BEVERLEY EDUCATIONAL SERIES 
 
 EDITED BY 
 
 W. W. CHARTERS 
 
 PROFESSOR OF EDUCATION 
 UNIVERSITY OF ILLINOIS 
 
SCHOOL STATISTICS AND 
 PUBLICITY ^f; 
 
 BY 
 
 CARTER ALEXANDER 
 
 FIRST ASSISTANT STATE SUPERINTENDENT OF PUBLIC 
 
 INSTRUCTION FOR WISCONSIN 
 
 SOMETIME PROFESSOR OF SCHOOL ADMINISTRATION 
 
 GEORGE PEABODY COLLEGE FOR TEACHERS 
 
 SILVER, BURDETT AND COMPANY 
 
 BOSTON NEW YORK CHICAGO 
 
\r^ 
 
 % 
 
 %^^ 
 
 •h^ 
 
 Copyright, 1919, by 
 CAETER ALEXANDER 
 
 All rights reserved 
 
 M":r 
 
 n -; 
 
 CI.A5L1824 
 
EDITOR'S PREFACE 
 
 We hear from the forum and pulpit that reconstruction 
 must follow a war of such magnitude as that just closing. 
 But the laws of habit still operate and, if permitted to do 
 so, the nation will return, in fact will prefer to return, to 
 those accustomed grooves of thought and action from 
 which they have been so vigorously shaken. 
 
 This return is not possible in those cases which have an 
 economic basis. The industrial world will be compelled 
 by the insistence of labor to change its ideals and practices, 
 and problems of finance, taxes, and revenues will force 
 reconstruction in the field of politics. But education will 
 not feel the insistent urge of economic forces unless some 
 obvious catastrophe, such as the spectacle of schoolrooms 
 unprovided with teachers because of public parsimony, 
 visualizes the crisis for the taxpayers. Situations less 
 obvious than this the'citizen does not see and understand. 
 It is difficult to show him that low salaries may result in 
 other evils as serious as the corporeal absence of teachers 
 from the classroom. He must be shown that not only 
 must teaching be good but that better buildings, better 
 sanitation, curriculum changes, compulsory education, 
 and vocational education must be provided. 
 
 Sonorous or staccato generalities about reconstruction 
 which, I venture to say, in less than a twelvemonth of 
 our swift-moving current of national life will have be- 
 come platitudinous, will not produce that sympathetic 
 attitude toward educational reconstruction which is neces- 
 
vi Editor's Preface 
 
 I 
 
 sary for its processes to be carried on. Antebellum habits 
 will wage a momentous warfare upon transitory emotional 
 ideals. . 
 
 And, even though the present moment should be preg-f 
 nant with sympathy for the improvement of the schools, 
 other obligations compete for a place in the necessarily 
 limited field of public attention. National taxes com-i 
 pete with school taxes, political conflicts are more promi-" 
 nent than school necessities, and industrial readjustments 
 obtain readier attention than educational reconstruction. 
 
 To meet this situation the man of the hour is the super- 
 intendent of schools. The problem of crystallizing the 
 liquid desires of the public for educational progress is as 
 squarely placed upon his shoulders as any human task 
 has ever rested upon any individual. For education is 
 personal. The nation is educated in community groups. 
 Just as the squad is the unit of the army, the community 
 is the unit of the nation, and just as the non-commissioned 
 officer who leads the squad is the ''backbone of the army,'' 
 so the superintendent who leads his community is the 
 backbone of the forces of education. The educational 
 army is just as dependent upon his intelligence and in- 
 dustry as the fighting army is dependent upon the cor- 
 poral. 
 
 The chief weapon for leading the people of a community 
 in educational activity is publicity. And this may be 
 obtained in three ways. One method is to develop ex- 
 cellent schools and let the work speak for itself through 
 satisfied parents, loyal teachers, and efficient children. 
 But this method as the sole method is open to the criti- 
 cism that parents and children cannot clearly distinguish 
 between good school systems and inferior systems, and 
 to the further criticism that if people are too well satisfied 
 
Editor's Preface vii 
 
 with their schools they are not sensitive to the need for 
 better schools. So, important as the maintenance of a 
 good system is, its presence does not insure knowledge of 
 its excellence and needs. 
 
 A second method is personal explanation of what the 
 school is doing and attempting, carried on by conferences 
 with the school board, by public meetings, and by private 
 conversation. But a third method to be added to these 
 is the superintendent's annual report and printed com- 
 munications, in which a wider audience is reached. All 
 three of these methods are used by those superintendents 
 who have been most successful in molding public opinion. 
 
 But, as Mr. Alexander points out, seventy per cent of 
 a group containing one hundred twenty-eight members 
 of two of the most distinguished organizations of an in- 
 telligent community stated that they did not read the 
 school board reports. And this proportion is small for 
 the nation as a whole. 
 
 Upon the problem of making the superintendent's 
 report readable by his community this text is directed. 
 The author attacks the whole problem from the collect- 
 ing of the data and their statistical treatment, to the 
 presentation of his findings in simple and graphic form. 
 It is, therefore, presented as a notable attempt to make 
 known to the public those inner workings of the school, 
 to the end that fluid educational interest may take on 
 stability of action directed toward progressive ends. 
 
AUTHOR'S PREFACE 
 
 The admitted ineffectiveness of our school statistics is 
 due mainly to two causes. First, many superintendents 
 do not know how to apply statistics as well as do their 
 equals in intelligence in other fields. Second, these super- 
 intendents do not know — because they have been too 
 busy to learn — how to present statistical matter to the 
 public effectively. 
 
 As the experience of publicists in other fields shows, the 
 ways of doing these things are simple. But not enough 
 advantage of the labors of these men has been taken, nor 
 have their results been adapted for easy and quick use 
 by the busy superintendent. There is pressing need for 
 a book which will do these very things. The aim of this 
 book, then, is to make available for superintendents and 
 classes in school administration the results of years of 
 study of statistical theory and its applications to school 
 data for publicity purposes, as shown in school reports 
 and surveys. 
 
 These results have all passed through the fires of criti- 
 cism of the large number of practical school men whom 
 it has been the author's good fortune to have as students 
 during the past six years. In particular, he wishes to 
 acknowledge his great indebtedness to the men of Ed- 
 ucation 245, his graduate course at Peabody, whose 
 generous aid and criticism have contributed much to 
 the practical helpfulness of the book. 
 
 Acknowledgments are also gratefully made to the fol- 
 
 ix 
 
X Author's Preface 
 
 1 
 
 lowing : Professor E. L. Thorndike of Teachers College, 
 who introduced the writer to statistical method ; the 
 editor, Professor W. W. Charters, who has given valuable 
 suggestions for modifying the form of the book; Mr. 
 W. C. Brinton and Professors J. F. Bobbitt and W. I. 
 King, on whose writings the author has drawn freely; 
 Dr. W. W. Theisen, supervisor of educational measure- 
 ments in the Wisconsin State Department of Public In- 
 struction, who has read part of the proof ; the various 
 writers and foundations who have given permission for 
 the reproduction of copyrighted charts; Messrs. C. H. 
 Moore, E. McK. Highsmith, and S. C. Garrison for careful 
 criticism of the manuscript. 
 
 A special feature of this book is that all but five of the 
 cuts are from drawings made by the boys in the Tech- 
 nological High School at Atlanta under the direction of 
 Mr. E. S. Maclin, at that time head of the drawing de- 
 partment there. Mr. Maclin, who was one of the men 
 in Education 245, has kindly given his services to produce 
 this practical demonstration of the possibilities in utilizing 
 high school students on school publicity. No doubt there 
 are some slight errors or irregularities to be found in the 
 cuts. But all such imperfections are only evidences of 
 the genuineness of the demonstration. 
 
 The school superintendent must be a publicist. He 
 must make reports to the public. In many places for 
 the next decade at least he must fight as hard as any officer 
 in the trenches to ward off the incessant and fierce attacks 
 made upon his school appropriations by politicians and 
 hard-pressed but unthinking tax-payers. For warding 
 off or beating back such attacks, his most effective weapons 
 will be reports containing simple but skillful statistical 
 devices for presenting the claims of the school children. 
 
Author's Preface xi 
 
 Unless he has such weapons, the enemy will be liable to 
 sweep over the schools and place them on a starvation 
 diet. 
 
 If this book in any material way helps the superin- 
 tendent of schools, active or in training, to arouse his 
 public and to secure adequate support for his school, it 
 
 will fulfill its mission. 
 
 Carter Alexander. 
 
SUGGESTIONS FOR USING THE BOOK 
 
 I. TO THE SUPERINTENDENT IN THE FIELD 
 
 The active superintendent, accustomed as he is to 
 adapting materials quickly to his own problems, of course 
 needs few "suggestions for using this book. But to save 
 his time, a few are given. 
 
 1. All of this book except Chapters V to VII can 
 be read with little effort by any active school man. Ap- 
 plications to his own work will constantly come to mind. 
 
 2. A beginner in statistical method can attain a fair 
 acquaintance with the matters involved in Chapters V 
 to VII by a careful reading of the text. But a usable 
 knowledge of these chapters, for one not previously familiar 
 with statistical method, can hardly be expected unless the 
 exercises or similar ones are done as indicated in the 
 suggestions for the instructor of future superintendents. 
 For such exercises, the superintendent generally has on 
 hand several administrative problems that would be 
 greatly simplified by adequate statistical treatment. 
 His annual report, in particular, will furnish material for 
 practically every exercise in the book. 
 
 I 3. The table of contents and the index will enable him 
 to use the book as a reference book for securing material 
 or suggestions for procedure on school problems necessitat- 
 ing statistical treatment. He can also quickly get ideas 
 and devices for presenting his results to the public. 
 
 4. If he has plenty of time, the procedure advocated 
 for students in training will be very profitable. 
 
 xiii 
 
xiv Suggestions for Using the Book 
 
 II. TO THE INSTRUCTOR ENGAGED IN TRAINING FUTURE 
 
 SUPERINTENDENTS 
 
 The experience of the writer with the active superin- 
 tendents and future superintendents in his own classes 
 indicates the following : 
 
 1. It is advisable to have a good supply of supple- 
 mentary material for practice work and concrete illus- 
 trations. At least one copy of each reference given Li 
 the bibliography in the back of the book is advisable. 
 In addition, one should secure before the class starts a 
 fairly complete collection of typical annual reports and 
 publicity pamphlets of the city and county superintend- 
 ents in the district, state, and section of the country from 
 which the students come. It is well to have several 
 copies of the better ones of these, but for most of them 
 one copy each is sufficient. These should be distributed 
 among the members of the class, each being allowed to 
 choose those he especially cares for and each being re- 
 sponsible for his own set. All students may be required 
 to become familiar with the better ones for which dupli- 
 cate copies are available. 
 
 2. Portions of the text may be prepared for class dis- 
 cussion as in any textbook course. But the instructor 
 may test the extent to which the students have mastered 
 the text and create much more interest by constantly 
 expecting them to bring in pertinent illustrations of the 
 points under discussion. These illustrations should be 
 obtained from the reports or other material for which 
 the student is responsible, from his previous experience, 
 or from current magazines and advertisements. 
 
 3. A mastery of the text, however, can be assured only 
 when students prove their ability to handle the exercises 
 
 ♦ 
 
Suggestions for Using the Book xv 
 
 in this book, or similar ones which the instructor may 
 easily make up or select from the references in the bibliog- 
 raphy. ''To know whether any one has a given mental 
 state, see if he can use it." 
 
 Answers to exercises have been purposely omitted. 
 Most students come to such exercises with a habit of 
 striving for an accuracy to several decimals in the answer 
 rather than of concentrating their attention on the prin- 
 ciples involved. The particular decimals secured in any 
 answer will often vary considerably according to the 
 grouping used, the extent to which decimals are carried 
 out, and so forth. Again, many administrative problems 
 involve original data based on approximate measures, 
 where attempts to secure absolute numerical accuracy 
 are a sheer waste of time for everybody. By checking 
 the procedure and answers of various students against 
 each other, a sufficiently accurate answer will be ob- 
 tained in the class. To secure this result, however, the 
 instructor should see that the exercises are worked out 
 in full and handed in. Judging by the writer's experi- 
 ence, it is hardly worth while to discuss any of the ex- 
 ercises that involve computation until they have been 
 completely written out by the students. The work of 
 both student and instructor will be greatly reduced for 
 exercises which occur on pages 38 to 316, by utilizing the 
 work already done on a problem, merely adding the work 
 for the particular point to be illustrated and handing in 
 the exercise again. 
 
 If there is time and the students are sufficiently mature, 
 the problem started in the exercises on pages 38, 43, 57, 
 61, 81, 89 may be continued throughout the course by 
 each student with excellent results. It is not continued 
 in this book because of the difficulty of keeping a group 
 
xvi Suggestions for Using the Book 
 
 together on such work in the usual time allotted to reci- 
 tations. But in the writer's own classes, his students 
 continue to work outside on such problems and make 
 oral or written reports at the end of the course. Prob- 
 lems of this nature interest students greatly. Any in- 
 structor can find many opportunities for them in the 
 local city school system, with resulting profit to the 
 system. 
 
 4. For the convenience of the instructor and a few of 
 the students who evince unusual interest or capacity for 
 the work, a few references are given at the end of various 
 chapters. It is not, however, advisable to use many of 
 these for class reading. In attempting to understand 
 statistical method, the student must, above everything 
 else, avoid confusion and keep his feeling of mastery over 
 what he has done. But he will find the various writers 
 taking up topics in different sequence and using different 
 definitions or different technical terms in a manner very 
 confusing to a beginner. Much better results will be 
 secured if the instructor will use most of the references 
 himself and adapt the additional material he wishes to 
 the plan of this book and the special needs of his students. 
 
 Experiments with the writer's own classes show that a 
 satisfactory working knowledge of the material presented 
 in this book can be comfortably acquired in from twenty- 
 five to forty-five class periods. The shorter time will, of 
 course, reduce the use of exercises. The longer time will 
 allow abundant opportunity for the use of the exercises 
 and special problems advocated. The book can thus be 
 used in a six weeks' summer school, in a corresponding 
 period during the regular session, or in whatever longer 
 period is available. 
 
I 
 
 TABLE OF CONTENTS 
 
 PAGE 
 
 Chapter One. Why We Need Better School Statistics 1 
 I. Common Errors 
 II. Wastefulness of These Errors 
 
 III. Indifference of the PubHc 
 
 IV. Progress Elsewhere in Using Statistics 
 
 V. Unsatisfactory State of Affairs in School Statistics 
 
 Chapter Two. Collection of Data 33 
 
 I. When to Use Statistical Method 
 II. How to Plan Statistical Treatment of Problems 
 
 III. How to Determine Units and Scales 
 
 IV. How to Do the Actual Collecting 
 
 Chapter Three. Technical Methods Needed in School 
 
 Statistics 90 
 
 I. Usual Views 
 II. Statistical Knowledge Needed for School Surveys 
 
 III. Statistical Knowledge Needed for Reading Educa- 
 tional Investigations 
 
 IV. Illustration of Value of Statistical Method to the 
 Superintendent 
 
 V. Statistical Method as a Form of Expression 
 
 Chapter Four. Scales, Distribution Tables, and Sur- 
 faces OF Frequency . . . . . . 100 
 
 I. Scales 
 II. Distribution Tables 
 III. Surfaces of Frequency 
 
 Chapter Five. Measures of Type 124 
 
 I. The Mode 
 II. The Median 
 
 III. The Average 
 
 IV. Which Measure of Type to Use in a Given Dis- 
 
 tribution 
 
 xvii 
 
XVIU 
 
 Table of Contents 
 
 Chapter Six. Measures of Deviation or Dispersion 
 I. Extreme Range Variation 
 II. Quartile Deviation 
 
 III. Other Percentile Deviations 
 
 IV. Median Deviation 
 V. Average Deviation 
 
 VI. Standard Deviation 
 VII. Deviations for Skew Distributions 
 VIII. Which Measure of Deviation to Use in a Given 
 Distribution 
 
 PAGE 
 
 149 
 
 164 
 
 Chapter Seven. Measures of Relationships . 
 I. Relationships Inside of One Group 
 II. Simple Relationships between Different Distributions 
 
 III. Coefficient of Variability or Dispersion 
 
 IV. Correlation 
 
 Chapter Eight. Supplement on Statistical Treatment . 187 
 I. Reliability of Statistical Results 
 II. Special Economies in Calculation 
 III. Combining Data Given in Rank Order Only 
 
 Chapter Nine. Uselessness of Statistics in Current 
 
 School Reports 200 
 
 I. The Situation 
 II. Causes of the Uselessness 
 III. Devices for Effective Presentation 
 
 Chapter Ten. Presenting Tabulated Statistics to the 
 
 Public 
 
 I. Possibilities of Using Tabulations of Statistics to In- 
 fluence the Public 
 II. How to Give a Bird's-Eye View through Tabulation 
 III. How to Make Up a Series of Tables of the Same 
 General Nature 
 
 Chapter Eleven. Graphic Presentations of School 
 Statistics Especially for the Public 
 I. Object of Graphic Presentations 
 
 II. How to Make Graphs for the Public from Statistical 
 Data 
 
 206 
 
 234 
 
Table of Contents xix 
 
 PAGE 
 
 III. How Graphs for the Public Differ from Those for 
 
 the Administrator 
 
 IV. Examples of Good Graphs on School Statistics for 
 
 the Public 
 V. Economies in Making School Graphs for the Public 
 
 Chapter Twelve. Translating Statistical Material 
 
 ON Schools for the Public .... 303 
 I. The Need of Translation 
 II. Suggestions for Good Translations 
 III. Examples of Good Translations of School Sta- 
 tistics 
 
 Selected and Annotated Bibliography .... 317 
 Index 323 
 
SCHOOL STATISTICS AND PUBLICITY 
 
 CHAPTER I 
 
 WHY WE NEED BETTER SCHOOL S'^ATISTICS 
 
 I. COMMON ERRORS 
 
 Many school superintendents have no doubt laughed 
 over some variation of the following : 
 
 A clerk in a department store asked for a raise in salary and got 
 this reply from the owner : 
 
 "Why, this is no time to ask for more wages. Times are too hard 
 and you do very little work. 
 
 "I will show you how little work you do for me in a year. 
 
 "The year has 365 days in it, and each day is 24 hours, divided 
 in three equal parts, 8 hours for work, 8 hours for sleep, and 8 hours 
 for play. 
 
 "Now just listen. Take 8 hours that you sleep in each day, 
 which is 122 days, from 365, and that leaves 243 days. 
 
 "You play 8 hours each day, which is another 122 days from 243, 
 and that leaves 121 days, — see? 
 
 "Now, there are 52 Sundays when you do not work; just take 
 these from 121, and that leaves 69 days. 
 
 "When the summer comes, you say: *I can't work, I'm all in, 
 and I want a vacation.' I give you 2 weeks off; take 14 days 
 from 69. This leaves 55 days. 
 
 "Then the store closes a half day every Saturday. This is 26 
 from 55, which leaves 29 days. 
 
 "Then you take li hours for your lunch each day, which is 28 
 days, from 29, or 1 day left. 
 
 "And I just remember that that day was the retailmen's picnic, 
 and you asked off to go to it." 
 
 1 
 
2 School Statistics and Publicity 
 
 The isolated statements seem fair enough on the sur- l\ 
 face when taken one at a time. What has been done 
 to manipulate the figures so that a patently erroneous 
 conclusion is reached ? Humorous as this is, it is not so 
 far beyond what can be found sometimes in the statistical 1 
 parts of school reports. 1 
 
 Let us examine samples of these errors, which for con- 
 venience may be regarded as arising from troubles with 
 
 1. Unanalyzed Totals. 
 
 2. Comparisons Employing Indefinite Units. 
 
 3. Comparisons Using Unsound Elementary Treatment. 
 
 4. Attempts at Too Great Accuracy. 
 
 5. Neglect of Technical Statistics. 
 
 6. Presentations of School Statistics to the Public. 
 
 These will now be discussed in order, but only for the " 
 purpose of showing the errors clearly. The remedies 
 for such troubles will be given in subsequent chapters. ^ 
 
 1. Unanalyzed Totals 
 
 Some commercial clubs and school systems print figures 
 on their letterheads giving the size of the schools, number 
 of teachers, number of children enrolled, number of build- 
 ings, size of plant, etc. A certain normal school in a 
 middle-western state uses this material on its letterhead : 
 
 A Teacher's College 
 
 Faculty, 46 — men, 19 ; women, 27 
 
 Enrollment, 1915-16, 2179 
 
 Average attendance, 874 
 
 1 For those who wish to use this book for reference purposes or 
 for review, cross references to later portions are given throughout 
 this chapter. But for those who intend to read the whole book and 
 who are going through the chapter for the first time, it is advisable 
 to ignore the cross references at this stage. 
 
