct once for all by the short-method stated below, as in Table 28. Table 27. To illustrate the Computation of Mean Devia- tion WITH Deviations stated in True Values, but in Units of Class-Intervals True-deviation of mid-points Class-interval / in units of class-intervals d' Id' 95.0-100 22 2.62 57.64 90.0- 94.99 68 1.62 110.16 85.0- 89.99 51 .62 31.62 80.0- 84.99 28 - .38 10.64 75.0- 79.99 47 -1.38 64.86 70.0- 74.99 33 -2.38 78.54 65.0- 69.99 21 -3.38 70.98 60.0- 64.99 9 -4.38 39.42 55.0- 59.99 6 -5.38 32.28 60.0- 54.99 2 -6.38 12.76 45.0- 49.99 1 -7.38 7.38 45.0- 44.99 1 -8.38 8.38 True median = 84. 38 iV = 289 N 524.66 289 d'= Sd'=524.66 = 1.816 = ikf.Z>. in units of class-intervals 1.816X5=9.08 = M.Z). in actual units MEASUREMENT OF VARIABILITY 163 Table 28. To illustrate the Computation of Mean De- viation BY Short Method. Deviations taken about the Assumed Median, in Units of Class-Intervals Class-interval / d fd 95.0-100. 22 3 66 90.0- 94.99 68 2 136 85.0- 89.99 51 1 51 80.0- 84.99 ,28:' 75.0- 79.99 47 1 47 70.0- 74.99 33 2 66 65.0- 69.99 21 3 63 60.0- 64.99 9 4 36 55.0- 59.99 6 5 30 50.0- 54.99 2 6 12 45.0- 49.99 1 7 7 40.0- 44.99 1 8 8 iV = 289 True median = 84. 375 Assumed median = 82. 50 c = 1.88/5 = .38 S/d = 522 Total correction = c, difference above and below true median = .38X7=2.66 Total deviations in units of class intervals, i.e., S/i = 522 + 2.66 = 524.66 M.D. S/tf_ 524.66 N~ 289 1.^16X5 = 9.08 = 1.816 Diagram 29 presents several of the class-intervals in en- larged form, together with the relative position of the true and assumed medians, and the relative sizes of the true and calculated deviations. The student is reminded again of the necessity for doing his thinking in terms of the scale of the frequency distribution. The diagram makes it clear that the deviation of each measure in any class-interval, when taken from the assumed median (a mid-point of a class-interval), is in error by that part of a class-interval that separates the true and assumed medians. For example, 164 STATISTICAL METHODS each of the 28 measures in class-interval 80.0—84.99, as- sumed to have a deviation of 0, actually has a deviation, from the true median (T.ifd.), of — .38 of an interval; each of the 51 measures in interval 85.0-89.99, similarly taken at C/055 /ntervaJs S5.0 -4 9ao Value of 650 - 600 - ISO _] ?s F/Teouency 68 *87.5 Trc/eDci^/of/on /n un/f3 of C/oss /nfcTfo/s c/=/ CZ S/ 1 ne'e// an -2S- c/=.6Z '7Zf TOO ^72.5 -?7 33 -^^''-^-> = ^'+^--^^'^ Q Q 180 STATISTICAL METHODS ILLUSTRATIVE PROBLEMS* 1. Find the quartile deviation, the mean deviation, and the standard deviation for each of the frequency distributions reported in the "illus- trative problems" of Chapter V. 2. Find the coefficient of variation for each of these problems by the Pearson formula and by the Thorndike formula. Given for Four Distributions Arithmetic Mean . . . Distribution A Number of words read per 3.9 1.4 Distribution B Percentile marks given pu- pils in drawing 77.8 19.3 Distribution C Number of arith- metic problems solved per min- ut? 12.4 2.9 Distribution D Marks given pupils in math- ematics 76.9 11.3 Questions: 1. In which of these distributions is the variability greatest? 2. Which may be compared directly by means of their measures of absolute variability? 3. Why? * Quoted from Rugg, H. O., Illustrative Problems in Educational Statistics, published by the author to accompany this text. (Univeisity of Chicago, 1917.) CHAPTER VII y THE FREQUENCY CURVE Third Method of describing a Frequency Distribution Summary of preceding work. We have been continually trying to find the best methods of describing a frequency distribution. We have tried the use of the *' range," or the distance on the scale between the lowest and highest values. It was noted that this number depends solely on two values of measures which are subject to great fluctuation, namely, the largest measure and the smallest measure. We have tried to typify distributions by various '* averages," but it was shown again that either the arithmetic mean, median, or mode can but partially describe the distribution. In other words, two distributions may vary widely in the way in which the measures are concentrated or scattered along the scale, at the same time that they present exactly the same "aver- age." So we have turned to the method of variability, and have discussed the use of measures to represent the amount of this "scattering" or "bunching" of measures. It was shown that a fairly adequate numerical representation of the two distributions in question could be obtained by giving both the average and the variability {e.g.., by the arith- metic mean and the standard deviation, or the median and the mean deviation, or the median and the quartile devia- tion, etc.). These could be supplemented, in cases where the units of the scale of the two distributions were differ- ent, by a coefficient of relative variability. Our sole aim in treating educational data by any of these devices is to organize a complex mass of material in such a 182 STATISTICAL METHODS way as to facilitate clear educational interpretations. It seems quite clear that the mind finds it difiicult to deal with whole frequency distributions, or with the original ungrouped measures themselves. The *' average'* and measure of "variability" help to condense the material and aid in in- terpretation. It was pointed out in Chapter IV that thor- ough use can be made of such measures only by the most experienced manipulator of statistical methods; that the student needs other methods of representing facts. It was shown that probably the greatest aid to sound interpreta- tion of statistical data can come from the graphic presenta- tion of the facts in question. Smoothing frequency polygons to approximate ideal ** distributions." It is suggested that at this point the student review the discussion of methods of plotting educa- tional data in the form of the frequency polygon and the column diagram (Chapter IV). It was emphasized there that, although we actually deal with but a small proportion of the total population of measures similar to those in ques- tion, our desire for educational interpretations of the data leads us to speak in terms of the frequency curve which is be- lieved to typify the law underlying our distribution. To be concrete : — Diagram 30 reveals clearly that it is drawn to represent a limited number of measures. If we had had an infinite number of measures, and the size of the class-intervals had been *'very small," the polygon of Diagram 30 would have become a *' smooth" curve, perhaps somewhat like those in Diagrams 30 and 31. The matter can be more clearly ex- plained from Diagram 31. Assume that we can refine our measuring scale so as to get class-intervals of, say, tenths or hundredths of a unit, instead of 5 units. Furthermore, assume that we increase the number of measures from 303 to some relatively large 6 11 16 21 26 31 36 41 46 51 56 Percentile Scores in 61 66 Imagery 71 76 81 86 91 96 45 40 35 30 25 20 15 10 'ft F iff- 2. 6- 1 6 H 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 Percentile Scores in Imagery Diagram 30. Frequency Polygon (Fig. 1) and Column Diagram (Fig. 2) to represent Distribution of Abilities of 303 College Students in Visual Imagery (Data in Table 30.) 184 STATISTICAL METHODS number, say 3000, or 30,000. The base of each rectangle be- comes ** infinitely " small, and the number of cases tends to be more continuously scattered. Thus we find that our '* rec- tangular histogram " approaches '* as a limit," some smoothed curve, perhaps having specific mathematical properties and capable of leading to generalized interpretations which the very particularized histogram does not. We say, — the column diagram represents the actual situation with this particular *' sample of 303 cases " ; the smoothed curve repre- sents what would be the most probable value of the measures at various points on the scale if we took the entire group of measures (from which we actually have but a small sample). It is very clear that a "law" could not be represented by the polygon or column diagram, but that the most probable definite curve must be sought to represent it adequately. The ** smoothing " process. Since in educational re- search we cannot work with all the cases in the entire popu- lation, we may be interested in '^smoothing** our polygons or column diagrams to approximate the ideal situation as far as possible. This can be done roughly by working on the assumption that the most probable value of a series of measures is the arithmetic mean of the series of values. This hypothesis can be applied to our problem of ** smoothing " by taking the arithmetic mean of small groups of adjacent measures on the scale. Thus if we let A, B, C, D, E, F, etc., be the actual values of the midpoints of the intervals, we may average the number of cases found at each three adjacent points by the formula: — 2A + B Smoothed value oi A = Smoothed value of B = Smoothed value of C = 3 A + B+C 3 B+C+D 2nd' 1st p. Smoothing /inVervals^ ." CO to - 5.99 •^ i!! «" 6-10.99 S S " 11-15.99 ^ Si- Pol:: 16-20.99 g K S 21-25.99 to. to to '^ i^ ^ ^ 26-30.99 CO. 1 ^ £2 31-35.99 1 3 n 36-40.9^ •^ fe §41-45.99 P ^ '^ 46-50.99 1 ^ e:51-55.9'9 §• 2 S S 56-60.99 1^ t^ -=« 61-65.99 ^■ <• en en 66-70.99 i:i •^ -^ « 71-75.99 •^ •« »« 76-80.99 00. •« 1^ *« 81-85.99 ii^ SJ ^ 86-90.99 JO,. U ^ - 91-95.99 43.35 43.36 43.35 Mean g 42.95 42.8 44.35 Median 48.2 48. 46. Mode ■ 1 7 A \ ^ ■ \ ■ V V • ^ '>>. ■ •iv^ r-^ ■ ^ - )\ / - / ^ ^ ■ ^ - . — >'' ^^ ■ l>^ y 7 / 1 II s ^ 7 rr Si < § 1 f p << ■S-? 1' ■ 1 :3 8 0! 3 3 a Diagram 31. Same Data as est Diagram 30, Comparison of "Actual' Frequency Polygon with Result of First and Second "Smooth- 186 STATISTICAL METHODS etc., throughout the series. It is seen that the "true" value of each point on the scale (except the two extreme values) is taken equal to the arithmetic mean of its value and the two adjacent values. In the case of each of the extreme measures, it is weighted by 2 and averaged with the adja- cent measure. The result of such a scheme of approximation is seen in Diagram 31, applied to the distribution of Table 30. It is sometimes necessary to repeat the process of smoothing several times. This will be true especially in those distributions revealing sharp irregularities. It is clear that in most educational distributions these irregularities or "peaks" in the curve will be explained either by scarcity of number of cases, or by lack of refinement in the process of measurement. The numerical and graphic results of the first and second "smoothings" are shown in Diagram 31. It should be noted that smoothing by this method will not change the arithmetic mean of the whole distribution. On the other hand, it may affect the median or mode considera- bly. The results of the different smoothings reveal that beyond a particular repetition of the process but little is gained in the way of smoothed refinement. I. Ideal Frequency Curves School-marking distributions. Fundamentally necessary to the advancement of all phases of school practice is ade- quate knowledge about the intellectual and physical capaci- ties of school children. The design of a course of study, planning of teaching methods, adapting of all such phases of school machinery as grading and classification of children, their promotion from grade to grade, marking systems, — all these questions rest back upon the possibility of being able to picture completely the distribution of abilities in our THE FREQUENCY CURVE 187 fioo.. Soo.. school population. For example, the design of a marking system, or of standardized tests for the measurement of ability in any subject of study, must rest upon clear-cut hypotheses as to the distribution of ability in the school popula- tion in question. Let us take a con- crete example, using data in the situation represented by Dia- gram 32; this gives the actual distribution of 5714 pupil marks in 15 high schools in plane geometry. The curve shows that over 30 per cent of the pupils were classed as being 90 per cent in ability or above, i.^., in the top fifth of 5 groups of ability. We are at once skeptical of the accuracy with which the teachers have judged the abilities of these pupils, all the more so when we note that the curves are concentrated at 75 and 90 and when we find that these points on the scale repre- sent the passing and exemption marks respectively. On comparing our data with those in Diagram 24 we are convinced that the marking machinery does not represent accurately the abilities of pupils. Here, we note that, as the result of careful testing of intelligence, arithmetical ability. Diagram 32. Distribution of 5714 Marks GIVEN IN Plane Geometry in Fifteen High Schools Compare this Diagram with Diagram 24. 188 STATISTICAL METHODS stature, and other anthropometrical measurements, the top fifth of our pupils is surely not more than 6 to 10 per cent of the total group. Certainly there is no reason to believe that even our high-school population is so badly "skewed" in ability that nearly one third falls closely together in the top fifth of the scale. Now, the administration in this particular system has rec- ognized recently that its marking is not fitted to the capaci- ties of pupils, and has faced the very real question: "With what relative frequency should pupil-ability be distributed in the various fifths of the marking scale .^ What per cent of the total group actually merit A, B, C, D, E?" To an- swer this question fully, this superintendent needs de- tailed objective evidence on the distribution of similar high- school pupils in large numbers. If he could secure it he would be perfectly justified in educating his teaching staff to the point where it would measure pupils' abilities roughly in accordance with this objectively-obtained distribution. The distribution of human traits. Complete figures on the abilities of high-school pupils are lacking, but he has avail- able many measurements on human intelligence, various mental traits, and a vast amount of evidence concerning the distribution of anthropometrical measurements on human beings. The student w^U be interested to note with what striking regularity they resemble a fairly symmetrical curve. In all such distributions, the measures are largely concen- trated very near the middle of the scale. Furthermore, they shade off in both directions from the middle high point, — the mode, — somewhat symmetrically. The student will note, furthermore, that in the case of those traits which are more subject to refined measurement, — e.g.^ heights of men, strength of grip, cephalic index, chest measurement and other physical measurements, and fairly refined psycholog- ical measurements, the curves the more closely approximate THE FREQUENCY CURVE 189 symmetry. In addition, we see that in those cases where very large numbers of measurements have been taken, as in Diagram 24, Fig. 4, heights of men, the curve strikingly approaches this symmetrical type. A century ago, the regularity of this accordance of the dis- tribution of human traits with definite symmetrical curves was noted by various observers. Quetelet, the Belgian scientist, made many such measurements, and early called attention to the recurring conformance of the shape of the curve of human measurements to the chance polygon got by plotting the coefficients of the separate terms in the binomial expansion. Especially close is the *'fit" in the case of such physical measurements as stature and girth of chest. Laws of nature show continuous distributions. With the agreement upon the shape of the distribution curve of hu- man traits there came a recognition of the need for the definite establishment of ideal curves which could be used in the case of interpretation of fairly limited numbers of ob- servations or measurements. Science demanded a means of generalization — a method of expressing *'the law.*' More and more they commented on the fact that laws of nature, as generalizations based on human experience, were inter- preted only in terms of continuous distributions. The dis- tribution of human measurements was checked further by the distribution of ** errors of observation" in refined meas- urement, — e.g., astronomy, surveying, etc. The plotting of such, refined measurements gave a distribution resembling, in a rather striking way, the shape of the curve of distribution of human traits, concentrated near a mode about the middle of the range, sloping off quite symmetrically in both direc- tions, and showing relatively few cases at the extremes. If the errors be plotted with the error at the middle and posi- tive and negative errors plotted on either side of this point, this may be interpreted partially by saying : first, that very 190 STATISTICAL METHODS small errors are most common (the error **zero" is really- most common) ; second, that positive and negative errors are about equally frequent; and third, that very large errors do not occur. This may be illustrated by a brief quotation from Merriman's Method of Least Squares (p. 13) : — For instance, in the Report of the Chief of Ordnance for 1878, Appendix S', Plate VI, is a record of one thousand shots fired deliberately (that is, with precision) from a battery-gun, at a target two hundred yards distant. The target was fifty-two feet long by eleven feet high, and the point of aim was its central horizontal line. All of the shots struck the target; there being few, however, near the upper and lower edges, and nearly the same number above the central horizontal liue as below it. On the record, horizontal lines are drawn, dividing the target into eleven equal divisions; and a count of the number of shots in each of these divisions gives the following results : — In top division 1 shot In second division 4 shots In third division 10 shots In fourth division 89 shots In fifth division 190 shots In middle division 212 shots In seventh division 204 shots In eighth division 193 shots In ninth division 79 shots In tenth division 16 shots In bottom division 2 shots Total 1000 shots It will be observed that there is a slight preponderance of shots below the center, and there is reason to believe that this is due to a constant error of gravitation not entirely eliminated in the sight- ing of the gun. The distribution of the errors or residuals in the case of direct observations is similar to that of the deviations just discussed. For instance, in the United States Coast Survey Report fbr 185Ii. (p. 91) are given a hundred measurements of angles of the primary triangulation in Massachusetts. The residual errors (art. 8) found by subtracting each measurement from the most probable values are distributed as follows: — THE FREQUENCY CURVE 191 Between + 6.0 and + 5.0 1 error Between + 5.0 and + 4.0 2 errors Between + 4.0 and + 3.0 2 errors Between + 3.0 and + 2.0 3 errors Between + 2.0 and + 1.0 13 errors Between + 1.0 and 0.0... 26 errors Between 0.0 and —1.0 26 errors Between —1.0 and —2.0 17 errors Between —2.0 and — 3.0 8 errors Between —3.0 and — 4.0 2 errors Total 100 errors Here also it is recognized that small errors are more frequent than large ones, that positive and negative errors are nearly equal in number, and that very large errors do not occur. In this case the largest residual error was 5.2; but, with a less precise method of observation, the limits of error would evidently be wider. The axioms derived from experience are, hence, the following: — 1. Small errors are more frequent than large ones. 2. Positive and negative errors are equally frequent. 3. Very large errors do not occur. II. The Normal Probability Curvb Resemblance of actual distributions to " chance " dis- tributions. Enough has been said to point out the very practical need in all the sciences for a distribution curve, from which generalizations could be made. It was early recognized by these workers that their distributions resem- bled in a striking way the shape of the frequency polygons obtained by plotting the frequency of various *' chances." Since the manipulation of the mathematical properties of the distribution of '* chance" leads to the ideal curve which we are seeking, we shall next turn to a very elementary dis- cussion of ** chance" and the ** probability" curve. Before doing so, let us state clearly the ultimate goal of the student of educational problems, in seeking an ideal curve against which he can check his actual distribution and from which 192 STATISTICAL METHODS he can generalize his experience. Expressed briefly, it is this : — 1. Knowing that human traits distribute closely enough for practical purposes in accordance with a particular ideal distribution, we wish to be able to locate easily the propor- tion of our total group (assuming it to be reasonably large) that should fall between any two points on the scale of our measurements. Concretely, our superintendent named above, wishes to know about how large a group of his pupils should get A's, B's, C's, D's, and E's. He also wants the process of this determination reduced to a minimum of arithmetic labor. In other words, our theory should lead to the preparation of tables by which the student can compare, easily and yet accurately, actual with ideal distributions. 2. Another important goal of the student of education in dealing with *' probability " is found in connection with his very real need for being able to establish the reliability of his data. He is measuring a relatively small "sample" of the total group, and has computed averages, measures of varia- bility, and perhaps of relationship. What dependence can be placed on the representativeness of the small sample? If he took other succeeding samples, would his measures of type and variability be practically what he has already found? Or can he feel assured that they would fluctuate much, and hence that from his data he can make no sound interpretation? It should be stressed here that adequate educational interpretations of the results of research must rest upon careful determination of the reliability of measures which have been computed. These two important needs of the student of education reveal the need for carefully ac- quainting ourselves with the way in which the "probabil- ity " distribution is found. We have pointed out that human traits are "combina- tions " and include many "arrangements" of a vast number THE FREQUENCY CURVE 193 of separate causes which may be assumed to be independent of each other. In deriving a theoretical curve of distribution for a set of many independent causes, we must recall the mathematical result obtained by combining and arranging such groups of causes. It is of interest to note that the results of such combinations accord so closely with certain mathematical schemes, namely, those of permutations and combinations, which, working under the laws of probability, may be studied and whose conclusions may be applied to the interpretation of our data. We shall next show the resemblance between the results of combining various arrangements of large numbers of inde- pendent causes and the straight mathematical theory of permutations and combinations. This leads to a statement of principles of GROUPING called combinations; and ar- rangements of same group or combination called permuta- tions. Use of permutations and combinations. From our ele- mentary algebra we will recall that, with a given number of things we can make only a definite number of groupings or "combinations," each combination of things being different from any other. For example, let ct, 6, c, d, represent four things. We may make four, and only four different combi- nations of these four things when we take them three at a time, namely: — ahcy abdf acd, bed. If we take but two at a time we can make six, and only six different combinations, thus : — ab, ac, ad, be, bd, ed. If we take four at a tirrfe, but one combination is possible, ahcd. Now, with each of these combinations we may make two or more arrangements or permutations. The permutation is determined by the order in which the things stand. For 194 STATISTICAL METHODS example, with any such combination as abcy we may make 6 permutations : — abc, acb, hoc, bca, cab, cba. Each thing here is combined with each remaining pair of things. It is seen that the number of arrangements of n things (4) taken r (2) at a time is n{n - 1); i.e., (4.3, or 12) : ab he cd da ac bd ca db ad ha cb dc Take three at a time: — abc bed cda dab ojcb hdc cad dba abd bda cdb dbc adb bad cbd deb acd bca cba dca ode bac cab dac n(n — 1) (n — 2), or (24), and so on. The number of permu- tation of n things, taken r at a time is, therefore, — w^r = 7i(n - 1) (n - 2) (n - r + 1). Thus, since with any given combination of, say, r things, we can combine every thing with every remaining group of things, we can make factorial r permutations of things from the combinations. (Factorial r is written r! or r and means 1, 2, 3, 4 r.) Therefore, as we take one combination after another of r things, with each combination we can make r! permutations. Hence, the total number of permutations of n things taken r at a time is equal to the number of combinations of n things taken r at a time, multiplied by r!, or p C rl P C VI T n r = n r ; or, n r = — - rl THE FREQUENCY CURVE 195 But, n^r = n(n - 1) (n - 2) (n - r + 1) n n{n- \) {n-St) {n -r -\- \) r! We have said that '* law " is but man's generalization from his experience. We are interested in seeing now in what way- he can check his experience against regularity of mathe- matical order. The above formula for the number of combi- nations of n things taken r at a time now enables us to fore- tell, in the case of a given number, n, of independent events working under ideal conditions, what is the probabihty of a stated number of them, r, happening or failing to happen. To illustrate the operation of the principle let us take the case of coin tossing, assuming a coin to be a homogeneous disc and equally likely to fall heads or tails. Suppose we throw out four coins at random on a table. According to the law of combinations and permutations what should be the number of heads or tails turning up when r takes values of 0, 1, 2, 3, 4? There is now a total of 16 possible arrangements of heads and tails. Taking four at a time, say all heads or all tails, we can make buton^ possible combination, ni, ^2, n^y n^'y taking three at a time, say three heads and one tail, or vice versa,>we can make /owr combinations: e.g., — r n{n-\) (n-r+1) 4 3 2 n r = or — — — = 4. . r! 123 Taking two at a time, two heads and two tails, we can make, 43 — = 6 combinations. Since each time we throw out 4 21 coins, it is possible to make these combinations of heads and tails, we can infer that, should we continue to throw, we ought in the long run to stand a chance of getting various combinations of heads and tails in about the ratio of 1, 4, 6, 4, and 1. Now if we plot a polygon, making the 196 STATISTICAL METHODS heights of the ordinates equal, to scale, to these figures, we note that we have a symmetrical polygon, with, it is true, but five ordinates. Let us take a larger number of cases, or coins, say 7. Now *'the chance of getting all heads," i.e., the number of combinations that it is possible to make of 7 things, taking 7 at a time (?2 = 7, r = 7) is 1; the chance of getting 6 heads and one tail at a time is 7, the number 5 at a time, 21, the number 4 at a time, 35, etc. Thus we find that in the long run, our "chances" ought to be about 1, 7, 21, 35, 35, 21, 7, 1. Plotting these "chances " we find a polygon, with more ordinates, a flatter slope to its sides, but still symmetrical in shape. Probability. Our discussion has now turned to the "chance " of this or that happening or not happening. It is possible then to extend our discussion in the form of general statements of probability, and thus establish an expression for the probability of any number of events happening or failing. To do that we must make clear what we mean by 'probability and establish certain fundamental principles. In defining probability we must recall that we are but trying to idealize our actual experience in order that we may establish what would be the most probable condition, in case our actual data could be made infinitely extensive. To take a familiar actuarial illustration first: what is the probability that a particular child will not live to be 21 years of age? We are forced to turn at once to the actual experi- ence of the human race under similar conditions. That is, we will find out what proportion of children actually have not lived to be 21, say 20 per cent or 1 out of every five. We idealize this experience by saying that since 1 in every 5 of a very large number of children fails to live to the age of 21, the probability that a child will fail to do so is l/5. Probabil- ity, then, is evidently to be defined as the ratio between the occurrence of a particular event and the very large group of THE FREQUENCY CURVE 197 events of which it is a part. Or, expressed in another way, it means a number less than 1 — taken to represent the ratio of the number of ways in which an event may happen to the total number of possible ways, — each of the ways be- ing supposed equally likely to occur. ^ For example : if we toss a coin there are two possible ways in which it may come down, heads or tails. Hence the prob- ability of its coming down heads is J, and of its coming down tails is J. The sum of the probabilities is of course unity, the mathematical symbol for certainty. For example, if the probability of hitting a target is 1/5000, the probabil- ity of not hitting it is 4999/5000. Now, if an event may happen in different independent ways, the probability of its happening in either of these ways is the S UM of the separate probabilities. To illustrate: if we put into a bag 12 green, 18 red, and 19 black balls, and draw out a ball, the probability that it will be green is 12/49 (the total number of balls is 49, and there are 12 green ones) ; that it will be red, 18/49; and that it will be black 19/49. But the probability that it might be either black or green will be 19 12 31 49 "^ 49 ~ 49 and the probability that it might be either black, green, pr "^^'^^ 19 12 18 — _I_ __L._ = 1. 49^49^49 If we let P represent the probability of an event happen- ing, and Q that of its not happening, then P=l-QorP+Q=l. Probability in educational research. Now, in our research we are dealing with ''compound" events; i.e., those pro- duced by the concurrence of a very large number of causes, 1 The writer has adapted to his uses here, Merriman's discussion of probabiUty and the binomial expansion in Methods of Least Squares, pp. 6-10. 198 STATISTICAL METHODS assumed to he independent of each other. For example, ** arith- metical ability " in a particular individual, may be said to be a complex resultant of a very large number of causes, e.g., those due to hereditary capacity, physical conditions of growth, and conditions of home and school training; e.g., absence from or regularity in school, outside activities, etc. We cannot isolate the specific unit causes, so hopelessly are they tangled up, but we can measure the effect of the combina- tion of this vast number of separate causes by the objective evidence; i.e., we measure the resultant human trait called, for convenience, ''arithmetical ability." Now it is a safe assumption that these many separate causes are independ- ent of each other, — at least they are not related in any definite way. Human events thus are assumed to be ''com- pounds," analogous in their determination to compound "chance" events of an ideal nature. That they show dis- tributions of somewhat similar shape is very evident from our foregoing discussions. We need statements, therefore, for the generalization of such compound events. What is the probability of the happening of a particular compound event ? The answer must be, — the product of the probabilities of the happening of the separate independent events. For example: if one of two bags contains 8 black balls and 9 red balls, and the other contains 3 black balls and 11 red balls, the probability of drawing 2 black balls 8 3 ' in 1 draw from each bag = — X — . In the same way we A i JLt! may extend this to any number of events. P^P^PzPa is the probability that all of four events will happen, and (1 _ p^) (1 -p^) (1 _ P3) (1 _ PJ is the probability that all will fail. Thus Pi (1 - P,) (1 - P,) (1 - P,) = the probability that 1 will happen and 3 will fail, etc. Probability expression. We are now in a position to es- tablish an expression for the probability of any number of THE FREQUENCY CURVE 199 events happening or failing. Assume "n" events, and as- sume that P + Q = 1. Then, 1. From the above, the probabihty that all events will happen is P1P2P3 ...Pn=P^. 2. The probability that 1 assigned event will fail and (n — 1) happens is P'^~^Q. Since this may happen in "n" ways, the proba- bility that 1 will fail and (n — 1) happen is np''~^q. 3. The probability that 2 assigned events will fail and (n — 2) fiffi 1 ) happen is P^~^Q^; since this may be done in — — — — ways (be- J. . /V cause the coefficient = the probability that 2 events out of the total will fail is ^^^-^^ P"-^). 4. The probability that 3 assigned events will fail and (n — 3) 1, • Dn^n3 O- *U- • ^(^ - 1) (n - 2) happen \s F Q . Smce this occurs m ways (be- cause the number of combinations that can be made of n things , . . n{n — 1) (n — r -{- 1)\ . 1 , .,• ^ taken r at a time is 1 the probabihty that two •11 . .1 , / ^x 1 . '^C'* — 1) (n — 2) _ o o will fail and (n - 2) happen is -^ — —^ p"~^gl Thus, if (P -f- Q)" is expanded by binomial formula, (P + QT = P" + nP^-' Q + ^'^l' ^- P"~V + ... n{n - 1) (n - 2). (n - r + 1) P"-''Q'" + The binomial expansion. But the first term of this ex- pression (called the binomial expansion), P" is the prob- ability that all events will happen; the second term of the expression is the probability that 1 will fail; the third term is the probability that 2 will fail, etc. Thus each successive term in the binomial expansion represents the probability of all events happening, all but one happening, all but two, etc., throughout the series. We thus have a general ex- pression to aid us in^delermining the probable frequency of 200 STATISTICAL METHODS occurrence of compound events contributed to by various assignable causes. To illustrate the method we ordinarily turn to such cases as coin tossing, or dice throwing, in which the chance of an event can be definitely assigned. If the chance of an event happening or failing is known, — as in coin tossing {i.e., if we let p = ^, q = i), n may be assigned any desired value and the separate constituent probabilities figured. For example, if we toss 7 coins, say 1280 times, and record the number of ** heads*' each time, we should get theoretically, from the binomial expan- sion: — 7H 6H 5E JiH SH 2H IH OH 1 ly _ j_ j_ ^ ^ ^ ^ _!L.JL 2"^2y ~ 128 "*" 128 "^128 "^128 "^128 "^128 "^128 128 The degree to which an actual distribution checks the theoretical expansion is shown by the following distribution of heads and tails obtained by 10 students, each tossing 7 coins 128 times. ^ 7 Heads 6H 5H iH SH 2H IH OH 1.1 7.0 21.6 36.8 33.3 20.2 6.9 1.1 It should be noted that in the tossing of the seven coins, nothing is more uncertain than that a particular coin will fall heads, but experience is found to check closely the theoretical statement that, in the long run, coins will fall heads and tails in proportion to the frequencies stated by the terms of the above expression. This checks the point made above, that while we expect great fluctuation in the sizes of par- ticular individuals selected from a total group of measures, if successive groups of considerable size are drawn out we expect constancy of average values. Recall here that we need these statements of probability because we are con- stantly dealing with selected samples of total groups of very ^ Data from H. L. Rietz, University of Illinois. THE FREQUENCY CURVE 201 large numbers, and are forced to make statements about them in terms of the most probable situation. For an ideal case like coin tossing, where P = Q = J, the binomial expansion becomes : — !+!Y.('!Y+,/!Y + *-»^'''" !)■ 2 27 V2y \y 1-2 \^ {n) {n - 1) (n-2) A V 1-2-3 W If n = 4, the probabilities are as stated on page 195. 1 iV _ J_ ± _5. ± _1 2"^ 2y " 16 ^ 16 ^ 16 "^ 16 "^ 16 A lY 1 6 15 20 15 6 1 , If 71 = 8 V2 2/ 25 256 256 256 256 256 256 256 256 256 Probable frequency polygons. In Diagram 33, we give a graphic representation of the distribution of the probable frequency of occurrence of various events when it is possible to assign values to j> and q. The student will perceive that making f and q equal, leads to a symmetrical distribution : with an odd number of terms, there will be one middle term with ordinates distributed symmetrically on both sides; with an even number, — two middle ordinates equal in size. Making p and q equal thus results in symmetrical polygons that seem to approximate the shape of distributions that have been found to fit various human traits. Each of the successive terms in these expansions repre- sents the chance of getting a given "combination*' of causes in contributing to a particular event. To make clearer what we have here, let us plot frequency polygons, as in Dia- 202 STATISTICAL METHODS gram 33. Here, the heights of the ordinates erected at equal intervals on the horizontal line (the X axis) represent, to scale, the relative probability of the various events happen- ^ J k I ! 4 / DiAGRAM 33. Polygons representing the Expansions {\-\-\Y, Height of mean ordinate taken equal to 8 units; other ordinates in proportion to relative sizes of coefficients of the expansions. Abscissae are approximated in length so as to make the polygons for different exponents similar. If the "normal curve" had been drawn the closeness of fit between it and (^ -f- ^)'2 would be evident. ing. For example, in the polygon for n = 8, the height of the extreme ordinate 1, represents a probability, — , th^t, say 256 8 heads (or all of 8 like human causes) might occur to- gether in one throw of 8 coins (or in one sample including THE FREQUENCY CURVE 203 these characters). That is, it is probable that, in the long run, once in 256 times all 8 coins would fall heads; the second Q ordinate 8 indicates that it is probable that, in — ths of ^ 256 the times, 7 will fall heads and 1 tail, etc., throughout the dis- tribution. It can be seen that, as we increase the number of independent contributing factors (n), our distribution con- tinually approaches a smooth curve as a limit. For example, the polygon plotted from the expansion (2 +i)^^ is shown by Diagram 33 to approximate very closely such a ''con- tinuous " curve. It can be seen that further increase of the number of cases refines very little, for practical purposes, the apparent continuity of the distribution. It should be noted at this point that the sum of the heights of all the ordinates in any one of our polygons represents the total number of measures. It also represents the sum of thf separate probabilities, which we found must be certainty, or 1, regardless of the value of n. The practical question^ now arises: How use the polygons plotted from various binomial expansions to help us in in- terpreting our actual frequency distribution? Is it possible to compute the terms of an expansion (and thus plot the polygon) comparable to the distribution of our actual data? We can answer at once : It is possible to do this, but to do so involves both a great deal of arithmetic labor (for example, the computation of many terms of a binomial series) and methods of approximation in computation. To check the interpretation of our data against an ideal frequency curve we certainly need shorter methods than would be involved in the use of "probability polygons." We need, for exam- ple, to replace the polygon by a continuous curve which will ^ For the mathematically trained student it should be pointed out that our binomial expansion is a ease of discontinuous variation, and we need a method of passing from such to a curve representing continuous variation. 204 STATISTICAL METHODS have ordinates approximately the same relative height, and which will be so built that the area between any two ordi- nates, say 2/1 and ?/2» will give the relative frequency of the measures between the two corresponding values of a;, say iCi, and X2. The normal, or probability, curve. The continuous curve which does this is known variously as: the probability curve; the curve of error; the normal frequency curve; the Gaus- sian curve, or the La Place-Gaussian curve, after Gauss and La Place who separately developed the equation for it. We shall refer to the curve hereafter as the normal curve or the probability curve. The equation of the curve is developed by certain investigators in accordance with criteria obtained from the binomial polygon (^ + 2)," and may be stated at once as : — ^ — £2 y = Voe^" In this equation e is the constant 2.71828, known as the base of the Napierian logarithm system, y and x are the two variables, x the distance taken on the base line of the curve from the mean to a given point, arid y the height of the ordinate erected at that point, y^ and a- are two very signifi- cant terms in the equation of the curve, a is the standard deviation of the distribution, which the student has already met in computing variability. Thus if the deviation of each measure is taken from the arithmetic mean, and called d. \ n and in Chapter VI, it was pointed out that it is a unit of » distance on the scale which can he used to describe the relative * In order to make definite use of probability and correlation methods the student will be forced to review slightly his elementary algebra. Chapter IX gives a discussion of equations and their plotting, which may also be help- ful at this point. THE FREQUENCY CURVE 205 amount of variability of the measures around the mean. The student should stress the fact that the unit of variabihty, o-, which he has learned to compute numerically, is exactly the same unit distance on the X axis, as now enters as a measure of variability of x in the equation of the curve. For an adequate comprehension of the graphical signifi- cance of o- the student must study the way in which the equa- tion of the curve is built. Note that any distance on the X axis {i.e., any ** a; ") is measured in units of o-. Familiarity with this is absolutely necessary. Note furthermore that as you let x take various values, y is always expressed as a proportional part of 2/o- That is, when x = o, e^ = 1, and y = yo- Thus y^ is the greatest or- dinate and all of the other ordinates of the curve will be expressed as fractional parts of 2/0. Furthermore the curve is symmetrical about the point x = Oy and the arithmetic mean, median, and mode coincide at this point. The term y^ may also be computed by the equation: N where A^ = the total number of cases, o- is the standard de- viation and TT is the constant 3.1416. Thus the complete equation of the curve, as written by followers of Pearson and measured in units adaptable to the data of educational re- search, is: N z±. y = — 7= e 20-^ This, then, represents the normal probability curve, taken to typify, approximately enough for practical purposes, many human traits in which educationists are primarily interested. Several practical questions must next be answered concern- ing the use of it to such students. 206 STATISTICAL METHODS ILLUSTRATIVE PROBLEMS i 1. Compute the first and second "smoothed" frequencies of the data in Problem No. 4, Chapter IV (distribution of monthly salary paid to teach- ers of science in 147 Kansas High Schools) . Plot a frequency polygon for (a) the original distribution; (6) the first smoothing; (c) the second smoothing. Tabulate the three sets of frequencies below the base line of each graph. 2. (Review problems on graphing). a. Plot frequency polygons for the data of Problems No. 1, 2, and 3 in Chapter IV. Arrange these three polygons on one sheet. b. Plot the data of above problems in column diagrams. Arrange the dia- grams on one sheet, making them as large as possible to fit the sheet. c. Show graphically on each of these graphs the position of the mean, the median and the mode (see computations on original problems) and rep- resent the value of the mean deviation, the quartile deviation, and the standard deviation. 1 Quoted from Rugg, H. O., Illustrative Problems in Educational Statistics, published by the author to accompany this text. (University of Chicago, 1917.) CHAPTER VIII USE OF THE NORMAL FREQUENCY CURVE IN EDUCATION Having established a type or ideal frequency distribu- tion, how may we make use of it? Four definite questions must now be answered : — 1. How is the normal curve plotted in general? 2. How may it be superimposed on any actual frequency polygon to permit of direct comparison of actual and theo- retical distributions? 3. How may the normal curve be used to determine the number or proportion of the individuals that ought to fall between any two selected values; e.g., in the marking prob- lem above, how many pupils theoretically ought to get A, B, C, etc.? 4. How may the curve be used to determine the probable reliability of the statistical results obtained from actual data? i. How to plot the normal curve To plot or graph a curve we need the equation of the curve. Having that given, e.g., 2/ = 4x + 8, our problem consists of three steps : — 1. Solving y for various assigned values of x; e.g.-. eta: = Then y = 1 12 2 16 S 20 4 24 etc. etc. STATISTICAL METHODS 2. Laying off on the axes of X and Y, corresponding values of X and y and plotting the points determined by them. 3. Connecting the points thus plotted, to give the graph of the line (e.g. in Diagram 34 the line ** represents" the equation y = 4a; + 8). To plot the equation of the normal curve — £9 evidently necessitates much more elaborate preliminary computation than is true of this simple illustrative problem. Furthermore, there are evidently two more terms, o- and yo, in the equation that need to have values assigned to them. Since the equation implies that all ordinates to the curve (erected at dis- tances from the mean equal to particular frac- tional parts of 0-) are constantly proportional to a fraction of i/o, our work would be much facilitated if we had a table in which were stated values of the or- dinates to the curve (y's) corresponding to assigned values of x. For example, in Table II it is noted that the ordinate erected at .la- from the mean =.995 of the height of the mean ordinate yoi that at .So- = .95Qyo; that at l.Oo- = .GOGi/o; that at 2.0ar = .1351/0, etc. The computation of values of y, then, for cor- 28 24 20 16 ,12 Y / / / / / / / . X 12 3 4 5 6 Diagram 34. Graph of the Line 2/=4x+8 EDUCATIONAL USE OF FREQUENCY CURVE 209 responding values of x can evidently be done once for all and the results compiled in a table. This has been done, and Table II gives the results. Note carefully that the computa- tion necessitated measuring x in units of o-, and y in units of y^. Recall here that c, the standard deviation of the distribu- tion, is the fundamental unit of distance on the scale (the a--axis); also that the equation of the normal curve is so stated that i/o is the ordinate of greatest height, and that it is a constant. Therefore, the table has been derived by letting o- and 2/0 both equal 1, with consequent values of both x and y represented as fractional parts of o- and 2/0- Steps in plotting the curve. To plot the curve then, our steps of procedure are clear: — 1. Lay off distances on the a;-axis equal to fractional parts of cr, say .lo", .2o-, .3o-, etc., out to, say S.Ocr. Note that the selec- tion of the magnitude of these unit distances is entirely arbi- trary, — hence that the exact shape of the curve will depend upon the units selected. 2. Select a unit of scale for the y's which will give a reason- ably steep curve, and erect at the middle point of the x-axis (i.e., X — 0) an ordinate equal to 2/0 {i.e., equal to 1). Note again that the actual length of yo {i.e., the unit of scale for the ys), is entirely arbitrary. Your aim should be to take such a unit on y, that, in connection with the unit on x, your final curve will be fairly steep. 3. At each of the selected points on the x-axis, .lo-, .2or, .So-, etc., erect an ordinate equal in length to the fractional part of 1/0 that is indicated in Table II. For example, for X = .lo- y = .995 y, ir = .20- y = .980 y. x= .30- y = .956 2/c etc. 4. Connect the tops of the ordinates thus erected, giving the normal frequency polygon desired. The student will 210 STATISTICAL METHODS note that the more closely together the ordinates are taken (i.e., the smaller the fractional parts of o"), the more closely will the probability polygon approach a smooth cm've. £. How to compare an actual frequency polygon with the normal frequency curve We have just seen that to plot a normal curve we need but two items, values of y for corresponding values of Xy and that these may be computed in fractional parts of yo and o-. In order to superimpose a normal curve on an actual fre- quency distribution, so as to permit comparison of the two, it is necessary to find elements common to the two distribu- tions. Examination will show: (1) that o- is common to both, that is, that the standard deviation can be computed and compared for ANY frequency distribution. Hence we can lay off distances on the ic-axis, which have been com- puted in fractional parts of an actually computed a-; (2) we can compute the height of 2/0 for our actual distribution from the formula : — N where N is the total number of measures in our distribution, -n- is 3.1416 and o- is the standard deviation. In addition we find: (3) that the origin of the normal curve, i.e., the point from which we begin to plot measurements, is at the mean' of the distribution. This is another element common to both theoretical and actual distributions, for we can compute the arithmetic mean of the actual distribution. Having 2/0 we can superimpose the two curves by putting the means to- gether, making a: = at the arithmetic mean of the actual distribution. The distances on the a:-scale may now be laid off by multiplying each successive fractional part of o-, say, .lo-, .2or, .3(r, etc., by the computed value of o- in the actual EDUCATIONAL USE OF FREQUENCY CURVE 211 distribution. Then, the length of the ordinates that are to be erected at these points may be obtained by multiplying the fractional part of i/o (read from Table II), correspond- X ing to the selected values of -, by the computed value of 2/0. cr Summary of steps necessary for the comparison of an actual frequency distribution with a normal frequency curve. To bring all the above steps clearly in mind let us list them in definite order: — 1. Plot the actual distribution, by methods already discussed in Chapter IV. 2. Set the mean point of the normal curve at the arithmetic mean of the actual distribution. Call this point a: = 0. 3. Compute unit distances in terms of a, that will be laid off on the a:-axis by multiplying fractional parts of 3o/xip/cyi' /rom tft9 * True <7«'tf /■«^« '73-A.. Diagram 37. "Normal" Distribution of "Errors" in Averages Computed for successive " samples," from " true average " of entire population. (Compare with Table 32.) plotted as in Diagram 37, that a distance on the scale equal to ^M will extend from — .2 to + .2. 68.26 per cent of the prob- able deviations theoretically will be included between ± fTMy i.e., ±.2. It was shown above that 99.73 per cent of the probable deviations will be included between ±3o-^, i.e., between ±.6. This is interpreted to mean that the chances are about 9973 to 27, i.e., about 365 to 1, that the average of any such sample selected at random will fall between 73.2±.6, i.e., between 72.6 and 73.8. (6) Unreliability of a standard deviation. In the same way we may express the probable deviation of a computed EDUCATIONAL USE OF FREQUENCY CURVE 229 standard deviation from the true standard deviation, using the formula: — ^ ^distribution V^N (2) The formula and method of computation may be inter- preted graphically in the same way as before, remembering now that the deviations to be plotted on the scale are prob- able deviations of the observed o-'s from the true o-. (c) Unreliabilty of a difference between two measures. Similarly, the unreliability of a difference between two quan- tities may be expressed in terms of the probable deviation of the true difference from the computed difference. It can be shown that the standard deviation of this probable de- viation of the difference between two measures equals the square root of the sum of the squares of the probable devia- tions of each true measure from its corresponding computed measure. That is, — ^difference between x and 1/ ~ '^ ^M of x "T O'if of y W/ (d) Unreliability of a coefficient of correlation. It will be shown in Chapter IX that the unreliability of a coefficient of correlation is ^deviation in r ~ / — \*) The graphic interpretation will be clear to the student pro- vided it is remembered that the scale of the base line of the curve is now, " deviations in the size of the computed correla- tion coefficient from the size of the true correlation coeffi- cient." B. Statement of Unreliability in Terms of the Probable Error It has been noted that there are two accepted unit measures of variability, the standard deviation (cr) and 230 STATISTICAL METHODS the probable error, P.E. The relation between the two can be shown to be P.E. = .67449 o- (5) This relationship can be made clear by turning to Table III, which states the fractional part of the area between the mean of the normal curve and ordinates erected at distances from the mean equal to successive increments of o-. For example, between the mean and o- will fall 34.134 per cent of the entire distribution. We define the probable error as that unit distance on the scale which, if laid off one way from the mean, will determine one fourth of the cases. Therefore we can determine from the table the fractional part of o- that one will lay oif from the mean to determine 2500/10,000 of the area of the curve. This proves to be .67449 o^, or approximately .6745 o". Because of the "common sense" meaning of the probable error (namely, that distance which if laid off both ways from the mean determines half the cases) it has become custom- ary to express the unreliability of measures in terms of the probable error instead of the standard deviation. Thus the above formulae become : — P.E. arithmetic mean = -^'^'^^^^^ distribution «3) ^/N- P. E. median = .84535o-^,v,^„,i„„ (7) ViV P'E.standard deviation = •67449o-^i,<„-6«(i<,n (8) Valvr P.E.coefficient of correlation = .67449 ^ _ ^2 (9) Viv It is convenient for the student to have in mind the follow- ing table of statements of unreliability of measures. The chances that the true value (of the average, stand- ard deviation, coefficient of correlation, etc.) lies within: — EDUCATIONAL USE OF FREQUENCY CURVE 231 ± P.E. are 1 to 1 (50 per cent of measures fall within ± P.E.) ±2 P.E: are 4.5 to 1 (82.26 per cent of measures fall within ± 2 P.E.) ±3 P.E. are 21 to 1 ±4 P.E. are 142 to 1 ±5 P.E. are 1310 to 1 ±6 P.E. are 19,200 to 1 • Thus, to insure a satisfactory degree of reliability of the com- puted measures conservative practice insists that the coeffi- cient be at least four times the size of the probable error. ILLUSTRATIVE PROBLEMS i 1. Make three different graphs of the normal probability curve to illustrate the differences occurring in the slope of the curve as distinctlv different scales are chosen for X and Y. Plot the three cm-ves on one sheet and use your own judgment in selecting the units for X and Y. 2. For the following data, plot a frequency polygon, choosing scales on X and Y that will give as large a graph as possible and a reasonably "steep" curve. Superimpose a NORMAL CURVE on this polygon to per- mit comparison of the actual distribution with the theoretical distribution. Distribution of Stature for Adult Males Born in Great Brit- ain. Report of Anthropometric Committee to the British Association, 1883, p. 256. (Quoted by Yule, p. 88.) Class-Inter- vals ARE presumably 57.99—58,99, etc. Height Number of Height Number of Height Number of (inches) men (inches) men (inches) men 57 2 64 669 71 392 58 4 65 990 72 202 59 14 66 1223 73 79 60 41 67 1329 74 32 61 83 68 1230 75 16 62 169 69 1063 76 5 63 394 70 646 77 2 Total 8585 3. Plot a normal curve on a base line extending from — 4 P.E. to + 4 P.E. Divide this base line into 5 equal parts and erect ordinates at the points of 1 Quoted from Rugg, H. 0., Illustrative Problems in Educational Statistics, published by the author to accompany this text. University of Chicago, 1917. 232 STATISTICAL METHODS division. Compute the exact proportion (to 2 decimal places) of all measures that should fall within each division of the total area under the curve. On the graph, letter the points of division in units of P.E. and the proportion of measures in each portion of the area. 4. Prepare a "probability table" for the normal curve on a base line extending from — 4 P.E. to + 4 P.E., in which the zero point of the table is transferred from the mean to — '4 P.E. State the percentage of measures that should fall between (which is set at — 4 P. £J.) and ordinates erected at successive intervals of .2 P.E. on the base line. This will give a table of 40 points of sub-division. erV \1 • // , 1 yl^HCl-^^c CHAPTER IX THE MEASUREMENT OF RELATIONSHIP: CORRELATION Practical need for measures of relationship. The pre- vious chapters have put before us the three methods of treating a single distribution of educational data: (1) that of picturing its status by computing some average to repre- sent it; (2) that of picturing its degree of concentration by computing some measure of variability or dispersion; (3) that of graphically picturing the entire distribution by plot- ting the frequency polygon, column diagram, or smoothed frequency curve that may be taken to represent the most probable statement of the true situation typified by our sample. It was found that if one desired to compare the status of two distributions he could use these methods of averages, dispersion, and frequency curves to give a com- plete picture of either distribution alone, or of the one com- pared with the other. It is probably true that most of the actual administrative problems faced by the practical school man may involve the use of these statistical methods, and only these. However, in the analytical experimental study of problems of learning and teaching, and of some admin- istrative problems, a new type of device is demanded, — namely, some method of determining the degree of causal connection exhibited by certain traits or activities in which we are interested. The measuring of physical, mental, and social activities constantly involves the study of causation or causal connection between two or more traits in ques- tion. The massing of data in this study of causation raises the necessity for statistical methods of computing degrees of causal connection. 