Why We Need Better School Statistics 3 
 
 These figures mean very little unless the reader knows in 
 addition whether the proportion of the number of the 
 faculty to the number in the student body is somewhere 
 close to that of the average normal school in the country ; 
 whether, in averaging the attendance, there was one 
 short term with a heavy attendance set over against 
 three long terms with a rather lean attendance, etc. 
 
 Another illustration of this error is the statement 
 sometimes made when medical inspection is installed 
 in a city, that the results show 90 per cent or over of the 
 children to be defective. This tends to alarm people 
 until they consider that there must be some fallacy in 
 it or there could be no such thing as normal children in 
 that city. The fact is that practically every child is 
 physically defective on some point. When children are 
 measured on ten to twenty physical points, each child 
 is certain to be defective on some one point, but many 
 of them are, on the whole, average or normally healthy 
 children. The unanalyzed total gives a misleading im- 
 pression. 
 
 2. Comparisons Employing Indefinite Units 
 
 Even if the error of unanalyzed totals is avoided, a 
 second weakness may creep in through foolish or mis- 
 leading comparisons. These may be of this nature 
 because of lack of suitable units for comparing the things, 
 or the trouble may arise because of unsound methods 
 in making the comparisons. 
 
 Enrollment. For example, a certain college at one 
 summer session announced an enrollment of 1108. At 
 the second summer session after this the figures were 
 1484. Apparently this was a gain of only 34 per cent. 
 In reality the school had gained about 60 per cent. The 
 
School Statistics and Publicity 
 
 ^ 
 
 enrollment the first time was padded by the issuing of 
 visitors' cards, even to wives of the faculty. The enroll- 
 ment the third time included only bona fide students, 
 owing to the need of impressing certain people with the 
 school's actual enrollment. The difference in units here 
 prevented the school from gaining the advertising value 
 of the really large increase of the second year. 
 
 Serious errors often occur in comparisons between 
 school systems because the number of children has not 
 been figured on the same basis. Thus, this number may 
 be calculated as the total enumerated of school age (school 
 census), as the total ever on the roll (enrollment), as the 
 total number in attendance that have not been absent 
 over three days (number belonging), or as the average 
 daily attendance. Also, the percentage of daily attendance 
 is often figured on the " number belonging " base. This of 
 course always gives a high percentage of daily attendance, 
 since absent pupils are rapidly dropped from the count. 
 It is in reality a measure of the holding power of the 
 schools for truants, rather than for all children of school 
 age. Sometimes the enrollment in a school system equals 
 or exceeds the enumeration, not because of the attraction 
 for enumerated children, but because of the enrollment 
 of non-residents, no mention of which may be made in 
 that connection. Or the enrollment may be greater be- 
 cause of transfers inside the system so that the same 
 pupil is enrolled in more than one ward school and con- 
 sequently is counted twice or more on enrollment. 
 
 School Expenditures. Erroneous conclusions are often 
 reached in attempting to compare the total expenditures 
 of one city school system with those of another city, 
 without proper units. These totals are vitally affected 
 by such factors as the number of people residing in the 
 
Why We Need Better School Statistics 5 
 
 city, the per capita wealth, the proportion of children 
 in the population, the rate of taxation for all purposes, 
 etc. A rich suburban city can be expected to expend 
 far more on its schools than can a city of the same size 
 made up mainly of factory workers or similar wage 
 earners. 
 
 Distribution of School Moneys. In some places in 
 the South, state school money is distributed on one unit, 
 the enumeration basis, and expended on another, the 
 number of children (more likely of white children) in at- 
 tendance. For example, some years ago in Alabama one 
 community is said to have had enough in this way to pay 
 the principal of the white school $1200 and give him 
 three assistants, all for an enrollment of forty-five white 
 children and an average daily attendance of not more 
 than thirty-five. This was of course achieved by spend- 
 ing on the white schools considerable money drawn on 
 negro census children. ^ 
 
 Age of Child. Another indefinite unit is the " age " 
 of a child in any given school system. Is a six-year-old 
 child one that is six and has not reached seven, or is it a 
 child from five and a half up to six and a half years old ? 
 In making the recent survey of Cleveland, Dr. Ayres 
 found that the age of a child in that city was determined 
 by whatever birthday fell nearest to September of that 
 particular school year. This meant that a child was the 
 same age for a whole year. Children entering in June 
 were set down as being the same age as they would have 
 been if they had entered the preceding September. This 
 made the average ages of the children somewhat younger 
 than they 'should be. It had the effect of making many 
 children that were over-age in the fall, no longer over-age 
 1 Reported by Mr. W. F. Puckett 
 
6 School Statistics and Publicity 
 
 1 
 
 when promoted at the mid-term. Thus at the beginning 
 of the second term there was always a large and instan- 
 taneous drop in the percentage of retarded children, al- 
 though the children in reality were relatively about as 
 they were in September. That is, they were a half year 
 older and a half year farther along in the grades. Be- 
 cause of this peculiar definition of age, the survey com- 
 mission could make no comparisons in acceleration, elim- 
 ination, and retardation, with other cities.^ Similar 
 difficulties confront a superintendent who figures retarda- 
 tion before promotions in his system one year, and after 
 promotions the next year. And when he compares his 
 school system with any other system on retardation, he 
 must be certain that he is taking it at the same time it 
 was figured in the other systems. Another example is 
 the custom reported from Louisville, Kentucky, of count- 
 ing withdrawals as of the age they were on the previous 
 September first. Many of those withdrawing after the 
 middle of the year are, of course, in a higher grade or half- 
 grade from what they were when their ages were taken. 
 Teachers' Salaries. It is not uncommon to publish 
 the salaries of teachers as being so much per month, with- 
 out any mention of the number of monthly pajrments 
 per year. Obviously the teacher has to live twelve months 
 in the year, and for purposes of comparison the total for 
 the year is the figure that must be used, especially if other 
 people are to be influenced. For instance, a Kentucky 
 superintendent in a small city reported that when he got 
 his first month's salary check for $125, the cashier of the 
 bank, who drew $90 a month, remarked jocularly that 
 his own salary should be raised. He considered that the 
 superintendent was earning more money. In fact, the 
 1 Cleveland Survey, volume on "Child Accounting," pp. 40-43 
 
Why We Need Better School Statistics 7 
 
 superintendent drew $125 a month for eight months or a 
 total of $1000 ; the cashier drew $90 a month or a total 
 of $1080. It is equally obvious that such a report on the 
 monthly basis is very much to the advantage of the sys- 
 tem that runs for a shorter term. 
 
 Another example is the published salary list of the 
 faculty in a certain southern normal school. Some men 
 were listed at $150 and others at $133J a month. But 
 as a matter of fact, the $150 men were paid for eleven 
 months and the $133J men for twelve months a year, 
 while all worked the same length of time, thus making 
 an average difference of only a little over $4 a month. 
 
 Teachers' Schedules. It is rather common to attempt 
 to compare the amount of work done by teachers within 
 the same high school, normal school, or college, mainly 
 by taking simply the number of recitations each has per 
 week. This assumes that one recitation means as much 
 work as any other and that teachers do no work except 
 that connected with recitations. It leaves out of account 
 such factors as : one recitation with a large class means 
 much more work than one with a small class ; two reci- 
 tations in different classes involve much more labor than 
 two sections of the same class ; teachers serving on cer- 
 tain committees do a great deal of work connected with 
 them and entirely apart from class duties. 
 
 Tax Rate. Some state superintendents and many city 
 superintendents have published comparisons of cities or 
 counties with other cities or counties, based on the tax 
 rate for school purposes. This is of no value unless one 
 knows the rate of assessment and can thus arrive at the 
 real rate of taxation. It is a well-known fact that there 
 is an enormous variation in the rate of assessments in 
 various cities; that no two are practically the same. 
 
8 School Statistics and Publicity ^^j 
 
 Yet how often do we see in school reports long lists of 
 figures giving the school tax levies of many cities chosen 
 almost at random. But these do not impress the tax- 
 payer. He knows whether his total school taxes are heavy 
 or not and will pay little attention to figures that merely 
 place his city low in the list of cities ranked according to 
 rate of school tax.^ 
 
 But even the actual rate of taxation is still of little 
 value unless one knows how heavily the taxpayer is bur- 
 dened with other taxes. For example, several years 
 ago at Arkadelphia, Arkansas, the taxpayers protested 
 against a school tax of seven mills (70 cents on the $100) ; 
 they said that if this were added to their state school levy 
 of three mills (30 cents on the $100), it would make their 
 total school tax ten mills (100 cents on the $100), or higher 
 than most places in the South. This was totally wrong, 
 for many of the communities with which they were com- 
 paring themselves paid a county school tax in addition to 
 the local and state school taxes considered by them.- 
 
 College Degrees. A similar trouble arises from the 
 custom in many schools of at least secondary grade, of 
 advertising the percentage of their faculty that hold de- 
 grees. But with such degrees undefined these figures 
 mean nothing. In all sincerity may not the public ask 
 why an A.B. from Harvard or Vanderbilt University is 
 rated just the same as one from a private college of junior 
 rank which has perhaps four second-rate teachers and 
 practically no equipment? The fact is that the public 
 does ask just such a question. For this reason thinking 
 people place very little value at the present time on the 
 mere announcement of a degree. They wish to know 
 what institution conferred it. As a consequence some 
 
 1 Se3 pp. 51-2 2 Reported by Mr. A. E. Phillips 
 
 I 
 
Why We Need Better School Statistics 9 
 
 editors, authors, and catalogues, when giving a man's 
 degrees, also indicate the institutions conferring them. 
 
 Unequal Things. In the preceding examples the lack 
 of suitable units has come about rather from thoughtless- 
 ness than from any definite intention. But a very com- 
 mon error in school statistics is caused by deliberately 
 considering as equal, things which a very little serious 
 thought would show are not equal. Professor Thorn- 
 dike 1 gives a good example of the seriousness of this 
 error. Dr. Rice had ranked a large list of words as of 
 equal difficulty to spell. Professor Thorndike gave the 
 same list to a group of children with these results : 42 
 children missed necessary; 37, disappoint; and so on, down 
 to 1, feather; none, picture. Making the most of the 
 possibility that the group of children found some of the 
 words easier than others because of recent drills, yet the 
 fact is unmistakably clear that picture is not so difficult for 
 these children to spell as is necessary. 
 
 An especially good example of considering as equal 
 things which are not at all equal is found in the 
 following excerpt from a recent circular published in 
 Virginia: ^ 
 
 The general administration of school affairs in Virginia has also 
 furnished striking evidence of sound economy as well as splendid 
 efficiency. This is best shown, perhaps, in the recent comparison 
 with the wealthy state of Minnesota, which has practically the same 
 population as Virginia. In 1916 Virginia with eight millions of rev- 
 enue enrolled 11,560 more pupils than Minnesota with twenty-six 
 millions of revenue. 
 
 This ignores the proportion of children in the two state 
 populations, the length of school term, or anything con- 
 nected with quality of education. The schooling given 
 1 Thorndike, E. L. : Mental and Social Measurements, p. 8 
 
 ft 
 
10 
 
 School Statistics and Publicity 
 
 the average child in Virginia certainly is nothing like 
 the schooling given the average child in Minnesota. 
 The latter is so much superior that it might con- 
 ceivably cost over twice as much and still be really more 
 economical. 
 
 A third instance is found in library reports of various 
 schools. These are often useless for comparative purposes] 
 because some schools report all bulletins, agricultural I 
 and census reports, and duplicate copies, as separate books 
 and of equal value. 
 
 A fourth example is that furnished by the vicious at- 
 tacks on the state university made in some states by un- 
 scrupulous politicians. In these the yearly cost of school- 
 ing for a child in a rural school is compared with the several 
 hundred dollars required to pay for a year's instruction 
 of a university student. For this comparison one child 
 in a rural school is deliberately considered to be equal to 
 one student in the university. ^ 
 
 Grading Standards. In this connection we must not 
 overlook the errors that arise from differences in the 
 grading standards of different teachers in the same school. 
 Superintendents for purposes of records, determination of 
 class honors, etc., generally assume that their teachers 
 grade on something like the same standard. But, in 
 fact, the almost countless articles and books on uniform 
 grading published in the last few years demonstrate that 
 all teachers if left to themselves will vary widely in their 
 standards of marking pupils. 
 
 In some schools elaborate statistics are kept to deter- 
 mine the standing of pupils by marks, especially in high 
 schools for class or graduating honors, etc. But these 
 awards are really doubtful because no effort has been made 
 
 1 See also p. 47 
 
Why We Need Better School Statistics 11 
 
 to have the teachers mark uniformly. It is only when 
 teachers mark on something like a uniform basis that the 
 valedictorian may be easily found by averaging marks 
 given by them. Without some system of uniform mark- 
 ing, a student getting an average of 95 may really be of 
 the same ability as another getting an average of 97. In 
 the University of Missouri, before the adoption of its 
 present system of uniform grading, it was well known 
 that certain departments were '' snap " departments 
 because of the high grades given in proportion to the 
 amount of work required of the student. Professor 
 Meyer in one of his earlier articles on uniform grading ^ 
 shows that in the department of philosophy at this time 
 55 per cent of all grades given were ''A," while in the 
 department of chemistry only one per cent were '* A." 
 Consequently, students of the university who wished to 
 get into the Phi Beta Kappa fraternity tended to choose 
 all their electives from the departments known to give 
 many high marks. It may be taken for granted that few 
 possessors of a Phi Beta Kappa key in this institution 
 about this time had taken any more than was required 
 in chemistry, but probably many of them had elected 
 work rather freely in philosophy. The application of 
 the same principle to the high school valedictorian needs 
 no elaboration. 
 
 Judging Contests. The same error is made in judging 
 debating or essay contests with from, say three to five, 
 judges, unless the judges are obliged to grade on the same 
 scale. Suppose we have ten contestants, and three judges 
 who grade in terms of their own judgment, the grades being 
 as follows : 
 
 1 Meyer, Max: "The Grading of Students," Science, 28: 246. 
 (Aug. 21, 1908) 
 
12 
 
 School Statistics and Publicity 
 
 
 Marked by 
 
 Marked by 
 
 Marked by 
 
 Average 
 
 Contestant 
 
 Judge A 
 
 Judge B 
 
 Judge C 
 
 Mark 
 
 1 
 
 55 
 
 82 
 
 90 
 
 75.6 
 
 2 
 
 95 
 
 80 
 
 94 
 
 89.6 
 
 3 
 
 70 
 
 82 
 
 92 
 
 81.3 
 
 4 
 
 85 
 
 88 
 
 94 
 
 89.0 
 
 5 
 
 70 
 
 82 
 
 91 
 
 81.0 
 
 6 
 
 84 
 
 86 
 
 96 
 
 88.6 
 
 7 
 
 80 
 
 86 
 
 96 
 
 87.3 
 
 8 
 
 75 
 
 84 
 
 96 
 
 85.0 
 
 9 
 
 75 
 
 80 
 
 96 
 
 83.6 
 
 10 
 
 60 
 
 82 
 
 92 
 
 78.0 
 
 Each of the judges has rated a different contestant as the 
 highest, and the proposition is made that an average be 
 taken of all the ratings. When this is done, it is found 
 that Contestant 2 has won, although Judge B has rated 
 him as one of the worst of the performers and Judge C 
 has ranked him as tied for fifth place. Contestant 4 
 loses although he is rated in first place by Judge B, second 
 place by Judge A, and as tied for second place in Judge 
 C's opinion. The reason he loses, although his combined 
 rankings are higher, is this : the judges did not grade on 
 the same scale. Judge A used a scale of much wider 
 variation than did the other judges, and he gave Contest- 
 ant 2 ten points more than any of the other contestants. 
 The whole range of Judge B's grades was only eight points ; 
 that of Judge C's only six points ; but Judge A's ranged 
 forty points. Therefore, his rating of Contestant 2 more 
 than overcame the combined judgments of the other two 
 men.^ 
 
 1 For further treatment of units and scales, see pp. 43-57. 
 
Why We Need Better School Statistics 13 
 
 3. Comparisons Using Unsound Elementary Treatment 
 
 Errors in the Use of Percentages. Of the unsound 
 methods of comparison, some common ones deal with 
 percentages. First, there is the kind that ignores the 
 real meaning of percentage. For example, a superin- 
 tendent may claim a decrease of 200 per cent in retarded 
 children or number of withdrawals. Such a statement is 
 of course meaningless, for there can be only 100 per cent 
 of anything on which to figure the decrease. What is 
 usually meant is that the present number is now only one 
 third of what it once was, a decrease of 66| per cent. 
 
 This error runs at once into the one which frankly loses 
 sight of the original data. Some years ago a prominent 
 city superintendent in Georgia had the charge brought 
 against him that illiteracy of adult whites had increased 
 100 per cent in his city, while in the state as a whole it 
 had decreased from 13 per cent to 10 per cent or a 23 per 
 cent actual decrease in the amount of illiteracy. (13 - 10 
 = 3 . y3_ := 23 % + . ) The facts were that this city of about 
 forty thousand population had, at the beginning of the 
 period in question, a certain small number of adult white 
 illiterates, say ten. During the period several families 
 of ignorant Greeks moved in, bringing at least ten more 
 adult white illiterates. Based on the original number 
 of illiterates, the increase was 100 per cent; but based 
 on the total population, the increase was negligible. Yet 
 some supposedly well-informed people in this city, not 
 knowing or thinking how few cases the misleading state- 
 ment was based upon, were inclined to criticize the super- 
 intendent of schools. 
 
 Especially may the original data be forgotten in figuring 
 percentages on parts of a small group. Thus, taking the 
 
 k 
 
14 School Statistics and Publicity 
 
 ^1 
 
 percentages of the various marks given by a teacher in a 
 class of less than twenty pupils is of doubtful value. Any j 
 unusual condition of any sort affecting one pupil only { 
 will affect the number of pupils receiving the same mark, 
 by more than one twentieth of the whole. If there are 
 ten pupils in the class, it will be affected 10 per cent and 
 so on. If two pupils out of ten received A, 20 per cent of 
 the class would get this mark. If one of these pupils for 
 any reason should do poorer work, the A group would 
 lose 10 per cent of the whole, but half, or 50 per cent, of 
 itself. 
 
 In similar fashion the Missouri School Journal some 
 years ago ran a comparison of state teachers' associations, 
 based upon the percentage of teachers in the state attend- 
 ing them. Rhode Island came first and Tennessee last. 
 This frankly ignored to a considerable extent the original 
 data with its lack of a unit for " teacher in attendance." 
 The Rhode Island teachers could attend their meeting 
 at Providence on a nickel street car fare or a dime at the 
 highest. But many teachers in the larger states could not 
 attend their state association for financial reasons. Thus 
 a low percentage of attendance in some states might mean 
 a much greater spirit and devotion than would a 100 per 
 cent attendance in Rhode Island. 
 
 Other comparisons attempt to relate directly groups of 
 data which should first have their component parts turned 
 into percentages. For example, on the wall in a certain 
 state department of education, the grades on certificate 
 examinations made in each subject are plotted in a graph 
 ranging from 70 to 100. The name of the subject is put 
 on the graph and the name of the person grading the 
 papers. The attempt is made to standardize the gi'ading 
 in the different subjects by comparing these graphs. But 
 
 • 
 
Why We Need Better School Statistics 15 
 
 because of the failure to turn the numbers of grades for 
 each group into percentages of the whole, the compari- 
 sons are either very difficult or erroneous. The compari- 
 sons could be made directly only if there were the same 
 number of papers in each subject, a thing which does not 
 often happen even approximately. 
 
 Carelessness in Securing Data. A second form of un- 
 sound comparisons frequently comes from carelessness in 
 securing data. For example, in comparing school sys- 
 tems on expenditures, the figures taken for one year may 
 be wholly misleading for at least one or two cities out of 
 any twenty. An unusual condition in some one city, such 
 as an epidemic or a great expansion of work, or the erec- 
 tion of a building a few years before, may make the ex- 
 penditures for that year wholly different from the usual 
 expenditures of that city. Again, unless great care is 
 taken, the total expenditures for one city may include 
 money spent on buildings and repairs, and duplicates, 
 while for other cities they cover current expenses only 
 without duplicates. Unless the figures for each city are 
 usual, or " average " ones, and unless they are taken for 
 all cities on the same basis, no amount of care later will 
 give sound comparisons. 
 