234 STATISTICAL METHODS Suppose, for example, that we were interested in the prac- tical problem of classifying pupils in school in terms of abilities. One of the questions for which we desire answers would be: Are *' school" abilities specialized, or general? Is it probable that a pupil who shows a high degree of achieve- ment in one subject of study, say mathematics, will show a high degree of achievement in another subject, say mod- ern languages? To illustrate the problem: the data of Table 33 represent the actual school marks given a class of 23 high-school pupils in mathematics and modern lan- guages. Each mark in the table is the average of three or more marks in the respective subject. Table 33. School Marks given a Class of 23 High-School Pupils in Mathematics and Modern Languages Average mark Ayeraqe mark Rank in, achieve- Rank in achieve- PupiU in mathemat- in modern lan- ment in mathe- ment in modem ics guages matics languages A 50 58 23 21 B 78 88 15 7 C 96 90 2 5 D 88 85 6 10 E 85 93 8 2 F 80 57 13 22 G 94 91 3 4 H 79 84 14 11 I 86 83 7 12 J 75 80 16 14 K 83 92 10 3 L 82 81 11 13 M 71 77 20 16 N 72 59 19 20 O 92 87 4 8 P 81 89 12 6 Q 84 76 9 17 R 74 75 17 18 S 69 78 21 15 T 97 94 1 1 U 73 86 18 9 V 66 72 22 19 W 90 50 5 23 MEASUREMENT OF RELATIONSHIP 235 To answer our question we now have for each pupil in the class a pair of records of achievement, i.e., his average mark in mathematics, and his average mark in modern lan- guages. If, now, there were absolutely perfect correspond- ence, or ^'correlation " as we shall call it, in the two abilities in question, and assuming for the time being that the school marks of these pupils adequately measure their respective abilities, each pupil should occupy the same relative posi- tion in the two series of marks; — i.e.y the pupil first in mathematics should be first in languages, the pupil second in mathematics should be second in languages, and so on through the list. Table 34 shows this situation by giving Table 34. Hypothetical Marks given to 23 Pupils; printed HERE to illustrate PerfECT "RaNK" CORRELATION Pupils Mark inmath- ematica Mark in modern languages Rank in achieve- ment in mathe- matics Rank in achieve- ment in modern languages A 97 94 1 1 B 95 93 2 2 C G3 91 3 3 D 90 90 4 4 E 89 89 5 5 F 88 87 6 6 1 G 87 86 7 7 1 ■ H 85 85 8 8 I 84 84 9 9 J 82 82 10 10 K 80 79 11 11 L 79 78 12 12 M 76 76 13 13 N 75 74 14 14 O 73 73 15 15 P 72 72 16 16 Q 71 70 17 17 R 70 69 18 18 S 67 67 19 19 T 66 65 20 20 u 65 60 21 21 V 64 55 22 22 w 63 50 23 23 236 STATISTICAL METHODS two hypothetical series of marks. This method of measur- ing the degree of correspondence between two traits obviously takes account only of the position of the various measures in the series. It neglects the absolute amounts of the measures. Not only should the position be the same for each pupil in the two series, but, in order that the correspondence be absolutely perfect, the actual proportional differences be- tween each two consecutive marks ought to be the same. It is clear that merely to rank the measures in the two series in order of size and to compare the corresponding ranks does not accurately measure the degree of corre- spondence; i.e., it does not take full account of the absolute value of each measure. Need of devices to show correspondence. For this reason we need devices for picturing the correspondence between the actual measures which will take full account of the actual amount of each one. For example, let us take the pairs of marks in Table 33. In Chapter IV it was pointed out that a distribution can be completely represented by graphic methods, — by plotting the data. Let us plot the data of Table 33. By what graphic methods can we now com- bine pairs of measures in the same diagram to show the correspondence between two varying traits? Recall here that in the preceding chapters we have been plotting single distributions by laying off the units of scale on the horizontal (X) axis, and the corresponding numbers of meas- ures on the vertical (F) axis. In the plotting of the single distribution, therefore, we deal with but two quantities, ■ — the value of magnitude of the measures, and the fre- quency with which each occurred. We now have two fre- quency distributions, each having a scale along which the measures are distributed, and a set of frequencies. It is possible to combine the two distributions, however, on two coordinate axes because they have one element in common MEASUREMENT OF RELATIONSHIP 237 — the frequency column. If, now, we construct a double- entry table, like that in Diagram 38, in which the x-axis represents, let us say, the scale of abilities in mathematics and the ^/-axis, the scale of abilities in modern languages, it is possible to represent on this squared table every pair of Vf /ii>///ry /r7 Ayjaf h erria ^/cs 9S ej So TS 70 6S 60 SO i t i i [s So sjr as 9o 9s 65 fo 7S 60 Diagram 38, Distribution of Correlated Abilities in Languages {y) and in Mathematics (x) Data of Table 33. Each point is plotted to scale to represent a pair of measures on one pupil. measures in Table 33. Furthermore, as we do this we can see that each measure is represented in accordance with both its absolute amount and position in the series. To illustrate : How show correlated abilities graphically. Standard usage in plotting pairs of measures involves two coordinate 238 STATISTICAL METHODS axes, one horizontal (OX), and the other vertical (OY), meeting at an "origin " or beginning point at the bottom and left. One of these axes, say OX, is chosen on which to lay off the scale of one of the traits in question, and the other axis to lay off the scale of the other trait. The selection of which trait to plot on a particular axis is left to the arbi- trary choice of the student. The units of the scale are now laid off from the origin to the right on OX, and upward on OY.^ It is now possible to represent to scale a pair of measures, by plotting the value of one on the a^-axis, and that of the other on the y-axis. Erecting perpendiculars to the X and y axes gives us a point as the intersection. When considered with respect to the distance which it is from either base line, X and F, this point represents the pair of meas- ures in question. For example, in Diagram 38 a point, determined by per- pendiculars erected at distances 50 units from the origin on OX and 58 units from the origin on F, represents the pair of marks given pupil A in Table 33, 50 in mathematics and 58 in modern languages. Similarly with pupil B, repre- sented by a point 78 units to the right of OF and 58 units above OX; and pupils C, D, etc. It should be noted, that in Diagram 38, although the scaling of distances is correct, the entire table down and over to the origin is not given. Theoretically, of course, each point on the table is referred to axes OX and OY, assumed to be at zero. Diagram 38 now becomes clear to us. Each point on the diagram represents a pair of measures on a pupil. All the points, considered together, typify the degree of corre- pondence of correlation between the two abilities. A glance at the table tells us these things : (1) in general, pupils who ^ This method of plotting is in contrast to those of many educational workers in statistical methods, but more consistent with standard algebraic practice. MEASUREMENT OF RELATIONSHIP 239 stand high or low in one abiUty stand high or low in the other; (2) there are three pupils in the class for whom very- low achievements in languages accompany high or moderate achievements in mathematics. For the remaining 20 pupils the correspondence is rather close. In clarifying the situa- tion for the investigator, however, the actual plotting of the table indicates at once, and in a much more definite way than does the ranking of Table 33, the absolute amount and relative position of each pair of measures. This method of treating the data points out that we are primarily inter- ested in changes in the size of one variable corresponding to changes in the size of the other. If we plot the data of Table 34 (rank correlation perfect) we have a distribution of pairs of measures as in Diagram 39. As we glance over the rank order of these 23 measures we note perfect correspondence in change of position of the pairs of measures in the two series; i.e., pupil N is 14th in both series, pupil A is first in both, W is last in both, M is 13th in both, etc. Diagram 39, though, gives this information con- cerning change in position and, in addition, shows the changes in magnitude of the various pairs of measures. It is noted, for example, that the four smaller measures beginning with 66, 65; 65, 60; 64, 55; and 63, 50; show a much smaller de- crease in the size of the x- variable (i.e., achievement in mathematics) than in the size of the ^/-variable (achieve- ment in modern languages). This type of eccentricity in distributions, which would not be revealed by. mere rank- ing methods, shows up clearly in the complete plotting of the table. Few- and many-pair correlations. Changes in the distri- tion of measures in a correlation-table which contains but relatively few pairs of measures, for example, 23 as above, can be comprehended rather easily. It is probable that a fairly adequate interpretation could be made of the general 240 STATISTICAL METHODS change in magnitude of these two variables, and expressed in word form. However, the expression certainly would be vague and consist in statements something like the fol- lowing: "Large achievements in mathematics seem to be accompanied by large achievements in modern languages. 65- SS Abi/i*-^ iri ^^at-^ertjay^/cs I Dc-^cx./^ r n . ¥ • V, ^ .-^ .;»;• TI- 70 • •^ — ;— - — - _- - ^ Diagram 40. Distribution of Correlate^ Abilities in Languages AND IN Mathematics for 130 College Students Crosses represent mean values of points in each column. represented by Diagram 38, we obtain a classified table like Diagram 41. In grouping in class-intervals, however, we must recall that the assumption is made that all measures in each square (representing an interval on each axis) are assumed to be grouped at the mean point of the square. This point is determined as the point common to the means of both axes, x and y, of the square. In Diagram 41 the measures of Diagram 38 are shown in their new grouping, each point having been moved to a position at the mean of the class-intervals. It will be noted by the student that 242 STATISTICAL METHODS the general shape of the distribution of the table is ap- proximately the same. In this particular case the material is somewhat more compact, the extreme points having been moved more closely together. 60 6-S ro 7S eo 95 9o 9s /oo • • • •. • • • • • , •• • • • % • .1) • / >> » • • /- a^l-5 • • fo es I.. -^0 Diagram 41. Data op Diagram 38 plotted under the Assumption that all Points are concentrated at the Mean Points of the "Compartments" or Class-Inter- vals OF the Table This diagram illustrates "grouping" of original data in class-intervals. To show relationship between traits. Having grouped the data in class-intervals, our next step is to tabulate in each square the number of points found to fall in that particular square. The correlation-table (Diagram 40), now becomes Diagram 42. The number in each square now represents the number of persons whose records in the two subjects fall in that particular class-interval: For example, 7 pupils MEASUREMENT OF RELATIONSHIP 243 received marks between 50-54.9 in languages and 75-79.9 in mathematics. Again, inspection of such a table enables general statements to be made concerning the degree of re- lationship between the two traits. From the general trend of the table it is evident that abihties in mathematics are directly related to abilities in languages. ^ I ^ J5- 40 - ^9 so- S9 59 6^ 7« 7S-. 79 BO- 9o - /oo /oo_ ?o Tf'~ as / / / o / Z S — t 6 s s J 6 / ^ 2 / / -a eo 7S 7o^ / / Diagram 42. Data of Diagram 40 tabulated under the Assump- tion THAT ALL MEASURES ARE CONCENTRATED AT THE MeAN PoINTS OF Compartments To illustrate second step in the computation of correlated and regression coefficients. Discovering laws of relationship. Inspection of a corre- lation table is not sufficient to tell us in a definite way, how- ever, to what degree the two are related. If for example we have two correlation tables, rather similar in "scatter," it is difficult to determine by inspection of the table in which case the correlation is the more perfect. Facing such a table upon which large numbers of measures are scattered we at once feel the need for some device for condensing the measures, — the need of devising an average or typical measure which will adequately represent the status of all the pairs of records 244 STATISTICAL METHODS taken together. Just as averages and measures of disper- sion typify a single distribution, so we wish a device which will succinctly and yet most completely describe the whole correlation table. This necessary device can be constructed by turning to the columns and rows of the table. Each column or row (either of which may be called an " array'') may be regarded as a separate frequency distribution, and as such may be typified by an average point on its scale. Remembering that the most probable value of a series of measures is the arithmetic mean of the series, we may take the arithmetic mean of each column to typify it. Doing this for each column, as in Diagram 40, we now have a fairly continuous series of mean points as we move up the table. Careful in- spection of the table will show the student that these points distribute themselves in close accordance with a straight line. Thus, in the line that will best fit these mean points we have a device for representing the entire table. The line of the means of the columns or of the rows may be shown to repre- sent the most probable law of relationship exhibited by the two variables. Nothing is of more importance to the stu- dent in studying this problem than the clear recognition of this point. "Law of relationship" implies regularity of change in the two traits, — as one grows larger or smaller the other grows larger or smaller, or vice versa. This may be typified by the line that most closely approximates the general scattering of the pairs of measures over the table. Now if a line can be drawn on the correlation table that will best describe or typify the law of relationship, our task is to find simple methods of dealing with such a line. MEASUREMENT OF RELATIONSHIP 245 METHODS OF DETERMINING RELATIONSHIP A. METHODS WHICH TAKE FULL ACCOUNT OF THE VALUE AND POSITION OF EVERY MEASURE IN THE SERIES I. The Case of Straight-Line Relationship 1. The first method of determining relationship Galton's graphic method. One's first tendency would be to deal with the graphic representation of the law — the line of relationship. That is what Galton did in his pioneer and suggestive study thirty years ago. In Diagram 43, drawn for the data of Diagram 42 (ability in languages and mathematics), the scales on the x and y axes have been so taken that Qs—Qi (for ability in languages) represents the same distance on ic-axis as Qs — Qi (for math- ematics) on the 2/-axis. The points Qz and Qi have been plotted by erecting perpendiculars to OX and OY from the respective Qs's and Qi's on X and on Y. The heavy hnes in the diagram represent coordinates drawn through the medians of both distributions. Their intersection is the median of the table. Under these conditions, and since the units of the scales on the two axes are the same, the line drawn through Qs, Qi is the line of perfect correlation. In this case, it is at 45° to the horizontal base line. Galton next drew a line to approximate as closely as possible the mean points of the actual pairs of measures in the columns (shown by the crosses). This line is seen to deviate from the hue of perfect correlation. Then in the figure any horizontal line AB, is drawn from the median line, cutting AB the two lines QiQz and 2)5. The ratio —^ measures the amount of correspondence in change in the two variables. For every point on the fine of perfect correlation^ a given 246 STATISTICAL METHODS change in the size of y is accompanied by a proportional change in the size of x. For every point on the line which best fits the means of the "arrays" a given change in the size of y is accompanied by a somewhat larger change in f/b///7y /n Languo^ es 55- 39 "to. 45- 50^ 5^ 55- S9 6o. 64 (.5 (.9 To 7H 75 79 80 85- ^89 9o 9a. "IS- loo /oo ^ 95 ? r= -4B AC / / .y' 94^ 90 .^ ^ ^ ^ ^, / <:^ / X A. 89 - ' 65 ^ ^ 8^ 60 Mc -C//C n / y/ < 79 15- ~V "A / 74 7o / •K / / e>9 6S / / -L 1 Diagram 43. A Galton Diagram for representing Correlation Graphically Data of Diagram 42. Scales on X and Y such that Q3 — Qi on F equal same distance as Q3 — Qi on X. Thus line BQi Qi, is line of perfect correlation, at 45° to horizontal. the size of x. It is clear that if the two lines in the diagram AB coincide, then — — ; equals 1 and the correlation may be said to be perfect and positive. If the line of the means is verti- AB cal, coinciding with the median, then —p^ becomes 0; i.e.. MEASUREMENT OF RELATIONSHIP 247 a given change in the size of y is accompanied by no change in the size of x. As the Hne of the means swings over to the left of the vertical median, and its direction becomes downward from left to right, the correlation evidently becomes nega- tive. That is, a given increase in the size of y results in a de- crease in the size of x and vice versa. Finally when the line AB falls at right angles to Q1Q3 —^ becomes — 1 and correla- te tion is perfect, but negative. Thus Galton's method enables us to measure graphically the degree of '' co-relation^' or correspondence between two traits. Galton applied his method to the measurement of in- heritance of stature by computing the coeflScient of "co- relation" between the stature of children and the stature of their parents (the stature of the two parents being aver- aged in each case to give the "mid-parent "). He found this coefficient (the ratio described in the foregoing paragraphs) to be -. This may be interpreted to mean that if the av- 3 erage stature of a group of parents is found to be, say y inches above or below the general average of the race, the average stature of their children will be only - y inches o above or below the mean of the race. Galton expressed this by saying that the mean heights of offspring tended to *' regress back toward the mean of the race." Since his time other workers in biological statistics have used his term '^ regression,'' and now it is common to speak of the line of the means of the correlation table as the line of regres- sion. The ratio described above has come to be called the coefficient-of-correlation, and is denoted by r. 248 STATISTICAL METHODS -7 -6 -S -«--3 -i -'' I -4> .J « T-t-t- 2. Second method of determining the law of relationship Finding the equation of a straight Une of regression. Re- fined comparative work in statistics demands a more accu- rate method of determining the law of relationship exhibited by two traits than that of graphic measurement. We said above that the law of relationship is described by the ^* best-fitting*' or ''most representative'' line of the table, i.e., by the line which fits most closely the mean points of the columns of the "arrays." Now, the most definite way by which we can describe a line is to write its equation. Since most of our educational in- vestigations give tables whose means approxi- mate closely a straight line, we shall confine our discussion for the time being to that type. In order to write its equation we must be able to put two vari- able quantities, say X and F, together in an algebraic expression in such a way that a given change in the value of one, say x, is accompanied by a proportional change in the value of the other, y. For example, in Diagram 44, a series of points are plotted, each of which represents a pair of measurements, and each of which ''satisfies" the equation of the line. That is, point P "represents," or is plotted from x = + 4, 2/ = + 2; point Q represents x = — 4, 2/= +3. The line £ Q. - /. O — . F> -r -6 -jf -* -^ -4 -/. -A -3 -t ii«j- iVor 2 the "slope" of the line. Now to simplify the final statement of the equation let us define r as 2 {x^ - x) (y^ - y )^ or, in terms of y and x as deviations^ — ? xy N 0-™ (j' X ^y Then the slope, and the final equation of the best-fitting line is: — y^-y = r—{xi-x) or y = r-^x o-x o-x (a) The significance of r — the coefficient of correlation. This brief mathematical statement has been given to permit us to make clear the real significance of r, the so-called "co- efiicient of correlation." r serves two specific functions in the determination of relationship. (i) It is a single index, a pure number, which measures the degree of "scatter" or of concentration of the data, by giving the mean product of the deviations of each of the meas- ures from the mean value of its ^^ array '^ when measured in units of the standard deviation. Stated in terms of such de- viations {x and y) r is more simply expressed as : — % xy r = N a-^a ^ MEASUREMENT OF RELATIONSfflP 253 We note that the deviation (x, or y) of each measure from its respective mean is measured in units of its respective standard deviation by dividing it by a-^ or a-y. %xy is evidently Bravais's product-sum of the deviations. Thus in this single numerical coefficient, r, we express relationship in terms of the mean values of the two traits by measuring the amount each individual deviates from its respective mean. The formula for r is generally called the product-moment formula. This will be explained later by reference to Dia- gram 46. (2) r is an intermediate device. In defining the second func- tion of r, we note that it is merely an intermediate numerical device, defined as it is for the purpose of bringing together, in one convenient expression, certain terms collected in the process of developing the law of regression. Thus, it is really only an intermediate expression in the ultimate math- ematical process of expressing the law of relationship in terms of the equation of the line of regression. On the other hand, it may have for the lay student a more definite connotation than the equational or algebraic expression for the line itself which really represents the relationship; e.g., — y = r— X To the mathematician this expression has a specific con- notation; to the non-mathematical student a vague and unsatisfying one. Largely for that reason, students of edu- cational research have neglected the equational expression for relationship, and have adopted the single numerical co- efficient r. It should be noted, however, that once having determined r, the regression equation of the line of the means can be expressed very simply by substituting the values of r, (Tx and a-y in the equation above. Furthermore, we said that such an equation was ex- 254 STATISTICAL METHODS pressed in the '*slope^* form, y = mx in which m is the "slope" or tangent of the angle that the line makes with the horizontal. Thus in the regression equation, — and this, in those cases in which the variability of the two traits is the same, becomes equal to r. r — is known as the regression coefficient of y on x, that is, the deviation of y corresponding on the average to a unit change in the type of X, and is represented by by^ or bi. In the same way r — is the regression coefficient of x on y, is represented by bj.y or &2» and means that deviation of x which corresponds to a unit change in the type of y. (b) What is the meaning of the coefficient of correla- tion and the regression coefficients? Statistical measures are computed only for the purpose of clarifying our interpreta- tion of complex masses of data. It has been pointed out re- peatedly in the foregoing chapters that such devices do not supply proofs of existing relationships, — rather that they are merely tools to refine our analysis of numerical situations, and that they are valuable only in so far as they agree with sound logical analysis. So it is with statistical devices for measuring correlation. The mind demands a tool for de- fining the extejit of correlation shown by a vast number of pairs of measures on the two traits in question. A coefficient designed to measure relationship is valuable to the extent that it does this. Our next problem therefore should be to show the com- mon-sense significance of the correlation coefficient, and of the regression coefficients, and to indicate the relative degree to which they aid us in interpretation of our data. Suppose from the example given in Diagram 46, — then. Then MEASUREMENT OF RELATIONSfflP 255 r = .48; a-y = 1.26; a^ = 0.89, r-^ = .68, and r— = 0.34. y = .68x and x = .My, In this problem we have three statements to aid us in the interpretation of the question: To what extent is abihty in shop practice accompanied by abiUty in drawing, or vice versa. In the first place, we may use the value of the corre- lation coefficient r = .48. The question arises: What does this mean? Is there a direct relationship between the two abilities? If so, is there an indication of considerable rela- tionship, little relationship, or no relationship? The sign of the correlation coefficient, which in this case is positive, answers the first question definitely. This positive sign means that any increase in one trait is accompanied by an increase in the other, and vice versa. Had the sign of r been negative, then an increase in, say x^ would have been ac- companied by a decrease in i/, and vice versa. To make clear the meaning of various values of r, suppose each series of measures had been ranked in order of size, as in Table 33. If the position of each measure were the same in both series {i.e., if pupil A were first in both series, pupil B second in both series, pupil C third, etc., throughout), then the correlation between the two traits would be perfect and positive, and r would be + 1. On the other hand, if the order of the pupils in the two series were exactly reversed {i.e.y the first pupil in one series should be the last in the other series, the second in one series should be the second last in the other series, etc.), then the correspondence ("correlation") again would be perfect but this time nega- tive, and r would equal — 1. Again, if there should be no correspondence in the position of the measures in the two ^5G STATISTICAL METHODS series, the value of r would be 0. Thus the value of r may range from — 1 to + 1. When between and 1 it will " express a tendency, greater or less according to r's size, for measures above the mean position in one series to be above the mean position in the other series. When r is between and — 1, it will express a tendency, greater or less, accord- ing as r is numerically greater or less, for the measures above the mean position of one series to be below the mean position in the other, and conversely." The exact degree of relation- ship is commonly inferred from the relative size of the co- efficient, r. Thus correlation may be spoken of as "high," "low," etc. It can be seen that the definite interpretation of correlation depends on the arbitrary placing of the limits of the values of r, which are to be called "high," "low," etc. " High " and " low " correlation. This definition of limits depends largely on the personal experience of the person making the interpretation. For example, it has been common for certain educational investigators to arbitrarily interpret a coefficient of .25 as an indication of "high" posi- tive correlation, and one of .40 as " very high." Others would interpret .25 as very low, and .50 as "marked" or "some- what high." Certainly, our educational conclusions must be colored by our arbitrary definition of such a coefficient. The experience of the present writer in examining many correlation tables has led him to regard correlation as "neg- ligible" or "indifferent" when r is less than .15 to .20; as being "present but low" when r ranges from .15 or .20 to .35 or .40; as being "markedly present" or "marked," when r ranges from .35 or .40 to .50 or .60; as being "high" when it is above .60 or .70. With the present limitations on educa- tional testing few correlations in testing will run above .70, and it is safe to regard this as a very high coefficient. The interpretation of the coefficient r = .48, in the above problem, would result in a general statement to this effect: MEASUREMENT OF RELATIONSfflP 257 " There is marked evidence that abiHties in shop practice and drawing accompany each other. Students above the average in one group will TEND to be above the average in the other. It is not known more specifically in what way the two abili- ties are centrally connected, or to what extent the presence of either one is an indication of the presence of the other." Except in the case in which the variability is the same, r does not enable us to foretell, for example, knowing the value of one trait, what, on the average, the value of the other will be. It does not enable us to say that for a given unit-change in abilities in shop practice, what changes should be expected, on the average, in drawing abilities. A more complete method of describing relationship. This very vagueness in the possibility of definition of r leads us to turn to the more complete method of describing the re- lationship, namely, the equation of the line. Taking that, we now find that, for the regression of y onx, — y = .68a;, and that for every unit deviation from the type of x (abili- ties in shop practice), it is most probable that there will be an accompanying deviation of .68 as much in y (abilities in drawing) . The tendency, in the past, has been to stop the analysis of the data at this point, the conclusion being drawn that the two abilities are very closely related. It must be remembered, however, that there are two regression lines, one for the means of the columns and the other for the means of the rows. The former shows the deviation in y correspond- ing on the average to a unit deviation in the type of ic, and the latter the deviation in x corresponding, on the average, to a unit deviation in the type of y. Thus, in our problem, X = .34?/; i.e.y it is probable that a unit deviation in y will be accompanied by a deviation of .34 as much in x. This explanation has made use of the ^'deviation ''formula 258 STATISTICAL METHODS (2). Using the formula (10) in which y and x are actual values instead of deviations, we can make this still clearer to the student. The equation now becomes 1 26 y _ 85.55 = 48.1 - — {x - S5.'25) .89 or, y - 85.55 = .68 {x - 85.25) In this case, it must be remembered that x and y are actual values of the two traits, abilities in shop practice and draw- ing, and for y and x have been substituted the values of their respective means, y =_85.55; x = 85.25. Expressing the equation of the line of the means now enables us to assign values to one of the traits, say Xy and compute the accompanying value of y. In Table 35, values decreasing by 5 have been assigned to x, and the i/'s computed. It will be noted that as each x decreases by 5 (90, 85, 80, 75, etc.), the corresponding decrease in the unit of y is .68 X 5 = 3.40. Table 35. Regression OF X ON y X y 95 92.18 90 88.78 85 85.38 80 81.98 75 78.58 70 75.18 60 68.38 Table 36. Regression OF 2/ ON a: y X 95 88.46 90 86.76 85 85.06 80 83.36 75 81.66 70 79.96 60 76.56 This should make clear the statement made above that a given deviation in x would be accompanied by .68 as much MEASUREMENT OF RELATIONSHIP 259 change in y. In the same way Table 36 gives corresponding values for y and x computed from the regression equation oi y on X : — ^ , X- S535 = .S4> (y- 85.55). jThe effect of the smaller regression coefficient (.34 instead of .68), is now seen in the relative values of y and x. As y de- creases steadily by 5 units, x decreases by only 1.70 units (.34 X 5 = 1.70). Reference to Diagram 46 will reveal the way in which differences in relationship between the two traits are partially described by the "slope" of the line of the means. The plotting of the equation of the line of the means of the columns, — y = .68.T gives a line of considerable steepness, CC. For given changes in x we have nearly proportional changes in y. The plotting of the equation of the line of the means of rows, — X = My, gives a line much flatter in slope, RiR^. For given changes in y we have much smaller changes in x. (c) How to plot the line ol the means. We are now in a position to draw the line of regression on our correlation table. There are two methods by which this may be done. The first is the rough method of drawing, from inspection of the mean points of the columns and rows, a line which most closely approximates them. This can be done by lay- ing a celluloid triangle, or a thread over the table, and ad- justing it by eye until it most closely fits the mean points of the columns and rows. The line may be drawn accurately, however, by first computing the equations of the lines of the means. Values may then be assigned to Xy and corre- sponding values of y can then be computed, exactly as in Tables 35 and 36. Since a straight line can be plotted from 260 STATISTICAL METHODS any two of its points, we can draw the line by plotting any two of the pairs of coordinates, x and y. For example, in Table 39, the line CC is determined by connecting C (which was plotted from x = 90, y = 88.88) and C (plotted from X = 60, y = 68.38). The remaining points of the table will fall on the same line, since their coordinates have been computed from the equation of this line.^ 3. Computation of the correlation coefficient and the regression coefficients It is now clear that statistical methods can supply us with a tool for estimating relationship in terms of the most probable values of two concurrently changing quantities. The determination of the law of relationship must lead to the computation of the regression coefficients. This in turn de- mands the computation of the correlation coefficient r, which, in itseK, will throw some light on the status of relationship. There are two principal steps in the computation of these coefficients: (1) the tabulation of the correlation table; (2) the computation of three devices, cr^, (Ty, and r, with the consequent substitution of these values in the regression equations. (a) The first step : the tabulation of the correlation table. The foregoing pages have made it clear that complete iii- terpretation of correlation demands the tabulation of each of the pairs of measures in the correlation table. The steps in the tabulation may be conveniently listed as follows: — (1) Decide on the size and position of the class-intervals in each distribution. This should be done in accordance with the principles laid down in Chapter IV, in the discussion ^ The more refined methods of fitting lines to plotted data, involving, as they do, the theory of ciirve-fitting, will not be taken up in this work. In the bibliography at the end of the book complete directions are given the mathematically trained student for finding the literature. MEASUREMENT OF RELATIONSHIP 261 of the classification of data in a single frequency distribution. The student must understand that he is now to tabulate pairs of measures which occur in two sets of class-intervals at the same time. (2) Write the limits of these class-intervals along the two axes of the table, assigning one trait to y and the other to x. Lay off these limits from a zero point, supposed to be at the bottom and left of the table, as in Diagram 45; e.g., 61-65, 66-70, etc., from bottom up, and 71-75, 76-80, etc., from left to right. (3) Having the original measures arranged in parallel series, as in Table 33, tabulate these pairs of measm*es in the appropriate rectangle in which they fall. It will be helpful to have the y-series on the left, and the x-series on the right in this pairing of the meas- ures. The caution stated in Chapter IV to define carefully the limits of class-intervals should be kept in mind in this work. More errors are made in the original tabulation of so c 1 a .S >, < Ability in Shop Practice 71 _ 75 76 _ 80 81 _ 85 86_ 90 91 _ 95 100 _ 96 / // / 95 _ 91 // III //// //// 77/r jnr II m- 90 _ 86 /// TtTr mr iiif ///I 71// JUT / mi MU /III Titr HUMI m 11/ 85_ 81 // //// III/ jm mr 1tir III! III! nil llll VTr Wr ■UU- Ml. mr mr -m 1 / 80_ 76 // m- nil nil wr mr III ' 75 _ 71 / //// 70 _ 66 / / / 65 _ 61 / Diagram 45. To illustrate the First Step IN Plotting a Correlation Table Checking pairs of measurements in appropriate compart- ments of the table. 262 STATISTICAL METHODS the correlation table than in any other one aspect of the work. The tabulation is illustrated in Diagram 45. (4) The pairs of measures having been checked on the table in pencil, next replace the checking by numbers, to give a table similar to Table 37. Table 37. To Illustrate Another Phase of the Second Step in the Tabulation of a Correlation Table Ability in Shop Practice 71-75 76-80 81-85 86-90 91-95 100-96 1 2 1 95-91 2 >8 22 5 90-86 3 21 31 8 Ability in drawing... 85-81 2 9 30 16 1 80-76 2 5 15 8, 75-71 1 6 4 70-66 ) 1 1 1 65-61 1 / (b) The second step ; the computation of the coefficient of correlation r and the regression coefficients, 6i and 62. Our task is to compute r from the formula Na-^a-^ and 61 and 62 from the formulae — ^ MEASUREMENT OF RELATIONSfflP 263 the final equations of the hnes of regression being — y = r-^ X (for the regression Hne of the columns), and a; = r— y (for the regression line of the rows). The work may be made clear by first listing the steps in the computation of r. The formula requires us to find the two standard deviations, o-^, for the total frequency columns of the i/'s, and o-^ for the total frequency rows of o^'s. The student's first difficulty in understanding the computation will be in comprehending clearly that o-y and a^ are the stand- ard deviations of the total frequency distribution of the columns and rows. Thus, in Diagram 46, the column and row headed fy and /a; mean respectively "total frequency of the 2/'s" and "total frequency of the x's." Thus, the standard deviations, a-y and o-a; are found from these two frequency distributions exactly as described in Chapter VI. Furthermore, the short method of computation can be applied to the two distribu- tions to cut down greatly the labor of computation, not only for the standard deviations but. also for ^xy. Steps in the computations. The entire steps in the compu- tation are as follows (compare Diagram 46 for illustrative references) : — 1. Total the measures in each distribution, giving A^. 2. Estimate the class-interval which contains the mean, e.g.y 86-90 for the 2/'s; 81-85 for the x's. 3. Tabulate the deviatioii in unit intervals, of the mid-value of each class-interval from that of the estimated mean, 1, 2, 3, etc., - 1, - 2» - 3, etc. 4. Multiply each frequency by its respective deviation; e.g.y for the 2/'s, 4 X 2 = 8, 37 X 1 = 37, etc., for the a:'s, 6 X - 2 = - 12, 26 X - 1 = - 26, etc. 71_ 76- 81- 86- 91- fv 75 80 85 90 95 100- 96 1 2^/ 4 1 4 95_ 91 2 2 8 /as 10 37 90- 86 3 21 V y 8 63 85 _ 81 4 2 9 9 ^ -16 16 1 58 80 _ 76 8 2/ y 10 5 /o 15/ -16 8 30 75_'- 71 1 18 6 / / * 11 70 _ 66 ■/ 1 -4 1 3 65 _ 61 10 1 / 1 •2. 6 26 80 80 15 207 fd fd' Sx'y. 8 16 , 8 45 37 30 ■1 -58 -3 —33 ■4 -12 -5 —5 . 26 10 -168 207)403 74 -5 45 1.S47 —5 —123 .35 69-5-207 = c2— .35 - + -6745x77 ^ ^..62 "^ N 14.39 ~ 14.39 Cu 1.26 »=^-^*= 48-o:89- *=-6&» .- <5x .89 y = My Diagram 46. To illustrate Computation of the Correlation Coefficient and the Regression Coefficients for the Case of Linear Regression (Adapted from form used by Dr. H. L. Rietz, of the University of Illinois.) MEASUREMENT OF RELATIONSHIP 2G5 5. Find the algebraic sum of such fd's, e.g., ^fdy = — 168 + 45 = - 123; 2/4= 110 - 38 = 72. 6. Divide 'Xfd by the number of cases, iV, to give the correction c; e.g. — -123 ^^ 72 = - .59; c^ = 207 207 ^y - ^^«, = - -59; c^ = ttt: = -35. 7. Square the corrections; e.g., c^ = .35; cj^ = .123. 8. Multiply each fd by c?, its corresponding deviation, to give column headed fd"^; e.g.,fd\ = 16, 37, 0, 58, 120, etc' fd\ = 24, 26, 0, 80, 60. 9. Find the sum of the/c?2; e.g., %fd\ = 403; ^Sd\ = 190. 10. Divide this sum by N, to give S"^ the square of the standard deviation of each distribution around the assumed mean; e.g., S/ = 1.947; SJ = .918. 11. Subtract the square of the correction from S'^;e.g., a-y^ = 1.947 - .35 = 1.597; crj- = .918 - .123 = .795. 12. Find the square root of cr^ giving o-; e.g., o-y = 1.26; 0-3. = .89. Note that these standard deviations are expressed in units of class-intervals of 1, and that to find the correla- tion coefficient, r, they may he left in these units, provided ^x'y' is computed in the same units. It will cut down the labor of computation greatly to do tHis. Note, furthermore, that the above twelve steps merely restate the steps in the computation of o- as given in Chapter VI. The formula ^x'y' Na-^fTy next demands that we compute the product-sum of the corresponding pairs of deviations from their respective means x^y^, x^y^, x^y^ for every point in the correlation table. Diagram 47 will make clear what is wanted. The two measures in the compartment y = 9Q—100, a; = 86— 90, each deviate from the mean of the x's, i.e., from ^ by 1 class- Ability in Shop Practice 71 _ 76_ |81_ J86_ ; 1 91- / d 75 80 1 85 ! 90 95 100_ 1 IxV-.i 96 h 4 2 95_ XzV=-2 91_ 9d_ "' 8 1 22 21 31 — 5 s V 37 1 11 -1 ; ! 63 Q 86 ' ] 1 ^ -- + - As sumed M.« an 85- ""T" 2 ' 1 30 j 16 1 58 -1 81 j so- 1 1 2 5 1 15 1 8 30 -2 ld 1 75 _ 1^ -18 ! 71 1 '< ' 4 • 11 -3 fc "> \ 70_ ! r:^ --i 66 1 1 < 1 3 -4 lif -4 -4 65_ 61 1 II 1 1 -6 / 6 i s 1 3 26 w < 80 80 15 20 d -2 ' -1 1 2 Diagram 47. A Product-Moment Diagram To illustrate the computation of Sx'j/'- For the data of Diagram 46. The computation is illustrated graphically for one compartment in each quadrant, x' and y' are deviations (or "moments") of the mean of the compartment from the respective assumed means of the •table. MEASUREMENT OF RELATIONSHIP 267 interval {=x') and from y hy 2 class-intervals ( = y'). That is, for each of these two measures x'y'=\ X 2. For the measures in the compartment 1/ = 96-100, x = 91-95, x'y' = 2X2; for the compartment 2/ = 76-80, x' = 96-90, ^x'y' % x' y' = 8 [1 X — 2] = — 16. Note carefully that the signs of the deviations must be taken account of. These signs are now determined by noting whether the measure in question is greater than or less than the mean of the total distribution. A measm-e greater than the mean will deviate positively; one less than the mean will deviate negatively. To expe- dite the work of the student the correlation table should be divided into four quadrants y as follows : — x= — 2/= + x= + y= + x= — y= - x= + y= - If the class-intervals have been laid off as suggested from left to right, and from bottom upward, the quadrants, with the signs of x and ?/, are as just given. Now, to compute '^x'y' for the whole table, going from compartment to compartment and summing the product of the pairs of measures, as shown above, will be a very la- borious task. The labor may be shgrtened very much by summing the x deviations of all the measures in one row, and multiplying 1x' once for all by y'. This method recognizes that all the measures in a given row, e.g., 2, 8, 22, 5, in row 91-95, have the same y\ namely, + 1. Treating the material in this way enables us to compute the deviations mentally 268 STATISTICAL METHODS and very rapidly. With this explanation we are now reaidy for step 13 in the computation of r. 13. Compute 'Xx'y', by finding the sum of the deviations of the measures in a particular row from the mean of the x's of the whole table {x) . This gives 2a;'. Multiply 5a:' by ?/', the devia- tion of this particular row from y the mean of the y's of the whole table. This gives "^x'y', which is the product-sum of the deviations about the two assumed means. Table 38. Columns corresponding to row 96-100 Row 96-100 81-85 86-90 91-95 Total Sx' x' = . + 1 + 2 n = 1 2 1 Sx' = + 2 + 2 + 4 y' = +2 SxY = + 8 Table 39. Columns corresponding to Row 76-80 71-75 76-80 81-85 86-90 Totals 2x' x' = -2 -1 +1 n — 2 5 15 8 Sx' = -4 -5 +8 -1 y' = -2 ^x'y' = + 2 The computation, presented here in tabular form, can be done mentally. In setting down the results of the Sx'i/' for each row, as 8, 30, etc., in Table 39, it may be more accurate for the beginning student to tabulate both the positive and negative 2a:' separately, summing them both separately to give the algebraic sum of the deviations. In the accom- panying problem. Diagram 46, the work has all been done MEASUREMENT OF RELATIONSHIP 269 mentally, the algebraic sum of the '^x'y' being tabulated in one column. This gives ^x'y' = .69. This product-sum is for deviations computed from the two assumed means, not the true means. Therefore, just as the means are in error by a correction Cy, or c^, so each deviation on y and on a^ is in error by the same amount. Thus, since we must apply corrections to find the true means and the true standard deviations, so we must apply a correc- tion to find 2x2/, the product-sum of the deviations about the true means. This means that we must multiply c^. and Cy together to get this correction. It has been shown that the formula for r, by this short method of computing the terms about the assumed mean, is : — N ^ocPy * Let E;c and Ey represent the estimated means of the two series, and Cx and Cy be corrections to be apphed to the estimated means to get the true means. Then the True Means, Mz and My are respectively Mg = Ex + Cx and My = Ey + Cy. Let X and y be deviations from the True Means, Mx and My. Let x' and y' be deviations from the Estimated Means, Eg and Ey. Thus, x' = X -{- Cx and y' =y -\- Cg. Therefore, '2.x' y' = S(x -f Cx) (y + cj = ^xy -h Cy^x + Cx^y + Sc^jCj,. Now, since Sx and 2y (the sum of the x and y deviations /row the TRUE MEAN) each = 0, then 2x'y' = 2xy + -EcxCy, or ^xy = ^x'y' - ^CgCy or, substituting this expression in the equation N U3 • '\ • • ^ .\ . . L< o • • • M .*. \ • . a • s .-. A* , \ 'a t^ • B • • • CO c- \ ' • A. o u to . • \ o , ^ y. •^ • \ » • s 1^ . \ , iz; g \ s. -^ * . N v,^ v.* lO . "V.^ • . • -■*-. o • (N • ^ ' a. <0 1 §i2 rSt tati cen r-i 5|.s > o a T-H (M CO '^ LO O t- 00 as o T— 1 1— 1 (M < P Ph o « P9 "^ w K CO g ° o to w * QQ i? -^ o W p^ " i ^ ^ 278 STATISTICAL METHODS relationship which, as before, will treat the separate columns of the table in terms of their arithmetic means. We shall be interested in finding how a deviation from the means of one trait (measured on, say, the x-axis) corresponds on the aver- age, to a unit deviation from the mean of the other trait, (measured on the other, the 2/-axis). The product-moment method involves finding the prod- X 11 . uct of the separate ratios of — , — , that is, of the devia- tions of every measure of the table from the mean of its distribution, measured in units of the standard deviation of the x's or y's of the whole table. We find the amount that each measure differs from the mean of its distribution. This is X for the a;-measures, and y for the ^/-measures. These x and y deviations can be made comparable by dividing each one by its respective standard deviation as a unit. Thus, in computing the correlation coefficient, "r," we measure each deviation x or y in units of its corresponding standard . X u » deviation a-^ or o-y, i.e., — and — . r is the ratio between O-x O-y the sum of the x deviations times the y deviations, each measured in terms of its standard deviation. Professor Pearson has suggested that the non-linear tables may be treated by finding the ratio of the standard devia- tion of the arithmetic means of each of the columns (or rows) of the table to the standard deviation of the whole table itself (the o-y for the columns and o-^ for the rows of the table). In symbols this means (letting the general expression be called the "correlation-ratio" =r]) S{n,{y.-yY) N where. MEASUREMENT OF RELATIONSHIP 279 rij. — total number of measures in any column; Vx = the arithmetic mean of any column; y = the arithmetic mean of all the y*s in the table; N — total number of measures; (Ty — the standard deviation of all the i/'s in the table. This is equivalent to saying \ the standard deviation of the means of the columns of the table. For, each (^j. — y) equals the difference between the arithmetic mean of a column {y^ and the arithmetic mean of the total frequency obtained from all the columns in the table. That is, each ^^ ~ V) is the "deviation" of the mean of a column from the mean of all the i/'s in the table. These are each squared and weighted by their corresponding fre- quencies, n^. Thus, it can be seen that the above formula is of the usual form of the standard deviation : — 1^ That is, in the above symbolism, n^ is. equivalent to /, the frequency; y^ — y; is equivalent \,o d\ S is equivalent to 2. Diagram 49 is supplied to make clear the use of the sym- bolism: Table 41 illustrates the method in detail. Summary of process. We may summarize the process of computing the correlation-ratio by listing the following steps . — 1. Tabulate correlation table in exactly the same way as in computing r. 2. Sum the columns {ux) or the rows {uy) of the table. (These correspond to/'s in the computation of r.) 3. Compute the arithmetic mean of all the ?/'s in the table. Call this y. (This is done in exactly the same way as in com- 280 STATISTICAL METHODS 40 45 j-o ss 60 6S 4 t 3i= (\l C^ '^S6Q< 5 II ^ ^ i 6 7 IN fs^!oik z s^^sm \-A ^133<5^ >• mmmmt 6 o/" ys = „ IN l>s 7 <0 10 ^< 1 U /'>^ 1 •0 ^^.^T-^ II X d / / 9 ^, 35.=A^r 10 / V. J J- 7 J S 7 DiAGEAM 49. Abstract from Table 41 To illustrate the fact, that, in the computation of the Correlation-Ratio (tj), iy^—y) is the deviation ((f) of the mean of the column from the mean of the table. puting the mean of any frequency distribution (see Chapter V). The distribution to use in this case is that of the total frequency column, headed %. The short method should be used, as before, using units of class-intervals instead of the original units. ^ ss«^^^§^§§§s 1 ^ ^ COOOOOo 'J'f-I b- rH „' t-: C^rH 3 9.33 3.35 11.22 33.66 ^ (M rHr-trH 5 10.8 4.82 23.23 116.15 § - 23 (M (N -H OS "= '-: ^ . CO CD O ^ - ^ooO§§ C^ r|5 Tji co-ii^iocot-oocio-Hc^jco II § i 1 ^ 8:)U8o ni nonB^ioa'ji rii^ « J, > + ^ 5 II CO ^ '»>:2 II + ^ ^---. 1 3 lis I&: fe; II •« , nlk ^ >^,« ^ biSi^Q W II 5?SlS§ §IS ^ II '^ II «SiL c. 282 STATISTICAL METHODS 4. Compute the arithmetic mean of the ?/'s in each column of the table. Call each of these yx- (Each of these can be left in the form of the correction to the true mean, or the difference be- tween the true and assumed means, provided y is expressed in the same way. To do this will cut down the arithmetic labor somewhat.) 5. Compute the square of the standard deviation of the y's in the whole table; call this a^^. (As in step 3, use the total frequency distribution, headed %.) 6. Subtract the arithmetic mean of the whole table y, from the arithmetic mean of each column, yx, to find the amount of deviation of the mean of each column from the mean of the table. That is, perform the operation (yx — y) for each col- umn of the table. (This corresponds to finding d in the case of the computation of the standard deviation of any distribu- tion.) 7. Square each of these deviations i^x — y), giving Q/x — vY' 8. Weight each of the deviations (squared) by the number of cases occurring in each column; that is, multiply each {yx — yY by its corresponding nx. (This corresponds to finding JiP' in the common standard deviation formula.) 9. Add the square of these weighted deviations. This gives ^[nxiyx-yYl (This is S/cZM 10. Take the square root of this quantity* and divide by N, the total number of cases in the whole table. This gives ^ \S[n^{yx- the standard deviation of the means of the columns, comparable to \| N 11. Divide S by cy, giving the correlation-ratio , tj. Since rj is the ratio between two standard deviations it is always positive, — that is, yj is always between and 1. The expression given here for rj is absolutely independent of the form of the distribution, whether it exhibits straight- line or curved-line relationship, and can be used for the MEASUREMENT OF RELATIONSHIP 283 computation of correlation for any kind of a table. To the writer's knowledge no published analysis has been made of educational distributions which have utilized both the pro- duct-moment and the correlation-ratio methods. Thus little comparative data are available for us at this point. We are interested to know: Under what conditions can we use the product-moment formula.? How can we determine whether or not a correlation table exhibits linear regression? For rough work, Blakeman^ has stated a criterion for linearity which we can use to aid us with most of our distributions. It is that Vn .-V,-^ .67449 2 must be less than 2.5. Applying this to our problem in Diagram 48, we get ^.lV(.83)»_(-.47)» = 6.169>2.5. In this case the table is obviously a non-linear table and the product-moment formula is inapplicable. Whenever the correlation table is not very linear the investigator should compute both v and r. Then the interpretation of the size of the coefficients v and r can be determined by the ap- plication of the criterion for linearity. B. METHODS WHICH TAKE ACCOUNT ONLY OF POSI- TION OF THE MEASURES IN THE SERIES I. Various Methods of Ranks and Grades From the discussion in the foregoing chapter the two methods of computing correlation which take account of the absolute value and position of each measure in the two * Blakeman, J. Biometrika, vol. iv, pp. 349, 350. £84 STATISTICAL METHODS series have been shown to be mathematically sound, but rather laborious in arithmetical work. It is clear that the student of educational psychology and education often has to content himself with comparatively few subjects, 10 to 30 being quite a common number. With such a small num- ber, the unreliability of the relationships, as shown by the size of the P.E., often would be so great as to vitiate the statistical results. Other things being equal, that index of correlation is best which gives the smallest P,E. With a small number of cases, however, it is clear that the probable error has little or no significance, and that we are unable to establish the reliability of coeflScients computed by any method. Spearman's method by " ranks " or ** position." At the same time we may desire a practicable formula for the cor- relation existing between two variables, easily computed and adapted to the conditions of psychological and edu- cational investigation. To supply this formula, Professor C. Spearman had empirically deduced a method of express- ing correlation in terms of "ranks'* or "position," rather than in terms of absolute quantity. ^ This method has been advanced and is coming into common usage, largely on two grounds: (1) the ease of computation of the rank index; (2) the belief that greater comparability of measures will be obtained through expressing the relationships which are found in psychological data in measures of position. Spearman suggests that a distribution of psychological measures may not be absolutely comparable at various points of the distribution, whereas measures obtained in physical and anthropometrical research may be statistically treated when regarded as being absolutely comparable at all points of the series. On the other hand. Professor William ^ Professor Pearson has since established mathematically the expression for this type of correlation by "grades." MEASUREMENT OF RELATIONSHIP 285 Brown and other psychological pupils of Pearson maintain that the measurement of the results of psychological ex- perimentation is physical measurement and that the meas- ures are objectively comparable. Spearman has attempted to show that we may turn the product-moment formula into the expression r = l-- -^(iV2-l) o or 6SD^ r=l- N{N^ - D' where (vi — v^) or D represents any difference in the rank of an individual in the two series, and where ^ N{N^ — 1) is the value that the sum of the D^'s would have by the operation of chance alone ^ This method is based on a very fundamental assumption, the validity of which is extremely doubtful, — namely, that the distribution of ability is rectangular in shape. This means that " the unit of rank is the same throughout the scale," — that is, that individuals are separated from each other at the end of the scale by the same distance (or increment of ability) by which they are separated in the middle of the scale. Our educational testing of mental abilities leads to the conclusion, however, that most mental abilities distribute themselves in a large school population in accordance with a curve in which the measures are largely concentrated near * See Brown, W., Essentials of Mental Measurement (1st ed.. Appendix), for proof of this statement. 286 STATISTICAL METHODS the central portion of the range. It has been shown, e.g., in measuring abihties by various mental tests that the shape of the distributions for each grade and on each test takes a form approximating a symmetrical curve. This curve, with the implications of its widespread use, has been dis- cussed in Chapters VII and VIII. That this relates to the problem of "rank-correlation" should be very clear. To assume a rectangular distribution is to assume that each individual in the series is the same distance from the adja- cent individuals, — throughout the series. A glance at the bell-shaped curve shows this to be incorrect. As Pearson says, "Between mediocrities, the unit of rank, ... is prac- tically zero; between extreme individuals it is very large indeed. Since we must assume a theoretical form of dis- tribution, the form in this case (referring to Spearman's rank-method of computing correlation) must be a rec- tangle, which is a most improbable one." Pearson's method by ** grades." It has been shown that the best assumption we can make concerning the distribu- tion of ability is that it is somewhat " bell-shaped," that is, resembles the normal curve. It happens, therefore, that Pearson has given us a method of computing correlation in which we can use the "grades" (which amount, practically, to "ranks" in actual computation) of each of the measures in the series. There are two points we should clear up, how- ever. (1) The " grade " of a particular individual in a series is measured by the number of individuals above him in the series. (The "rank" indicates the position only.) (2) The theoretical distribution of the measures by which the method is worked out is assumed to be that of the "normal" or "probability" curve. This accords more closely with the actual distribution of abilities than Spearman's assumption. It is possible, therefore, to assume a normal distribution and deduce an expression for r (not Spearman's p or R) MEASUREMENT OF RELATIONSHIP 287 measured in terms of the '' grade^^ of an individual in the series. The expression for the correlation by grades may now be set down as: — (A) r = 2 5mf— pj where (^^ ^=1- iV(iV^-l)^^^ = ^"iV(iV^-l) It will be noted that formula (A) is a sound expression for r, and is unlike Spearman's empirical formula for r, which is (C) r = ^s{n(^p In this expression it must be remembered that -N (iV2_i) 6 is the value that SZ)^ would be under the operation of chance alone. The expression = 2sinl-pj for the correlation of grades, measured in terms of the sum of the squares of the differences of the ranks of all of the measures in the two series, can be shown to be replaced by the following expression when the grades are measured in terms of the sum of the positive differences between the grades in the two series. The formula for r now becomes * For the mathematical development of the theory underlying these expressions the student is referred to the original memoirs by Pearson and his colleagues. (See Appendix.) STATISTICAL METHODS (D) r = 2cos^' "(i^) or in which (E) r = 2cos ^ (1- R)-l o (F) i? = l- '^^ N'-l We thus have two complete formulae for r, when com- puted for *' grades,'* which are sound mathematically and may be applied, providing the distributions of the traits which are being correlated are approximately ** normal,'' These are formulae (A) and (E) above. It is clear that the computation by either one may be shortened a great deal by reducing the work as far as pos- sible to the use of tables. It is evident that this can be done for the transmutation of p and R into r. Tables VII and VIII (see Appendix) are given herewith for that purpose. Having computed p by formula (B), the student can read from Table VII the value of r corresponding to the com- puted value of p. Similarly, for any value of R, the corre- sponding value of r can be read from Table VIII. Steps in the computation of r by ** rank " methods. Re- ferring to the illustrative problem in Table 42, let us list the steps in the computation of r by these so-called rank- methods. 