 Omission of Important Factors. A particularly un- 
 fortunate form of comparison is that which presents data 
 related to each other and then draws from them conclu- 
 sions that are entirely erroneous because certain impor- 
 tant factors have been left out of consideration. In other 
 words, it is the old fallacy of arguing from insufficient 
 data. For instance, take the case of the state superin- 
 tendent in the South who some years ago claimed credit 
 for the increase in school revenues in his state. Both 
 from the platform and in press reports, he claimed that he 
 
16 School Statistics and Publicity 
 
 had done more towards adding money to the state school 
 fund than had any of his predecessors. He even issued 
 a bulletin in which he compared the amount per capita 
 for schools during his regime with that of five of his prede- 
 cessors in the office. In his closing sentence he styled 
 himself " the wizard of finance." As a matter of fact, 
 the state in question through industrial development had 
 largely increased its wealth so that the same rates for 
 school taxation brought larger school revenues. The 
 wealth had increased faster than the population, and so 
 the per capita spent on schools rose. However capable 
 a state superintendent of schools may be, he can hardly 
 legitimately claim much credit for increasing the wealth 
 of his state in a few years, especially at a rate faster than 
 the population. 
 
 Another example is the argument sometimes used 
 against compulsory education. This cites that illiteracy 
 has decreased faster in certain southern states that have 
 no compulsory education laws than in certain northern 
 states that have had such laws for years. But it leaves 
 out of account the fact that the smaller northern decrease 
 in illiteracy is due to recent immigration of foreign illit- 
 erates, a class of which only a small number come to the 
 South. 
 
 4. Attempts at Too Great Accuracy 
 
 Cost Figures. Occasionally time and effort are wasted 
 by the superintendent's going to great extremes to show 
 accuracy in his figures. This is particularly unfortunate 
 at times, because what is considered great accuracy may 
 be only unnecessary work that can never make things 
 accurate. For instance, a superintendent may go to 
 great length to calculate the cost of instruction per day 
 
Why We Need Better School Statistics 17 
 
 per child in his system, for comparison with similar figures 
 for other systems. He runs his figures out to hundredths 
 of a cent. But owing to the trouble in units previously 
 mentioned, his results are really worth very little. One 
 system has taken the school census for the number of 
 children; another, the enrollment; and still another, 
 the average daily attendance. Unless he knows that the 
 same base has been taken for each system, no amount of 
 mechanical accuracy later will give exact results. 
 
 In many public presentations of school statistics, much 
 of the effect is lost by attempting to be too accurate in 
 giving the figures to the cent or the fraction thereof in all 
 places. The mind of the average citizen, or even of an 
 expert for that matter, cannot take in too many details. 
 The exact figures add nothing to the impression in his 
 mind, and indeed detract from the main things. As an 
 illustration of this point, take Professor Bobbitt's expres- 
 sions in the San Antonio Survey that " English costs in 
 the neighborhood of $210,000 " and " spelling costs in 
 the neighborhood of $40,000." ^ These are far more ef- 
 fective than if he had run the sums out to the exact num- 
 ber of dollars and cents obtained from a very long and 
 tedious computation. 
 
 Standards. A superintendent may make an unwise 
 recommendation when he advocates that his school 
 should reach the precise figure on an '' average " or 
 " middle figure " of a certain table, comparing his school 
 with other systems. In view of the well-known chances 
 for inaccuracy in the original figures, it is far better to 
 take for granted that such hair-splitting accuracy is not 
 desirable and to use a rather wide standard. Professor 
 
 1 Bobbitt, J. F. : Survey of San Antonio Public Schools, pp. 99, 
 103 
 
18 
 
 School Statistics and Publicity 
 
 Bobbitt 1 does this in his " zone of safety '^ standard 
 which includes the middle half of the group. As an il- 
 lustration of the workings of his standard, we quote 
 Table I on the costs of instruction in high school mathe- 
 matics per one thousand student hours, in certain cities 
 of the country. 
 
 Table 1. Bobbitt Table Showing Cost of Instruction per 
 1000 Student Hours (Mathematics) 
 
 Cost per 
 Name of school 1000 stu- 
 
 dent hours 
 
 University High . . , . $169 
 
 Mishawaka, Ind , . . . . 112 
 
 Elgin, 111 100 
 
 Maple Lake, Minn » . . . 100 
 
 Granite City, 111 88 
 
 East Chicago, Ind 82 
 
 De Kalb, 111. . T~~ 74 
 
 San Antonio, Tex 69 
 
 Harvey, 111 69 
 
 Waukegan, 111 63 
 
 South Bend, Ind 62 
 
 East Aurora, 111 61 
 
 Rockford, 111 59 
 
 Booneville, Mo 58 
 
 Brazil, Ind 56 
 
 Leavenworth, Kan 56 
 
 Greensburg, Ind 54 
 
 Morgan Park, 111 53 
 
 Noblesville, Ind 52 
 
 Norfolk, Neb. \ . ~. ~. ! '. ] '. \ ] ] ] ] ] ] 42 
 
 Washington, Mo 41 
 
 Bonner Springs, Kan 38 
 
 Russell, Kan 34 
 
 Junction City, Kan 33 
 
 Mt. Carroll, 111 30 
 
 1 Bobbitt, J. F. : "High School Costs," School Review, 23 : 505-534. 
 (Oct., 1915) 
 
Why We Need Better School Statistics 19 
 
 From this table, Professor Bobbitt would not say that 
 the cost of teaching high school mathematics should be 
 exactly $59; but that it should be somewhere between 
 $52 and $74. ^ This measure must for another reason 
 be used with still more caution. In this same study by 
 the same method, Professor Bobbitt found the '' zone of 
 safety " for Latin to be from $54 to $92, while that for 
 English ranged only from $43 to $67. That is, the zones 
 differ for different subjects, and one cannot at first glance 
 be sure that one ought to strive for a difference in costs 
 in. the subjects. Again, this measure is not good for 
 publicity purposes unless the superintendent's system is 
 below the '' safety zone." In such cases it is very effec- 
 tive. But if his system is above this zone, citizens are apt 
 to rest contented or even to contemplate reducing school 
 
 taxes. 
 
 5. Neglect of Technical Statistics 
 
 Averages. Even when the data are accurate, errors 
 may creep in because of neglect of the elements of tech- 
 nical statistics. A frequent example of this arises from a 
 very loose use of the average to typify a group, especially 
 in regard to teachers' salaries. In a recent investigation 2 
 of salaries paid by fifteen of the best-known colleges for 
 women, it was found that the average lowest salary paid 
 was $700 and the average highest was $1500. The 
 average would give a very erroneous impression, as far 
 more of the instructors received the lower salaries than 
 the higher. The maximum amount reported was $3000 ; 
 the lowest, $100 and home. It is obvious that the high 
 salary would exert a greater influence on the average than 
 the lower one, since it is twice as far away. 
 
 1 To be still more accurate, between $47 and $78. See p. 130. 
 
 2 Journal of Pedagogy, 19 : 185 
 
20 School Statistics and Publicity ^^B 
 
 This point may be further illustrated from a report of 
 a state superintendent. ^ In one table he gives the highest 
 and the lowest salary paid per month to male and to 
 female teachers, for both races, separately. The table I 
 following, in the same way, gives the average salary of 
 the various groups. For example. County A reports 
 the highest salary for white male teachers, $175 per month ; 
 the lowest, $40; the average, $73.07. The fact un- 
 doubtedly is that in County A more white male teachers 
 get less than $73.07 than get more. This table is made 
 worse by averaging the salaries paid both white teachers 
 and colored. This average means little for two reasons : 
 First, there are fewer colored teachers by far in Florida 
 than white ; second, the colored teachers do not get any- 
 thing like so much salary as the white. The result is 
 that the average is higher than all or most of the salaries 
 paid colored teachers, and lower than all or most of the 
 salaries paid white teachers, and so is significant for 
 neither group. 
 
 In the summaries of another report of the same super- 
 intendent of the same state for 1913-14, this error appears. ^ 
 The average salary per month paid teachers in the ten 
 highest and the ten lowest states in the United States is 
 given. Wisconsin stands at the top of the lowest ten. 
 Florida does not appear, so we suppose that she is some- 
 where between the highest ten states and the lowest ten ; 
 hence the apparent conclusion that Florida pays a better 
 monthly salary than does Wisconsin. But from another 
 table we learn that the average school term in Wisconsin 
 is 175.7 days ; for Florida, 122.2 days. In other words, 
 Wisconsin pays on the average for 53.5 more days than 
 
 1 Report of State Superintendent of Florida, 1912 : 471-472 
 
 2 Report of State Superintendent of Florida, 1913-14 : 37 
 
Why We Need Better School Statistics 21 
 
 does Florida. Now if these averages had been expressed 
 in years instead of months, the chances are very likely 
 that Wisconsin would be higher up in the list of states 
 than Florida. 
 
 A very inaccurate but common way of taking the aver- 
 age is illustrated by the following procedure which came 
 under the writer's notice : In a study for school purposes 
 of the negroes of Texas, the percentage of the negro pop- 
 ulation to the total population of the state was desired. 
 Three counties with a high percentage, three with a mod- 
 erate percentage, and three with a low percentage were 
 taken for the whole state. Then the average of these nine 
 counties was determined. This is a very rough and in- 
 accurate method. If the number of counties in the state 
 were divided into groups having the same ratio to each 
 other as the counties taken in the study, and if these latter 
 were then chosen at random or equally spaced in their 
 groups, the results would be fairly accurate. But such a 
 state of affairs could hardly be hoped for in popular 
 sampling of this kind. 
 
 A similar error is liable to occur in the common practice 
 of judging the work of a class by examining one of the 
 best notebooks, one of the medium group, and one of 
 the worst. Unless care is taken, the extremes may re- 
 ceive an undue amount of emphasis. The medium group 
 is always so much larger than the extreme groups that it 
 ought to have at least two samples to each one for the 
 other groups. 
 
 Variations. Often bare averages are given, when varia- 
 tions from the average are the significant thing. Suppose 
 that two boys in school have the same average for their 
 grades in different subjects, say 85. But one boy made 
 these grades, 84, 86, 86, 88, 81, and the other made 97, 98, 
 
22 School Statistics and Publicity 
 
 90, 80, 60. The deviations from the average in these two 
 instances show that one boy is an '' average " boy in all 
 his studies while the other is very fine in some, possibly 
 those he likes, but poor in the others. 
 
 In the illustration of the salaries paid teachers in col- 
 leges for women, previously given, the significant thing 
 is not the average but the deviation from that average. 
 
 Professor Bobbitt in his San Antonio survey has one 
 rather incomplete treatment in this matter of deviation, 
 occurring in what is otherwise a most admirable use of 
 variations.^ He desires to make the point that the 
 problem of heating the schoolrooms in the city is of minor 
 importance and gives a table of mean hourly temperature 
 of the city for all school hours, month by month. This 
 of course allows for deviations. These figures are for the 
 winter months not a tremendous distance below the 68 
 degree standard for comfort. This would indicate that 
 the city was always mild in winter. However, the sig- 
 nificant thing is not this mean temperature of San An- 
 tonio, but the deviations from it. To get a mean temper- 
 ature of say 55, there must be days below it, possibly a 
 considerable number. If there are only a few days each 
 month, it is clear that the school buildings must be 
 equipped to keep children warm on those days or else 
 school must be dismissed until it gets warmer. 
 
 Sampling. An extremely common error in school 
 statistics arises because of ignorance of " sampling." 
 This corresponds to the error in logic arising from con- 
 clusions drawn from one case, from too few cases, or from 
 cases not properly selected. In many school statistics 
 the samples are too few, or they are not impartially taken. 
 Especially is this true in most questionnaire or straw 
 1 San Antonio Survey, pp. 226 j^. 
 
Why We Need Better School Statistics 23 
 
 ballot investigations. A questionnaire is generally an- 
 swered chiefly by those especially interested in the subject. 
 Any conclusions from such data should not be interpreted 
 as applying to any save that class. The methods of taking 
 the average described on page 21 are of course illustra- 
 tions of bad sampling. 
 
 Incorrect sampling appears often in school advertising. 
 Thus, a leading southern university gives prominence in 
 its advertising to the number of its graduates who have 
 been members of Congress, governors of states, United 
 States senators, or even President. In many ways of 
 course such advertising is thoroughly justifiable, but not 
 when it seeks to convey the impression that the typical 
 student at the university will later reach such prominence. 
 Again, a certain southern university appears to admit to 
 graduate standing all graduates of a certain normal 
 school which does about two years of college work. This 
 graduate standing is granted because at one time three or 
 four students from the normal school in question were 
 taken on trial at this university and did good graduate 
 work. That is, all students from the normal school are 
 assumed to be as capable as the original three or four. 
 Practically every school system, especially a high school 
 striving for recognition, cites the performance of its best 
 students in creating a general impression of its work. 
 But such sampling needs to be taken with considerable 
 salt. We might with equal justice expect all Chinamen 
 to be so many Confuciuses, or all Americans to be so many 
 Woodrow Wilsons. 
 
 The process of sampling is treated at length later on.^ 
 But briefly, where sampling is resorted to, it must be done 
 purely at random. And enough cases must be taken to 
 
 1 Pp. 62-71 
 
24 School Statistics and Publicity 
 
 insure an approximately accurate impression of the entire 
 distribution that is being sampled. 
 
 Number on a Scale. A final technical weakness, one 
 requiring great care to avoid, comes from a wrong inter- 
 pretation of what a number means on a scale. For ex- 
 ample, a score of 6.25 on any test should mean one-fourth 
 of a step above the "6" step's starting point. If this 
 step runs from 6 to 7, obviously the score of 6.25 is just 
 what we need. If '' 6 '' is regarded as running from 5.5 
 to 6.5, then 6.25 may really mean 5.75. A superin- 
 tendent may measure his work on a scale by one method 
 and compare it with work measured by another method 
 on the same scale, and feel much elated over beating the 
 other man by half a step, when as a matter of fact the 
 half step advantage is due solely to differences in counting 
 on the scale. 1 
 
 6. Presentations of School Statistics to the Public 
 
 When it comes to presenting school statistics to the 
 public, the errors are even greater. There is often little 
 or no tabulation, or there are such complications that 
 they can be understood only by experts. Many of the 
 presentations make no use of graphs, and others contain 
 such intricate graphs or charts that laymen cannot easily 
 understand them. Some of the charts have the zero line 
 cut off so that the effect on the reader is apt to be totally 
 misleading.2 In other graphs the scales used may give a 
 false impression. 2 
 
 II. WASTEFULNESS OF THESE ERRORS 
 
 The mere errors of school statistics are not their worst 
 feature ; it is the amount of time and energy put into such 
 
 1 See p. 49 2 See p. 283 ^ g^e p. 284 
 
Why We Need Better School Statistics 25 
 
 useless things. Some school executives apparently de- 
 vote much time and effort to the collection and publica- 
 tion of educational statistics of this useless sort, pre- 
 sumably with the idea of doing some good. Professor 
 Hanus ^ in a recent study of superintendents' reports 
 found this to be true of one city : The report of the super- 
 intendent contained 35 tables, 23 of which contained 
 statistics for the year 1913-14; the other 12 presented 
 comparative statistical summaries covering the period 
 from 1906 to 1914 ; but only two of the latter group of 
 tables had any relation to the first-named group. Forty- 
 two per cent of all the reports studied by Hanus con- 
 tained *' few or no comparative statistics, and very 
 little or no satisfactory interpretation of statistics." 
 The significance of such statistics is practically nil, he 
 concluded. 
 
 Snedden and Allen, in a book published sorhe years ago,^ 
 indicate that in their investigation of the school reports 
 of the best city systems of the country, much useless 
 statistical material was found . This material they roughly 
 classified as of two kinds : First, material that would be 
 useful if properly related and explained ; second, material 
 useless in itself, either because it has no significance what- 
 ever, or because it is in detailed form, when only the sum- 
 mary is the significant thing. 
 
 The protests against such ineffectiveness in school 
 statistics arise also from other sources. Superintendent 
 Maxwell of New York frankly says : " There are some 
 ways in which the efficiency of a school may be deter- 
 mined with an approach to accuracy and without the 
 
 1 Hanus, Paul : "Town and Gity Reports," etc., School and Society, 
 3 : 145-155, 186-198. (Jan. 29, Feb. 5, 1916) 
 
 2 School Reports and School Efficiency, pp. 1-10 
 
26 School Statistics and Publicity 
 
 assistance and without the retardation of time-wasting, 
 energy-destroying statistical research. 
 
 '' There may be ways in which the so-called scientific 
 surveys or investigations, when stripped of past and 
 present absurdities, will help in determining efficiency." ^ 
 
 Again, the writer of a very thought-provoking satire 
 on present-day educational theory offers the following 
 '' statistics on the different classes of teachers with re- 
 spect to ' pedaguese ' or the scientific terminology em- 
 ployed by educational theorists " 2 ; 
 
 Use and think they understand it 12 % 
 
 Have used and thought they understood it, but don't now . 2% 
 
 Think they understand it, but don't use it 6% 
 
 Use it but don't understand it 9% 
 
 Don't use it, don't understand it, but esteem with awe those 
 
 who do 51% 
 
 Think it is rot 20% 
 
 100% 
 III. INDIFFERENCE OF THE PUBLIC 
 
 Moreover, the public, particularly the influential 
 portion of the public, have a distrust of these school 
 statistics and are little affected by them. It may be that 
 the average citizen has a wholesome respect akin to awe 
 for any statistical presentation, as claimed by some 
 writer in the Unpopular Review.^ In this article appear 
 some very illuminating examples of misleading statistics 
 in general fields. Errors similar to these can be found 
 in many school reports. But the writer's experience 
 and results from conferring with many practical school 
 men point in the other direction. Many influential 
 
 1 Maxwell, W. H. : "How to' Determine the Efficiency of a School 
 System," American School Board Journal, May, 1915 
 
 2 Henrick, Welland : A Joysome History of Education, p, 54 
 8 May- June, 1915:352-366 
 
Why We Need Better School Statistics 27 
 
 people who are not especially trained in statistics, are 
 frankly skeptical of any statistical report on schools. At 
 the same time they may be vitally interested in the schools 
 and have confidence in the superintendent. 
 
 School statistics are usually included in large reports 
 which do not reach the public to any appreciable extent. 
 Professor Hanus sent 250 letters to the members of the 
 Harvard Club, and 250 to the members of the Chamber 
 of Commerce in Boston. He asked these people if they 
 had seen a report of the school board, including that of 
 the city superintendent, within the last two years. He 
 received 128 replies from members of the club and 83 
 from members of the Chamber of Commerce. About 
 70 per cent of these answered ''No." Snedden and Allen 
 assert that the reports of city systems are not read and 
 give as the reason that they are not so arranged as to be 
 intelligible to the ordinary citizen. The common practice 
 of issuing summaries or abstracts of school reports or 
 surveys, indicates the same thing. 
 
 The indifference of the public is further shown by the 
 fact that editions of school reports are usually very small. 
 Only a few years ago the proposition was made to place 
 copies of the New York City report in the hands of each 
 of the 15,000 teachers, so that more intelligent and ear- 
 nest work would result. It was objected to on the ground 
 that it would cost too much. But the cost of the reprints 
 would have been insignificant compared with the $36,000,- 
 000 spent on the public schools that year. Another 
 illustration comes from Mobile. This city has a debt of 
 $4,000,000, and an annual expenditure of about $464,000 
 with $110,000 spent on schools. But the board hesi- 
 tated about paying the $500 necessary to print the recent 
 school survey. 
 
28 School Statistics and Publicity 
 
 IV. PROGRESS ELSEWHERE IN USING STATISTICS 
 
 While these and similar errors continue in statistics 
 of many school systems with resulting indifference of the 
 public, remarkable improvements and extensions in the 
 use of statistical method have been made in other fields 
 and in some schools. Whether we look in economics, 
 sociology, census reports, insurance, cost accounting, 
 biological sciences, or the best school reports, wherever 
 scientific method is used, that is, practically, wherever the 
 experimental method is used, there has been a refinement 
 of statistical method and an application of it to the prob- 
 lems in question. 
 
 Let us list at random some of these problems as they 
 come to mind : 
 
 Life insurance calculations, by which the average number of years 
 a man will live after any given age, is known. 
 
 Weather conditions (rain, temperature, etc.) over any given area 
 for a long period of time, by which it may be known whether it will 
 be profitable to grow certain crops in that region. 
 
 The adaptation of the average to measuring and sampling processes, 
 especially of grain. 
 
 The movement from country to city and its results. 
 
 The tenancy problem, its causes, types, and effects on general 
 social status of the region under consideration. 
 
 Relative attainments of city and country born children. 
 
 Relative strength of various European belligerents. 
 
 The relation of men and women in various mental processes. 
 
 The effect of factory labor on children. 
 
 The fight against preventable diseases. 
 
 Comparison of the intellect of the white man with that of the negro. 
 
 The influence of woman suffrage in national elections. 
 
 Within the field of educational investigation itself, it 
 is significant to scan the list of publications at a graduate 
 school of education like Teachers College (Columbia 
 University) and see that from only a small number of 
 
Why We Need Better School Statistics 29 
 
 dissertations embodying a few statistics simply handled, 
 there has been a steady drift toward investigations in- 
 volving the comprehensive use of statistical method of 
 greater and greater accuracy. The School of Education 
 at the University of Chicago requires a course in statis- 
 tical method very early in the program of every graduate 
 student. 
 