1. Rank the measures in order of size, beginning with the smallest or largest. 2. Subtract the rank of each measure in the first series from its corresponding rank in the second series. Call this D, the dif- ference in rank. Tabulate these as positive, negative, or 0. 3. If formula (B) is used, square each of these differences, giv- ing the column headed D"^. If Formula (F) is used, treat only the positive differences, the gr's of formula (F). 4. Sum the D-'s (or the g's) giving SD^ or '^g. MEASUREMENT OF RELATIONSfflP 289 5. Multiply 2Z)2 or ^gr by 6. 6. For formula (B) divide 6'^D^ by N{N^ - 1). N = total number of measures. In the same way for formula (F), divide 6% by N' - 1. 7. Subtract the quotient in either case from 1. This is p for the first method, R for the second. 8. Transmute p into r by reading proper value from Table VII. Transmute R into r by reading proper value from Table VIII, In the illustrative problem it is noted that r= .732 by formula (F), and .717 by formula (B). The conclusion drawn from either one would be the same. In general it may be said that the two formulae give fairly comparable results, and that from the standpoint of ease of computation the "Footrule*' formula 6% R = l- N'-l may well be the one chosen for use. For small values of iV, the only cases after all in which the rank methods are to be used, they lead to as sound conclusions as any of the more accurate methods, the product-moment or correlation- ratio. Discussion of rank methods of computing correlation. The^r^^ and principal criticism of Spearman's rank method has been indicated above, namely, that it assumes a rec- tangular distribution and an equal unit of rank throughout the scale. These assumptions are inadmissible. Second, Pearson has shown that when the number of cases is small, Spearman's R retains the same value for very wide variations in p. Third, he has shown that the probable error of a zero correlation obtained by Spearman's R is considerably larger than that obtained by his r, — hence that "rank" correla- tions are less accurate than "product-moment" correlations. He says, " In particular it requires about 30 % more obser- 290 STATISTICAL METHODS Table 42. Comparison of Expenditures per Pupil in Aver- age Daily Attendance for Various Specific Kinds of Educational Service. Computed from the records of the United States Bureau of the Census (Financial Statistics of Cities) and United States Bureau of Education (An- nual Report) for the year 1912 * To illustrate Computation of Correlation by " rank " methods. Salaries of Teachers Expenditure per pupil Rankin' expenditure Difference in rank D CUy •^1 II 1^ ^1 + - 2D» Baltimore Boston Cleveland Detroit Jersey City Kansas City . . . Los Angeles .... Milwaukee. ... Minneapolis Newark New Orleans . . . Philadelphia . . . New York Pittsburgh San Francisco . . Seattle St. Louis 22.43 32.13 23.50 28.38 25.24 26.49 33.77 29.91 31.30 20.17 22.17 24.07 36.15 31.59 32.63 34.32 26.30 21.76 29.18 28.57 28.91 23.96 25.43 41.14 31.41 31.33 28.32 22.90 22.80 30.66 21.03 32.44 39.58 28.66 15 5 14 9 12 10 3 8 7 17 16 13 1 6 4 2 11 16 7 10 8 13 12 1 4 5 11 14 15 6 17 3 2 9 1 2 1 2 2 5 11 -4 -1 -2 -4 -2 -6 -v2 -1 -2 1 4 16 1 1 4 4 16 4 36 4 4 25 121 1 4 24 -24 246 6 22)2 = 1 .30 .70. = 1 6.246 _ 1476 17 (288) 4896 From Table VII, for p = .70, r 6.24 _ _144 288 288 From Table VIII, for R = .5, r .72. l-.5= 5. Compare r = .72 and r = .73 obtained by the two methods. * Rugg, H. O. Public School Costs and Business Management in St. Louis. (Report of the St. Louis School Survey, 1917.) \ MEASUREMENT OF RELATIONSHIP 291 vations by the R method to obtain r with the same degree of certainty when r is 0." Fourth, Spearman's transmutation formula r = Sini^ . R) was obtained empirically from 111 correlations with only 21 cases (N = 21). Brown suggests that the chance that the formula thus selected empirically with but 21 cases was the best one, could not have been great. Many like formulas would have fitted equally well. We should use that formula which has a sound mathematical basis. In general we may say, that with 30-100 cases or more, that where accuracy is desired in relationships the product- moment method should be used. It gives definite averages (means) and measures of variabiHty, and when tabulated in table form gives a definite perspective of the distribution of measures themselves. In the interpretation of the co- efficient it is of great value, — in fact is positively necessary to the adequate interpretation of r. Furthermore, by the use of the correlation table the correlation ratio, v can be com- puted, which is a necessary step in determining the line- arity of regression. Again, ranking the measures introduces a "spurious homogeneity" which may effect the accuracy of our later interpretation and conclusions. We can thus lay down a rule: USE THE RANK METHOD ONLY WHEN N is small (say, less than 30). In such cases the means and the standard deviations are of little value, owing to the size of the P.E.'s. The result in cases of this sort can at best only indicate the EXIS- TE NCE of correlation and Not the Closeness of the Rela- tionship. Therefore we must be extremely cautious in our interpretation of rank correlations, or of any correlations computed for a small number of cases. 292 STATISTICAL METHODS Summary Outline of Methods of Determining Relationship It will pay us, at this point, to summarize in outline form the methods discussed to date, indicating their proper func- tions : — I. Methods of Computing Relationship between Series of Meas- urable Quantities, (Statistics of Variables.) 1. Methods which take FULL ACCOUNT of the ABSO- LUTE VALUE and POSITION of every measure of the series. A. The case of Linear Regression, i.e., the line best ** representing" the mean points of the individual columns of the correlation table is a straight line. The proper method with N larger than, say, 30 to 50, is the product-moment method ^~~i^ Na-xO-y with the consequent regression equations of the lines of the means of the columns and rows y = r — X, and x = r — y. Cx o- B. The case of Non-Linear Regression, i.e., the case in which the line that best represents the mean points of the correlation table is not approximately a straight line. The proper method is the "correlation- ratio,"' rj, method of Pearson: — V=-orV=^ iV 4 2 ]S{vAy^-y)'] ay Methods which take account only of the position of measures in the series. A. Various methods of Ranks and Grades. a. Ranks. 1 . Spearman's Method of Rank Differences. MEASUREMENT OF RELATIONSHIP 293 _ 6SZ)2 ^~^~ N{N'-l) 2. Spearman's "Footrule" for Correlation. b. Grades. Spearman's Transmutation formulae are not correct, so we need : — 1. Pearson's Method of Correlation of Grades. (a) r = ^sin{^-p) m which P = l- ^^^,_^) (6) r = 2cos'^(l-R)-l in which i2=l- ^ ^ 3 ' iV2_i Use Tables VII and VIII (see Appendix) for transmutation to r. Rough approximation methods. The methods discussed in the foregoing sections have been of two types: (1) refined methods which take full account of the absolute value and position of each pair of measures; (2) those which take ac- count only of the rank or position of each pair of measures. There is available to the student, however, a group of rough methods even more approximate in character than the methods of "ranks." These methods take account of posi- tion of the measures very roughly by classifying the meas- ures with reference to some average point in the two series. We list these methods next, in this outline, prior to discuss- ing them. B. Various methods of Fourfold Tables. 1. Pearson's: _ r = cos , 7= TT Vad+Vbc 2. Sheppard's Method of Unlike-Signed Pairs: U r = cos- — —IT L+U 294 STATISTICAL METHODS The methods aheady discussed in the book have dealt with the statistics of variables, — with problems involving measured quantities of the continuously varying type. It was pointed out in Chapter IV that the student would meet types of problems in which the presence or absence of cer- tain traits would be noted (counted) and in which the cor- relation methods adapted to statistics of variables would not be applicable. These problems were pointed out under the name "statistics of attributes." Various attempts ^ have been made to devise coefficients which would measure relationships in these types of problems. Most successful of all has been Pearson's coefficient of mean-square-con- tingency with which we shall close the discussion of relation- ship. Thus, to complete the outline: — II. Methods of Computing Relationship between Series of Non- Measured Traits. (The Statistics of Attributes.) 1. Pearson's Method of Contingency. B. Methods op Computing Relationship for Fourfold Tables 1. Pearson's cos "^ method The correlation between the two series of (17) measures in Table 42 was computed by taking account of the relative position, or rank, of each measure in the two series. In this work there was no attempt to measure relative changes in value of the measures, except as these were gross enough to change relative ranks. It is evident that a still shorter 1 Yule has devised a "coefficient of association," Q, for fourfold tables. (See Yule, G. U., An Introduction to the Theory of Statistics, chaps, ii, iii, IV, V.) Pearson, K., and Heron, D. {Biometrika, vol. 9, pp. 159-3L5) have shown that this coefficient is unstable and rarely leads to sound measures of rela- tionship. Its use is not recommended to the student. MEASUREMENT OF RELATIONbx. method of computing the extent of relationship coula ,. vised by finding an average of each series of ranks, and coiix paring the position of each pair of measures with respect to being above or below that average in each series. To do this results in turning the ranking of the two series of measures into a "fourfold table." Tables 43 and 44, and Diagram 50 illustrate this fact. Table 43. Rank OF Measures IN Two Se- ries .^ .^ § "«. OS City •S. .S »«•« S £. |l C^^ A 1 6 B 2 2 C 3 1 D 4 3 E 5 7 F 6 17 a 7 5 H 8 4 I 9 8 J 10 12 K 11 9 T. 12 13 M 13 15 N 14 10 15 16 P 16 14 Q 17 11 Table 44. Rela- tive Positions OF EACH Pair of Measures with Reference to Average of both Series 1 ll .1 il i1 •S-S |i (a) A B C D B G H I id) J L M N P Q (b) F (c) K 8 7 1 1 Diagram 50. Illustrating Grouping of Measures e K ^ §p abode FGHI* ABODE GHI*K Median JKLM NOPQ FJLM NOPQ * With an odd number of cases the middle case must arbitrarily be placed either above or below the median. Condensing the measures into the number of cases and remembering that in Table 44 — STATISTICAL METHODS number of cases above the average in both series, ^a) = number of cases below the average in both series, (b) = number of cases above in first series and below in the second series. (c) = number of cases below in first series and above in the second series, we have : — + + a= 8 + - c= 1 - + b= 1 d= 7 It is clear that such a method of finding correlation takes inadequate account of either position or value of the meas- ures in those cases in which the form of the two distribu- tions is not closely the same. For those cases in which the measures are distributed over the scale in approximately the same way, this rough method will supply an adequate measure of correlation, provided a single index can prop>erly be devised for the amount of relationship. Pearson's formula is Vbc Vad + Vbc 7r* Applying this formula to the problem in Table 43, we have VT 1 r = cos , — 7= TT = cos V5^ _j_ Vl 8.48 = cos 21.24° = .932 .118^ * TT = 180°; the student should have a table of natural trigonometric functions, from which to read the value of r for various values of the angle. This is supplied in the Appendix. MEASUREMENT OF RELATIONSHIP 297 It will be remembered that r by the rank methods gave .717, and .732 by the product-moment method. In general, such approximate methods should be used for only rough preliminary examination. 2. Sheppard's method of unlike signs Sheppard has suggested an approximate formula for roughly measuring relationship in fourfold classification in terms of the percentage of cases that are of like or unlike "signs" in the two series of measures. To get this expression, substitute in Pearson's formula Vbc r = cos -y= 7^ Vad + Vbc for the square root of the product of the be cases, the percent- age of cases having unlike signs (call this U) ; and for the square root of the ad cases, the percentage of cases having like signs in the two series (call it L), This gives at once Sheppard's formula U (N) r = cos L+U Now, Z + Z7 always is 100, and tt is 180°. Hence we may reduce the formula to r = cos U 1.8°. Whipple ^ points out that U must lie between 50 and for positive, and 50 and 100 for inverse correlations, and that therefore it becomes possible to prepare a table from which values of r for any integer of U may be read directly. This table is given herewith as Table IX, Appendix. The P.E. of this * American Journal of Psychology, vol. xviii, pp. 322-25. 298 STATISTICAL METHODS On account of the very large P.E. involved in its use, the method of unlike signs must not be used in important correlation work unless the correlations are high (exceeding .50); the classes are very jSne, and the number of cases fairly large. Its real function is one of preliminary investiga- tion only. On the other hand, since it involves arranging, in order of size, all the measures in the series, the device is hardly serviceable with large numbers of cases (say 70 to 100 and upwards) . For series of 30-50 measures, it might well be used as a method of preliminary investigation of relation- ship. To illustrate the employment of these methods, Whipple cites an example in which the correlation is desired between the accuracy with which 50 boys can cancel e from a printed slip, and the accuracy with which the same 50 boys can can- cel Qy r, 5, and t from a similar slip. The results of each test are first arranged in order, the least accurate boy first and the most accurate last. We can either determine the average, in which case all the boys that rank below the average are minus and all that rank above the average are plus, or we can take the median value and consider the first 25 boys in each array as minus and the second 25 as plus cases. The following values were obtained : — a = 18; 6 = 11; c = 8; cZ = 13. Hence U = 38. By the use of either short formula, r = .37 with a P.E. of .26. By using Pear- son's product-moment method we obtain, for the same arrays, r = .47 with P.E. of .06. By actual timing, after the distribution had been made, the first method occupied eight minutes and the second two hours and fifteen minutes, even with the adding machine and the tables previously mentioned. On the other hand, it will be noticed that in the above problem the correlation of .37 with P.E. of .26 has abso- lutely no significance at all, whereas the product-moment value of .47 with P.E. of .06 is satisfactory. Furthermore, MEASUREMENT OF RELATIONSfflP 299 it should be pointed out that practice in the tabulation of double-entry tables and computation of r by the short method will cut down the time of computation very mark- edly. Thirty to forty -five minutes should be ample for the computation of r in the above problem. III. Methods of Measuring Relationship between Series OF Attributes 1. Pearson's coefficient of mean square contingency (C) In the foregoing sections methods have been described for treating two kinds of data. The first type was data which have been collected in the refined measurement of human traits (known as Statistics of Variables). Both refined and approximate methods of treating such measures have been discussed; i.e., detailed regression methods, and approximate rank and fourfold methods. The second type of data is that in which we merely count the presence or absence of traits (as when pupils in school pass or fail, are tall or short, are normal or feeble-minded) or in which at the most we classify the data in several groups, without specific quantitative measurement (such as is illustrated by the tables showing relationship between mental age and pedagogical age, in Chapter IV) . These kinds of statistics have been called the Statistics of Attributes. It is clear that the methods designed to describe relationship between measxu*ed quantities are not applicable to the statistics of non-measured traits. The coarsest method of measuring relationship between such traits is to classify them in a fourfold table, and to treat them by Pearson's or Sheppard's fourfold methods. The weaknesses of these methods already have been pointed out. We need methods which will take, cognizance of the classification of measures into several classes and which will be mathematically consistent (as we increase the fine- 300 STATISTICAL METHODS ness of classification) with the estabhshed theory of the re- lationship between variables. Pearson's coefficient. Such a method is supplied by Pear- son's coefficient of mean-square contingency — It is built up by reference to the theory of probability, and measures relationship in terms of the difference between the numbers of measures actually found in the various com- partments of the cor- relation table (or "contingency" ta- ble more generally) , and the numbers that might be expected there by pure chance. In Diagram 51 and Table 45 let nr represent the total number of measures in any row of the table, nc rep- resent the total number in any column, N repre- sent the total num- ber in the table, ^. A/ tuo//?(/rnb number ^h To/b/Z/ihere er/n co/np orrment expecfecf ha nee '^c N Diagram 51. To illustrate nc Wr, n, Urc, rir Tic N IN THE Computation op the "Contingency Coefficient" C and n^c represent the number in the compartment determined by such a row and column. Our first task is to state the number of measures that ou^ht to fall in any compartment (say the one determined by the row marked Uf and the column marked n^) by pure chance. MEASUREMENT OF RELATIONSHIP 301 This can be stated by first stating the probability that any one measure will fall in that particular compartment. Now, the probability that a particular measure will fall Tl anywhere in the row marked n, is -^ and the probability that a particular measure will fall anywhere in the column marked Tl Tic is — ^. Hence the probabiUty that any one measure will fall n n in both this row and this compartment will be -^ (the probability of a compound event happening is the prod- uct of the probabilities of the separate events.) But we wish the NUMBER of measures that ought to fall in this particular compartment. Since there are A^ measures in the table, this must be N times the probability that any one will fall there. Thus the number that might be expected to fall there by pure chance is Since n^ represents the number that actually fell in that compartment, the difference between the two is . UrTlc A = „..- — Pearson suggests that a coeflScient can be built which will measure relationship by finding the ratios of the differences between the number that actually fall in any compartment and the number that might be expected to fall there by pure chance to the number that might be expected to fall there by pure chance. That is by summing the ratios — rtrric ^^^"Iv" (1) N 302 STATISTICAL METHODS We cannot simply add the differences together, for the sum of the values of A must be zero (some A's are negative, and some are positive), and so we square each of the dif- ferences and sum them. If, then, we compute A for each compartment, square it, and compute the ratio of each A^ to the corresponding value which is to be expected by pure chance, we can write Pearson's expression for " square-con- tingency" which will be represented by X^, thus: — ^^<^ (nrnc\' rirUc To give Pearson's mean-square-contingency y <^^ divide this expression by A^ — (3) we must ^ N ( nrnc^" N (3) In terms of x^ Pearson's coefficient of square-contingency is In terms oi ^ his coeflBcient of mean-square-contingency is, since <^^ ~N'^~\1 <^2 + <^2 (5) It is evident that C is if the two traits are not correlated, and that it approaches more nearly towards unity as x^ increases. C is always positive, and no sign should be at- tached except for conventional purposes. Yule shows ^ that such coefficients, when " calculated on 1 Yule, G. U., An Introduction to the Theory of Statistics, pp. 65 and 66. MEASUREMENT OF RELATIONSHIP 303 different systems of classification, are not comparable with each other. It is clearly desirable, for practical purposes, that two coefficients calculated from the same data, classified in two different ways, should be, at least approximately, identical. With the present coefficient this is not the case: if certain data be classified in, say, (1) 6 X 6-fold, (2) 3X3- fold form, the coefficient in the latter form tends to be the least. The greatest possible value is, in fact, only unity if the number of classes be infinitely great ; for any finite number of classes the limiting value of C is the smaller the smaller the number of classes." Yule then shows that Pearson's coefficient of mean-square- contingency may be replaced by another which is easier of computation, thus : — N which may be written X^= J ""-^^^^rsL { -N (6) For simplicity of statement let the expression (7) UrUc N be represented by S. Then x'^ S-N Then yiN + x' \N + S-N \ S-N (8) 304 STATISTICAL METHODS This expresses C in terms much easier of computation, and formulas (7) and (8) should be used by the student in com- puting the relationship between two traits by ** contingency." Yule next shows that if we deal with a t X t-fold classi- fication of data in which the relationship is perfect, "all the frequency is then concentrated in the diagonal com- partments of the table, and each contributes N to the sum S. The total value of S is accordingly t N and the value of = t-1 (9) This is the greatest possible value of C for a symmetrical t X ^-fold classification, and therefore, in such a table, for t = 2 C cannot exceed 0.707 t = 3 « << 0.816 t = 4 (( « 0.866 t = 5 « « 0.894 t = 6 " " 0.913 t = 7 « i< 0.926 t = 8 (( « 0.935 < = 9 " " 0.943 t= 10 « « 0.949 It is well, therefore, to restrict the use of the * coefficient of contingency ' to 5 X 5-fold or finer classifications. At the same time the classification must not be made too fine, or else the value of the coefficient is largely affected by casual irregularities of no physical significance in the class-fre- quencies." Steps in the computation of the coefficient. Taking for- mula (8) C = sJ' S-N S we next make clear the steps in the computation of the coefficient. The arithmetic work reduces to four main MEASUREMENT OF RELATIONSfflP 305 steps: (1) finding S; (2) subtracting A^ from S; (3) divid- es — iV ing S — N hy S; (4) extracting the square root of — . The detailed procedure is as follows : — A. Find s N This involves four steps: (1) Square the number found in each compartment of the table: (rircy [e.g., 4, 49, 9, 1, 1, for the first row of Table 45.] (2) For each compartment in the table multiply the total number in its column by the total number in its row, (nrHc) and divide each product by (A^), the total num- ber in the table. For example, for the illustrative problem for the compartments in the lowest row : — 82 '^=- 82 ■-^,"— "X"_.„ . ♦ 82 It will probably save time and reduce errors of computation to tabulate these results separately as given by Table 46 below. (3) For each compartment divide the result of doing (1) by the result of doing (2). For example, for the top row, — 4 TTTTc = 1-41 2.83 49 _^ = 10,etc. 306 STATISTICAL METHODS (4) Sum each of the results obtained by doing (3). This gives S. B. Subtract (A^), the total frequency of the table, from S, giving S- N. C. Divide S-NhyS. S-N D. Extract square root of — ~ . This gives C, the coeffi- cient of mean-square-contingency. Table 45. Relation between Mental Age and Pedagogical Age {Computed by coefficient of mean- square-contingency) Mental Age in Years 9 10 11 12 13 U 15 Totals p e d a g Retarded 2 years Retarded 1 year 1 2 4 9 7 3 2 1 11 18 g Normal 3 8 4 1 16 i c a 1 A Accelerated 1 year 5 10 6 2 23 Accelerated 2 years 7 3 1 1 14 g e ' 2 13 16 21 16 11 3 82 nrUc For each compartment compute — , giving the data in the convenient form shown in Table 46. MEASUREMENT OF RELATIONSHIP ' 307 nrUc Table 46. Data giving Results of Computing iV FOR EACH Compartment of Table 45 Mental Age 9 10 11 12 13 U 15 Fed a Retarded 2 years Retarded 1 year 2.85 2.82 4.61 3.51 1.48 2.42 .40 .66 gog Normal 3.12 4.10 3.12 2.15 cal Age Accelerated 1 year Accelerated 2 years .34 3.65 2.22 4.49 2.73 5.89 3.59 4.49 2.73 These are computed as follows, for the top row: — 21X11 82 To compute = 2.82 21 = Wc n = nr 82 = iV (nrc)'' WrWc N ¥2.82 =1.42 2%.65 = 6.85 4%.48= 33.14 100,4.49 = 22.27 Mo = 10 3%.89 ^4.49 = 6.11 = .891 ¥2.85 = 0.351 i%.6i = 3.471 y.34 = 11.735 ^Va.Bi = 23.08 *%.22 = 22.07 %.42 = 3.727 %.73 = 3.295 V66 = 1.515 ¥3.59 ¥2.73 = .279 = .367 %.12= 2.88 «%.io =15.61 Total = =s = 174.656 i%.i2 = 5.13 N = 82. ¥2.15 = 0.465 s- -N = 92.656 / 92.656 / 174.656 c = \ V.5305 = .728 308 STATISTICAL METHODS ILLUSTRATIVE PROBLEMS i 1. (a) Plot to scale on cross-section paper the following pairs of meas- ures which show the relation between ability of pupils in each of two tests in first-year algebra. Plot Test II on X and Test I on Y. Arrange the work so that this problem (1) and the next problem (2) can be placed on one cross-section sheet. Test 1 27 27 27 16 27 18 27 9 15 15 21 20 26 10 22 24 16 13 23 Test II.... 20 18 14 3 13 3 16 3 3 7 8 17 2 9 20 2 6 9 16 25 22 14 17 25 22 5 6 12 11 3 7 17 4 2 Test 1 15 22 20 17 21 20 20 15 22 23 27 Test II.... 2 11 6 8 19 7 9 5 8 16 18 Test 1 25 18 27 20 18 27 24 24 24 22 21 20 20 20 24 Test II.... 15 4 23 12 5 22 10 9 12 10 10 13 13 14 13 (b) Plot these same pairs of measures having grouped them in class- intervals of 2 units each. (c) Turn this "point representation" of the pairs of measures into a corre- lation-table, with totals stated on both axes. Use another cross-section sheet for this table, and arrange the work with the tabulation in the upper left- hand corner. '2. (a) Plot to scale on cross-section paper the following pairs of measures, which show for United States history the relation between the cost of instruction per 1000 student-hours and the average size of class. Plot the costs on Y and the size of class on X. Cost 134 114 26 35 25 62 55 47 46 49 48 55 56 59 61 72 106 Size class... . 11 10 38 37 36 23 22 25 24 25 24 22 23 25 24 15 14 Cost 87 91 114 111 47 53 57 09 35 42 58 31 39 44 105 65 62 Size class.... 15 14 12 13 20 21 20 21 27 26 27 29 28 28 17 16 17 Cost 88 165 137 61 65 72 77 50 38 43 30 40 49 70 Size class.... 12 13 15 19 18 19 18 25 24 30 33 32 33 20 (&) Plot these measures having grouped them in intervals of 2 units. (c) Turn this "point representation" into a "correlation-table." Ar- range in upper left-hand corner of the page. Use separate cross-section sheet for (3). Put (1) and (2) on one sheet. 3. Plot the "lines of regression" of the columns and rows for each of the correlation tables plotted in Problems 1 and 2 by the approximate method; (i.e., compute the means of the columns and rows and draw the lines of re- gression by "cut and try.") * Quoted from Rugg, H. O., Illustrative Problems in Educational Statistics, published by the author to accompany this text. (University of Chicago, 1917.) MEASUREMENT OF RELATIONSHIP 309 4. Compute the correlation between the following pairs of measures (scores made by pupils in two a'lgebra tests) without tabulation in a cor- relation-table, by xy Test 1 27 27 27 16 27 18 27 9 15 15 21 20 26 10 22 24 16 13 23 Test II.... 20 18 14 3 13 3 16 3 3 7 8 8 17 2 9 20 2 6 9 Test 1 15 22 20 17 21 20 20 15 22 23 27 16 25 22 14 17 25 22 5 Test II 2 11 6 8 19 7 9 5 8 16 18 6 12 11 3 7 17 4 2 Test 1 25 18 27 20 18 27 24 24 24 22 Test II.... 15 4 23 12 5 22 10 9 12 10 5. For the above data compute the coefficient of correlation by the Spearman "Rank-Coordination" and by the "Foot-Rule" methods. 