 Statistical method has been used to get at the truth of 
 many educational problems, ones that could not other- 
 wise be solved, for example : 
 
 Do women teachers drive boys out of high schools? 
 
 Is the South taxing itself as high for schools as the North? 
 
 Is the record on college entrance examinations as good an index 
 of the student's later achievement in college as is his high school 
 record or the college record of his brother? 
 
 What is the relation between reasoning and fundamental opera- 
 tions in arithmetic? 
 
 What are the most economical ways of memorizing? 
 
 Do teachers get more salary for each year of training beyond the 
 high school? 
 
 What are the causes of elimination of children from school? 
 
 A similar progress has been made in these other fields 
 in presenting statistical information to the public. Po- 
 litical parties, labor leaders, Y. M. C. A. leaders, phil- 
 anthropic workers, various foundations, bureaus of mu- 
 nicipal research, evangelists, advertising agencies, and 
 corporations are showing in their reports great progress in 
 pictorial and graphic ways of presenting statistical mate- 
 rial so as to influence the layman. 
 
 V. UNSATISFACTORY STATE OF AFFAIRS IN SCHOOL 
 
 STATISTICS 
 
 In view of the progress in the use of statistics in other 
 fields and in special educational investigations, our present 
 unsatisfactory state of general school statistics cannot be 
 
30 School Statistics and Publicity 
 
 allowed to continue. Statistical methods apply wherever 
 things are to be counted or measured. Nearly all the 
 problems of the school executive involve numerical data 
 and cannot be adequately handled without statistical 
 method. For example, what problem would a super- 
 intendent have that did not relate to one of the following 
 general fields? 
 
 1. Children to be educated or changed. 
 
 2. The aims of education, or the nature and amount of change to 
 be produced in these children. 
 
 3. The agents of this education — teachers and others. 
 
 4. The means of this education — buildings, books, laboratories, 
 etc. 
 
 5. The methods by which these agents use these means. 
 
 6. The changes resulting from various combinations of these with 
 the first, which is the big thing in education .^ 
 
 Every one of these fields affords numerical data for the 
 solution of problems, and they cannot be solved without 
 handling such data. In such fields '' the number of use- 
 ful studies to be made is, for all practical purposes, in- 
 finite." 2 
 
 The school executive has as great need to appeal to the 
 layman with statistical matter on schools, as do any of 
 the publicists in other fields. In particular, he has prac- 
 tically to make the same appeal for funds as do workers 
 in these other fields, especially the religious and philan- 
 thropic ones. Consecjuently, the school superintendent 
 needs for school purposes any valuable methods or dis- 
 coveries on statistics worked out in other fields. 
 
 1 Thorndike, E. L. : "Quantitative Investigations in Education," 
 School Revietv Monographs for College Teachers of Education, No. I, 
 pp. 31-32 
 
 2 Ibidem, p. 34 
 
 I 
 
Why We Need Better School Statistics 31 
 
 In the preceding, the aim has been to set forth clearly 
 typical errors in school statistics and to indicate briefly 
 the causes for such ineffectiveness. The rest of this book 
 shows in detail how to work up school statistics properly 
 and how to present them effectively to the public. 
 
 Chapter Two deals with the collection of data. Chap- 
 ter Three discusses the need for technical methods in han- 
 dling school statistics. Chapters Four, Five, Six, and 
 Seven show how to apply these technical methods to school 
 facts that are to be presented to the public, — Chapter 
 Four dealing with scales, distribution tables, and surfaces 
 of frequency; Chapter Five with the three measures of 
 type ; Chapter Six with the measures of deviation or dis- 
 persion; Chapter Seven with measures of relationship. 
 Chapter Eight is a supplement on statistical treatment, 
 dealing with additional problems which the superintendent 
 will encounter in working up his school statistics to present 
 to the public. Chapter Nine treats of the ineffectiveness 
 with the public of the statistics in many school reports. 
 Chapter Ten shows how to tabulate school statistics for 
 the layman. Chapter Eleven deals with graphic presenta- 
 tions of school statistics for the public. Chapter Twelve 
 gives effective ways of translating school statistics into 
 words, for popular consumption. 
 
 EXERCISES 
 
 1. When may a child be counted as a six-year-old? 
 
 2. Just what is the justice of a state university's rating a high 
 school on the performance of its former graduates in the first two 
 years at the university? 
 
 3. Precisely what does a boast in the state teachers' journal that 
 a certain girl graduate in a given high school made an average of 98.5 
 for the four years, mean to other high school teachers? 
 
 4. How valuable is the practice of some teachers of keeping 
 
32 School Statistics and Publicity 
 
 records of the work of pupils for only the last few days before the end 
 of the term ? 
 
 5. How significant is the claim of a superintendent that his high 
 school is better than that of a neighboring superintendent because 
 he has in it 6 teachers as against the other's 4, and enrolls 60 non- 
 residents as against the other's 40? 
 
 6. Of what worth is the statement of a superintendent that he 
 has increased the size of the graduating class in his high school 75 
 per cent in two years? 
 
 7. How could you compare in the same high school the cost of 
 teaching one period a day for the principal who teaches 4 periods a 
 day, with that for the Ej;)glish teacher who has 6 periods a day; with 
 that for a science teacher who has 4 classes with double periods each, 
 a day? 
 
 8. What examples of the errors given on pages 2-24 have you come 
 across in your previous experience with school statistics, and in just 
 what did the trouble consist? 
 
 9. Take some school report or some article in your state teachers' 
 journal that contains considerable statistical material. Locate the 
 errors, if any, in it and show in just what they consist. 
 
CHAPTER II 
 
 COLLECTION OF DATA 
 
 I. WHEN TO USE STATISTICAL METHOD 
 
 1. When Statistical Method Is Profitable 
 
 Which school problems may be subjected with profit 
 to statistical treatment? The superintendent needs to 
 know this, for we have seen how easy it is for him to 
 waste his time and make errors in school statistics. 
 These troubles frequently come from two ever-present 
 dangers: starting on the details of the task before it 
 has been clearly thought out; and attempting to use 
 statistical method where it is not applicable or will not 
 yield results worth the effort. In other words, two 
 questions at once arise : 
 
 1. Which problems of the superintendent may profitably be sub- 
 jected to statistical treatment? ' 
 
 2. How may he know whether a given problem can profitably be 
 so treated? 
 
 In reply to the first of these questions, it may be said 
 that practically all of the superintendent's problems may 
 be statistically treated, because on all of them he can 
 work with data that may be counted or measured more 
 or less accurately. The quotation from Professor Thorn- 
 dike, page 30, shows how wide a range of problems can 
 be thus treated. But to be still more specific, the follow- 
 ing list of problems ever present for the superintendent 
 must be so treated : ^ 
 
 1 Made up chiefly from Snedden and Allen : School Reports and 
 School Efficiency, pp. 118-127 
 
 33 
 
 m 
 
34 
 
 School Statistics and Publicity 
 
 Buildings ^| 
 
 Is the number of sittings adequate? WM 
 
 What is the value per child that can be comfortably seated? " 
 
 How much has the city spent for buildings per $1000 of taxable 
 property as compared with other cities for a period of years? 
 
 How much waste space is there in the buildings? 
 
 What is the proportionate cost of upkeep and repairs, per capita 
 average daily attendance? 
 
 What has been the cost of equipping special buildings or rooms, as 
 laboratories, manual training, etc., per capita average daily at- 
 tendance, as compared with other years and other systems? 
 Receipts and expenditures 
 
 What has been the per capita expenditure, expressed in terms of 
 average daily attendance of the system as a whole for several years? 
 
 What has been the per capita cost, per school, of such items as 
 salaries, general administration, fuel, building and repairs, 
 special classes, etc.? Also what have been the relative costs? 
 
 What do the different classes of schools, — high, junior high, ele- 
 mentary, special, — cost per capita? 
 
 Are receipts keeping pace with the increase in number of children? 
 Census 
 
 What are the numbers of children for each year within the school 
 age limits in the city ? 
 
 How many of these are within the compulsory school age ? 
 Attendance 
 
 What is the number of children of compulsory school age attend- 
 ing public schools? 
 
 How many children of compulsory school age are attending private 
 schools, special schools, or are otherwise satisfactorily accounted 
 for? 
 
 How many are in voluntary attendance, per school, and has the 
 number increased during the past several years? 
 
 What is the number of persistent attendances, i.e., children who 
 attend 160 days out of a possible 200, etc.? 
 
 What is the character of the absences? 
 Elimination 
 
 How many children drop out each year, by grades and schools? 
 
 Why do these children drop out — transferred to other schools, 
 non-promotion, irregular attendance, over-age, etc.? 
 
 Is the percentage of elimination increasing? 
 
Collection of Data 35 
 
 RetardoMon 
 
 What are the percentages of promotion and non-promotion by- 
 grades and schools? 
 
 Would retardation be lessened by flexible promotion schemes? 
 By promotion by subject in the upper grades? 
 
 Are retardation and elimination increased or lessened in schools 
 where industrial work is given? 
 
 What is the relation between retardation and over-age? 
 Special classes 
 
 What is the per capita cost, average daily attendance, of special 
 classes, evening schools, etc.? 
 
 How does this cost compare with that in other cities? 
 Medical inspection 
 
 How many children have been treated ? How many defects have been 
 remedied as a result of this treatment ? How many homes visited ? 
 
 What is the per capita cost of this work ? 
 
 Is it adequate? 
 Truancy 
 
 How many cases were reported? 
 
 What disposition was made of these cases? 
 
 What were the causes of these cases of truancy? 
 
 What has been the per capita cost of the truancy department ? 
 
 How does it compare with other cities in amount of work done and 
 in expense? 
 
 How m.uch time on the average elapses before a reported truant is 
 returned to school ? 
 Supervision 
 
 Which of several methods of teaching a subject produces the best 
 results with the given teachers and conditions of the system? 
 
 Which teachers or schools are doing the best work, in what subjects, 
 and just how much better? 
 
 How many teachers can a supervisor most profitably work with in 
 the given system? 
 
 How successful is the scheme of the system for rating teachers ? 
 
 How effective for producing better teaching are the methods em- 
 ployed for improving teachers in service? 
 
 Are the teachers overburdened with routine and clerical work? 
 
 Are the classes too large for good work? 
 
 Which parts of the curriculum ought to be eliminated or given re- 
 duced time allotments? 
 
36 School Statistics and Publicity 
 
 2. When Statistical Method Is Unprofitable 
 
 Sometimes statistical method cannot be used, because 
 it is impracticable to get suitable data. Lack of records, 
 of suitable help, of money to pay for such assistance, or of 
 time on the part of the superintendent, may bring this about. 
 
 There are cases also in which the effect aimed at with 
 a statistical presentation could be secured more quickly 
 and easily with some argument from analogy, some emo- 
 tional appeal, etc. For example, one of the Cleveland 
 reports some years ago wished to emphasize the improve- 
 ment in school children that could be produced by op- 
 erations for adenoids and similar troubles. Instead of 
 using statistics, it simply showed the picture of a boy 
 afflicted with adenoids before the operation, and another 
 picture of him some months afterward. The difference 
 in expression in the two pictures, especially as regards 
 intelligence, probably was more effective than any 
 possible statistical presentation. 
 
 Then there are cases of individuals which, as Professor 
 King so aptly puts it, " statistics cannot and never will 
 be able to take into account. When these are important, 
 other means must be used for their study." ^ The second 
 of these statements is really true only on the assumption 
 that the superintendent has the right conclusion to present. 
 Often this statistical treatment would show him that his 
 presentation was untrue or at least needed material 
 modification. 
 
 Finally, there are some school problems that cannot be 
 treated statistically with profit, for example : 
 
 Cost of subjects sure to enroll few students compared with the cost 
 of those sure to enroll many. 
 
 Getting a city already spending money freely to do better. 
 1 King, W. I. : Elements of Statistical Method, p. 35 
 
Collection of Data 37 
 
 Selection of texts ; cheap books usually mean inferior quality. 
 
 Percentages of elimination and retardation where it is impossible 
 to estimate fairly accurately the number of children entering in a 
 given year. 
 
 Worth of teaching methods outside of those that may be objec- 
 tively measured. Superintendent Maxwell in the article previously 
 cited gives a few such cases, for example, the items of character- 
 building, development of reasoning ability (not yet measured to any 
 appreciable extent), the motive to good, hard work, and all problems 
 involving tact.^ 
 
 3. How to Decide Doubtful Cases 
 How shall the superintendent decide quickly which 
 problems shall be statistically treated and which not? 
 Conditions vary often with each problem. But in 
 general, if the big elements in a given problem involve 
 numbers or can be expressed or measured in numbers, 
 statistical treatment will be applicable. The cost of 
 subjects sure to enroll few students cannot profitably be 
 statistically compared with the cost of subjects sure 
 to enroll many students, because the big factor in the 
 situation cannot be stated in numbers. This factor is 
 the relative value of the two subjects as parts of educa- 
 tion. One may correspond to salt, of which all of us 
 need a few grains each day, preferably some at each 
 meal, and for which there is no substitute. English is a 
 good example. The other may correspond to protein, 
 which all of us also need, but which does not have to be 
 taken at every meal, or even every day, and of which 
 there are various forms. Science with its numerous 
 alternative forms is a good example. Each subject is as 
 important as the other for perfect educational health, 
 but this relative importance can hardly be profitably 
 
 1 Maxwell, W. H. : "How to Determine the Efficiency of a School 
 System," American School Board Journal, May, 1915, p. 11 
 
38 School Statistics and Publicity 
 
 treated or measured with numbers. Before rushing 
 into any statistical treatment of a school problem, the 
 superintendent, then, should first try to analyze out the 
 big factors, considering whether the most important ones 
 can be satisfactorily treated with statistical method. 
 
 EXERCISES 
 
 1. Which of the following problems, or which parts of them, may 
 be profitably subjected to statistical treatment? Which may not? 
 Give your reasons precisely for each one. 
 
 (a) To what extent can science be profitably taught in the 
 grades ? 
 
 (6) What percentage of the funds available for library pur- 
 poses in a high school should go to each department in 
 a given year? 
 
 (c) Should home teachers be paid markedly lower salaries than 
 
 those from a distance? 
 
 (d) Should the passing mark be 65 or 70 on a scale of 100? 
 
 (e) Should a given child in a given grade be promoted? 
 (/) Is a given teacher marking too hard or too laxly? 
 
 (g) What salary as superintendent in a given state may a com- 
 petent man reasonably look forward to? 
 
 2. State in the form of definite questions at least three school 
 problems in which you are interested that might profitably be subjected 
 to statistical treatment. Give your specific reasons for so listing them. 
 
 3. Do the same for at least three school problems in which you are 
 interested that cannot profitably be subjected to statistical treatment. 
 
 II. HOW TO PLAN STATISTICAL TREATMENT OF 
 
 PROBLEMS 
 
 Careful planning in statistical work is always a sine 
 qua non for success. " Each hour spent in carefully 
 arranging the work is likely to save a score of hours in 
 trying to straighten out the confusion due to a hasty and 
 ill-advised program." ^ It is equally true that " one 
 ^ King, W. I. : Elements of Statistics, p. 47 
 
 I 
 
Collection of Data 39 
 
 of the peculiarities of statistical work is that practically 
 everything must be anticipated in advance, all possible 
 sources of error detected and guarded against, and even 
 the general results estimated." ^ The saving of time in 
 statistical work becomes all the more necessary when we 
 remember that the superintendent is at best a very much 
 overworked man. Careful planning means time saved 
 for the really big things in the statistical work he does, — 
 the results and their meaning for his school work. He is 
 also able to get a larger number of statistical processes 
 done in a given time, which in turn means that more 
 results and meanings will be available. In a word, 
 careful planning of statistical work permits a larger use 
 of statistical method by the superintendent in the time 
 at his disposal. 
 
 To save time, the main cautions to be kept in mind in 
 planning statistical work will be given briefly and dogmat- 
 ically with only necessary explanation. 
 
 1. Decide Precisely What Is to Be Found Out or Proved 
 in the Statistical Work 
 
 Indefinite phrasing of the problem means indefinite 
 thinking, with the inevitable wastes of time that accom- 
 pany it. The best device the author has ever found for 
 compelling one to make a sharp and clean-cut statement 
 of the problem, is to state it in the form of a very definite 
 question, the adjectives, adverbs, subordinate phrases, 
 etc. of which indicate the sub-questions or minor problems. 
 This device has been found to be very serviceable on many 
 problems not involving statistical treatment. It is 
 just as serviceable on those that do need statistical 
 
 1 King, W. I. : Elements of Statistics, p. 47 
 
40 
 
 School Statistics and Publicity 
 
 method. The usual topical statement of the problem is 
 one of the surest guarantees of loose thinking at the 
 start. The interrogative form of statement accentuates 
 the problem effect. A superintendent may simply state 
 his problem as '' School Costs in Blankville." How much 
 better it would be to state it thus : '' Are we paying all 
 we can possibly afford for schools and are we getting our 
 money's worth? " The same thing holds true for sub- 
 ordinate problems. For example, compare the ordinary 
 superintendent's statement of his problem with that of 
 Superintendent Spaulding in the 1912 school report for 
 Newton, Massachusetts. 
 
 Usual Statement 
 
 Superintendent Spaulding's 
 Statement 
 
 Report of Blankville Public 
 
 Schools 
 Statement of aims 
 
 The Newton Schools : what are 
 they trying to do? 
 
 Attendance and progress of 
 pupils 
 
 Are they doing what they are 
 trying to do ? 
 
 (Taken for granted) 
 
 Do you approve of their policy? 
 
 Expenditures of the system for 
 the current year 
 
 Is their policy carried out eco- 
 nomically? 
 
 Administration 
 
 Course of study 
 
 Reports of various supervisors 
 
 Is it administered efficiently? 
 
 Recommendations for next 
 year's work 
 
 Can we afford to continue it? 
 Can we afford not to continue 
 it? 
 
Collection of Data 41 
 
 2. Plan to Collect only Data for Which One Can Point 
 
 Out in Advance Specific Ways in Which They 
 Will Be of Value to Him 
 
 This does not of course mean that one can know in 
 advance all the ways in which the data will be of value 
 to him. The collection of data that do not seemingly 
 answer some of one's problems or promise to buttress 
 certain of his arguments, is simply a gambling proposition. 
 And the odds are ten to one that the data will never be 
 of any material use to him. It is true that sometimes 
 unexpected uses for data will appear after the work of 
 collecting them has begun. For example, the writer and 
 a graduate student collected considerable data on the 
 cost of instruction in southern normal schools. But they 
 had worked for some weeks before they discovered that 
 the same data would give material for answering very 
 important questions about size of classes. However, 
 such a valuable by-product cannot be counted upon in 
 all cases. 
 
 3. Plan the Whole Procedure Through to the End, 
 Trying It Out on Sample Data to Be Sure That the 
 
 Units, Blanks, Processes, etc. Will Work 
 
 Here is the place where one hour of good work will save 
 twenty later on, as Professor King says. The units 
 chosen should be carefully tested to see if they are prac- 
 tical. The blanks should be drawn out in detail and the 
 actual operations attempted with them. 
 
 In particular, what is known as the question of " group- 
 ing " must be decided. This means that if the data 
 are to be considered in groups, the exact range of each 
 group must be determined beforehand. For example, 
 
42 School Statistics and Publicity 
 
 if one is studying days' attendance, are the children to be 
 grouped as those attending 0-19, 20-39, 40-59, etc., or 
 0-9, 10-19, 20-29, etc. ? This cannot be treated in detail 
 here, but is discussed fully on pages 106 and 107. 
 
 If the blanks are to be filled out by outside persons, 
 some of the actual people, or preferably similar but less 
 intelligent people, should be used to test out the blanks. 
 The errors that these persons make should be noted and 
 the blank revised accordingly. Thus, in sending out a 
 blank for teachers to fill out, it is advisable to submit the 
 rough draft to several average teachers, see how they can 
 fill it out, and revise as necessary. 
 
 A very helpful device at this point is to make a " brief " 
 of just what is to be found and of the methods to be used. 
 This can be elaborated from the material accumulated 
 under the suggestions in i and 2. 
 
 The actual processes for handling the data should be 
 tested on the blanks themselves. Thus in making out 
 a blank, if percentages are later to be calculated, the 
 numbers from which they are to be calculated should come 
 in adjacent columns if possible. Actually calculating 
 such percentages on sample blanks will insure that an 
 economy of this nature is cared for. Again, in making 
 out blanks in series, the same fact should appear if possible 
 in relatively the same column in different blanks. This 
 will make all manipulation and calculation much easier. 
 But such placing is almost sure to be overlooked unless 
 the calculation with simple data method is carried through. 
 