6. For the above data compute the coefficient of correlation by the " cos it" method, and by Sheppard's method of " unlike-signed-pairs." CHAPTER X USE OF TABULAR AND GRAPHIC METHODS IN REPORTING SCHOOL FACTS Studying vs. reporting facts. Each chapter of this book has pointed out specific uses of graphic and statistical meth- ods in school practice. Chapter I, especially, gave atten- tion to the use of such methods in the current attempts to solve school problems, by giving typical examples. The use of statistical and graphic methods was shown : in the con- struction and use of standardized tests; in the preparation of forms for recording school statistics; in the supervision of the teaching of school studies; in the detection of weak- nesses in the course of study, and in teaching methods by means of studies in failures; in the comparative method of studying school costs, as shown by Bobbitt's and by Upde- graff's early devices; in the use of the probability curve in marking pupils and in standardizing school tests; in the dis- tribution of intelligence in the public schools; and in the use of correlation methods. Throughout the book, the ap- plications have been illustrative of the more refined statis- tical and graphic methods that can be used in the carrying on of school research, and in the reporting of results to readers technically trained in statistical methods. The school man, however, having made use of various fairly refined methods in studying his problems, faces the problem of reporting the status of his schools to a public that is, in part, neither trained in the rudiments of statis- tical method nor familiar with the general conditions of public school administration to-day. USE OF TABULAR AND GRAPHIC METHODS 311 It has been decided, therefore, to conclude the discussion in this book by presenting, in outhne form, a representative selection of examples of the application of various tabular and graphic methods of reporting school facts. There is available in print no systematic statement of such methods, brought up to date. Now that school men are beginning to study problems of school administration scientifically, — now that they really are beginning to build up a quantitative knowledge about school conditions, — they are recognizing at last the definite need of ways and means of reporting the facts to the public. School men face no greater problem to-day than that of determining best ways to tell the non- teaching public about the status of schools, and to make clear to them the necessity for doing something about it which will conduce to the definite advancement of school practice. In reporting school facts to the public we must therefore distinguish the interests and technical equipment of the persons to whom we are reporting our facts. That is, we must recognize that the methods that we should use in re- porting experimental and statistical studies to a technically trained group of school people must necessarily differ from the methods with which we should report facts concerning school practice to the general lay citizenship. The most immediate technical agency (aside from news- papers) for acquainting the public with school conditions is the annual school report. The remainder of this chapter will, therefore, be devoted to a discussion of the form and content of the annual city school report. School reports are planned and printed to reach three classes of people. These three classes are: (1) administra- tive officers, teachers, and other school employees, w^hq should be informed of the conduct of school affairs through- out the entire system for the year just finished; (2) pro- 312 STATISTICAL METHODS fessional school officers (interested in either the educational or business aspects of school administration) in other school systems, bureaus, foundations, and professional schools, who are active in studying comparatively the various problems of school administration; and (3) the board of education itself and the more intelligent lay public, whose general insight and educational interest can be depended on to support campaigns for the betterment of the public schools. Kinds of material that should be included. This clearly must be determined by the aim in mind in attempting to reach the various classes of people to whom the report is to go. It undoubtedly will be agreed that a school report should supply: (1) those facts that can be interpreted and used so as to improve school practice directly, by contributing to the betterment of instruction; (2) those facts that can be interpreted and used so as to improve school practice in- directly, by contributing to the improvement of the work of a non-educational department (buildings or supplies, for example) ; (3) those facts that will be comprehended by and will stimulate an interest on the part of the general public in the community, and will result in the support of better schools; (4) those facts which will acquaint the public, in accordance with law, with the condition of school property and of school finance in the city. It can be seen, therefore, that the criteria of interpreta- hility and of use should largely govern the content of a school report. The questions to be asked in making up the report should be: Can this statistical table be interpreted so as to improve some phase of school practice .f* Does it provide comparative material of which other school systems or students of school administration can make use? Can these data be understood by the public, and has the interpretation and explanation been made so complete and clear that this report will operate as a means of "educating" — as well USE OF TABULAR AND GRAPHIC METHODS 313 as informing — the public to active support of the public schools? Again it undoubtedly will be agreed that a school report should contain material of three distinct types : (1) Current material : It should report the local situation in sufficient detail to explain the significant developments of the cur- rent year and the present condition of the public schools. (2) Historical material : It should present enough historical statistics concerning the growth of important phases of the schools' work to permit a discussion of particular aspects and of relative efficiency of the activity of certain depart- ments. (3) Comparative material : It should contain com- parative data of the procedure of other school systems working under similar conditions. Lacking an absolute standard for judging the efficiency of school practice, the common practice of many cities may well serve as a check upon the methods employed in any one. Statistical material must be interpreted by descriptive material. To include pages of statistical material with no interpretations or comparisons is, for the layman at least, a waste of printer's ink. All tabular and graphic data should be interpreted clearly, either by the officer who publishes the material, by the superintendent of the schools, or by some other officer especially appointed in the system to study ways and means of improving the conduct of school business, — for example, the director of the " bureau of school research and efficiency." Thus, school reports which have been very largely "statistical" and "informational" should become "educational" in the widest community sense. The school report in a city system can be made a valuable instrument for the promotion of school work in the city. To become that, however, it at least must conform to the foregoing fundamental criteria. Important criteria concerning the form of the school 314 STATISTICAL METHODS report. We may discuss briefly the more important questions arising in connection with the form of the annual school report. The first question to be settled is this: Shall the school report be one volume appearing annually, biennially, or less frequently, or shall it be published in the form of a series of short monographs, each of which discusses one phase of school work? The traditional school report is a com- posite volume made up of general descriptive articles by edu- cational officers of the system, put together in one portion of the report, and followed by a large mass of statistical data, as a rule completely uninterpreted and, on the whole, uninterpretable by the general public. Such a volume, in- quiry has shown, is almost never read by any portion of the pubhc. A great advance has been marked out by the recent inno- vation begun by Superintendent F. E. Spaulding, while he was Superintendent of Schools at Minneapolis, Minnesota, in the publication of a series of monographs describing in clear language, and pertinently illustrated by graphic and comparative statistical methods, the status of educational activities of the city of Minneapolis. In the quotations of this chapter, we shall make several references to this ex- cellent practice. We cannot decide the question of the general form of the school report, however, without taking account of the question of the frequency with which school facts need to be reported. Should all data regarding school practice be reported annually, or are there types of facts which may well be published but intermittently.^ Classes of school facts. We can distinguish school facts, therefore, in two classes : — First, those that are reported annually. These may be summarized as follows: (1) facts that either state or munici- pal law requires must be published each year, concerning the USE OF TABULAR AND GRAPHIC METHODS 315 extent and condition of school property; (2) current and his- torical local statistics concerning the financial condition of the board of education, the distribution of pupils according to ages and grades, the enumeration of children of school- census age in the city, and the detailed reporting of statistics on the teaching staff; (3) facts concerning the progress of educational experiments or innovations that have been es- tablished prior to the current year, and in which the public will have a definite interest: for example, new methods of detecting defects in children, and of providing for them; special forms of instruction; new developments in voca- tional education; etc.; (4) information concerning the es- tablishment of new educational experiments, — important and far-reaching changes in the administration of the local schools, etc. Second, school facts that are reported intermittently. It frequently occurs that it is necessary for the superintend- ent and the board of education to give the public detailed and specific information concerning needed enlargements, greater financial support of the schools, etc. For example, our larger cities are all feeling the need for increased revenues for permanent improvements to the school plant. School populations are increasing, and the consequent demands on our public school facilities are likewise increasing, usually more rapidly than are the revenues made possible under state law by the increase in real wealth in the community. City school boards are finding it imperative, therefore, to go to the people for authority to bond the school district in order to finance the additional school plant which is needed. This necessitates an educational campaign, and this in turn demands a special kind of school report. This report may well give facts to the public that ordinarily will not need to be given each year. For example, a detailed compara- tive analysis of the status of school finance in this particular 316 STATISTICAL METHODS city with that in other comparable cities, together with an analytical study of the historical development of various aspects of school finance may well be needed. We shall point out, later on, illustrative methods of reporting such studies. School facts that should not be printed at all. Careful examination of current city school reports reveals the pub- lication of many types of statistical and descriptive material that ought not to be printed at all. This can be illustrated partially by listing specific types of non-usable statistics published in the annual report of one of our largest and most progressive school systems: (1) tables of total values of sup- plies delivered to various types of schools (of little value unless reduced to some unit basis, and presented histori- cally); (2) analyzed statement of total cost of transporta- tion by schools for current year; (3) a table, twelve pages long, giving itemized amounts of each particular kind of supplies delivered to each building in the system; (4) list of textbooks lost or destroyed in district schools, giving names of the books, number, and price of each; (5) number and money value of condemned books, together with rebound books, by specific title, number, price, etc. ; (6) list of text- books, giving name of book, number in usable condition in all public schools, price, value, etc. (16 pages) ; (7) names of pupils graduating from various schools; (8) names and facts concerning all teachers and other officers on the staff; (9) detailed statement of total expenditures for particular activ- ities for each building in the system (as "totals" the table is uninterpretable; it might be condensed to small fraction of present size, 44 pages; it ought to be reduced to a per- pupil basis); (10) detailed statements concerning cost of particular activities and special schools, giving totals and itemized expenditures, etc., — might well be condensed and published as "unit" costs. USE OF TABULAR AND GRAPHIC METHODS 317 I. Content of the Annual School Report: Sugges- tive Examples of Tabular and Graphic Methods The foregoing introductory discussion can now more im- mediately be focused upon the specific organization of the content of the school report. As we proceed with the dis- cussion, in each case we shall point out whether the material should be annually reported or reported at intervals of several years. 1, Legal ba^is of the local school system Form of organization. The introductory statements should contain a table of contents, with a pertinent list of subheadings, to make clear to the reader the important points discussed in the report. This should be followed by a brief text statement describing the legal basis of the system. The reader should be told the important facts concerning the origin, development, and present legal status of the city school district, exactly how its functions are affected by those of city civil district, and what important changes have come about in this legal status. A clear statement should be given concerning the present board of education — its size, how members are selected, the specific powers and duties of the board, the committee organization, tenure, compensa- tion of board members, etc. Legal basis of school finance. This should be pointed out very clearly, answering such questions as : Does the board of education have complete tax-levying power.^ If not, by what agency are its budgets reviewed .^^ ^Vhat are the legal limits of school revenue? Are permanent improvements and current school expenses financed out of taxation? What is the legal status of bonding the school district for school purposes, and of borrowing for temporary purposes on short- term notes? 318 STATISTICAL METHODS The detail into which the annual discussion of the legal basis of school finance should go must be determined by the financial condition and by the current financial powers of the board. A brief statement of the latter is all that is required in an annual report in which special efforts are not being made to effect a change in taxing powers, taxing limits, etc. In case it becomes necessary to make a special plea to the people, the report should go into the legal status carefully. If the critical change needed is to give the board of education complete taxing power, and the board wishes to show the effects of having its budgets reviewed by another govern- mental body, a table such as Table 47 and a diagram such as Diagram 52 might be used, with proper textual explanations. Table 47. Comparison of the Board of Education and Common Council Budgets of Grand Rapids, Michigan, together with Amounts spent for Permanent Improve- ments, 1910-11 TO 1915-16 INCLUSIVE. {Data from Official Proceedings of the Board of Education) Total amount Amount included in com- Year Board of educa- Common council spent for per- mon council budget to tion budget budget manent im- be devoted to payment provements of bonds and interest 1910-11 $201,443.79 $107,897.11 $404,466.14 $49,860 1911-12 183,160.50 121,166.50 245,751.97 78,792 1912-13 103,785.50 97,055.50 157,159.14 80,577 1913-14 100,089.00 100,089.00 89,880.59 64,095 1914-15 273,792.00 126,792.00 249,594.73 101,292 1915-16 233,310.00 98,960.00 545,771.48 77,960 2. Presentation of facts concerning school revenues and expenditures The superintendent of schools annually will wish to make clear to his community the following facts concerning school finance : — USE OF TABULAR AND GRAPHIC METHODS 319 (a) Comparison of total possible school revenue and actual school revenue. This would mean the total possible tax levies for school purposes (computed from the assessed property valuation and the legal limit, in mills on the dollar Thousands of Dollars 800 276 260 200 175 160 125 100 75 50 25 Board of Education Budget Common Council Budget 1910-11 1911-12 1912-13 1913-14 1914-16 1915-16 Diagram 52. Comparison of Board op Education Budget for Permanent Improvements with Budget approved by Common Council of assessed valuation), compared with the actual tax levy for school purposes, and covering a series of years. Throughout the entire school report the presentation should distinguish definitely between school finance for current 'purposes and for permanent improvements. To get the situation clearly 320 STATISTICAL METHODS before the reader, diagrams such as Diagrams 53 and 54 can be used effectively. If tables are to be given, the head- ings and data can be organized somewhat as follows : — Year Assessed property valuation No, of mills Possible tax levy Actual tax levy For current purposes For permanent improvements For current purposes For permanent improvements 1906 1907 1915 In discussing the relation between school-taxing capacity and the degree to which the city is taking advantage of it, a brief table such as the following will make clear the most probable taXative possibilities in future years : — Table 48. Comparison of Estimated Possible School Tax- ing Capacity for Years 1920 to 1930, with Probable Actual Tax Levies * Year Assessed valuation Actual school tax No. of mills possible Amount No. of mills 1920 1925 1930 $203,000,000 243,000,000 283,000,000 $1,001,000 1,276,000 1,551,000 4.98 5.26 5.48 6 6 6 * Example quoted from Rugg, H. O., Cost of Public Education in Grand Rapids, p. 369. (1917.) (b) Sources and amounts of revenue. A table should be printed each year that will present concisely the sources and amounts of revenue during a series of recent years, classified under such headings as: (1) balance on hand; (2) received Hundred Thousands of Dollars 10 / / / / J 1 Possible Levy ( Actual l^evv / / / \ / / / / / 1 .... ^•^^ / .-"- --"'' ■'•''''' / \ / \ / \ / ^ 1906 -07 -08 -09 -10 -11 -12 -13 -14 1915 Diagram 53. Comparison of Curve of Possible Taxation fob General Purposes with Actual Tax Levy, 1906-15 322 STATISTICAL METHODS from state sources; (3) from county sources; (4) from city sources; (5) from miscellaneous sources; (6) total income for annual maintenance; (7) from sale of bonds; and (8) total receipts. It might be well to add another short table giving percentages that each source contributed to the total re- ceipts. Each of these items should be shown for a series of years, at least ten, so as to admit of comparisons being easily made. (c) Relation of revenue receipts to current expenditures. Each year it would be well to publish a table giving the facts for a series of years relating to receipts and expenditures, both for current maintenance and for permanent improve- ments, and showing the financial condition of the board of education by comparing the receipts with expenditures for each and showing the surplus or deficit each year for a number of years. (d) Methods of financing permanent improvements. If a table showing the source of all receipts for a series of years and such charts as Diagrams 53 and 54 have been presented, the degree to which the city is paying for its Table 49. School Bonded Indebtedness in Minneapolis ADDED during THE LaST FiVE YeARS 1911 $1,116,700 1912 500,000 1913 775,300 1914 825,000 1915 1,125,000 Total for five years 4,342,000 Bonds redemmed in same five years 80,000 Net increase 4,262,000 School Bond Issues from 1889 to 1916 School bonds outstanding December 31, 1889 $542,500 School bonds issued, 1889-1916 6,825,000 School bonds redeemed during 26 years 292,500 School bonds outstanding December 31, 1915 7,075,000 USE OF TABULAR AND GRAPHIC METHODS 323 Table 50. Minneapolis Bonded School Debt compared WITH School Debts of Other Cities, December 31, 1915 ^ S "K >- ~--e 1 1 3f Cities of 200,000 popula- *2 "1 ^ 1 ii 1°°?- ►1*^ tion or more — 1915 0*0 B 'Q c 1-s estimate 1-a ill 1 8 1 l5 J3. •2 !; £ "^ (^ E^ a^ OS New York. . . . .5,468.190 $123,425,000 $22.57 $991,219,000 $181.27 12.45 Chicago .2,44^045 39,423,000 16.11 .... .... Philadelphia.. .1,683,604 13,827",000 8.23 104,823,000 62,25 13.19 St. Louis - 745,938 .... 19,579,000 26.24 .... .... Boston . 745,139 16,227,000 2L78 84,423,000 113.34 19.22 Cleveland.... . 6.56,905 6,946,000 10.57 56,242.000 85.61 12.35 Baltimore . 584,685 3,300,000 5.54 67,064,000 114.64 4.96 Pittsburgh... . 571,914 10,703,000 18.71 42,923,000 75.04 24.93 58.53 Detroit . 554,760 8,767,000 15.80 17,563,000 31.64 49.91 60.45 Los Angeles . . . 475,337 8,635,000 18.18 45.696,000 96.20 18.89 65.63 Buffalo . *461,305 7,411,000 16.07 38,095,000 82.64 19.46 70.90 San Francisco . 448,502 5,779,000 12.90 42,172,000 93.92 13.70 Milwaukee... . 428,062 3,537,000 8.27 11,921,000 27.85 29.66 Cincinnati . . . . 406,706 4,588,000 11.27 61,170,000 150.30 7.48 Newark . 399,000 8,922,000 22.36 30,864,000 77.35 28.91 86.15 New Orleans. . 366,484 37,088,000 101.33 Washington . . . 358,679 Bonds not iss specific pur 7,075.000 ued for 6,287,000 17.56 Minneapolis. • 353.460 poses 20.04 19,906,000 56.3Q 35-54 85.45 Seattle . 330,834 4,750,000 14.35 21,807,000 65.88 21.78 Jersey City... . 300,133 4,492,000 14.97 19,397,000 64.66 23.16 Kansas City.. . 289,879 7,823.000 26.98 10,733,000 37.00 72.89 80.51 Portland . 272,833 849,000 3.11 15,980,000 58.53 5.31 Indianapolis . . 265,578 2,007,000 7.55 6,369,00 23.94 31.52 43.00 Denver . 253,161 2,946,000 11.64 Rochester 250,747 1,428,000 5.69 17,996,000 71.70 k'.M 33.49 Providence . . • 250,025 2,097,000 10.79 11,138,000 44.55 24.22 02.28 St. Paul 241,999 1,990,000 8.22 11,359,000 46.94 17.52 50.90 Louisville . 237,012 937,300 4.08 11,995,000 50.61 7.81 30.33 Columbus ... . 209,722 1,457,200 6.94 11,260,000 53.62 12.94 General average of the per capitas. 10.85 66.48 Average of percentages 21.41 01.14 * Estimate, 1914. school property "as it goes" will have been shown. The policy of the city in the use of school bo^ds, and the condi- tion of school and city indebtedness should be shown, es- pecially if a special campaign is being carried on for funds. Tables 49 and 50 show how Superintendent Spaulding pre- Hundred Thousands of Dollars w 9 Possi — Actus ILevi a 1 1 1 1 7 1 1 1 1 1 1 1 I 6 I 1 1 1 5 4 3 / /" ^^^^^ ^^^ -» 2 ■»^ / 1 •^ / 1906 -07 -09 -13 -14 1915 Diagram 54. Comparison of Curve of Possible Taxation for Permanent Improvements with Actual Tax Levy, 1906-15 USE OF TABULAR AND GRAPHIC METHODS 325 sented the data in one of his 1916-17 School Monographs.^ Columns might well have been added giving the rank of the cities in question. Thousands of Dollars 5,000 4,500 ,000 3.500 3,000 2,500 2,000 1,500 1,000 y City indebtedness (including scV lool) 1 OpU^^i ;^^«K*-^^v,^^,^ 1 1 ^v / 1 ^-- « X ''* / y^ \'' / ^ y ^ " — ~ — ^ 500 1890 91 92 93 94 95 96 97 98 991900 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 Diagram b5. Total City and School Bonded Indebtedness, 1890-1915 It is probable that a line diagram of the nature of Dia- gram '55 will do much to clarify such a presentation. If possible the data ought to be given for each year, as in that diagram. The present indebtedness of the school city can be cleared up further by a table giving the total out- standing bonds maturing each year. Table 51 suggests the form.^ 1 For excellent descriptive and graphic methods of reporting such facts see three monographs published by the Minneapolis Board of Education: Financing the Public Schools; A Million A Year; The Price of Progress, 25c each. 2 From Annual Report of Business Manager (1915), Grand Rapids, Mich- igan. 326 STATISTICAL METHODS Table 51. Total Amounts of Outstanding Bonds Maturing Each Year, 1916 to 1930 {Data from 1915 Report of the Board of Education) Year — June SO to July 1 Principal Interest Total 1915-16 $35,000.00 63,000.00 75,000.00 75,000.00 75,000.00 75,000.00 75,000.00 100,000.00 50,000.00 75,000.00 70,000.00 64,000.00 75,000.00 58,000.00 $41,935.00 39,912.50 37,002.50 33,727.50 30,352.50 26,977.50 23,602.50 19,752.50 16,490.00 13,702.50 10,440.00 8,865.00 7,425.00 4,297.50 1,305.50 $76,935.00 1916-17 102,912.50 1917-18 112,002.50 1918-19 108,727.50 1919-20 105,352.50 1920-21 101,977.50 1921-22 98,602.50 1922-23 . ... 119,752.50 1923-24 66,490.00 1924-25 88,702.50 1925-26 80,440.00 1926-27 8,865.00 1927-28 71,425.00 1928-29 79,297.50 1929-30 59,305.50 Total outstanding. $965,000.00 $315,788.00 $1,280,788.00 (e) Capacity of the city to support schools, and degree to which it is doing so. This can be shown by stating the city expenditures, — first, per inhabitant, second, per $1000 of real wealth in the city, and, third, per pupil in aver- age daily attendance. This further calls up the question of comparing expenditures in the local city with those in a group of comparable cities. How often should such a comparative analysis be made? It is probable that lack of clerical assistance will prevent the compilation each year of original data, and the computation of unit costs with consequent "ranking " of cities. It certainly should be done every few years. If it can be done. Tables 52 and 53 ^ and Diagram 56^ suggest the type of comparison that can be made to establish the point at hand. 1 Clark, E., Financing the Public Schools, pp. 27 and 29. 2 Jbid., p. 33. USE OF TABULAR AND GRAPHIC METHODS 327 Table 52. Expenditure per Inhabitant for Operation and Maintenance of Schools in Cleveland, and in 17 Other Cities of from 250,000 to 750,000 Inhabitants, 1914 Estimated popula- inl9H Expenditure for operation and maintenance City Total Per in- habitant pendilure per inhabitant Baltimore Boston Buffalo Cleveland Detroit Indianapolis . . . Jersey City Kansas City . . . Los Angeles. . . Milwaukee .... Minneapolis . . . Newark New Orleans. . . Pittsburgh San Francisco. . Seattle St. Louis Washington.. . . 579,590 733,802 454,112 639,431 537,650 259,413 293,921 281,911 438,914 417,054 343,466 389,106 361,221 564,878 448,502 313,029 734,667 353,378 $1,954,670 5,516,762 2,449,533 3,569,504 2,553,488 1,409,504 1,421,147 1,761,389 3,706,519 1,794,796 2,147,856 2,699,239 1,097,552 3,602,303 1,879,187 1,750,988 4,084,693 2.391,976 $3.37 7.52 5.39 5.58 4.75 5.43 4.84 6.25 8.45 4.30 6.25 6.94 3.04 6.38 4.19 5.59 5.5Q 6.77 17 2 12 9 14 11 13 7 1 15 6 3 18 5 16 8 10 4 Average .... $5.59 Other graphic methods. The four diagrams which follow show means which may be used by superintendents to reveal facts to their constituencies, using graphic instead of tabular methods of presentation. A little thought given to devising such graphic representations at the time of pre- paring the annual report will be time well spent. Chapter II discusses in detail the sources and validity of such comparative school statistics. Cities should always be 328 STATISTICAL METHODS Table 53. Expenditure per $1000 of Wealth for Opera- tion AND Maintenance of Schools in Cleveland, and in 17 Other Cities of from 250,000 to 750,000 Inhabitants, 1914 Estimated true value of all property assessed Expenditure for operation and maintenance Rank in ex- penditure per City Total Per $1000 of property assessed $1000 of properly assetssad Baltimore Boston Buffalo Cleveland Detroit Indianapolis . . . Jersey City .... Kansas City. . . Los Angeles .... Milwaukee Minneapolis Newark New Orleans. . . Pittsburgh San Francisco . . Seattle. St. Louis Washington. . . . $723,800,340 1,489,608,820 494,200,459 756,831,185 598,634,198 363,413,650 257,644,605 371,191,014 836,604,260 511,720,797 639,258,841 383,864,182 314,086,036 789,035,200 1,247,391,284 473,174,995 1,125,308,749 538,389,607 $1,954,670 I 5,516,762 2,449,533 3,569,504 ' 2,553,488 ! 1,409,504 1,421,147 : 1,761,389 • 3,706,519 1,794,796 : 2,147,856 2,699,239 1,097,552 3,602,303 1,879,187 1,750,998 4,084,693 2,391,976 $2.70 3.70 4.96 4.72 4.27 3.88 5.52 4.75 4.43 3.51 3.36 7.03 3.49 4.57 1.51 3.70 3.63 4.44 17 11 3 5 9 10 2 4 8 14 16 1 15 6 18 12 13 7 Average .... $4.12 selected for ranking purposes which are comparable as to (1) population, (2) geographical location, (3) wealth, and (4) legal status. (/) Extent to which city supports schools as compared with way in which it supports other city departments. This can be presented if careful study shows the necessity. The data can be found in an annual publication of the United States USE OF TABULAR AND GRAPHIC METHODS 329 Bureau of the Census {Financial Statistics of Cities). If the data are used, three comparative tables should be given stating: (1) amount spent per inhabitant for various city 1 1 1 2 2 2 3 3 3 4 4 4 5 pKSi 5 B 6 6 7 , 7 7 e 8 6 ^E^H 9 9 10 10 10 11 11 11 12 12 pKiH 13 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 Ea^enditure Ejqjendlture Bqjcnditur© per per $1,000 of per child in Inhabitant taxable wealth average daily attendance Diagram 56. Rank of Cleveland in Group of Eighteen Cities in Expenditure for Operation and Maintenance of Schools Given per inhabitant, per $1000 of taxable wealth, and per child in average daily attendance. (From Ayres, L. P., The Cleveland School Survey, 1916.) 