 In all probability the reader by this time is asking this 
 question : " But how can one keep open-minded if so 
 much planning is to be done ahead? " This is a natural 
 question. So is the usual one as to whether such planning 
 will not tend to the buttressing of preconceived opinions 
 
Collection of Data 43 
 
 rather than to the discovery of anything new. In the 
 attempt to find the truth about anything, the question 
 method of outlining the plans is undoubtedly the surest 
 mechanical device to aid in keeping one open-minded. 
 Note that Superintendent Spaulding does not suggest 
 the answer to any one of his questions. Beyond this, 
 probably no device will be of much service to a person 
 who is determined to prove a certain thing by statistics 
 whether or no. 
 
 Furthermore, much of the superintendent's statistical 
 work is for the purpose of demonstrating or proving to 
 others what the superintendent already knows to be 
 true. Here the element of keeping open-minded does not 
 enter, and planning ahead is unquestionably very helpful. 
 
 EXERCISE 
 
 Take any one problem from your list in Exercise 2, page 38. Out- 
 line in question form precisely what you would wish to find or prove. 
 
 III. HOW TO DETERMINE UNITS AND SCALES 
 
 In planning statistical work, the need of units and 
 scales early becomes apparent. ^ The superintendent is 
 constantly called upon to pass judgment upon the worth 
 of many school matters. He generally does this by 
 merely placing the thing judged in its proper place in a 
 gi'aduated scale of values of such things. For example, a 
 superintendent passing judgment on the work of a teacher 
 merely puts her in her proper place in the list of teachers 
 in his school system, ranged from high to low, or in a list 
 made up of all the teachers he has ever observed. This is 
 
 1 See pp. 3-11 for examples of errors arising from a lack of proper 
 units and scales. 
 
44 School Statistics and Publicity 
 
 indicated in the very language he uses : '' Best I ever 
 had/' *' worst I ever saw," " hopeless," '' practically 
 perfect," etc. If he expresses his judgment in letters, 
 as A, B, etc., or in figures as 85, 90, 100, etc., he is merely 
 substituting symbols for such word estimates. 
 
 1. Subjective and Objective Scales 
 
 It is generally recognized that the superintendent 
 will vary at different times in his judgments of the same 
 teacher, engaged in the same kind of work. He may be 
 suffering from a severe headache, or be perturbed over a 
 recent business reversal. But his readings of a standard 
 thermometer, when it was at the same temperature, 
 would vary little from time to time. 
 
 Let us now consider the reasons for this difference. 
 The thermometer is graduated into constant definite units 
 that measure the same amount of heat in the room always 
 in the same way. Not so in the case of judging the 
 teacher. The superintendent's scale of teaching ability 
 has no definite units that always measure the same amount 
 of teaching efficiency. 
 
 These two scales represent the extremes of the kinds of 
 scales that the superintendent must use. The units in 
 the scale for judging teachers are in the superintendent's 
 mind. Granting that he can transmit fairly clear ideas 
 of his scale to others, there will be great disagreement 
 among those using it. If they agree, they may easily 
 be unaware of the fact, for the same descriptive words 
 mean very different things to different persons. Since in 
 a scale of this kind there are no units that can be made to 
 mean exactly the same thing to different people, such a 
 scale is said to be a '' subjective " one. In the case of 
 
Collection of Data 45 
 
 the scale for measuring temperature, the units are not 
 concealed in the mind of the person using the scale; 
 they are external to every one who wishes to use it. For 
 this reason, such a scale is called " objective." The 
 chances for error or difference of opinion in reading 
 the units of an objective scale are slight as com- 
 pared with those arising from the use of a subjective 
 one. 
 
 Between these two tjrpes of scales lie others with vary- 
 ing degrees of subjectivity and objectivity. For example, 
 the Thorndike handwriting scale is made up of samples 
 of handwriting rated from to 18, as determined by the 
 combined judgment of a considerable number of com- 
 petent judges. This scale is objective in that any one 
 can see the sample of handwriting grades, say No. 8. 
 But its use is subjective in that all people do not agree 
 that No. 8 is worse than No. 9, nor would all judges rate 
 any other sample of handwriting at the same place on 
 the scale, say No. 12. If the number of judges were 
 large enough, the variation in placing such a sample 
 might range from 10 to 17. 
 
 It is highly desirable in planning any statistical work 
 to try to secure units and scales that shall be as objective 
 as possible and that shall have a minimum of harmful 
 subjective elements. 
 
 Thus, the step between 94 and 95 in the marking 
 system of two teachers does not mean at all the same 
 thing. Again, merely because several people think two 
 words are equal in difficulty, it does not make them so. 
 Professor Thorndike, as noted on page 9, quotes Rice as 
 counting that disappoint is equal to feather in difficulty 
 in spelling, or as proceeding as though it were. But by 
 actual experimentation in a 5A grade, twenty-four times 
 
46 School Statistics and Publicity 
 
 as many girls and thirteen times as many boys missed 
 disappoint as missed feather. In measuring arithmetic 
 work, it is much better to take examples from the Courtis 
 or Stone tests, because the practical worth of these has been 
 demonstrated by the actual achievements of thousands 
 of pupils. 
 
 One reason why subjective scales are so often un- 
 desirable is that the zero points on them are unknown. 
 On an objective scale, such as length, 90 inches is just 
 three times as long as 30 inches, or it is just three times 
 as far from zero length. But in the grading or giving of 
 marks by two teachers, there is no assurance that each 
 regards 90 as just three times as good as 30 or just three 
 times as far removed from utter failure. One teacher 
 on a test may grade the worst student in the class at 30 
 and the best at 90. Another teacher might grade these 
 same students on the same test at 60 and 90. The teachers 
 would obviously be grading from different zero points. 
 In the standard scales for grading composition, hand- 
 writing, and so on, the zero points have been determined 
 by a procedure too complicated to be given here.^ This 
 is why such scales, even when they involve many sub- 
 jective elements, are so superior to the attempt of a 
 novice at making his own scale. 
 
 If we must take subjective estimates as units and make 
 our own scale, it is better to pursue the following treat- 
 ment : 
 
 1. Avoid choosing estimators with known or probably marked 
 prejudices. 
 
 2. Have all these persons estimate the worth of the problems in 
 terms of a separate problem, which for convenience is to be consid- 
 
 1 See Thorndike : Mental and Social Measurements, Revised Edi- 
 tion, p. 16^. 
 
Collection of Data 47 
 
 ered worth so much, say 10. (That is, all call this problem the value 
 of 10.) 
 
 3. For the value of any other problem, take the average of the 
 estimates given by the different persons.^ 
 
 2. The Jingle Fallacy 
 
 The superintendent must beware especially of con- 
 sidering things equal because they are called by the same 
 words. This is known as the " jingle " fallacy.^ Thus, 
 one child does not equal another child as a matter of 
 school expenditure, if the first child is in the primary 
 grades and the second child is in the last year of the high 
 school. The cost of educating the latter for one year is 
 much more than in the case of the former. The differ- 
 ence between the ability to do one problem and the 
 ability to do two problems in the Courtis tests is not the 
 same as the difference in ability to do fifteen problems 
 and the ability to do sixteen problems. Any one who 
 can do fifteen can fairly easily work up to sixteen. But 
 if a child can barely do one, it is a tremendous task to 
 work up to doing two. The '' jingle " fallacy usually 
 results from neglecting to define units or to consider the 
 zero points. 
 
 3. Essentials of a Valid Scale 
 
 The construction of a good scale for many lines of 
 school work demands considerable technical knowledge 
 and experience. The superintendent in general had 
 
 1 Professor Thorndike on pages 9 and 10 of Mental and Social 
 Measurements has a much more complicated method for utilizing sub- 
 jective estimates. 
 
 2 Professor Thorndike borrows this term from Professor Aikins. 
 See Mental and Social Measurements, p. 10. 
 
48 School Statistics and Publicity 
 
 better plan to use scales already worked out.^ Beyond 
 this we may for our purposes summarize the essentials 
 of a valid scale from Thorndike : ^ 
 
 1. The scale must be as objective as possible. 
 
 Its meaning must be such that all competent judges will agree 
 as to what it is. 
 
 2. The series of facts used in making up the scale must be of the 
 
 same sort of thing or quality. 
 
 3. The steps in the scale should be clearly defined. 
 
 It is better if they are equal ; if unequal, let the steps be de- 
 fined as definitely as possible. However, a scale in which 
 only the order or rank of the various facts making it up is 
 known, is often very useful. 
 
 4. The zero point must be defined if possible. ^ 
 
 4. Discrete and Continuous Series 
 
 It is impossible to use a scale properly unless one knows 
 whether the facts it is to measure are in a discrete or a 
 continuous series. A series is said to be discrete if it is 
 regarded as broken up, i.e., the different items are separate 
 or there are gaps between them. On the other hand, if 
 the series is capable of any degree of subdivision, that is, 
 if the items are regarded as strung out along the scale, 
 and running into each other, the scale is said to be con- 
 tinuous. The table of the costs of instruction in mathe- 
 matics, page 18, is an example of a discrete scale. In this 
 table every item is regarded as an integer and there are 
 gaps between the items. 
 
 A good example of a continuous series is Table 2, made 
 up from data worked out by the writer. 
 
 iSee "Descriptive List of Standard Tests," by W. S. Gray, Ele- 
 mentary School Journal, 18: 56. (Sept., 1917.) 
 
 2 Thorndike, E. L. : Mental and Social Measurements, pp. 11-18. 
 
Collection of Data 49 
 
 Table 2. Continuous Series Showing Fifth Grade Achieve- 
 ments WITH Courtis Tests in Addition, in a Western City 
 
 Number 
 
 of 
 
 Number of children 
 
 >lems attempted 
 
 making each score 
 
 1 
 
 
 
 
 2 
 
 
 2 
 
 3 
 
 
 10 
 
 4 
 
 
 24 
 
 5 
 
 
 32 
 
 6 
 
 
 35 
 
 7 
 
 
 34 
 
 8 
 
 
 54 
 
 9 
 
 
 25 
 
 10 
 
 
 27 
 
 11 
 
 
 24 
 
 12 
 
 
 4 
 
 13 
 
 
 4 
 
 14 
 
 
 3 
 
 15 
 
 
 6 
 
 16 
 
 
 9 
 
 17 
 
 
 2 
 
 18 
 
 
 2 
 
 In such a series as this one, the 32 children who at- 
 tempted five problems are not all regarded as being pre- 
 cisely at the point " five problems attempted " on the 
 scale, but as distributed from " five problems attempted " 
 to " six problems attempted/^ For this reason it is 
 very important to know what a given number means 
 on a scale. That is, does 6 mean from 5.5 to 6.5, or from 
 6 to 6.99, or nearer 6 than either 5 or 7? The second of 
 these methods, that of measuring in terms of the point 
 last passed, is often the natural way and saves labor in 
 all sorts of measurements. ^ This method is the one to use 
 where it is possible to say authoritatively that a given case 
 
 1 See Thorndike : Mental and Social Measurements, p. 22. 
 
50 School Statistics and Publicity 
 
 is beyond a certain point on the scale, but " the how much 
 beyond " cannot be easily determined. Obviously it is a 
 good method for Table 2, since a given case has attempted 
 say five problems, but one cannot easily tell whether it has 
 just started on five, is half way through the fifth, or is 
 practically ready to start on the sixth. The other method 
 can be used with a scale like the hand- writing scales, where 
 a given case is said to be nearer a given sample on the 
 scale than anything else. A case would be called "9,'^ 
 for example, without being definitely located as either 
 below or above that point. Here '' 9 " would mean from 
 8.5 to 9.5. 
 
 5. How to Use Scales 
 
 In actual practice the superintendent can measure the 
 worth of his work in whole or in part, on one of three 
 kinds of scales : 
 
 a. He can 'place the thing measured in its relative position 
 in a scale of items (school systems, rooms, classes, etc.), 
 all considered from the same viewpoint and without the 
 use of units. 
 
 This is the method used by the superintendent or school board of 
 Blanktown, when he announces that his town has the best schools 
 in the state, or that So-and-So makes this statement. It of course 
 carries no weight whatever unless we know that the judgment of the 
 one making the statement is sound. If one could read all the small 
 town papers of any given state for one year, he would probably find 
 three-fourths of them claiming that their home town had the best 
 schools in the state. The same method with all its weaknesses is 
 often used by a teacher in regard to the value of his pet method 
 of teaching, his favorite mode of discipline, or the particular class 
 he happens to be teaching at this time. 
 
Collection of Data 
 
 51 
 
 Table 3. Ranks of Certain Cities on Real Wealth and 
 
 Assessed Wealth behind Each $1 Spent on Schools 
 
 (Adapted from* Portland Survey, pp. 80, 304) 
 
 
 Real Wealth 
 
 
 Assessed Wealth 
 
 
 City 
 
 behind Each 
 $1 for Schools 
 
 Rank 
 
 behind Each 
 $1 for Schools 
 
 Rank 
 
 Newark, N. J. ... 
 
 $165 
 
 1 
 
 $165 
 
 11 
 
 Worcester, Mass. 
 
 
 180 
 
 2 
 
 180 
 
 15 
 
 Toledo, 
 
 
 184 
 185 
 
 3 
 
 4 
 
 110 
 185 
 
 2 
 
 New Haven, Conn. . 
 
 
 18 
 
 Paterson, N. J. . . 
 
 
 185 
 
 5 
 
 185 
 
 19 
 
 Lowell, Mass. . . . 
 
 
 194 
 
 6 
 
 194 
 
 20 
 
 Fall River, Mass. 
 
 
 196 
 
 7 
 
 196 
 
 21 
 
 Syracuse, N. Y. . . 
 
 
 202 
 
 8 
 
 180 
 
 17 
 
 Cambridge, Mass. 
 
 
 204 
 
 9 
 
 204 
 
 22 
 
 Grand Rapids, Mich. 
 
 
 207 
 
 10 
 
 166 
 
 12 
 
 Dayton, 0. ... 
 
 
 208 
 
 11 
 
 125 
 
 4 
 
 Washington, D. C. 
 
 
 212 
 
 12- 
 
 148 
 
 8 
 
 Scranton, Pa. . . 
 
 
 216 
 
 13 
 
 173 
 
 18 
 
 Jersey City, N. J. 
 
 
 218 
 
 14 
 
 218 
 
 24 
 
 Columbus, 0. . . 
 
 
 221 
 
 15 
 
 133 
 
 6 
 
 Rochester, N. Y. . 
 
 
 225 
 
 16 
 
 180 
 
 16 
 
 Denver, Colo. . . 
 
 
 231 
 
 17 
 
 116 
 
 3 
 
 Albany, N. Y. . 
 
 
 234 
 
 18 
 
 234 
 
 26 
 
 Providence, R. I. 
 
 
 256 
 
 19 
 
 256 
 
 31 
 
 Bridgeport, Conn. 
 
 
 276 
 
 20 
 
 276 
 
 34 
 
 Kansas City, Mo. 
 
 
 280 
 
 21 
 
 140 
 
 7 
 
 Minneapolis, Minn. 
 
 
 294 
 
 22 
 
 132 
 
 5 
 
 New Orleans, La. 
 
 
 314 
 
 23 
 
 236 
 
 27 
 
 Louisville, Ky. 
 
 
 326 
 
 24 
 
 228 
 
 25 
 
 Nashville, Tenn. . 
 
 
 350 
 
 25 
 
 263 
 
 33 
 
 Omaha, Neb. 
 
 
 352 
 354 
 
 26 
 
 27 
 
 53 
 177 
 
 1 
 
 Oakland, Cal. . . 
 
 
 14 
 
 Seattle, Wash. . . 
 
 
 364 
 
 28 
 
 164 
 
 10 
 
 Spokane, Wash. . 
 
 
 370 
 
 29 
 
 152 
 
 9 
 
 St. Paul, Minn. . 
 
 
 407 
 
 30 
 
 204 
 
 23 
 
 Indianapolis, Ind, 
 
 
 408 
 
 31 
 
 245 
 
 29 
 
 Memphis, Tenn. . 
 
 
 449 
 
 32 
 
 247 
 
 30 
 
 PORTLAND, ORE. 
 
 
 456 
 
 33 
 
 260 
 
 32 
 
 Birmingham, Ala. 
 
 
 479 
 
 34 
 
 240 
 
 28 
 
 Richmond, Va. 
 
 
 536 
 
 35 
 
 402 
 
 35 
 
52 School Stat'* xs and Publicity 
 
 A refinement of the same method is used by a judge in a contest, 
 when he ranks the contestants in order of merit only. He then 
 gives the best contestant the rank of l,.the next best the rank of 2, 
 and so on. But he wisely refrains from attempting to say how much 
 better is the first.^ 
 
 h. He can compare his own school with other schools on 
 a scale of his own making, all schools being measured with 
 definite units. 
 
 Thus, he may wish to compare his community with others on the 
 basis of the amount of money it really pays, wealth considered, for 
 schools. The Portland Survey Commission found in comparing the 
 wealth and school expenditures of that city with thirty-six others 
 nearest it in size, that there was a vast difference between the 
 assessed and the real wealth in many cities. To show this point 
 more clearly the table on p. 51 has been adapted from the Portland 
 Survey. 
 
 It may be seen from this table that it is very important to have as 
 the unit, the number of dollars of real wealth in the city, not the 
 number of dollars of assessed wealth. If assessed valuation were 
 taken as the unit, Omaha would be 1 instead of 26 ; Denver 3 instead 
 of 17 ; Lowell 20 instead of 6, etc. 
 
 It is evident that any one seeing this or a similar scale must agree 
 to the ranking of each city or item as shown in the first column, 
 provided the original data for the calculation are correct and the unit 
 is a reasonable one. Then there remains only the question as to 
 whether the cities selected for the scale are representative ones for 
 fair comparison in the matter under discussion. 
 
 c. He may compare his own school with other schools hy 
 means of a standard test, and then place his school on a 
 scale of cities made up as in h, or merely compare it with 
 the standards of the makers of the test. {This amounts to a 
 scale.) 
 
 The advantage of a scale of this kind is that the units have been 
 proved equal or approximately equal, and there can be no question 
 of the relative positions of samples in a given scale. And as time 
 
 1 See pp. 11, 192-198 
 
 I 
 
Collection « ' Data 53 
 
 goes on, very authoritative scales will appear. Thus, we now have 
 scales of this sort in the Thorndike or the Ayres handwriting scales, 
 the Hillegas or the Harvard-Newton composition scales, the Ayres 
 spelling scale, etc. Furthermore, by taking the achievements of 
 school systems as measured on these scales or with standard units, a 
 superintendent can easily make a scale of such achievements and 
 see where his school system comes on it. 
 
 6. Practical Examples of Units and Scales for 
 Superintendents 
 
 The best practical examples of units and scales for a 
 superintendent, of course, appear in the recent school 
 surveys. A superintendent wishing to get up a scale 
 or find units on a given problem can get them very quickly 
 by utilizing the following table. The particular survey 
 is denoted by the name of yie city, and the numbers 
 refer to pages. 
 