330 STATKTICAL METHODS departments, including schools; (2) per cent of total govern- mental cost payments devoted to various city departments; and (3) rank in per cent of total governmental cost payments devoted to various city departments. Diagram 57 is re- d 4 10 11 1 1 2 _}_ 4 5 7 8 9 10 11 1 1 2 _3_ 4 5 6 G 9 10 11 • 1 • 1 2 3 4 5 6 10 11 t 1 2 3 _4 5 6 7 8 9 11 M S • 1 2 3 4 5 6 7 8 9 11 10 5 n r 1 2 2 3 3 4 4 5 5 6 6 7 7 8 6 9 9 Z 10 i Diagram 57. Rank of Cleveland among Eleven Large Cities in per capita Expenditures for each Principal Kind of Municipal Activity Numbers in black circles show Cleveland's rank, (From Ayres, L. P., The Cleveland Sthoel Survey, p. 26.) USE OF TABULAR AND GRAPHIC METHODS 331 produced ^ to illustrate the departments considered and the method. (g) How the board of education spends its money. The publication of current total and per capita expenditures are of little value, unless they are accompanied by historical and comparative statistics. The school report should give : — First. The total amounts spent, and the amounts spent per pupil in average daily attendance, for (1) all current expenses, and (2) permanent improvements. This can be pictured clearly by a chart after the form of Diagram 53. Second. The total and per pupil (in average daily attend- ance) 2 expenditures for all educational purposes, as con- trasted with all business purposes and the per cent devoted to each. Ranks of the cities should be given for each table. A five-year table giving the relative expenditures in the local city might well be included. Third. The degree to which the board supports different kinds of educational service: (1) for the larger aspects, such as administration, supervision and instruction, operation of plant and maintenance of plant; and (2) for specific kinds of service, such as board of education office, superintendent's office, salaries of supervisors and their clerks, salaries of principals and their clerks, salaries of teachers, stationery and educational supplies, wages of janitors, fuel, water, light and power, and repairs. For each of these items the reporting should be done in terms of (1) total amount spent; (2) amount spent per pupil; (3) per cent of total expendi- tures devoted to each; and (4) rank of all cities in the list, for the expenditures for each item, in order to compare the local city with other cities of its class. ^ ^ Ayres, L. P., The Cleveland School Survey, p. 26. 2 For items to include under each see Clark, E., Financing the Public Schools, p. 65; or Grand Rapids School Survey, p. 388. Detailed tables and forms are given in the latter. ^ For suggestions see Clark, E., Financing the Public Schools ; or Rugg, H. O., Cost of Public Education in Grand Rapids. 332 STATISTICAL METHODS Fourth. Total expenditures and expenditures per pupil for capital outlay. Because of the fluctuations from year to year in expenditures for permanent improvements, such ought to be reported both for the ciu'rent year, and for an average of four or five years. If clerical assistance makes it possible, this should also be compared with the other cities in the list. It involves very laborious computations, if done for many years. Fifth. The degree to which the board supports different kinds of schools, — elementary and secondary. Table 54 suggests a comparative method of reporting this aspect of Table 54. Distribution of Current Expenditures for Elementary ai^d Second ajry Schools — 17 Cities, 1915* {Data from United States Commissioner's Report, 1915, vol. 2) Per cent of total current expend- itures devoted to Rank in per cent of total current expenditures de- voted to Expenditure" per pupil in average daily attendance Rankof 17 cUies in expenditure per pupil in av- erage daily at- tendance for City . 11 3" ■H-2 1" 1. 11 («5 ll J1 77.44 81.78 88.39 75.29 72.37 77.25 79.86 76.65 82.08 74.72 82.25 82.07 82.82 80.10 83.G2 81.26 73.52 22.56 18.22 11.61 24.71 27.63 22.75 20.14 ^M 25.28 17.75 17.93 17.18 19.90 16.38 19.73 26.48 11 14 17 12 10 '! 15 4 6 3 9 2 8 16 7 11 17 4 1 6 8 ,1 3 14 12 15 .1 10 2 35.69 23.71 26.01 36.35 29.85 33.66 34.92 40.45 31.37 27.49 24.37 32.46 27.65 22.24 31.51 27.75 44.&4 70.56 49.10 54.95 63.58 63.77 51.17 87.32 87.36 47.27 65.42 56.57 84.02 51.88 56.73 50.66 66.70 94.74 4 16 14 3 10 6 5 2 9 13 15 7 12 17 8 11 1 5- Birmingham Bridgeport Cambridge Dayton 16 12 9 8 Pes Moines Fall River Grand Rapids-. Lowell 14 3 2 17 Lynn 7 Nashville New Bedford.... Paterson Richmond San Antonio Scranton Springfield 11 4 13 10 15 6 1 * Rugg, H. O., Cost of Public Education in Grand Rapids. (1917.) USE OF TABULAR AND GRAPHIC METHODS 333 school finance to the pubUc. It presupposes the publication of the total expenditures for elementary and secondary schools, together with the per cent cf all expenditures de- voted to each, and the unit expenditure per pupil in aver- age daily attendance. Ranks of all cities are given for both sets of data. 3. The reporting of facts concerning the teaching staff Data to be reported. The numerical status of the city's teaching staff, in each of its various departments, should be reported each year. It may be presented compactly, together with various historical data, in a table such as Table 55. Line diagrams of the sort shown for enrollment in Diagram 60 may well be drawn to picture the status more clearly. It is desirable to present the facts on the distribution of the teaching staff according to ranks and salaries, as com- Table 55. Distribution of School Officers and Teachers IN Different Grades of Schools, 1910-1915 inclusive* Year High school principals and assistants Elementary schools Kindergarten Manual training 1 1 1 i Sept. 05 t 1 2 11 1^ . i 1 ■^ 1' 1 CO II to e5 1 1 1 1910.. 1911.. 1912. . 1913.. 1914.. 1915.. 64 77 79 81 92 114 4 4 *2%+ 3% 3% 4% 34 31 33V3 34^3 36% 361/6 205 300 314 329 334% »49y2 10 16 21 23 20 I8V2 1V2 35 35 35 35 36 34y2 16 17V2 20 26 30 31 2 25 28 18 25 26 33 4 6 6 6 8 12 3 2 6 7 8V2 14^2 * Rugg, H.' O., Cost 0/ Public Education in Grand Rapids. 334 STATISTICAL METHODS Table 56. Showing Distribution of Teachers' AND Salaries Ranks School and rank ■^ 1 2 CQ *" 4 ^1 5 -.1 1*. 6 II on ■S a i5 11 1 3 7 8 9 10 Supervision SUPERVISORS 2150 2150 2150 2150 2120 2150 1400 1800 1 1640 1 2300 2300 .... 2300 2300 2300 2300 1500 2400 2466 2466 2466 2466 2400 1640 2500 2566 2566 2566 2566 2566 1726 2700 2700 2700 1 2700 1 2700 1 2700 1866 2850 2856 2856 2856 2856 2850 .1 1900 3000 1 3000 1 3000 3666 3666 3666 2666 3 Woman Kindergarten Woman. Man Man.... Man Pbvsica;! TraiiniuG'. Man Women School Gardens (12 months) Man Special Schools HIGH SCHOOLS 2150 2300 1 2040 7 6 1700 15 10 1180 3 32 1020 1 10 1300 1*266 2 740 2400 2166 1 1806 5 2 1240 6 3 1400 1366 'soo 1 700 2500 2180 26 1966 3 5 1300 4 6 1566 2 1400 3 900 "866 2700 2666 19 3 1360 8 8 1600 1500 4 1000 '966 2850 1466 22 13 1766 1 1072 2 972 3000 3 1526 1 2 1806 3 ii32 6 1032 1580 3 7 Men Head Assistant 2000 4 1640 4 7 1120 1 7 980 2 9 1072 1 1200 iioo 1 700 Men Second Assistant Men 1640 18 11 Substitute Assistant Men Women Substitute (2 A Gr.) Woman Instructor Phys. Tr. (Boys) Men Instructor Phys. Tr. (Girls) Women Clerk 2 A Gr. +$100 (for 12 mos.) Cl^rk, Summer, 2 A Gr 600 640 1 Woman ELEMENTARY SCHOOLS • Head Assistant 1180 16 920 1240 1 1020 1300 47 1072 123 700 77 3 1126 173 800 75 1 '900 79 '972 72 1632 616 Women First Assistant Women Second Assistant 600 11 110 8 640 88 17 2 Women (Substitutes) Permanent Temporary USE OF TABULAR AND GRAPHIC METHODS 335 pactly as is consistent with clearness. Table 56, from the 1914-15 Report of Board of Education in St. Louis, does this very suggestively by indicating in one table the number of years in the salary schedule for each position, the corre- sponding salary for each year and for each grade, and the number of men and number of women who draw the sala- ries stated for each grade. ^1050 4 I 1000 I I 950 IZ I 900 3) I 850 ZSO I 800 60 Diagram 58. Showing Number of Elementary Teachers receiving Various Salaries Showing the salary situation. To picture the general sal- ary situation clearly the Rochester School Report (1911-13), p. 66, makes use of a bar diagram to good effect, as repro- duced in Diagram 58. The growth of salaries in the system, as shown by ave- rage salaries paid and by the corresponding percentage of increase for each grade of position during past years, may similarly be shown in a table. The general level of teaching salaries may also be brought out by some such tabular representation as that shown in Table 62. 336 STATISTICAL METHODS Table 57.. Showing the Genera.l Level of Salaries IN A City Salary Teacher $3000 or over I 2000 to $2999 4 1500 to 1999 24 1000 to 1499 30 800 to 999 30 600 to 799 238 400 to 599 8 200 to 399 8 343 The years of teaching experience should be reported, first, for total experience, and, second, for years of experience within the local system. Table 58 suggests a compact tabular arrangement for such data, which can be used to show either type of statistical information. Table 58. Showing the Years of Teaching Service of all Teachers employed in 1915-16 Grades H.S. Total Beginners 11 6 17 2 years 3 to 4 years 5 to 9 2 4 2 1 1 2 5 3 10 to 14 2 , , 2 15 to 19 2 . . 2 20 to 24 2 1 3 25 to 29 Total 1 26 ~9 1 35 The training of the teachers. This should be reported by the same sort of tabular arrangement as that showing the salary distribution. Diagram 59 suggests a graphic method ^ by which this, as well as many other kinds of school facts may be reported. 1 Jessup, W. A. The Teaching Staff, p. 58. (Cleveland Education Sur- vey Monographs.) USE OF TABULAR AND GRAPHIC METHODS 337 Diagram 59. Per cent op Elementary Teachers, High-School Teachers, and Elementary Principals in Cleveland who are Home-trained and not Home-trained (After Jessup, 1916.) Size of classes. The size of classes within the local sys- tem should be reported annually. Table 59 shows in simple form the size of classes in the school system as a whole, while Table 60 shows the size of classes in each main division of the school system. Table 59. Showing the Number of Pupils per Teacher, Elementary Grades, December, 1910 23 teachers had over 50 pupils 90 45 to 49 63 40 to 44 56 35 to 39 14 30 to 34 . 8 below 30 Table 60. Showing the Number of Pupils per Teacher IN Different Classes of Schools Auxiliary school High school Grammar grades Primary grades Kindergarten 1909-10.... 68 22.2 34.0 36.0 32.4 1910-11.... 70 19.7 32.2 35.8 30.3 1911-12. . . . 96 19.8 31.5 35.9 32.3 1912-13.... 93 19.2 27.5 31.7 27.6 1913-14.... 150 19.5 27.2 32.3 27.2 1914-15 •• 23.4 33.9 35.3 35.0 Often it is desirable to show the size of classes in the city, compared with those in other city school systems of the same 338 STATISTICAL METHODS size or class. In such cases Table 61 gives a good form of table for displaying such information. Table 61. Number of Pupils in Average Daily Attend- ance PER Teacher in Elementary Schools in 19 Amer- ican Cities, 1915 (Data from Annual Report, United States Commissioner of Education, 1915, vol. 2) Albany Birmingham. . Bridgeport . . . . Cambridge Dayton Des Moines. . . Fall River Grand Rapids Kansas City. . Lowell Lynn Memphis Nashville New Bedford. Paterson Richmond San Antonio. . . Scranton Springfield . . . . No. of teachers Average daily No. of pupils Rank employed attendance per teacher 320 9,427 29.5 6 571 17,781 31.1 8 388 15,093 38.9 18 386 12,255 32.0 10 433 13,242 30.6 7 486 13,021 27.0 2 499 12,899 25.8 1 Prim. 32.3 10 471 12,909 Gram. 27.2 4 337 11,026 32.7 12 264 9,665 86.6 15 285 10,793 87.8 17 450 14,070 31.3 9 314 14,135 44.7 19 853 11,466 32.5 11 462 17,362 87.6 16 599 20,142 33.6 14 373 10,253 27.5 5 550 18,014 32.8 13 490 13,296 27.2 3 ^. The reporting of facts concerning the pupil Data needed, and forms. There are eight types of fact that the annual school report should give the public and school officers about the pupil: (1) the number of children of school census age in the city; (2) the total enrollment in all schools in the city; (3) the total enrollment in public schools; (4) as closely as possible, the estimated enrollment in parochial schools; (5) the total and the average enrollment and aver- age daily attendance in each of the various grades, kinder- garten to last grade in high school inclusive; (6) the distribu- USE OF TABULAR AND GRAPHIC METHODS 339 tion of children in each grade according to age; (7) the dis- tribution of children in each grade according to number of years spent in the grade; (8) distribution of children in each grade according to the facts concerning "promotion.'* Tables 62 to 67 suggest tabular arrangements of these data.^ Picturing the holding power of the schools. It will be desirable to use graphic devices to picture the efficiency with which the school machinery holds pupils in school, grades and classifies them, and promotes them through the various grades. Diagram 60 represents a good method of presenting to the people the degree to which the public schools are educating the children of school age in the city. Diagram 61 is an excellent pictorial device, taken from the 1914-15 St. Louis School Report^ for showing the increase in persistence of children in school. Such a diagram is clear and is easily comprehended by citizens. Diagram 62 ^ suggests a method of illustrating the "holding power" of the schools. Diagrams 5, 6, and 7, in Chapter I, give graphic methods of studying and reporting failures in the schools, by grades and by subjects. 5. Reporting fads as to the school plant School buildings. The following topical hst of points should be covered in reporting facts as to the school build- ings in use : — 1. Number of buildings, — elementary, intermediate, sec- ondary, covering a period of years. 2. Number of classrooms in use at stated time, covering comparison of several years. 3. Valuation of school property; historical, several years. ^ The writer is indebted to Dr. L. P. Ayres for the material in Tables 62 to 67 inclusive. 2 Ayres, L. P., Child Accounting in the Public Schools, p. 19. (Cleveland Education Survey Monographs.) J / ^^ 2 } f 1 1 1 S. /I §i 3 \s 1 & S Schoo Ce ISUS ? S « S S Si ^ 1 1 M s 1 /- 1 £^ S S -3r4^'' 1 ^ EnrolIii!,(^j7~Si ^ y^^ S s i o 1 i i i 2 2 s f;- _ ^ 12 :ii taent in Pi. blic Total Enrol) f? « ¥ i %a o ". 1 2 s oJ <£ Irades- -__u. "Enroli ment U y^u 1 n V>Kr> " 1 i 5? ! ^^^ 1 s 2 ^-<^ •Estim rolhnen S?^^ ^gi^^ramm r G •ades .-^ - . 5 1 : iP- f s ya § m "" 1 1 g ? S •i S S \ \ 1 1 ^ '^- High Schoo Grad ■s Ei tollmeni. - 30,000 29,000 28.000 27,000 26.000 25,000 24.000 23.000 22.000 21,000 20,000 19.000 18,000 17,000 16,000 15.000 14.000 13,000 12.000 11,000 10,000 9.000 8,000 7.000 6,000 5.000 4.000 3.000 2.000 1,000 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 Diagram 60. Showing for a Series of Years the Degree to WHICH Public Schools are educating the Children of School Age est the City {.Grand Rapids School Report, 1915.) USE OF TABULAR AND GRAPHIC METHODS 841 4. Cost of new buildings. A tabular form for such data is suggested by Table 68. 5. Standards used in judging buildings. For such facts a graphic form is shown in Diagrams 69 and 70. Standards may be set, as shown in Diagram 70, against the individual buildings of the system to permit of a judg- ment as to the present condition of the school plant. «-^NTARY GrToes HIGH 5CHOOLS Diagram 61. Persistence of Attendance at School (From the St. Louis School Report, 1914-15.) Numbers on top of columns show the number of pupils out of each 100 entering the second grade in the years indicated who were enrolled in the several grades in the succeeding years. 342 STATISTICAL METHODS Table 62. Number of Children of School Census Age - Illustrative Form Age Public schools Private schools Parochial schools In no schools Total 5 6 7 8 9 10 11 U 13 14 15 16 17 Total* * Data between dotted lines refer to children of compulsory school age. Table 63. Total Enrollment, Average Enrollment, and Average Attendance, 1916-17 Grade Total enrollment Average enrollment Average attendance K 1 2 3 4 5 6 7 8 Total El. I II III IV Total High Night II 00 00 l> eo 'f* I> o »o •* Is fe o< «o ■^ ■* «5 -<*< GO ft^§ %> ^ »o «o o «o "* ■^ W5 ,_^ l>. o l> '^ 00 r- 00 GO 00 fe !"• r-* I-H r^ r-< o o _^ Oi o< GO »o (-5 ■* ^ 05 5 CO GO o ^ ©< i> i> 00 e5 GO ■* '^ ^ ^ GO GO o< CO ?^ I-H »o CO to rH Tft CO ot o GO GO »o W5 I-H 1-H "tH «5 CO J> '"' O^ CO . 05 OS -<^ I-H fc- 1> o l> l> 00 o< »<0 o o I-H ^ 2 Q< i^ l> »o >* t' I- '-< »o 00 I-H •* 1> GO 00 1-H CO •» I-H GO §> ^ >-< ••« no 1> 00 o o CO o o« 1> »o 00 ^ o GO GO o Oi o CO ^ ©< 00 CO M ^ «> Oi 1> o> o< CO • -* 05 ©< ■"* GO 00 05 1-H CO 00 ^ ^ ^ J> GO o« ®< 00 »0 I-H ^ 1> »< o> r-l ^ -S g rl ©« CO '^ »o CO t- 00 1 "il P ! H 1 S vi »J^ ^ o M5 Oi CD 00 ss-i «s* CO 'S* CO 00 00 CO ftn g t^ o o »o o W5 to CO ■* 1 ^ o CO CO r-l 05 ^ CO I-H 05 00 a 5 Oi ©* CO «5 o "<^ ^ Ol GO o CO •o -* c^< 1> i> 00 CO ^ CO ■* "* "* Tjt CO CO CO CO o< •<^ o I— 1 CO CO o o O) 00 05 00 l-H ^ ^ 1—1 00 0< ©< ©< ^ o* o< to O) o GO CO t>. I— 1 o* CO o 00 GO CO I— 1 «5 r- }^ iSi ■^ 1 CO e «o I—t o © Oi f- to ^ ©^ '^ CO o to to o £ l-H CO ^. »o l-H o* J> ,_( I^ J^ CO Q» o Ol 05 i> CO t- CO -* Q< »iO O CO »o o< o* 05 o< »o o <5< tP <»3 i> o< o 1-H CO CO o 00 00 G^ ■* ©4 t- ^ 1> CO CO CO CO CO CO o< CO »H ©* »o ^ ©< o CO CO CO ■^ 5 C 3 5 ©< CO '*! »o CO i> « H USE OF TABULAR AND GRAPHIC METHODS 345 Table 66. Showing Attendance in I^ementary Schools DURING 1916-17 Pupils Days attended Pupils Days attended 0- 9 100-109 10-19 110-119 20-29 120-129 30-39 130-139 40-49 140-149 50-59 150-159 60-69 160-169 70-79 170-179 80-89 180-189 90-99 190-199 200 Total Total Table 67. Showing Promotions for School Year ending June, 1917 Grade On June promotion list Uncon- ditionally promoted Con- ditionally promoted Left behind Promoted more than one grade Special promotion between September and June Number who were promoted and dropped back K 1 2 3 4 5 6 7 8 Total I II III IV Total O § m i i IE g H <«5 Eh O .,■8 1 pi.'S. 00 Oi ©< ,_H »o ■«f< o CO CO ,^ o CO en ^. ,_| i> CO CO CO l:s o< ot ©< o< o< Ci^ CO o« ^§ 0& t 1^ GO !_, «5 ©* CO o Oi CO GO »o GO CO 05 00 05 o CO ^ "S s (X o< r-t I-H a< o< o< o e. ^ C 11 1—1 CO GO GO I— c CO <3^ CO "^ «o S 05 )->. 05 CO o % fe ■^. "* I— 1 O 05 o< rH CO J> o 2 GO* O l> CO* l> i> OD t-* r- €© 1 >>> ■§ S |« s o O* 00 pH 00 »o ,_^ ^ Ci ^ lO ©< b- ^ GO Q£ ^ ill o d c co^ CO* CO co' CO* ^11 1— 1 G< »^ c» o J> «5 j^ 00 CO >* ^ ^ ^.S GO CO ^ i> oc CO 05 CO 05 wS CO *H 05 05 oc t* o o §^ ■^' cc CO ©< 00 o< oo' a oo ^J Oi 1> l> «o CO o I-H rH l—t €© ^ 00 ©< ^ t- GO 8 b- CO 1> W5 1—1 co GO »o 05 i> »o '^ «5 CO GO b- o* •»o a. "^ "e =0 ■§i >o t- -* b- •^ s< b- ■^ CO a,o CQ «- 5 S b- "* «3i| «C l> o ©« CO o S£ I— 1 G^ ■* « 1> a o: o o Ci CD rH a o 2 *E 1 4. 1 ^ (/ tr X o CQ 'c 5 a I cc t- . 1 12 . 1 1 1 a USE OF TABULAR AND GRAPHIC METHODS 347 6. Reporting miscellaneous educational information The foregoing sections have presented, in outhne, definite suggestions for the content and form of the school report on five principal phases: (1) the legal basis; (2) school finance; 15000 - LEGAL SCHOOL AGE lUOOO 13000 12000 liooo 10000 9000 6000 7000 6000 5000 Uooo 3000 2000 1000 ^6 7 8 9 10 11 12 13 lU 15 16 17 16 19 20 Diagram 62. Showing the Holding Power of the Schools The columns represent the children enumerated by the school census as of each age from 6 through 20. Portion in outline represents children in public schools. Portion in black represents those not in public schools. {Cleveland Education Survey Report, 1916.) (3) the teaching staff; (4) the pupil; (5) school buildings. The primary aim has been to illustrate, in compact form, suggestive tabular and graphic means for setting forth ef- fectively such information.^ At the same time the actual facts needed in the report have been sketched. In addition, there are many other types of miscellaneous school facts that ^ A very complete compilation of Graphic Methods for Presenting Facts has been published by W. C. Brinton in a book by that name. (Engineering Magazine Company, New York, 1914.) The preparer of graphic reports, in whatever subjects, will receive very great aid from consulting this book. 848 STATISTICAL METHODS © © Under aga and rapid progress Normal age and rapid progress Over age and rapid progress (-') (^ Under age and normal progress Normal age and normal progress Over age and normal progress ® (■) Under age and slow progress Normal age and slow progress Over age and slow progress Diagram 63. Per cent of Children in Each Age and Progress Group in Elementary Schools at Close of Year 1914-15 (Clevelajid Education Survey Report, 1916.) s 10 11 13 tl4l 7 9 10 12 13 :?14^ 7 8 10 11 12 13 13 15 ■ 7 8 9 11 12 14 7 8 9 10 _1L 11 12 13 M^^. 6 ' 8 9 10 12 13 Wm 6 7 9 10 11 12 13 14 -'^^y': 6 7 8 9 10 11 12 14 15 m. 6 7 8 9 10 11 12 13 15 16 17 1 18 6 7 8 9 10 11 12 13 14 15 16 1 17 1st 2nd. 3rd 4 th 5 th 6 th 7 th 8 th I II III IV Diagram 64. Progress of Ten Typical Pupils through the School System Each square represents one child. The number represents his age. As they advance through the grades, they advance in age. The shaded squares repre- sent those who drop out. (Cleveland Education Survey Report, 1916.) USE OF TABULAR AND GRAPHIC METHODS 349 should be tabulated and graphed, the full presentation of which must be left to a volume devoted to the specific prob- lem of this chapter. It may be of some service to school men. D 21year» lOP.M Diagram 65. The Environment of a Minor during the Principal Periods of his Growth (From Perry, C. A., Educational Extension, p. 35.) EAGLE SCHOOL TREMONT SCHOOL. II 14 n ■ 20 ■ 22 ■ 17 lO 23 ■ Albanian [0 3 1 Armenian 1 2 1 Bohemian ji 10 26 ■ English. iHiH^^^HBHHl276 I French 6 I German ■■■^■^■B 202 9 I Greek 22 ■ Hebrew 14 ■ Hungarian 1 Italian 1 Lithuanian 3 1 Norse 23 ■ Polish BHHHBJJ^HHHI^HHiiHHHi^SS 3 1 Roumanian [ 16 ■ Russian ■■■■■■■■■■■■■■■I 443 2 1 Ruthenian i I Scotch "0 1 Servian 116 ■■■■i Slovak *" 4 1 (Slovenian 01 Spanish 89 HI^BI Syrian 21 Welsh 2 1 Yiddish Dlagram 66. Showing the Distribution of Pupils by Nationalities IN Two Elementary Schools (Cleveland Education Survey Report, 1916.) 266 350 STATISTICAL METHODS S3nm.onSBhool/>vnfises tfui46miii/nReacfin9 lfir.48mih.inkfftng IhKS^nin.infbrk J7 ft^ ^tirs4£m/n.uti^rk ^hrs^eminon the street. B/irs.44nttn./n/%ay Diagram 67. How 915 Children spent their Spare Time on Two Pleasant Days in June (Johnson, G, E., Education through Recreation, p. 47. Cleveland Education Survey RepoH, 1916.) 411631 I70I7T 62 32 37 58 43 52 85 24 33 13 54 75 83 23 64 16 30 39 10 45 14 48 Ml 67 60 89 92 95 94 71 65 64 65 66 67 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 Diagram 68. Average Scores made in Spelling by the Ninety-six Elementary Schools The figures below the di^gram show the percentages, and the ones in the diagram show the numbers of the schools. (From Judd, C. H., Measuring the Work of the Public Schools, p. 84. Cleveland Education Survey Report, 1916.) USE OF TABULAR AND GRAPHIC METHODS 351 SQUARE FEET OF FLOOR SPACE FOR EACH CHILD CUBIC FEET OF AIR SPACE FOR EACH CHILD PER CENT WINDOW AREA IS OF FLOOR AREA SQUARE FEET OF PLAYGROUND AREA FOR EACH CHILD NUMBER 07 BOYS PER URINAL NUMBER OF BOYS PER ~ TOILET SEAT NUMBER OF GIRLS PER TOILET SEAT CHILDREN PER DRINKING FOUNTAIN Diagram 69. Some Standards used in judging School Buildings (Ayres, L. P., School Buildings and Equipment, p. 54. Cleveland Education Survey Report, 1916.) S59, STATISTICAL METHODS /ERY POOR FAIR JgoodJJI^ Bat Jo of glass area to floor atea'fn percentage for actual cfass rooms of grade schools. 54 Kobcrl. Fulton 31 Jbhin Ericsson 8 Caih.tf.,n. Note;- The Minnesota Department of Education requires a minimum glass area equal to 20.°o of the floor area for elementary class rooms. Diagram 70. Ratio of Glass Area to Floor Area (From A Million a Year. Minneapolis Board of Education, 1916.) USE OF TABULAR AND GRAPHIC METHODS 353 however, in closing this volume, to bring together in Dia- grams 71 to 80, inclusive, a few striking pictorial methods of presenting such miscellaneous school facts which have been used effectively by school men, in presenting information to the people of their school city. Diagrams 74 to 78 inclusive, and Diagram 80, have been quoted from Help- Your-Own- School Suggestions, Bulletin No. 31, Feb. 21, 1914, of the Bureau of Municipal Research, New York City. The others People of the State represented in the Legislature uusiness Committees of the Board X state Superintendent of Public Instruction iperintendent Educational Committees of the Board Business and Office Clerk I Stenographer 1 Kindergai-ten Teachers Diagram 71. Plan of Educational Organization in a Small City This illustrates construction of "organization charts," which superintendents often desire to show. (From Cubberley, E. P., Public School Administration, p. 167.) 354 STATISTICAL METHODS are properly credited to the report from which they have been taken. Machine l.Q Steno- graphers MEN WOMEM. '•^1 General Clerical 10,7 Book- keepers 11 .U Clerks 67.8 Machine workers 23 t^ Steno- graphers 36.2 General clerical IS.U Book- keepers 20*6 Clerks I. If Diagram 72. Percentage Distribution of Non-Adminis- trative Positions in Office Work As held by men and women in Cleveland, 1912-15, 1955 positions for men and 2747 for women. (From Stevens, B. E., Boys and Girls in Commercial Work, p. 26. Cleveland Education Survey Report, 1916.) USE OF TABULAR AND GRAPHIC METHODS 355 Cabinetmakers guilders an d bu iIJaing contractors Paintereand glaziers 42; 29 Sheet metal workers and tinsmiths 42 Foreign born I Native born of foreign parents P mm Native born of native parents D Diagram 73. Percentage of Workers in Building Trades THAT are Foreign-Born, Native-Born of Foreign Par- ents, AND Native-Born of Native Parents (From Shaw, F. L., The Building Trades, p. 33. Cleveland Education Survey Report, 1916.) 356 STATISTICAL METHODS SPELLING MISFITS ductile. \ communicant ^ accessiiic ^ ^ amphibious muralgta infrangible' arottOZS terrestrial ^ schedule y'^ eJtymdlog}/. ^ ^yoyancy " 5 ■• -f^^.'^m^^ Diagram 74. Illustrating Spelling Difficulties USE OF TABULAR AND GRAPHIC METHODS 357 IMPORTANCE O F AFTER SCHOOL ACTIVITIE, Ways of using hours after 3 o'clock Diagram 75. Illustrating the Importance of After-School Activities Adjustment oi desks and seats C494 examined) ?^t*l^?^?^ "^ A ^ A' A A JtJHDWiJfnrLJyx MSI^ ml ml ml tSf W^"^ proportion badly ptaczd , 36J'7t> Diagram 76. Illustrating Seating Conditions in the School 358 STATISTICAL METHODS Scale of Minutes Library /i MU51C If Opcnin|, Exercises J< Penmanship ^a Physical Trammg Geograpky 'V. History Drawing AritKmctic 5S3. English Tirrii. as per- C^ fecial Sc, Dzyotu ' in PS. i izdulz (^ Diagram 77. Illustrating the School Program: Of Every 100 Failures 7b, pupils 30 made by normal pupils 63 Tnadc by overage -pupils 97 occrr in grades 1 to 6 grades Promotiond Sc non-promctions, first scme;5tei;l^I2''13 Failure 107. Promoted Regularly YY^ Promoted I twrct more 13^ Diagram 78. Illustrating Promotion and Failures USE OF TABULAR AND GRAPHIC METHODS 359 OVER 200 5Q.FT.PCR CHILD 170-200 SQ.FT. PER CHILD 100-130 SQ.rT. PER CHILD LESS THAN 100 SQ.FT. PERCHILO DiAGBAM. 79 Showing the Pekcentage op Children having Play- grounds OF Various Sizes (Salt Lake City School Survey Report, 1915.) 360 STATISTICAL METHODS ^a^mmr-... ..:-... ... .,.-, ^B^^PIB--:;. '"'.'.<'.' ' ^^^^^^^^^H- .- ^^^^^^^^^^B : -~ ^^^^^^^^■'"). . Hj^H^^IH >: ^ . ■■^^■B .:.....- '..v7;v>V(;:;>''; ^^'.^: ^^^^^^^^Myt--r.'-i-ii':-yi' ■r:.-'^i^ ^^^^^^^^^^■l'''>:^''V^O^: .'.-' >'.' . ^^^^^^^^^^HiMHtaMMMMMlH ^H^^^^^^^H^^^H^^I ^^^^^^^^^^^^^^^^t^^^^^^M ^^^^^^^^^^^^H^^^^H ^^^^^^^^^^^^^^^^^^^^^^^1 H^^^^^^^^^^^^^^^^H ^^^^^^^^^I^^^^^^^^^^^HI ^^^^^^^^^^^^^^^l^^^^^^^l ^^^^^^^^^^^^^^^^^^1 ^^^^^^^^^^^^^^^^^^^^^^^1 ^^^^^^^^^^^^^^^^^^1 ^^^^^^^^^^^^^^^^^^^^^^^1 ^^^^^^^^^^^^^^^^^^1 ^ -::$—:::_:_::__:_::::::_::__:::—:_::: H not examined D examined; found perfect B found defective and treated ■ found defective but not treated DiAGBAM 80. What the School Records relating to Medical Examinations show BIBLIOGRAPHY OF THE QUANTITATIVE STUDIES or SCHOOL ADMINISTRATION ^or Tabular Key see Plate I. I. STUDIES ON THE COURSE OF STUDY 1. Ayres, L. P. A Scale for the Measurement of Spelling Ability. Russell Sage Foundation, Division of Education, 1915. 2. Bagley, W. C, and Rugg, H. O. The Content of American History as Taught in the Seventh and Eighth Grades. Univer- sity of Illinois, School of Education, Bulletin No. 16, 1916. 3. Baker, J. H., Chairman. Economy of Time in Education. 106 pp. United States Bureau of Education, Bulletin No. 38, 1913. 4. Bobbitt, J. F., and others. "Literature in the Elementary Curriculum." Elementary School Teacher, December, 1913, and January, 1914. 5. Cleveland Education Survey Reports, 1916. These reports can be secured from the Survey Committee of the Cleveland Foundation, Cleveland, Ohio, and from the Division of Edu- cation of the Russell Sage Foundation, New York City. 1. Child Accounting in the Public Schools — Ayres. 2. Educational Extension — Perry, 3. Education through Recreation — Johnson. 4. Financing the Public Schools — Clark. 5. Health Work in the Public Schools — Ayres. 6. Household Arts and School Lunches — Boughton. 7. Measuring the Work of the Public Schools — Judd. 8. Overcrowded Schools and the Platoon Plan — Hartwell. 9. School Buildings and Equipment — Ayres. 10. Schools and Classes for Exceptional Children — Mitchell. ^^1^- 11. School Organization and Administration — Ayres. 12. The Public Library and the Public Schools — Ayres and McKinnie. 13. The School and the Immigrant — Miller. 14. The Teaching Staff — Jessup. 15. What the Schools Teach and Might Teach — Bobbitt. 16. The Cleveland School Survey (summary) — Ayres. 17. Boys and Girls in Commercial Work — Stevens. 18. Department Store Occupations — O'Leary. 19. Dressmaking and Millinery — Bryner. 20. Railroad and Street Transportation — Fleming. n. The B,ilding Trades — Shaw. 22. The Garment Trades — Bryner. £3. The Metal Trades — Lutz. 24. The Printing Trades — Shaw. 25. Wage-I jrning and Education (summary) — Lutz. m BIBLIOGRAPHY 6. Cook, W. A., and O'Shea, M. J. The Child and His Spelling. Bobbs Merrill Co., Indianapolis, Indiana. 7. Iowa State Teachers' Association, Bulletin of the, November, 1916. Elimination of Obsolete and Useless Topics and Materials from the Common Branches. G. W. Wilson, Chairman. 8. Jessup, W. A., and Coffman, L. D. The Supervision of Arith- metic. 9. Jones, W. F. Concrete Investigation of the Material of English Spelling. Published by the University of South Dakota, Vermilion, South Dakota, 10 cents. 10. Koos, L. V. The Administration of Secondary School Units. University of Chicago Press, 1917. No. 3 of the Supple- mentary Educational Monographs. 