 Units and Scales that a Superintendent may Profitably Use 
 
 Description Where found (Surveys) 
 
 Playgrounds 
 
 r Salt Lake City, 222 
 Square feet per child < Rockford, 7 ; Ashland, 11 
 
 I Denver, 11, 122 
 Buildings 
 
 Square feet per child ( f " ^'^,?"^' ''' 
 
 t Ashland, 11 
 
 Cubic feet per child Ashland, 11 
 
 Space per teacher and child Leavenworth, 48 
 
 Number of sittings per room Snedden and Allen, 29 
 
 Total seating capacity by 
 
 buildings Snedden and Allen, 29 
 
 . . I,- r ^ f San Antonio, 315 
 
 Average cost per cubic foot l ^ . ^ , , Ar. no 
 
 \ Sprmgiield, 22, 23 
 
 Average cost per pupil San Antonio, 315 
 
 Average cost per classroom San Antonio, 315 
 
 Same for fuel Denver, 1, 55 
 
54 
 
 School Statistics and Publicity 
 
 Same for repairs 
 Lighting by candle power 
 
 Janitor's salary per room 
 
 Same per hour 
 
 Same per 1000 cubic feet 
 
 Valuation per room 
 
 Teaching staff 
 
 Number of children per super- 
 visory officer 
 
 Same on average daily at- 
 tendance 
 
 Number of children, average 
 daily attendance per teacher 
 
 Training of teachers 
 Years of experience 
 Years of training 
 
 Teachers' salaries 
 
 On yearly basis 
 
 On monthly basis 
 
 On weekly basis 
 
 Based on enrollment 
 
 Based on years taught 
 
 Maximum and minimum sal- 
 aries 
 
 Principal's salary based on 
 number of rooms in building 
 Proportionate expenditures 
 
 Percentages of school expendi- 
 tures for different purposes 
 
 Denver, 1, 62 
 
 Salt Lake City, 235 
 / San Antonio, 249 
 \St. Louis, IV, 117-120 
 
 Ashland, 11 
 
 Ashland, 11 
 
 San Antonio, 250 
 
 Salt Lake City, 39 
 
 Oakland, 26 
 f Salt Lake City, 53 
 
 Louisville, 33 
 
 Newton, 1913, Table IX 
 . Oakland, 24 
 
 South Bend, 198 
 South Bend, 200 
 
 f South Bend, 101 
 
 Leavenworth, 50 
 
 Butte, 120 
 
 Newton, 1913, V 
 
 Janesville, 43, 44 
 
 Bridgeport, 17 
 
 Vermont, 225 
 . Springfield, 61 
 
 Ashland, 14 
 
 Vermont, 225 
 
 Baltimore, 74 
 
 Salt Lake City, 55, 56 
 
 Portland, 75 
 f Cleveland, 97 
 \ Springfield, 61 
 
 I Janesville, 74 
 
 \ St. Louis, IV, 55, 56 
 
Collection of Data 
 
 55 
 
 Percentage of salaries for high 
 
 schools 
 Percentage of city expenditures 
 
 for schools 
 
 Per capita costs 
 Total population 
 
 For each of population over 15 
 
 For each adult male 
 
 High school costs per person 
 
 in population 
 Average daily attendance 
 
 Same for fuel 
 
 Enrollment 
 
 Average number belonging 
 
 Cost of instruction 
 Student hour 
 
 Per pupil 
 
 Per pupil enrolled 
 
 Per 1000 student hours 
 
 Miscellaneous costs 
 
 Expenditures for whole cities 
 
 on medical inspection 
 Evening schools per session 
 Per wagon, rural consolidated 
 
 schools 
 Part of each $1000 spent on 
 
 instruction in each subject 
 
 Janesville, 82 
 
 J Janesville, 68 
 \ St. Louis, IV, 32 
 
 - Butte, 143-4 
 
 Bridgeport, 21 
 " Janesville, 68 
 
 Oakland, 44 
 I Baltimore, 34 
 
 Portland, 407 
 
 Portland, 407 
 
 Springfield, 95, 96 
 
 In most surveys 
 ' Butte, 82 
 
 Oakland, 44 
 I Kansas City, 82 
 [ Birmingham, 36 
 
 Houston, 83 
 
 Sneddon and Allen, 17 
 
 ' South Bend, 204 
 Vermont, 227 
 
 I Leavenworth, 51 
 
 . Springfield, 114, 97 
 / Rockford, 111 
 \ Janesville, 75 
 Springfield, 114, 97 
 
 r San Antonio, 215 
 \ Denver, 1, 60 
 
 I Janesville, 90, 100 
 
 South Bend, 177 
 Newton (1913), 41 
 
 Texas, 33 
 
 San Antonio, 213 
 
56 
 
 School Statistics and Publicity 
 
 Expense for attendance officer 
 per 1000 pupils enrolled 
 Time spent on each subject 
 
 Minutes per week 
 
 Part of each 1000 hours spent 
 on each subject 
 
 Hours of recitation and di- 
 rected study in reading and 
 history 
 Population 
 
 Number per 1000 in certain 
 occupations 
 
 Same by lOO's 
 
 Races by lOO's 
 
 Family for nearly everything 
 
 Wealth 
 
 Portland, 390 
 
 f Salt Lake City, 76 
 I Leavenworth, 54 
 [ Houston, 83 
 
 San Antonio, 214 
 Cleveland, 121, 125 
 
 South Bend, 141 
 - r Salt Lake City, 17 ' 
 . \ Cleveland, 21 
 
 Cleveland, 21 
 
 Red River, 42, 48, 82 
 
 Per capita population 
 Same for real wealth 
 
 r South Bend, 211 
 < Janesville, 68 
 [ St. Louis, IV, 18 
 r Salt Lake City, 19 
 1 Cleveland, 25 
 Taxable wealth behind each / Oakland, 43 
 
 child in school 
 Same for child in average daily 
 
 attendance 
 Real wealth behind each $1 
 
 spent on schools 
 Possible revenue per child en- 
 rolled 
 Tax rate 
 
 Mills on assessed valuation 
 
 1 Maryland, 128 
 Janesville, 52 
 Portland, 414 
 Salt Lake City, 48 
 
 Same on real valuation 
 
 Same per $100 real valuation < 
 
 Illinois, 262 
 r Portland, 108 
 \ Leavenworth, 17 
 ' South Bend, 215 
 
 Janesville, 60 
 
 Salt Lake City, 313 
 
 Rockford, 119, 49 
 
Collection of Data 57 
 
 Per $100,000 real wealth Cleveland, 25 
 
 Rate necessary to produce esti- 
 mated per capita support for 
 
 schools on actual wealth Salt Lake City, 311 
 
 Enrollment 
 
 Increase in children per week 
 
 (5 years) Salt Lake City, 36 
 
 On basis of 1000 children in 
 kindergarten Ogden, 9 
 
 Parts of 100 pupils in public, 
 private, and parochial 
 schools Cleveland, 28 
 
 Same, failures, and promotions Denver, 1, 70, 2, 77 
 
 EXERCISES 
 
 1. Discuss the value for the superintendent of the units used in 
 each of the examples given on pages 53-57, and of the scales that 
 could be made up from such units. Just how would you make up 
 these scales? 
 
 2. What is the value of the question for measuring the efficiency of 
 teachers, counting the number of questions they ask in a given time ? 
 Why? 
 
 3. Which is better for measuring the preparation of high school 
 teachers, and why? 
 
 (a) The number of years they were in college. 
 (h) The number of years beyond the elementary school they 
 spent in study. 
 
 4. Precisely what is the value of each of the following methods of 
 instructing judges in a contest, and why? 
 
 (a) Mark on a scale of 100. 
 (6) Mark on a scale of 30. 
 
 (c) Mark the best 1, the next best 2, etc. 
 
 (d) Mark the best 100, the worst 70, and the others where 
 
 they should come in between. 
 
 (e) Mark on a scale of 100, allowing 50 for content and organ- 
 
 ization, 30 for English, and 20 for delivery. 
 
 5. What units and scales would you plan to use in studying the 
 statistical problem selected in the exercise on page 43, and just why? 
 Precisely how would you plan to use the scale or scales chosen ? 
 
 k 
 
58 School Statistics and Publicity 
 
 IV. HOW TO DO THE ACTUAL COLLECTING 
 1. Records in One's Own School 
 
 Most of the superintendent's data must come from his 
 own school system. But on many problems the matter 
 of giving advice about the collecting may be like that of 
 Holmes, when he said that one should always exercise 
 great care in the selection of one's grandfather. The 
 superintendent can at any rate collect far more data by 
 seeing that his records are so kept as to show the desired 
 facts later, than he can ever suddenly exact from teachers 
 who have never thought of keeping or giving out infor- 
 mation on this point. For example, the disputes and 
 troubles in the studies of retardation and elimination a 
 few years ago arose mainly because of the way school 
 records had been kept. It was impossible to tell from 
 existing records how many of the given pupils had entered 
 the first grade at any given time years before, or to find 
 all the significant facts about a pupil brought together 
 in one place. 
 
 Inasmuch as the facts in the superintendent's own school 
 are often meaningless unless they can be compared with 
 similar facts in other school systems, he must as far as 
 possible use records similar to those of his fellow school 
 men. Therefore, it is advisable that he use the records 
 and reports recommended by the Committee of the 
 National Education Association on Records and Reports, 
 and by the United States Bureau of Education. He 
 should also try to get the State Department of Education 
 in his state to use blanks that will fit in with this system. 
 If a decalogue for superintendents should be written, 
 one of the first commandments undoubtedly should be: 
 " Thou shalt keep thy records as nearly as possible by 
 
Collection of Data 59 
 
 the uniform system of the National Education Associa- 
 tion." 1 
 
 2. Other Sources of Data 
 
 The data from other school systems are obtained 
 usually by the use of questionnaires, from printed reports, 
 school surveys, magazine articles, etc. Aside from the 
 question of selection, the questionnaire method is often 
 practically worthless for collecting statistical data. 
 School men are too busy to answer large numbers of 
 questions, to work out the object of the questionnaire 
 when this is not clearly stated, to hunt up much infor- 
 mation on former conditions, to puzzle out involved or 
 ambiguous questions, and sometimes too careless to give 
 information definite enough to be of any service. As a 
 result, few replies will come in on a given problem, 
 and not all of these will be complete. This means few 
 opportunities for comparison. 
 
 It is always better to make use of printed statistics 
 where possible, taking care to be sure of the units and 
 processes used in compiling the statistics taken. The 
 mere fact that they are part of a printed report or formal 
 presentation, often required by law, practically insures 
 much more accurate figures than can be secured by the 
 questionnaire method. A second advantage is that the 
 superintendent can see that all his figures are on the same 
 basis, a thing impossible with a questionnaire because of 
 the inability of many people to understand or follow 
 directions on an inquiry which they have never seen 
 before and never expect to see again. 
 
 1 See Bulletin U. S. Bureau Education, 1912, No. 3, or get particu- 
 lars from such companies as the Library Bureau or the Shaw-Walker 
 Co. 
 
60 School Statistics and Publicity 
 
 ^ 
 
 For busy superintendents the following suggestions as 
 to where material for statistical comparisons may be 
 found, are appended : 
 
 1. Report of the United States Commissioner of Education, Volume 
 II, each year contains numerous tables on city school systems. It 
 includes such material as enrollment, number of teachers, aggregate 
 days' attendance, average daily attendance for both elementary and 
 secondary schools, length of session, number of children of census age 
 in private schools, number of buildings, number of sittings, itemized 
 receipts and expenditures of all school systems in cities of 5000 or more. 
 
 2. The bulletins of the United States Bureau of Education often 
 contain much valuable material on special topics. Especially good 
 are such ones as the following : 
 
 A comparative study of the salaries of teachers and officers. 
 1915, No. 31. 
 
 Ayres, L. P. : Provision for exceptional children in the pub- 
 lic schools. 1911, No. 14. 
 
 Boykin, Jas. C. and King, Roberta: The tangible rewards 
 of teaching. 1914, No. 16. 
 
 Deffenbaugh, W. S. : School administration in the smaller 
 cities. 1915, No. 44. 
 
 Frost, Norman : A statistical study of the public school sys- 
 tems of the southern Appalachian Mountains. 1915, No. 11. 
 
 Monahan, A. C. and Dye, C. H. : A comparison of the salaries 
 of rural and urban superintendents of schools. 1917, No. 33. 
 
 Morse, H. N. : Educational survey of Montgomery County, Md. 
 1913, No. 32. 
 
 Public, society, and school libraries. 1915, No. 25. 
 
 Statistics of certain manual training, agricultural, and industrial 
 schools. 1915, No. 19. 
 
 Strayer, G. D. : Age and grade census of schools and colleges. 
 
 1911, No. 5. 
 
 Thorndike, E. L. : The elimination of pupils from school. 
 1907, No. 4. 
 
 Thorndike, E. L. : The teaching staff of secondary schools in 
 the U. S. . 1909, No. 4. 
 
 Updegraff, Harlan : A study of expenses of city school systems. 
 
 1912, No. 5. 
 
 1 
 
Collection of Data 61 
 
 Updegraff, Harlan : Public and private high schools. 1912, 
 No. 22. 
 
 Updegrafif, Harlan and Hood, Wm. : Urban and rural school 
 statistics. 1912, No. 21. 
 
 3. The reports of the State Department of Education within the 
 state where the superintendent lives. 
 
 4. Strayer, G. D. and Thorndike, E. L. Studies in Educational 
 Administration. (Contains the cream of many dissertations pub- 
 lished at Teachers College prior to 1912.) 
 
 5. Publications of the United States Census Bureau, especially 
 special reports on cities and abstracts of each census. 
 
 6. School surveys of all classes. 
 
 7. Bulletins issued at various times such as those started by Super- 
 intendent Spaulding at Minneapolis in September, 1916. 
 
 8. Dissertations from leading schools of education in universities. 
 The publications of the superintendent's own state university in this 
 line will be easily available. In addition, the dissertations and theses 
 from Teachers College, Columbia University, and the School of Edu- 
 cation at the University of Chicago, are very valuable. 
 
 9. Reports of investigations appearing in standard educational 
 magazines. The best for this purpose are : 
 
 American School Board Journal, Milwaukee 
 Educational Administration and Supervision, Baltimore 
 Elementary School .Journal, University of Chicago Press 
 Journal of Educational Psychology, Baltimore 
 School Review, University of Chicago Press 
 School and Society, New York 
 
 10. After the foregoing was written. Professor H. 0. Rugg's book 
 on Statistical Methods Applied to Education appeared. It has much 
 more extended references, especially on pages 28-39, 361-375. The 
 material is admirably classified for ready reference. 
 
 EXERCISE 
 
 For the special problem you have selected, jot down as carefully 
 as you can at this time : 
 
 (a) The places in your own school system (or one in which you 
 are interested) from which you might secure statistical 
 data on it. 
 (6) Other likely sources of statistical data on it. 
 
62 School Statistics and Publicity 
 
 3. Sampling ^ 
 
 As soon as the sources of data are determined, the 
 question arises as to what can be done in the way of 
 " sampling " with a view to cutting down the inevitably 
 large amount of work. " Sampling/' of course, means 
 working from selected or tjrpical specimens rather than 
 with the whole mass of data. 
 
 The superintendent usually considers sampling be- 
 cause he must answer one of three questions : 
 
 1. How many measures are needed of an item to be sure that the 
 item is fairly well represented? 
 
 2. How many cases and which ones need to be treated in a large 
 mass of data to be sure that the results will be approximately true of 
 the whole ? 
 
 3. In case the superintendent at best can get only a small list of 
 items for comparative purposes (say only a dozen towns that are 
 really comparable to his town on school expenditures), how is he to 
 choose these items? 
 
 Let us now take these up in order. 
 
 Number of Measures of One Item Needed. No 
 arbitrary rule can be laid down for the number of measures 
 needed on any one item. But it is safe to say that often 
 there should be more than one in order to insure a reliable 
 average measure for the item. For example, in comparing 
 cities for school expenditures, it is often very unfair to 
 get the expenditures for one year only. There may be 
 very unusual conditions for that- year in several cities, 
 say fires, cold spells that sent up fuel bills, epidemics that 
 necessitated much medical expense, etc. In such studies 
 it is customary to take the average of two years for each 
 city, as Strayer does in his City School Expenditures, 
 
 1 See also pp. 22-23 
 
Collection of Data 63 
 
 Superintendent Spaulding, in a recent monograph on 
 expenditures in Minneapolis, takes the average for five 
 years for all cities studied. Another good example is 
 seen in comparing teachers on their ways of marking 
 students. It is very unfair to judge a teacher by the 
 marks she gives at any one term examination, or in any 
 one class. Professor Max Meyer at the University of 
 Missouri, for instance, never attempted to pass judgment 
 on the marking of a member of the faculty there until he 
 could get at least five hundred marks given by that teacher 
 in all his classes. A teacher may rightly object to a 
 rating given her by the superintendent on one short 
 visitation in one subject only. In practice, of course, she 
 will not object to such rating if it is highly favorable 
 to her, but it is probably about as far from the exact truth 
 as an unfavorable rating would be if made on the same 
 visitation. The writer some years ago, in inspecting high 
 schools for the University of Missouri, found that the 
 inspection was hardest in this one particular. It was im- 
 possible to visit a school oftener than once a year for a 
 day, and this made it extremely difficult to pass judgment 
 on four or five teachers in the course of the six hours or 
 less of teaching, judgment to which the teachers or their 
 superintendents would really subscribe. 
 
 The more measures we take of an item, provided they 
 are not all chosen with the same bias or cause for error, 
 the more reliable will be the final average measure taken 
 for that item. But time is too valuable to permit going 
 on indefinitely getting measures of one item. The safe 
 procedure is to take as few measures of it as may reason- 
 ably be expected to represent it fairly. 
 
 Selection of Samples Ordinarily. The amount of 
 work in any statistical treatment is so great that the 
 
64 School Statistics and Publicity 
 
 question of cutting down the number of items by sampling 
 is very important. But it is equally important that the 
 samples be so selected as really to represent the whole. 
 Neglect of this is responsible for the worthlessness of 
 many laborious pieces of school statistics. 
 
 The selection of samples should be absolutely at random, 
 and if there are groups of data, the same percentage of 
 samples should be taken from each group. Thus, a su- 
 perintendent who was making a study of outside reading 
 done by students in his town, could not get very trust- 
 worthy results by asking the pupils in five wealthy families, 
 in five families in comfortable circumstances, and in five 
 poor families. There would be so many more in the 
 second group than in the first, and so very many more in 
 the third than in either of the other two groups that his 
 results would be untrustworthy. To get at anything like 
 the truth, he would have to take, say, twenty from the 
 second group and possibly a hundred from the third. It is 
 equally erroneous to attempt to obtain results on the effect 
 of negro population on schools in Texas by ranging the 
 counties in order of percentage of negro population and 
 then taking three of the counties most free from negroes, 
 three from the middle group, and three of those having 
 the largest numbers of negroes. The results are wholly 
 unreliable until we know how many counties are in the 
 lowest natural group (that is, one without gaps in the 
 percentages), how many in the middle group, and how 
 many in the highest group, and that we have taken the 
 same proportion of samples from each. Again, a teacher 
 in giving a grade to a pupil ought not to take the work of 
 the pupil during the last few days of the term. It is not 
 a fair sampling of the student's work. Nor should 
 she take the sample grades for her private book at any 
 
Collection of Data 65 
 
 stated time known to students. If they know she always 
 grades them on Friday, they will do well on that day and 
 slow down on other days. 
 
 Unfair selection explains why questionnaire methods 
 of getting results are often so unsatisfactory and mis- 
 leading. The people who answer questionnaires are 
 often very selected and biased, and do not represent the 
 whole group at all fairly. For example, a superintendent 
 may wish to know what parents think of his school and 
 may send out a questionnaire for this purpose. It will 
 be sure to be answered mainly by those who are favorable 
 or who wish to make him think they are favorable, and 
 possibly by a very few opponents who are stirred up 
 enough to come out frankly, but who are probably se- 
 riously or unfairly prejudiced against him. If he sends 
 out a questionnaire to other superintendents, as a general 
 rule only the superintendents who think they can make 
 a better showing than he can, will send in results. So 
 very serious is this defect of the questionnaire method 
 that the phrase of President Kirk of the Kirksville, 
 Missouri, Normal School, " a questionable question- 
 naire/' is often justified. The most casual examination 
 of the statistical outpourings of the questionnaire type 
 in the last few years will show that very often only a small 
 percentage of the persons receiving questionnaires ever 
 answer them. This small percentage is almost certain to 
 be selected on some peculiar bias and is getting smaller 
 because so many foolish and needless questionnaires have 
 been sent out to superintendents in the last few years. 
 Many school men confess to throwing nearly all question- 
 naires into the waste basket or turning them over to 
 clerks or pupils to answer. 
 
 A good method of sampling is the familiar one of taking 
 
66 School Statistics and Publicity 
 
 every fourth or fifth ease of the items arranged alphabeti- 
 cally, or in order of magnitude, so that there can be no 
 prejudice in the matter. This, of course, insures getting 
 the same percentage in each of any possible groups. Thus, 
 if it is a case of consulting citizens, every fifth name in 
 the telephone book would give the superintendent a good 
 sampling of men able to have telephones. But it would 
 not represent all citizens. To get all represented, he 
 had better take every fifth name in a directory of the 
 city, or in a list of the registered voters. If it is a case of 
 children, arranging them alphabetically by grades, per- 
 haps boys and girls separately, and then taking every 
 fifth name would give him a good set of samples for boys 
 and another good set for girls. If a teacher has kept daily 
 grades of pupils, he can get a good sample set of grades 
 by taking, say, every seventh grade for each pupil, or 
 something of this sort, just so long as the grade taken 
 does not fall on the same day of the week every time. 
 
 Sampling is often resorted to in giving standard tests 
 to large numbers of children where the labor of grading 
 the papers from all would be very great. One of the best 
 sampling schemes coming to the writer's notice is the one 
 employed in the San Francisco survey in 1916. Four 
 tests — arithmetic, spelling, penmanship, and reading — 
 were given. Two classrooms in different grades in the 
 eighty-one elementary schools in the city were chosen at 
 random for a test in some one of the four subjects. No 
 teacher or principal knew in advance what rooms had been 
 chosen or what subject would be given in the room selected. 
 