11. Minnesota Education Association, March, 1914. Report of the Committee on the Elementary Course of Study. 12. National Education Association. 1916 Report of the National Council of Education Committee on Standards and Tests of Efficiency. Fifteenth Yearbook of National Society for the Study of Education, 1916. Part i. 13. The Fourteenth Yearbook of the National Society for the Study of Education, 1915. Part i: Minimal Essentials in the Elementary School Subjects — Standards and Current Practices. 14. The Sixteenth Yearbook of the National Society for the Study of Education, 1917. Part i: Second Report of the Committee on Minimal Essentials in the Elementary School Subjects. 15. The Fifteenth Yearbook of the National Society for the Study of Education, 1916. Part iii: The Junior High School, A. A. Douglas. 16. Payne, B. R. Elementary School Curricula. Silver, Burdett & Co., 1895. 17. Rice, J. M. Scientiftc Management in Education. Hinds, Noble & Eldridge, New York, 1913. 17a. Rugg, H. O., and Clark, J. R. "Standardized Tests and the Improvement of Teaching in First- Year Algebra." School Review, February, March, May, and October, 1917. 18. Studley, C. K., and Ware, Allison. Common Essentials in Spelling. Chico State Normal School, Chico, California, Bul- letm No. 7, 1914. BIBLIOGRAPHY 363 II. QUANTITATIVE STUDIES OF THE TEACHING STAFF AND TEACHING EFFICIENCY (Studies dealing with the social status, training, experience, tenure, selection, and salaries of teachers.) 19. Ballou, F. W. Appointment of Teachers in Cities. Harvard Studies in Education, vol. ii, 1915. Harvard University Press. 20. Boice, A. C. Methods of Rating Teaching Efficiency. Part ii. Fourteenth Yearbook of the National Society for the Study of Education, 1915. (See also "Qualities of Merit in Secon- dary School Teachers," Journal of Educational Psychology, March, 1912.) 21. Coffman, L. D. The Social Composition of the Teaching Popu- lation. Teachers College, Columbia University, Contribu- tions to Education, 1911, No. 41. 22. Cubberley, E. P. The Certification of Teachers. Part ii. Fifth Yearbook of National Society for the Study of Educa- tion. University of Chicago Press, 1906. 23. Elliott, E. C. A Provisional Plan for the Measure of Merit of Teachers. Department of Public Instruction, Madison, Wis- consin, 1910. 24. Hood, W. R. Digest of State Laws Relating to Public Educa- tion. United States Bureau of Education, Bulletin No. 47, 1915. 25. Judd, C. H., and Parker, S. C. Problems Involved in Standard- izing State Normal Schools. United States Bureau of Educa- tion, Bulletin No. 12, 1916. 26. Littler, S. "Causes of Failure Among Elementary School Teachers." School and Home Education, March, 1914. 27. Manny, F. A. City Training for Teachers. United States Bureau of Education, Bulletin No. 47, 1914. 28. Moses. "Causes of Failure Among High-School Teachers." School and Home Education, January, 1914. 29. National Education Association. Report of the Committee on Salaries, Tenure and Pensions, 1905. 30. National Education Association. Report of the Committee on Teachers' Salaries and Cost of Living, January, 1913. 31. Ruediger, W. C. Agencies for the Improvement of Teachers in Service. United States Bureau of Education, Bulletin No. 3, 1911. 364 BIBLIOGRAPHY 32. Ruediger, W. C, and Strayer, G. D. "The Qualities of Merit in Teachers." Journal of Educational Psychology, May, 1910, vol. I, pp. 272-78. 33. Salaries of Teachers and School Officers, A Comparative Study of. United States Bureau of Education, Bulletin No. 31, 1915. 34. The Tangible Rewards of Teaching. United States Bureau of Education, Bulletin No. 16, 1914. Compiled by J. C. Boykin and Roberta, King for the Committee of the National Edu- cation Association on Teachers, Salaries, and Cost of Living. 35. Thorndike, E. L. The Teaching Staff of Secondary Schools in the United States. United States Bureau of Education, Bulle- tin No. 4, 1909. 36. Updegraff, H. Teachers'' Certificates Issued Under General State Laws and Regulations. United States Bureau of Educa- tion, Bulletin No. 18, 1911. III. QUANTITATIVE STUDIES DEALING WITH THE PUPIL A. ELIMINATION AND RETARDATION 37. Ayres, L. P. Laggards in Our Schools. Russell Sage Founda- tion, Division of Education, New York City, 1909. 38. Bachman, F. P. Problems in Elementary School Administra- tion. World Book Company, Yonkers, New York, 1914. 39. Blan, A. B. A Special Study of the Incidence of Retardation. Teachers College, Columbia University, Contributions to Education, No. 40, 1911. 40. Dearborn, W. F. "The Qualitative Elimination of Pupils from School." Elementary School Teacher, September, 1910. 41. Holley, C. E. The Relationship between Persistence in School and Home Conditions. Part ii. Fifteenth Yearbook, National Society for the Study of Education, 1916. 42. Keyes, C. H. Progress through the Grades of City Schools. Teachers College, Columbia University, Contributions to Education, No. 42, 1911. 43. Strayer, G. D. Age and Grade Census of Schools and Colleges: A Study of Retardation and Elimination. United States Bureau of Education, Bulletin No. 5, 1911. 44. Thorndike, E. L. Elimination of Pupils from School. United States Bureau of Education, Bulletin No. 4, 1907. BIBLIOGRAPHY 365 45. Van Den Burg, J. K. Causes of Elimination of Students in Public Secondary Schools of New York City. Teachers Col- lege, Columbia University, Contributions to Education, No. 47, 1911. B. SIZE OF CLASS AND EFFICIENCY OF INSTRUCTION 46. Breed, F. S., and McCarthy, J. D. "Size of Class and Effi- ciency of Teaching." School and Society, vol. iv, No. 104, December 23, 1916, pp. 965-71. 47. Boyer, P. A. " Class Size and School Progress." Ps^/c/wZo^ricaZ Clinic, vol. viii, 1914. 48. Cornman, O. P. " Size of Classes and School Progress." Psychological Clinic, vol. iii, 1909. 49. Harlan, C. L. " Size of Class as a Factor in School-Room Efficiency, " Journal of Educational Administration and Super- vision, vol. I, 1915. C. GRADING, CLASSIFICATION AND PROMOTION OF PUPILS 50. Bunker, F. F. Reorganization of the Public School System. United States Bureau of Education, Bulletin No. 8, 1916. 51. Burk, F. Monograph A, A Remedy for Lock-Step Schooling. 15 cents 1913. Monograph C, Every Child vs. Lockstep Schooling. 15 cents. 1915. San Francisco State Normal School, California. 52. Quantitative and descriptive historical summaries on "pro- motion plans" in Annual Reports of the United States Com- missioner of Education: (1) 1890-91, p. 991; (2) 1891-92, pp. 600-32; (3) 1898-99, p. 330. 53. Dearborn, W. F. The Relative Standing of Pupils in the High School and University. University of Wisconsin, Bulletin, No. 312, High School Series No. 6. 54. Frailey, L. S., and Grain, C. M. "Correlation of Excellence in Different School Subjects Based on a Study of School Grades." Journal of Educational Psychology, March, 1914. 55. Holmes, W. H. School Organization and the Individual Child, Grading and Special Schools. The Davis Press, Worcester, Massachusetts, 1912. 5Q. Van Sickle, J. H., Witmer, L., and Ayres, L. P. Provisions for Exceptional Children in the Public Schools. United States Bureau of Education, Bulletin No. 14, 1911. 366 BIBLIOGRAPHY D. ON TEACHERS' MARKS AND MARKING SYSTEMS 57. Rugg, H. O. "Teachers' Marks and Marking Systems." Journal of Educational Administration and Supervision, Feb- ruary, 1915. Complete bibliography and summary of pub- lished literature to February, 1915. IV. STUDIES OF PUBLIC SCHOOL COSTS AND BUSINESS MANAGEMENT A. PUBLICATIONS RELATED TO CITY SCHOOL COSTS (a) Studies based on facts collected by question-blank methods 58. EUiott, E. C. Some Fiscal Aspects of Public Education. Teachers College, Columbia University, Contributions to Education, No. 6, 1905. 58a. Monroe, W. S. The Cost of Instruction in Kansas High Schools. Emporia State Normal School, Bulletin No. 2, 1915. 59. Strayer, G. D. City School Expenditures. Teachers College, Columbia University, Contributions to Education, No. 5, 1905. (6) Studies based on facts collected personally from records of city systems 60. Updegraff, H. Study of Expenses of City School Systems. United States Bureau of Education, Bulletin No. 5, 1912. 61. Hutchinson, J. H. School Costs and School Accounting. Teachers College, Columbia University, Contributions to Education, No. 62, 1914. (c) Repcyrts of surveys of city school systems 62. Clark, E. Financing the Public Schools. Cleveland Educa- tion Survey Monographs, Russell Sage Foundation, Division of Education, New York City. 63. Rugg, H. O. Cost of Public Education in Grand Rapids, Mich- igan. Chaps, xiv and xv of Grand Rapids School Survey. Board of Education, Grand Rapids, Michigan. 64. Rugg, H. O. Public School Costs and Business Management in St. Louis. (To be published, 1917, by Board of Education.) 65. Schroeder, H. H. Cost of Public Education, Peoria, Illinois, 1915-16. Board of Education, Peoria. BIBLIOGRAPHY 367 (d) School reports as a means of presenting financial facts. 66. Spaulding, F. E. 1912 and 1913, Newton, Massachusetts, School Reports. (Out of print.) 67. Spaulding, F. E. Three Monographs on School Finance in Minneapolis. Board of Education, Minneapolis, Minnesota: (a) A Million a Year. (b) Financing the Minneapolis Schools. (c) The Price of Progress. (e) General descriptive articles on financial practices of cities 68. Baker, G. M. ** Financial Practices in Cities and Towns below Twenty-five Thousand." American School Board Journal, October, November, December, 1916: January, February, March, May, June, 1917. 69. Clark, E. "The Indebtedness of City School Systems and Current School Expenditures." American Scliool Board Jour- nal March, 1917, p. 17. B. PUBLICATIONS RELATING TO THE STUDY OF BUSINESS MANAGEMENT OF THE PUBLIC CITY SCHOOLS 70. Byrne, J. T. Report of School Survey. Vartiv: The Business Management. School Survey Committee, Denver, Colorado. 71. Proceedings of National Association of School Accounting Officers, published for the years 1913, 1914, 1915, 1916. Ad- dresses reprinted in American School Board Journal for those years. 72. Shapleigh, F. E. "The Compensation of School Janitors." American School Board Journal, December, 1916, p. 23. 73. Hanus, P. H. "Town ancf City School Reports, more particu- larly Superintendents' Reports." School and Society, Janu- ary 29, and February 5, 1916. See also complete discussions in Nos. 63 and 64, Rugg, H. O. C. PUBLICATIONS RELATING TO STATE AND COUNTY SCHOOL FINANCE 74. Cubberley, E. P. School Funds and their Apportionment. Teachers College, Columbia University, Contributions to Education, No. 2, 1915. 75. Swift, F. H. A History of Public Permanent School Funds in the United States. Henry Holt & Co., 1911. 368 BIBLIOGRAPHY 76. Cubberley, E. P., and Elliott, E. C. State and County School Administration. The Macmillan Company, 1915. 77. Cubberley, E. P. State and County Educational Reorganiza- tion. The Macmillan Company, 1914. 78. MacDowell, T. L. State vs. Local Control of Elementary Edu- cation. (Finance.) United States Bureau of Education, Bulletin No. 22, 1915. V. STUDIES OF CENTRAL ORGANIZATION AND ADMINISTRATION 5-11. Ayres, L. P. See Nos.'5-ll. Bobbitt, J. F. See No. 103 — Part I. 79. Chamberlain, A. H. The Growth of Responsibility and En- largement of Power of the City School Superintendent. 158 pp. University of California Publications. Education, vol. Ill, No. 4, 1913. 80. Douglass, A. A. The Junior High School. Part iii. Fifteenth Yearbook of the National Society for the Study of Educa- tion. 157 pp. The Public School Publishing Co., Blooming- ton, Illinois, 1917. 81. Shapleigh, F. E. "Commission Government and the Admin- istration of City School Systems." American School Board Journal, November, 1915, p. 11. 82. Shapleigh, F. E. " School Administration in Non-Commission- Governed Cities." American School Board Journal, Decem- ber, 1915, p. 11. SUPERVISED STUDY 83. Breslich, E. R. Supervised Study as a Means of Providing Supplementary Individual Instruction. Thirteenth Yearbook of the National Society for the Study of Education, Part i, 1914. 84. Hall-Quest, A. L. Supervised Study. The Macmillan Com- pany, 1916. 85. Minnich, J. H. "An Experiment in the Supervised Study of Mathematics." School Review, December, 1913, pp. 670-75. 86. Reavis, W. C. "Factors that Determine the Habits of Study in Grade Pupils." Elementary School Teacher, October, 1911, vol. XII, pp. 71-81. BIBLIOGRAPHY SQ9 GENERAL SUPERVISION OF INSTRUCTION 87. Twelfth Yearbook of National Society for the Study of Edu- cation. Part i: The Supervision of City Schools, 1913. 88. Twelfth Yearbook of National Society for the Study of Edu- cation. Part II : The Supervision of Rural Schools, 1913. 89. Jessup, W. A. Social Factors Affecting Special Supervision. Teachers College, Columbia University, Contributions to Education, No. 43, 1911. VI. A BIBLIOGRAPHY OF SCHOOL SURVEYS A. SURVEYS OF CITY SCHOOL SYSTEMS 90. Atlanta, Georgia. Report of Survey of the Department of Edu- cation, 1912. By New York Bureau of Municipal Research. 90a. Baltimore, Md. Report of the Commission Appointed to Study the System of Education in the Public Schools of Baltimore. U.S. Bureau of Education, Bulletin No. 4, 1911. 91. Blaine, Washington. A Survey of the Blaine Public Schools. By Lull, H. G., MiUay, F. E., and Kruse, P. J. The Exten- sion Division, University of Washington, Seattle, Washing- ton, 1914. 92. Boise, Idaho. Expert Survey of Public School System. By Elliott, E. C, Judd, C. H., and Strayer, G. D. Board of Education, Boise, Idaho, 1913. 93. Boise, Idaho. Special Report of the Boise Public Schools, 1915. 94. Boston, Massachusetts. The Finance Commission of the City of Boston. Report on the Boston School System. City of Boston, Printing Department, 1911. 95. Boston, Massachusetts. Report of a Study of Certain Phases of the Public School System of Boston, Massachusetts. Sold by Teachers College, Columbia University, New York City, 1916. 96. Bridgeport, Connecticut. Report of the Examination of the School System, 1915. By Van Sickle, J. H. 97. Buffalo, New York. Examination of the Public School System of the City of Buffalo. By the Education Department of the State of New York. Albany. University of the State of New York, 1916. 98. Butte, Montana. Report of a Survey of the School System of 370 BIBLIOGRAPHY Butte, Montana. By Strayer, G. D., and others. Board of School Trustees, 1914. 99. Chicago, Illmois. Report of the Educational Commission of the City of Chicago, 1897. 100. Cleveland, Ohio. The Cleveland Education Survey. See Bibli- ography in Section on Course of Study. Published as 25 sep- arate monographs. Ayres, L. P., Director. Russell Sage Foundation, New York City. 101. Dallas, Texas. Report of the Public Schools of the City of Dallas, Texas. Wilkinson Printing Co., Dallas, 1915. 102. Dansville, New York. A Study — The Dansville High School. By Foster, J. M. F. A. Owen Publishing Co., Dansville, New York. 103. Denver, Colorado. Report of the School Survey of School Dis- trict Number One in the City and County of Denver. Part i: General Organization and Management. Part ii: The Work of the Schools. Part iii: The Industrial Survey. Part iv: The Business Management. Part v: The Building Situation and Medical Inspection. The School Survey Committee, Denver, Colorado, 1916. 104. East Orange, New Jersey. Report of the Examination of the School System of East Orange, New Jersey, 1912. By Moore, E. C. Issued by the Board of Education, 1912. 105. Grafton, West Virginia. Report of the Survey of the Grafton City Schools. By Deahl, J. N., Rosier, J., and Wilson, O. G. Department of Schools, Charleston, West Virginia, 1913. 106. Grand Junction, Colorado. A Survey of the City Schools of Grand Junction, Colorado, District No. 1, Mesa County. By Clapp, F. L., and others. The Daily News Press, 1916. 107. Grand Rapids, Michigan. School Survey. Grand Rapids, Michigan, 1916. 108. Greenwich, Connecticut. The Book of the Educational Exhibit of Greenwich, Connecticut, 1912. Conducted by the Russell Sage Foundation. 109. Hammond, Indiana. Some Facts Concerning the People, Indus- tries, and Schools of Hammond and a Suggested Program for Elementary, Industrial, Prevocational and Vocational Educa- tion. By Leonard, R. J. Hammond, Indiana, Board of Educa- tion, 1915. 110. Leavenworth, Kansas. Report of a Survey of the Public Schools of Leavenworthy Kansas. Bureau of Educational Measurement BIBLIOGRAPHY 371 and Standards, Kansas State Normal School, Emporia, Kansas, 1915. 111. Minneapolis, Minnesota. Report on the Survey of the Business Administration of the Minneapolis Public Schools. By Bureau of Municipal Research of the Minneapolis Civic and Com- merce Association, 1915. 112. Montclair, New Jersey. Report on the Programme of Studies in the Public Schools of Montclair y New Jersey^ 1911, By Hanus, P. H. 113. Newburgh, New York. The Newburgh Survey. Department of Surveys and Exhibits, Russell Sage Foundation, 128 East Twenty-Third Street, New York City, 1913. 114. New York City. Report of Committee on School Inquiry. By Hanus, P. H., and others. Separate Reports reprinted as single volumes of World Book Company's School EflSciency Series, Yonkers, New York. 115. Oakland, California. Report of a Survey of the Organization, Scope, and Finances of the Public System of Oakland, California. By Cubberley, E. P. Board of Education, Bulletin No. 8, 1915. 116. Ogden, Utah. Report of Ogden Public School Survey Commis- sion. By Deffenbaugh, W. S. State Department of Educa- tion, 1914. 117. Peoria, Illinois. Cost of Public Education, 1915-1916, Peoria, Illinois. By Schroeder, H. H. 1916. 118. Portland, Oregon. The Portland Survey. By Cubberley, E. P., and others. 1913. 119. Port Townsend, Washington. A Survey of the Port Townsend Public Schools. By Lull, H. G. University of Washington, University Extension Division, 1915. 120. Rockford, Illinois. A Review of the Rockford Public Schools, 1915-1916, Issued by the Board of Education, Rockford, Illinois, 1916. 121. Salt Lake City, Utah. Report of a Survey of the School System of Salt Lake City, Utah. By Cubberley, E. P., and others. Board of Education, Salt Lake City, 1915. 122. San Antonio, Texas. The San Antonio Public School System. By Bobbitt, J. F. The San Antonio School Board, 1915. 123. San Francisco, California. Some Conditions in the Schools of San Francisco. By Steinhart, Mrs. J. H., and others. School Survey Class, San Francisco, California, 1914. 372 BIBLIOGRAPHY 124. Springfield, HHnois. The Public Schools of Springfield, Illinois. By Ayres, L. P. Division of Education, Russell Sage Founda- tion, New York City, 1914. 125. South Bend, Indiana. Superintendent's Report of the School City of South Bend, Indiana, and School Survey. By Depart- ment of Education of the University of Chicago, 1914. 126. St. Louis, Missouri. Report of Survey of St. Louis School Sys- tem. (To be published, 1917, by Board of Education.) 127. Syracuse, New York. Report of Investigations for the Associ- ated Charities of Syracuse, New York. Made by the Training School for Public Service. Conducted by the Bureau of Mu- nicipal Research, New York City, 1912. 128. Waterbury, Connecticut. Help Your School Surveys. By Brittain, H. L. New York Bureau of Municipal Research. Published together with report on instruction in St. Paul, Minnesota, by A. W. Farmer. B. STATE SCHOOL SURVEYS 129. Connecticut. Report of Education Commissiony in Report of the Board of Education of the State of Connecticut, 1909. 130. Colorado. A General Survey of Public High-School Education, in Colorado. University of Colorado Bulletin, vol. xiv. No. 10, 1914. 131. Colorado. Report of an Inquiry into the Administration and Support of the Colorado School System. Department of Interior, United States Bureau of Education, Bulletin No. 5, 1917. 132. Illinois. Illinois School Survey. Coffman, L. D., Director. Published by J. A. Browne, Bloomington, Illinois, for the Illi- nois State Teachers' Association, 1917. 133. Iowa. State Higher Educational Institutions of Iowa. Commis- sioner of Education, Department of Interior, United States Bureau of Education, Bulletin No. 19, 1916. Washington, Government Printing Office, 1916. 134. Kansas. Survey of Accredited High Schools and Professional Directory. By Josselyn, H. W. Bulletin of the University of Kansas, vol. xv. No. 16, 1914. 135. Maryland. Public Education in Maryland. A Report to the Maryland Educational Survey Commission. General Educa- tion Board, New York, 1916. 136. Massachusetts. Report of the Industrial Commission of the BIBLIOGRAPHY 373 State of MassachmettSy 1907. Reprinted by Teachers College, Columbia University, New York City. 137. North Dakota. Report of the Temporary Educational Com- mission to the Governor and Legislature of the State of North Dakota. 1912. 138. North Dakota. State Higher Educational Institutions of North Dakota. Department of Interior, United States Bureau of Education, Bulletin No. 27, 1916. Washington, Government Printing Office, 1917. 139. Ohio. Report of the Ohio State School Survey Commission. By Campbell, M. E., Allendorf, W. L., and Thatcher, C. J. 1914. 140. United States. A Comparative Study of Public School Systems in the Forty-eight States. Russell Sage Foundation, Division of Education, 1912. 141. Vermont. Education in Vermont. The Carnegie Foundation for the Advancement of Teaching, Bulletin No. 7, 1914. 142. Virginia. Report of the Virginia Education Commission to the General Assembly of the Commownealth of Virginia. 1912. 143. Washington. A Survey of Educational Institutions of the State of Washington. United States Bureau of Education, Bulletin No. 26, 1916. 144. Wyoming. Educational Survey of Wyoming. By Monahan, A. C, and Cook, K. M. United States Bureau of Education, Bulletin No. 29, 1916. C. RURAL SCHOOL SURVEYS 145. Alabama, Three Counties. An Educational Survey of Three Counties in Alabama. Department of Education, Montgom- ery, Alabama, Bulletin No. 43, 1914. 146. Colorado. The Rural and Village Schools of Colorado. By vSargent, S. C. Colorado Agricultural College, Fort Collins, Colorado, 1914. 147. Indiana, Porter County. Rural School Sanitation. By Clark, T., Collins, G. L., and Tread way, W. L. Treasury Depart- ment, United States Public Health Service, 1916. 148. Maryland, Montgomery County. An Educational Survey of a Suburban and Rural County. By Morse. H. N., Eastman, F., and Monahan, A. C. United States Bureau of Education, Bulletin No. 32, 1913. 149. Missouri, Saline County. A Study of the Rural Schools of 374 BIBLIOGRAPHY Saline County, Missouri. By Elliff, J. D., and Jones, A. University of Missouri Bulletin, vol. 16, No. 22. University of Missouri, Columbia, Missouri, 1915. 150. Reports in Westchester County, New York : A Study of Local School Conditions, 1912. By Inglis, A. J. 151. Texas. A Study of Rural Schools in Texas. By White, E. V., and Davis, E. E. University of Texas, Austin, Texas, 1914. 152. Texas, Travis County. A Study of Rural Schools in Travis County, Texas. By Davis, E. E. University of Texas, Austin, Texas, 1916. 153. Virginia, Orange County. Sanitary Survey of the Schoob of Orange County, Virginia. By Flannagan, R. K. United States Bureau of Education, Bulletin No. 17, 1914. D. VOCATIONAL AND INDUSTRIAL SURVEYS 154. Denver, Colorado. Report of the School Survey of School Dis- trict Number One in the City and County of Denver. Part iii: Vocational Education. 1916. 155. Minneapolis Survey for Vocational Education, Report of the. National Society for the Promotion of Industrial Education, Bulletin No. 21, 1916. 156. New York City. Seventeenth Annual Report of the City Sur- perintendent of Schools, 1914^-1915. Survey of the Gary Pre- vocational Schools. Department of Education. City of New York. 157. Richmond, Virginia. Vocational Education Survey of Rich- mond, Virginia. United States Department of Labor, Bureau of Labor Statistics, 1916. 158. A Survey of Manual, Domestic, and Vocational Training in the United States. Hackett, W. E. Public Schools, Department of Practical Arts, Reading, Pennsylvania, 1914. MISCELLANEOUS 159. Some Foreign Educational Surveys. By Mahoney, J. United States Bureau of Education, 1915. 160. Community Action through Surveys. By Harrison, S. M. Department of Surveys and Exhibits, Russell Sage Founda- tion, New York City, 1916. 161. Williams, J. H. Reorganizing a County System of Schools, United States Bureau of Education, Bulletin No. 16, 1916. BIBLIOGRAPHY 375 Vn. REFERENCES CONTAINING DETAILED BIBLIO- GRAPHIES OF VARIOUS PHASES OF SCHOOL ADMINISTRATION 162. Cubberley, E. P. Public School Administration. Houghton Mifflin Company, Boston, 1916. 163. Strayer, G. D., and Tliorndike, E. L. Educational Adminis^ tration. The Macmillan Company, New York, 1913. 13, 14, and 15. Fourteenth, Fifteenth, and Sixteenth Yearbooks of National Society for the Study of Education. See 13, 14, 15. 164. Holmes, H. W., and others. A Descriptive Bibliography of Measurement in Elementary Subjects. Harvard University Press, Cambridge, 1917. 41. Holley, C. E. The Relationship between Persistence in School and Home Conditions. See 41. 80. Douglass, A. A. The Junior High School. See 80. 84. Hall-Quest, A. L. Supervised Study. See 84. 165. Rugg, H. O. "Summary of the Literature on Public School Costs and Business Management." Elementary School Jour- nal, April, 1917. 57. Rugg, H. 0. Teachers' Marks and Marking Systems. See 57. APPENDIX A SELECTED BIBLIOGRAPHY ON STATISTICAL METHODS A. HISTORY OF STATISTICS Meitzen, A. "History, Theory and Technique of Statistics." Annals, American Academy of Political and Social Science. Philadelphia, 1891, Part i. Translation by R. P. Falkner. B. GENERAL AND NON-MATHEMATICAL BOOKS Elderton, W. P., and E. M. Primer of Statistics. A. & C. Black, London, 1910. King, W. 1. The Elements of Statistical Method, The Macmillan Company, New York, 1915. Zizek, F. Statistical Averages. Translated by W. M. Persons. Henry Holt & Co., New York, 1913. C. GENERAL BOOKS INVOLVING NO CALCULUS Bowley, A. L. Elements of Statistics. P. S. King & Son, London; Charles Scribner's Sons, New York, 1907. Bowley, A. L. An Elementary Manual of Statistics. McDonald & Evans, London, 1910. Yule, G. U. An Introduction to the Theory of Statistics. C. GrijBSn & Co., London, 1912. D. BOOKS AND PAMPHLETS SUMMARIZING STATISTICAL METHODS AND GIVING TABLES Davenport, C. B. Statistical Methods. (Especially adapted to the study of biological variation). Summarizes all mathematical formulae with brief descriptions of methods. Complete and useful tables, containing probability integrals. Beta and Gamma func- tions, logarithms, etc. J. Wiley & Sons, New York, 1904. Kelley, T. L. Tables: To Facilitate the Calculation of Partial Co- efficients of Correlation and Regression Coefficients. University of Texas, Bulletin No. 27, 1916. Austin, Texas. Rietz, H. L. Appendix to E. Davenport's Principles of Breeding. Ginn & Co., Boston, 1907. APPENDIX 377 Rietz, H. L. Bulletins 119 and 148 of the University of Illinois Agricultural Experimentation Station. Summaries of statistical methods involving little advanced mathematics. Whipple, G. M. Manual of Mental and Physical Tests. Vol. 1, Simpler Processes, chap. iii. A brief summary of the statistical treatment of measures as applied to problems of educational psychology. Warwick & York, Baltimore, 2d edition, 1914. E. CALCULATING TABLES Barlow's Tables of Squares, Cubes, Square-roots, Cube-roots and Reciprocals of all Integers, Numbers up to 10,000. E. Spon, New York. Cotsworth, M. B. The Direct Calculator. Series 0. R. M. B. Cotsworth, Holgate, York, England. Crelle, A. L. Rechentafeln. G. Reiner, Berlin, new edition, 1907. (Cotsworth and Crelle give products to 1000 by 1000.) Elderton, W. P. Tables of Powers of Natural Numbers and of the Sums of Powers of the Natural Numbers from 1 to 100. Biometrika, vol. II, p. 474. Peters, J. Neue Rechentafeln fiir Multiplikation und Division. G. Reimer, Berlin. F. BOOKS ADAPTING STATISTICAL METHODS TO EDUCA- TIONAL AND PSYCHOLOGICAL INVESTIGATION Brown, The Essentials of Mental Measurement. Cambridge Uni- versity, 1911. Thorndike, E. L. An Introduction to the Theory of Mental and Social Measurements. Teachers College, Columbia University, New York, 1913. G. BOOKS ON GRAPHIC METHODS Brinton, W. C. Graphic Methods for Presenting Facts. Engineering Magazine Company, New York, 1914. H. FOR SUMMARY OF MATHEMATICAL THEORY UNDER- LYING CORRELATION AND TYPE CURVES OF DIS- TRIBUTION, see Elderton, W. P. Frequency Curves and Correlation. C. & E. Layton, London, 1906. 378 APPENDIX I. FOR A TREATMENT OF LEAST SQUARES, see Merriman, M. A Textbook on the Method of Least Squares. J. Wiley and Sons, New York. J. THE ORIGINAL CONTRIBUTIONS TO THE MATHEMATICAL FOUNDATION OF STATISTICAL METHODS WILL BE FOUND PRINCIPALLY IN Journal of the Royal Statistical Society; Philosophical Transactions of the Royal Society; Biometrika; Draper's Company Research; Memoirs; Philosophical Magazine. K. A COMPLETE BIBLIOGRAPHY OF THE ORIGINAL MEMOIRS WILL BE FOUND IN Yule, G. U. Introduction to the Theory of Statistics. C. Griffin & Co., London, 1912. L. PROBLEM BOOKS IN EDUCATIONAL STATISTICS Rugg, H. O. Illustrative Problems in Educational Statistics. Published by the author. University of Chicago Press, 1916. APPENDIX B SUMMARY OF FORMULAE AND SYMBOLS USED IN THE TEXT CHAPTERS IV AND V / = frequency of measures 771 = a measure A^ = total number of cases d = deviation (used for deviation in units of class-intervals) ,_ . , . , S/m M = arithmetical mean = —rr N Md = median (= Q2) Mo = mode c = correction applied to assumed mean to obtain true mean H = Harmonic mean, t; = - ^ ( ~" ) Tave = average rate tave = average time n Mo= geometric mean = y/ Ml . m2 . 7113 . m^ w„ CHAPTER VI i Qi = first or lower quartile point Qs = third or upper " " Q = quartile deviation or semi-interquartile-range = ~ — — 2 a- = stan dard deviation (sometimes represented S.D. or e) = \| N P.E. = Probable error = .6745-y_'"»