 If a questionnaire has to be used, it is generally advisable 
 for the superintendent to pick out a reasonably small list 
 by one of the preceding methods, and then to devote his 
 energies to seeing that approximately every name on the 
 
Collection of Data 67 
 
 list sends in a fairly accurate answer. The " personal 
 questionnaire " filled out in person by an '' interviewer " 
 is sometimes an excellent device. It relieves the person 
 interviewed of much drudgery and insures correct inter- 
 pretation of the questions. But it tends to embarrass 
 the person giving the information, often to the point of 
 stopping easy conversation. To avoid this, the inter- 
 viewer is apt to encourage or accept rough estimates in 
 place of accurate data. In any case, the best way for the 
 superintendent to get answers is first to be sure that his 
 investigation will be of value to some one beside himself, 
 and especially to those he requests to fill out his question- 
 naire. Then he should promise to give all those answering 
 it a copy of the results. If he ever intends to do any more 
 investigating, he must faithfully keep this promise, 
 aside from his moral obligation to keep it.^ 
 
 Selection of Very Few Samples. Often the superintend- 
 ent knows that he cannot by any possibility get more than 
 a small number of items. His problem then is how to 
 select these items so that he may be sure that his com- 
 parisons will not be absurd. For example, if he wishes 
 to know whether his community is spending enough 
 money on its schools, is he to take all cities of that size 
 in the United States or in his state ? Neither procedure 
 will do because the same conditions manifestly do not 
 obtain in all of these cities. The same would be true if 
 the cities were selected on the basis of the number of 
 children of school age, the number of children in school, 
 etc. The towns certainly ought not to be compared 
 
 1 For those who, in spite of the preceding, find it necessary to use 
 a questionnaire, pp. 40-56 of Rugg's Statistical Methods Applied to 
 Education will be very valuable. This contains the most practical 
 treatment of the questionnaire that the author has yet seen. 
 
6S 
 
 School Statistics and Publicity 
 
 mercilessly unless their wealth per capita is something 
 like the same. They cannot be compared on current 
 school expenditures unless we know that their school 
 debts are something like the same relatively, and even 
 this may not be enough We may have to know their 
 present taxes for other needs, the city's indebtedness for 
 other purposes, etc. 
 
 Mr. F. 0. Seymour in writing his master's thesis at 
 Peabody in 1916 had the same problems in making a 
 study of school costs for Amarillo, Texas. Table 4 is 
 adapted from his work. 
 
 Table 4. Showing School Statistics for Certain Cities of 
 ABOUT the Same Population and General Situation 
 
 
 . 
 
 Per 
 
 Per 
 
 
 
 
 Per cent 
 
 Town 
 
 Popu- 
 lation 
 
 cent of 
 pupils 
 to total 
 popu- 
 lation 
 
 cent of 
 pupils 
 
 in 
 school 
 
 Wealth 
 
 per 
 capita 
 
 City 
 
 debt 
 
 per 
 
 capita 
 
 City 
 
 tax 
 
 per 
 
 capita 
 
 of city 
 
 revenue 
 
 going to 
 
 schools 
 
 Coffeyville, Kan. 
 
 13,687 
 
 24.0 
 
 86 
 
 $635 
 
 $62 
 
 $30 
 
 48.1 
 
 Dennison, Tex. 
 
 12,632 
 
 19.4 
 
 69 
 
 501 
 
 22 
 
 14 
 
 41.9 
 
 Pearsons, Kan. 
 
 12,363 
 
 18.3 
 
 71 
 
 815 
 
 31 
 
 15 
 
 52.9 
 
 Sherman, Tex. 
 
 12,412 
 
 20.9 
 
 71 
 
 608 
 
 20 
 
 15 
 
 39.1 
 
 Guthrie, Okla. 
 
 11,654 
 
 20.1 
 
 72 
 
 474 
 
 68 
 
 14 
 
 34.2 
 
 Marshall, Tex. 
 
 11,452 
 
 19.4 
 
 61 
 
 409 
 
 51 
 
 13 
 
 33.4 
 
 Paris, Tex. 
 
 11,269 
 
 27.2 
 
 85 
 
 815 
 
 51 
 
 19 
 
 36.5 
 
 Palestine, Tex. 
 
 10,480 
 
 18.1 
 
 59 
 
 508 
 
 19 
 
 10 
 
 40.3 
 
 Cleburne, Tex. 
 
 10,464 
 
 24.3 
 
 78 
 
 508 
 
 31 
 
 14 
 
 40.8 
 
 San Angelo, Tex. 
 
 10,321 
 
 15.2 
 
 49 
 
 566 
 
 20 
 
 10 
 
 44.8 
 
 Amarillo, Tex. 
 
 9,957 
 
 16.8 
 
 61 
 
 549 
 
 26 
 
 9 
 
 45.7 
 
 It may be noted concerning these cities which Mr. Sey- 
 mour has selected for purposes of comparison with 
 Amarillo that : (1) They are of practically the same 
 population; (2) they are all in the same section of the 
 
Collection of Data 
 
 69 
 
 country; (3) the various items given in the table are on 
 the whole rather close together for the different cities. 
 Had he included some eastern, northern, or extreme 
 southern cities of the same population, undoubtedly the 
 variations in these items would have been much greater. 
 Examples of Bad Sampling. Failure to pick out 
 samples that really represent the whole group may lead 
 to some very fallacious conclusions. For instance, 
 Professor Bobbitt in his article on the cost of instruction 
 in high schools, for purely illustrative purposes, compares 
 results from various cities. ^ The population of the cities 
 he uses varies greatly. Figures could not be obtained 
 for all, but Table 5 gives enough to show the variations. 
 
 Table 5. Variations in Population of the Cities Used by 
 Professor Bobbitt in His Study of High School Costs 
 
 City 
 
 Population 
 in 1910 
 
 City 
 
 Population 
 in 1910 
 
 Mishawaka, Ind. 
 
 Elgin, 111 
 
 Maple Lake, Minn. 
 Granite City, 111. . 
 East Chicago, 111. 
 De Kalb, 111. . . . 
 San Antonio, Tex. . 
 Harvey, 111. . . . 
 Waukegan, 111. . . 
 South Bend, Ind. 
 East Aurora, 111. . . 
 Rockford, 111. . . 
 
 11,885 
 25,976 
 
 9,903 
 
 8,102 
 96,614 
 
 7,227 
 16,069 
 53,684 
 29,807 2 
 45,401 
 
 Booneville, Mo. . . . 
 
 Brazil, Ind. .... . 
 
 Leavenworth, Kan. . 
 Greensburg, Ind. . . 
 Morgan Park, 111. . . 
 Noblesville, Ind. . . 
 Norfolk, Neb. . . . 
 
 Washington, Mo. . . 
 Bonner Springs, Kan. . 
 Russell, Kan. . . . 
 
 Junction City, Kan. 
 Mt. Carroll, 111. . . . 
 
 4,252 
 9,340 
 19,363 
 5,420 
 3,694 
 5,073 
 6,025 
 3,670 
 
 5,598 
 1,759 
 
 1 Bobbitt, J. F. : "Cost of Instruction in High Schools," School 
 Review, 23 : 505-534. (Oct., 1915) 
 
 2 Population for Aurora. There are two high schools, one in East 
 Aurora and one in West Aurora. 
 
70 
 
 School Statistics and Publicity 
 
 But Professor Updegraff has shown that the cost of 
 instruction is higher in large cities than in small ones. 
 See Table 6. 
 
 Table 6. Variations in Cost of Instruction and Supervision 
 PER Capita (Population) in Cities of Different Sizes ^ 
 
 
 
 Per capita 
 
 Per capita 
 expenses of 
 
 
 Median 
 city in 
 
 Population 
 
 expenses of 
 teachers' 
 salaries 
 
 salaries and 
 expenditures 
 
 for 
 supervisors 
 
 Total 
 
 Group I 
 
 300,000 and over 
 
 $50.98 
 
 $1.18 
 
 $52.16 
 
 Group II 
 
 100,000 to 200,000 
 
 36.15 
 
 3.26 
 
 39.41 
 
 Group III 
 
 50,000 to 100,000 
 
 36.93 
 
 3.39 
 
 40.32 
 
 Group IV 
 
 30,000 to 50,000 
 
 29.25 
 
 3.38 
 
 32.63 
 
 As to costs in schools in cities below 30,000, no such figures 
 are known to the author. It is entirely probable that they 
 are higher than the lowest figures cited here because of not 
 having such full classes in high school. But the main 
 point is that there are always great dangers in selecting 
 samples that cannot reasonably be regarded as coming 
 from the same class. Professor Bobbitt was, of course, 
 perfectly aware of this difficulty and on page 506 of his 
 article indicates that his tables are valuable as patterns 
 of work mainly. 
 
 Another example of the results of unwise selection 
 of samples is furnished by Superintendent Spaulding's 
 monograph on the cost of the Minneapolis schools.^ 
 
 1 Adapted from Updegraff, Harlan : "A Study of Expenses in City- 
 School Systems," Bulletin U. S. Bureau Education, 1912, No. 5, 
 pp. 7, 86. (Computed on enrollment of pupils) 
 
 2 Spaulding, F. E. : Financing the Minneapolis Schools. (Board 
 of Education, Minneapolis, Sept., 1916) 
 
Collection of Data 71 
 
 He has a chart on page 46 showing the expenditure per 
 child for ordinary maintenance of the elementary schools. 
 But in this chart he has included three southern cities, 
 Louisville, Birmingham, and New Orleans. All three 
 have in their population large negro elements which 
 seldom pay taxes and add little wealth to the community 
 because they are not able to do so. At the same time 
 they have relatively large numbers of children to be 
 provided with schooling. Consequently these southern 
 cities have to take care of a large number of negro children 
 with no corresponding increase in revenue. Therefore, 
 they should not be placed in comparison with Minneapolis. 
 Their presence in this table has the effect of making 
 Minneapolis appear more extravagant in her expenditures 
 for schools than would be the case if only northern cities 
 were considered. 
 
 Summary of Rules for Sampling. These may be 
 given briefly: 
 
 1. Be sure that the measures of any one item represent its usual 
 state. 
 
 2. Select samples absolutely at random. 
 
 3. If there are groups of data, take the same percentage of samples 
 from each group. 
 
 4. Avoid using a questionnaire if possible. If it must be used, be 
 careful to discount the results for classes liable to be selected by it. 
 
 5. If only a small number of samples are obtainable, select them 
 with unusual care. 
 
 4. Blanks and. Tabulating 
 
 Devices for Blanks. The actual collection of data 
 should be made on some form for tabulating, called for 
 convenience a '' blank." Rules for making blanks which 
 hold good for many specific problems are extremely hard 
 to lay down, but the following devices are often helpful : 
 
72 
 
 School Statistics and Publicity 
 
 a. Plan the blanks so as to get a maximum amount of 
 information with a minimum of space. 
 
 This reduces chances for error, because the more nearly a person 
 can see all of the blank at one look, the greater mastery of it and 
 of the relations of its parts he will have. But this is not to be inter- 
 preted as advocating extreme condensation resulting in eye strain, or 
 the elimination of all space for calculations. Nor does it mean that 
 setting all the material on one sheet of paper is sufficient. A blank 
 may be so long or so wide as to be hard to understand at one look. 
 
 6. Use " double distribution " tables where possible. 
 
 By these are meant " two-way " tables, — tables that classify data 
 two ways, across the page on one classification, and down the page 
 on another. They thus enable one to get the data from at least two 
 separate tables condensed into a convenient form in one. Tables 7, 
 8, and 9, with slight modifications, are taken from Snedden and 
 Allen's School Reports and School Efficiency ^ to illustrate double dis- 
 tribution tables. 
 
 Table 7. Blank for Showing Number op Pupils Making 
 Given Number op Days' Attendance 
 
 First grade . 
 Second grade 
 Third grade 
 Fourth grade 
 Etc. . . . 
 
 From 
 19 days 
 
 From 
 
 20 to 
 
 39 days 
 
 From 
 
 JfO to 
 
 59 days 
 
 From 
 
 60 to 
 
 79 days 
 
 From 
 
 80 to 
 
 99 days 
 
 Etc. 
 
 Explanation: This table provides for a distribution of pupils' 
 
 attendance in two ways: (1) by the grade; (2) by the number of 
 
 days. Hence it is called a ''two-way" or "double distribution" 
 
 table. 
 
 1 Pp. 130, 132 
 
Collection of Data 
 
 73 
 
 Table 8. Blank for Showing Distribution of Pupils in 
 Each Grade by Ages, Records of Ages Being Made 
 
 
 First grade 
 
 Second grade 
 
 Totals 
 
 
 Males 
 
 Females 
 
 Males 
 
 Females 
 
 Males 
 
 Females 
 
 From 5 yr. 
 6 mo. to 6 
 
 
 
 
 
 
 
 yr. 6 mo. 
 From 6 yr. 
 6 mo. to 7 
 
 
 
 
 
 
 
 yr. 6 mo. 
 Etc. . . . 
 
 
 
 
 
 
 
 Note : This is a very convenient table, since it gives a distribution 
 of pupils both by ages, and by grade and sex. 
 
 c. Strive to have all the printing on the blank so that it 
 can he read from one position while filling in the data. 
 
 Table 7 was originally printed like Table 9. 
 
 Table 9. Blank for Showing Number of Pupils Making 
 Given Number of Days' Attendance 
 
 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 
 
 
 CO 
 
 S^ 
 
 ^ 
 
 STi 
 
 >5 
 
 
 
 ?r5 
 
 ;3 
 
 e 
 
 !3 
 
 S 
 
 
 
 e 
 
 ^ 
 
 -^ 
 
 ^ 
 
 ^3 
 
 
 
 'XS 
 
 05 
 
 Oi 
 
 Oi 
 
 Oi 
 
 
 
 Oi 
 
 <50 
 
 ^ 
 
 t^ 
 
 Oi 
 
 
 
 •»-s 
 
 O 
 
 o 
 
 o 
 
 o 
 
 
 
 O 
 
 "*-^ 
 
 '*^ 
 
 -»^ 
 
 -K) 
 
 
 
 -»o 
 
 o 
 
 o 
 
 O 
 
 O 
 
 
 
 H|(N 
 
 (^ 
 
 -^ 
 
 ^ 
 
 oo 
 
 
 
 g 
 
 g 
 
 s 
 
 g 
 
 g 
 
 
 
 Q 
 
 o- 
 
 o 
 
 o 
 
 o 
 
 «:> 
 
 
 ^ 
 
 ^ 
 
 
 V. 
 
 
 
 
 fcn 
 
 c^ 
 
 ^ 
 
 !*H 
 
 t^ 
 
 uq 
 
 First grade . . . 
 
 
 
 
 
 
 
 Second grade . . 
 
 
 
 
 
 
 
 Etc 
 
 
 
 
 
 
 
74 
 
 School Statistics and Publicity 
 
 Note: This is exactly the way the blank should not be made up. 
 The proper form is given on page 72. With care, the space usually 
 can all be utilized either way. If all the titles and headings can be 
 read from one position, that is, without having to turn the paper 
 around, the chances for error are much reduced. 
 
 d. Put full or plainly abbreviated title, not merely an 
 arbitrary symbol or a number, above each column, if any 
 one besides the superintendent is to use the blank. 
 
 4. NATIONAL SCHOOL RECORD SYSTEM 
 
 TRANSFER CARD 
 
 To be filled out for the Attendance Offi- 
 cer in case of transfer to any other 
 school, either in or outside city. . . . 
 
 1. Last name 
 
 2. First name and initial 
 
 6. Name of parents or 
 guardian 
 
 8. Residence before 
 discharge 
 
 7. Occupation of parent or 9. Date of discharge 
 guardian 
 
 8. New residence (or name 
 of private or parochial school 
 if pupil is transferred to one) 
 
 10. Age when discharged 
 Years Months 
 
 d. 
 
 Grade 
 
 e. 
 
 Room 
 
 f. 
 
 Days Present 
 
 Health 
 
 h. 
 
 Conduct 
 
 i. 
 
 Scholarship 
 
 Date of last attendance 
 
 School 
 
 District 
 
 Teacher 
 
 Principal 
 
 Remarks on other side 
 
 THE SHAW-WALKER CO., MUSKEGON FORM NO 4 
 
 Some superintendents get out blanks of the kind condemned here. 
 But it is very doubtful if such blanks will ever secure accuracy from 
 anyone except the person who thought out the key (even he may 
 forget it if it is at all intricate) and the small percentage of teachers 
 who may be called "hyper-patient." Teachers dread such blanks 
 for precisely the same reason that most readers dread having to refer 
 constantly to footnotes in something they are trying to read. 
 
Collection of Data 75 
 
 If the data are to be copied or read to some one else, a key num- 
 ber should appear in each column immediately under, or in front of, 
 the title. The reader or copyist can catch this key number much 
 more quickly and easily than he can the heading. The key number 
 is also preferable for calling or reading off to some one else. 
 
 A good example of the use of key numbers and letters with titles 
 is found in the National System of Uniform Records, the transfer 
 card of which is reproduced on page 74. 
 
 e. If the same item appears in different blanks of a 
 series, always give it the same key number and the same 
 relative position to closely connected items. 
 
 The report of the Committee on Uniform Records and Reports, 
 previously referred to, recognized this principle. The order of the 
 various items appearing on the pupil's report card is the same as 
 that appearing on the lower half of the attendance and scholar- 
 ship record kept in the teacher's loose-leaf register, which is later 
 filed in the principal's office. Other cards in the system which 
 contain these items use the same key letters and relative position. 
 
 /. If data are to be summarized from figures furnished by 
 many persons, make the summary with as little copying as 
 possible. 
 
 Copying always tends to error, and the checking necessary to avoid 
 such errors is very laborious and time-consuming. The special econo- 
 mies here are things like the following: (1) In a building, the data 
 for a report on the table given on page 72 may be secured by hav- 
 ing each teacher call at the principal's office and put her data on 
 the proper horizontal line. The completed blank will then be the 
 desired summary. (2) If the data are to come from different build- 
 ings, each teacher can put hers on the proper line on a blank and 
 send it to headquarters. There such blanks can be gathered, the 
 line clipped off each blank (keeping necessary identification marks of 
 course), and pasted on a sheet of paper, thus forming the summary 
 at once. In very extensive or frequent tabulations of this sort, 
 blanks with perforations where the lines are to be cut apart may be of 
 service. Perforated blanks are, however, very expensive and usually 
 
76 
 
 School Statistics and Publicity 
 
 unnecessary. For most purposes, cutting and pasting, or copying 
 and checking, will be sufficient. 
 
 g. Figure or summarize on the blank itself if time can be 
 saved thereby, as is often the case. 
 
 Sabjact P^WSt'cS 
 
 Class 
 
 Teacbe 
 
 r Li^S 
 
 Co Vvi 
 
 Ir 
 
 Sex 
 
 Al 
 
 Tears Erperience, before 1913-14_ 
 Head of Department? 
 
 :^5'-l05 ; 90.94 : 80-89 : 65-79 : 150-701 ; 7SA : Below 65 : Total 
 
 Boys : 
 
 / = 
 
 \ 
 
 -ffH- '- 
 II 
 
 t 
 
 , / '■ 
 
 
 ; ; 
 
 
 
 1^ 
 
 1 
 
 1 
 
 ® 
 
 1 — ^ 
 
 
 Girls J / 
 
 
 /// 
 
 . // 
 
 
 ; 
 
 1 • , • 
 
 ; / 
 
 ;<2) 
 
 : ^ 
 
 
 
 o 
 
 
 
 = / 
 
 • 7 
 
 ■ If 
 
 (D 
 
 : / 
 
 Id \31 
 
 Data for Grading System - Hume -Fogg High Sobool 
 Grades for 1913-14. 
 
 FiQ. 1. — Example of Scoring and Figuring on the Blank Used for Collect- 
 ing Data. 
 
Collection of Data 77 
 
 In summarizing the material collected as advised in section /, the 
 additions can be made in ink (often red ink) on the bottom of the 
 sheet where all the data have been collected. A good illustration 
 of this point is afforded by a blank which was used by a class of 
 graduate students making under the writer's direction a study of 
 the marks given in the Hume-Fogg High School in Nashville.^ (See 
 Figure 1.) 
 
 In Figure .1 original scores are indicated by the marks above the 
 line. The plain figures below each line are the numerical equiva- 
 lents of the scores. The figures in rings represent the approximate 
 percentage values of the numbers just above them. On the original 
 blank such percentages could be indicated by red ink without using 
 rings. All percentages are calculated here to show the method. But 
 in this particular problem, because of the small number of cases in 
 each group, such percentages have little value. See pages 13, 63. 
 
 h. In case only one copy of a blank can be made, use 
 cross-section paper, or ruled blank book. 
 
 Cross-section paper saves an enormous amount of time in ruling 
 paper and avoids eyestrain. Such paper may be purchased from a 
 good scientific supply or drawing company, in sheets of various sizes. 
 That particular kind characterized by heavy lines every five or ten 
 small squares is the better kind. The heavy lines serve as columns, 
 while the small squares help to keep figures under each other. 
 
 A ruled blank book or account book, especially the kind extending 
 over two adjacent pages, is similarly useful because of the vertical 
 columns and horizontal lines. 
 
 i. Put at the top of the blank the item which determines 
 the place of the blank in a series or which identifies it. 
 
 The observance of this caution will make for ease in handling the 
 data. For example, in the case of cards that are to be filed alphabet- 
 ically according to the name of the student, this name should appear 
 at the top of the card where it will be the first thing the eye of the 
 one going through the records will light upon. Yet many school men 
 
 1 The results of this investigation appeared in Educational Ad- 
 ministration and Supervision, 1 : 648 (Dec, 1915), from which article 
 this illustration is adapted. 
 
78 School Statistics and Publicity 
 
 get out blanks with the name of the school, the name of the card, the 
 name of the town, and other insignificant items at the top, which is 
 the choice position for identifying a blank quickly, especially if it is 
 ever filed. Such material is very seldom handled except by persons 
 who use the blanks more or less constantly. For these, this material 
 may as well come at the bottom of the card and often even in fine 
 print. Even the transfer card of the National School Record System 
 shown on page 74 could be improved upon in this respect. The 
 card is to be filed alphabetically by the name of the pupil. This 
 name should come on the top line, and the title of the card with the 
 caution to the attendance officer should be placed at the bottom. 
 
 j. Always use blanks cut to standard sizes. 
 
 This is essential for easy handling of data, filing for ready refer- 
 ence, and storage for future use. Folders and filing cases can easily 
 be procured for standard sizes, and the printer or supply house can 
 furnish paper and cards for these sizes at much lower prices than for 
 odd sizes. The standard sizes are 3x5, 4x6, 5x8, and 8|xll inches. 
 
 Examples of Good Blanks. The busy superintendent 
 may often save much time by getting good blanks used 
 in other systems and quickly modifying them for his own 
 purposes. Such blanks may be found in : 
 
 1. Snedden and Allen : School Reports and School Efficiency. 
 (Macmillan) 
 
 This discusses the good and bad points in actual blanks from 
 city school systems, on such topics as the school plant, expend- 
 itures, the census, attendance, attendance and ages, promotions, 
 summaries, etc. 
 
 2. Report of the National Education Association Committee on Uni^ 
 form Records and Reports, published as Bulletin U. S. Bureau Educa- 
 tion, 1912, No. 3. 
 
 It gives model blanks for the following in elementary schools : 
 
 1. Principal's term report : (a) enrollment ; (6) promotions, 
 non-promotions, by grades; (c) distribution of withdrawals 
 
Collection of Data 79 
 
 by ages and causes ; (d) distribution of attendance ; (e) grad- 
 uates by years in schools; (/) non-promotions by grades and 
 causes ; (g) failures by studies and grades ; (h) distribution of 
 leavings and withdrawals by ages and grades ; (i) ages of grad- 
 uates; (j) enrollment and attendance; (k) distribution of 
 whole-time teachers. 
 
 2. Teacher's term report: (a) enrollment by divisions; 
 (6) non-promotions by grades and classes ; (c) failures by grades 
 and studies ; (d) enrollment and attendance ; (e) distribution 
 of enrollment by ages ; (/) distribution of withdrawals by ages 
 and causes ; (g) distribution of leavings by ages ; (h) beginners 
 by training; (i) beginners by ages. 
 
 There is also a set of blanks for high schools similar to these. 
 
 3. Strayer and Thorndike: Educational Administration. (Mac- 
 millan) 
 
 This contains extracts from statistical researches on educa- 
 tion made at Teachers College, Columbia University. It con- 
 tains some of the blanks recommended by the National Educa- 
 tion Association Committee as well as very valuable table 
 forms on other things. 
 
 4. Rugg, H. O. : Statistical Methods Applied to Education, pages 
 28-87. (Houghton Mifflin) 
 
 5. Certain Surveys. 
 
 The Butte Survey, for example, contains good suggestive 
 blanks on these phases : census, attendance, absences, educa- 
 tion and experience of teachers, enrollment, promotion and 
 failures, size of classes, ages of children at beginning of first 
 semester, receipts and expenditures. 
 
 One Blank versus Card Index. In many pieces of 
 statistical work, one must early face the question as to 
 whether it is better to put all the data on one large 
 blank, allowing one line to each case or group, or to enter 
 each case or group on a separate card. 
 
 One of the best arguments for the latter method is that 
 the data in the columns cannot show their full meaning, 
 
80 School Statistics and Publicity 
 
 as a rule, unless arranged in order from lowest to highest. 
 If the data are all on one large sheet, the classes may come 
 in order for the data in the first column, but this order 
 will not be the same in any one of the other columns, 
 which may be several in number. If these columns are 
 to be accurately studied, the data in them must be 
 copied off and rearranged, with many chances for error. 
 On the other hand, if no large table is made, the cards 
 may be taken up in order easily for one column. The 
 data may then be copied off for a table on this one item. 
 Next, the cards may be rearranged for column two and 
 the corresponding table made. This process is much 
 easier mentally and more accurate than the first one 
 mentioned above. It is the procedure used in the Courtis 
 tests, where each child has a folder containing his own 
 work. These folders are taken up and sorted into piles 
 of differing achievement for attempts in addition, and 
 from these a distribution table is made ; they are re- 
 distributed for attempts in subtraction, and a second 
 distribution table is made, this time for subtraction ; the 
 process is repeated for multiplication and division. Then 
 the start is made on ''rights," and the four sortings take 
 place on this basis and the four resulting tables are 
 copied off. 
 
 To summarize, the advantages of each plan are : 
 
 Large Sheet Card Index Plan 
 The data are not easily lost. Less chance for error in copy- 
 It is impossible to lose a part, ing or rearranging data. 
 
 as can easily happen in the use Easier mentally. 
 
 of cards. 
 
 All data are before the eye at Any parts of the data may be 
 
 one time, and thus a better pre- separated from the remainder at 
 
 liminary grasp of the situation any time, 
 
 may be had. 
 
Collection of Data 
 
 81 
 
 Less work in compiling at The data may be easily shifted 
 
 first. to any order. 
 
 At the beginning data may be Pupils can handle data better, 
 
 copied in any order. Often it will be advisable to give 
 
 portions to different workers. 
 
 The disadvantages of these two plans are largely the 
 converse of the advantages, but they may be emphasized 
 through the statement of them: 
 
 Card Index Plan 
 If card is lost, it cannot be re- 
 placed, nor can the loss be easily 
 discovered. 
 
 A card may stick to another 
 and be easily overlooked. 
 
 Cards for easy handling must 
 ordinarily be kept in alphabetical 
 order. Hence if shifted for pur- 
 poses of making tables, etc., 
 must be returned to regular order. 
 
 Much mechanical work is neces- 
 sary in examining data. 
 
 Large Sheet 
 
 Data from large buildings or 
 groups not easily brought to- 
 gether on one sheet. 
 
 Additional data (often come 
 late) are hard to insert. 
 
 Data in all columns not ar- 
 ranged from highest to lowest 
 nor can they be so arranged with- 
 out recopying. 
 
 More chances for error in re- 
 copying and rearranging. 
 
 Will stand less wear and tear. 
 
 Averages and summaries are 
 hard to make unless data are ar- 
 ranged in order from the start. 
 Often this cannot be determined 
 when data are copied on large 
 sheet. 
 
 On any given problem, a little forethought or practice 
 on sample data will usually indicate which is the better 
 plan. However, it is very difficult to be certain in this 
 matter until the whole statistical procedure has been 
 thought through. 
 
 EXERCISE 
 
 For your special problem : 
 (a) Decide just where you would use a card index and just where 
 one large blank, giving full reasons. 
 
82 School Statistics and Publicity 
 
 (b) List the blanks, with very definite titles, that you now think 
 
 you would need. 
 
 (c) Draw off in complete detail, accurate to size of paper, at 
 
 least one of these blanks. 
 
 5. Miscellaneous Economies in Collecting Data 
 
 a. Cross-section paper for scoring. Many times it is necessary to 
 keep track of the number of cases, the data for which may come from 
 widely separated places. Keeping track mentally is very difficult 
 and inaccurate. It is far better to adopt some method of putting 
 down a mark for each case as soon as it is located, and later count 
 the marks. The cross-section paper device is one for such marking. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 < 
 
 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 )0 
 
 -1 
 
 0( 
 
 ) 
 
 1 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 I 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 e 
 
 10 
 
 "~ 
 
 8J 
 
 ) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 2. — Scoring on Cross-Section Paper. 
 
 Suppose that a study is being made of the marks given by various 
 teachers to pupils. A sheet, column, or whatever space is desired, 
 may be assigned to each teacher. One horizontal row of large squares 
 may be labeled 90-100 or A, according to the grading system ; another 
 80-89 or B, etc. Each mark between 90 and 100 as soon as located 
 would be represented by a check mark in one of the little squares in 
 
Collection of Data 83 
 
 the row of large squares representing grades 90-100, under the column 
 for the proper teacher or subject. For such work, always begin at 
 the left upper corner of the first big square on the left and fill in reg- 
 ularly to the right, row by row. Suppose, when the work has been 
 finished, that the 90-100 column looks like the illustration given. 
 One glance at the squares in Figure 2 shows that there are 33 grades 
 between 90 and 100 given by the teacher during the time under in- 
 vestigation. 
 
 b. Scoring by fives on plain paper. This is the old device used in 
 counting the votes in elections, etc. The first four cases of every five 
 
 WW 
 
 m 
 
 THi 
 
 ^^ or M m w tm mi rtu in 
 
 m 
 
 Ill 
 
 Fig. 3. — Scoring on Plain Paper. 
 
 are given a straight perpendicular mark each ; the fifth is made hori- 
 zontally, tying the other four, as in Figure 3. 
 
 This makes 33 cases, as in the preceding method of scoring. For 
 most people the counting is much easier if the groups of five are under 
 each other and not in a horizontal line. 
 
 The author also has made much use of the scheme of putting down 
 a mark for every case in the group as best he could for speed, on any 
 
 Fig. 4. — Scoring on Plain Paper. 
 
 paper handy. Then rings are drawn around the marks with five in 
 each ring. It is easy to count the rings. The chief merit of this 
 
84 
 
 School Statistics and Publicity 
 
 scheme for a single worker is that the eye does not have to be shifted 
 quickly and accurately, with the resulting strain. Putting down the 
 dots roughly requires no eyestrain. Putting the rings around the 
 dots is not hard, as the eye is kept focused on the particular part of 
 the paper. Thus, the same 33 cases might be scored as in Figure 4. 
 c. Checking on blanks. Where possible, tabulate by check marks 
 in appropriate column on a blank, as in the United States Govern- 
 ment stock device. Thus, a superintendent could study physical 
 defects of school children with a blank like Table 10. 
 
 Table 10. Blank Illustrating Method of Checking in 
 
 Entering Data 
 
 
 
 Physical Defects 
 
 
 
 
 
 Sight 
 
 Hearing 
 
 
 
 
 Pupil No. 
 
 
 
 Ade- 
 noids 
 
 Tubercu- 
 losis 
 
 Efr 
 
 
 
 
 
 
 
 Right 
 
 Left 
 
 Right 
 
 Left 
 
 
 
 
 1 
 
 
 
 X 
 
 
 
 
 
 2 
 
 
 X 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 
 
 5 
 
 X 
 
 X 
 
 
 
 X 
 
 
 
 6 
 
 
 
 
 X 
 
 
 
 
 7 
 
 
 
 X 
 
 ' 
 
 
 
 
 d. Some entry for each case. In some instances accuracy is in- 
 creased by the device of making some sort of entry for each case. It 
 is very easy to omit a few cases from a large number if entries are 
 made only where data actually exist. But if numbers are put down 
 for actual data, zeros if it is known that nothing is done, and "n. d." 
 (for "no data") where it has been impossible to secure data, the 
 results will be likely to be much more accurate. The necessity of 
 having to account for each case in a positive form, reduces the chances 
 for omission, a fact long ago discovered by insurance companies, 
 which require their agents to make some report on each item on a 
 blank. 
 
Collection of Data 
 
 85 
 
 e. Rider strip for printed reports. Often it is necessary for the 
 superintendent to get figures from two or three widely separated 
 columns in a table of fine print. There is great eyestrain and many 
 chances for inaccuracy in trying to copy them off directly, especially 
 where the tables run over two pages, as do those in the government 
 bulletins. The best way to avoid this is to cut out a strip of paper 
 which may be placed over the table so that the desired figures will 
 stand out clearly and quickly in the angles of the paper. Table 11 
 indicates how to work the scheme if the superintendent wishes to 
 compare his system with the cities in the table in the matter of ex- 
 penditures under the headings, Board of Education and Business 
 Offices, Superintendent's Office, and Other Supplies. Government 
 offices use specially constructed rulers for the same purpose. 
 
 Table 11. Illustration of Use of Ruler Strip Device on 
 A Page, in the Report of the U. S. Commissioner of 
 
 Education 
 
 Cities 
 
 B'd of Ed. 
 
 S^Bus. 
 
 Offices. 
 
 Supf's 
 OffTce. 
 
 Sal. 
 ^Exp 
 
 of 
 super- 
 visors. 
 
 Sal 
 
 of 
 ?rm. 
 
 Sal.of 
 Teachers 
 
 Text 
 Bool<5 
 
 Other 
 Supplies 
 
 ALABAMA 
 
 $3631 
 128163 
 
 $10750 
 43801 
 
 
 
 
 
 $5967 
 J05600 
 
 Birmingham 
 
 CALIFORNIA 
 
 L,osAn3etes 
 
 J 
 
 1 1 
 
 1 1 
 
 
 1 .. ^ 
 
 
 
 
 
 
 
 Another very convenient form of ruler strip is made with two 
 strips of zinc adhesive or bicycle tape and some thin, tough paper. 
 The strips of adhesive are pasted across the paper, but a little apart. 
 Then the top strip is cut across, the cuts corresponding to the lines in 
 the table to be used. Then some of the resulting flaps may be pulled 
 back and pressed back to show the figures in the desired columns, 
 as in Figure 5. 
 
86 
 
 School Statistics and Publicity 
 
 The advantages of this form are that it can be used for various 
 combinations of columns, and that it is very durable. 
 
 /. One operation at a time. The idea is to carry the same step 
 or operation all the way through without stopping to do something 
 else. That is, if one is preparing a table from data found in different 
 tables, he should copy all the data from each table in turn, and not 
 skip from one to the other. Or if one has several groups of data to 
 
 Papi 
 
 Fig. 5. 
 
 /ap« 
 
 - 
 
 43801 
 
 I 1 
 
 1 1 
 
 1 1 
 
 105600 
 
 
 
 
 
 -——-"— 
 
 
 
 
 
 
 
 
 
 
 Illustration of Use of Adhesive Tape Ruler Strip on Figures of 
 Table 11. 
 
 rearrange, he should finish each group before going to the next. This 
 gives the practice effect for that operation, insures greater accuracy, 
 and is much easier mentally. 
 
 g. Using high school students to gather data. For gathering many 
 of the data, high school students can do as well as any one else. A 
 class can collect, classify, and check a great deal. Of course the 
 question at once arises as to whether this is a legitimate use of the time 
 of high school students, who supposedly go to school for their own 
 benefit and not to help work out school statistics. But many of these 
 are later to do clerical work of various sorts. Much of the statistical 
 work in school affords the finest sort of clerical practice for such stu- 
 dents. Their welfare will be properly cared for if the work is done 
 under the careful supervision of the superintendent. He will natu- 
 rally discover a great deal of valuable knowledge about the vocational 
 aptitudes of his individual students for satisfactorily doing clerical 
 work or for directing others in such work. Accordingly it may be 
 considered thoroughly sound preliminary training for them, as Pro- 
 fessor Bobbitt has pointed out.^ '^ 
 
 But there is more direct evidence of the value of such work than 
 any such theoretical statement. In 1915 one of the author's graduate 
 students, Mr. S. J. Phelps, later professor of secondary education at 
 the University of Vermont, directed the work of the students of the 
 
 1 San Antonio Survey, pp. 32, 33. 
 
Collection of Data 87 
 
 Gallatin, Tennessee, High School in making a study of four problems 
 connected with their school system.^ These problems were : 
 
 1. Cost of maintenance of the system. 
 
 2. Age-grade distribution of the pupils. 
 
 3. Variations in marks given by high school teachers. 
 
 4. Study of the lighting facilities in each room. 
 
 Clear and concise instructions were placed by Professor Phelps in the 
 hands of the various teachers engaged in the work. The answers of 
 the students were carefully checked over in class and then submitted 
 to the inspection of a graduate class in school administration at George 
 Peabody College for Teachers. Most of these men had had much 
 experience in administrative work and all were specializing in survey 
 work. They unanimously agreed that the work was surprisingly 
 accurate. 
 
 In answer to the question : Is not such work an exploitation of the 
 students? Professor Phelps answers: 
 
 "All are agreed that in mathematics, especially, it is necessary 
 for a high school student to do a large amount of drill work. Now 
 in doing this, which is the more profitable and practical for a high 
 school student who is studying, for instance, eleventh grade civics 
 or arithmetic, to do as outside work : To study the costs of oper- 
 ating his own school system, compared to similar costs in other 
 towns, or to find out the number of days it will take A, B, and C 
 working together to do a piece of work which A can do in three 
 days, B in four days, and C in five days? 
 
 "How would a study of costs in his school compare with a paper 
 which he might, after much delving, prepare on the source of some 
 abstract principle of governmental costs? Isn't this a place where 
 the much-talked-of subject, Community Civics, could get some 
 practical problems ? 
 
 "Which seems more practical and profitable for a class in algebra 
 studying the graph, — to make a graph of these same costs, or to 
 graph the profile of a river bed, or perhaps an extract from the 
 table of American Mortality Experience? 
 
 "In which would a student in percentage be expected to show 
 . the more interest, — in a study of the percentage distribution of 
 
 1 Phelps, S. J. : Master's thesis at George Peabody College for 
 Teachers, 1915, on file in library. 
 
88 School Statistics and Publicity 
 
 the marks given by his own teachers, among which he has a mark, 
 or in studying the percentage composition of some compound, 
 perhaps a fertilizer, which he has never seen and in which he cannot 
 be expected to show a passing interest or curiosity? 
 
 "Would another student in practical measurements get more 
 from computing the surface and volume of the earth, or from find- 
 ing how many spheres of a certain diameter could be placed in a 
 cylindrical cup of certain dimensions, than he would get from 
 studying the ratio of lighting space to floor space and air space per 
 pupil in the same room?" 
 
 From such suggestive questions as these it may be surmised that 
 work of this sort, instead of "exploiting" the high school student, 
 would be of great practical benefit to him. 
 
 A report of similar work from Wisconsin is as follows : 
 
 HIGH SCHOOL CLASSES 
 GRAPH CONDITIONS 
 
 Over-Age and Failures Studied 
 
 In connection with the algebra work 
 in the Frederic High School, graphic rep- 
 resentations in colors are made showing 
 conditions in the number of students re- 
 tarded, and other school problems. One 
 percentage graph compares the percentage 
 of students over age, showing a distinct 
 decrease in retardation during the past 
 four years. Other graphs include mate- 
 rial on students dropped, failed, and pro- 
 moted in various subjects. Teachers are 
 also compared with respect to the number 
 of students failed by each of them. 
 
 Incidentally, such work as this gives the 
 algebra pupil practical work to do, illumi- 
 nates the general subject of graphic analy- 
 sis, and makes mathematics interesting.^ 
 
 1 Wisconsin State Department of Education : Educational News 
 Bulletin, Jan. 1, 1917. 
 
Collection of Data 89 
 
 EXERCISE 
 
 Which of the miscellaneous economies would be applicable to your 
 problem, and just how would you use them? 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 King, W. I. Elements of Statistical Method, Chapters IV-IX. 
 Report of the Committee on Uniform Records and Reports. U. S. 
 
 Bureau of Education Bulletin, 1912, No. 3. 
 Rugg, H. 0. Statistical Methods Applied to Education, Chapters II, 
 
 III. 
 Thorndike, E. L. Mental and Social Measurements, Chapter II. 
 
CHAPTER III 
 
 TECHNICAL METHODS NEEDED IN SCHOOL 
 
 STATISTICS 
 
 I. USUAL VIEWS 
 
 So far we have considered only statistical matters 
 that are plain to any experienced school man. With the 
 suggestions previously given, such a man could success- 
 fully collect statistical data on most of his school prob- 
 lems. But the working up of the data and the proper 
 interpretation of them would be altogether different 
 and much more difficult. He would at once face 
 such questions as these: Does the superintendent need 
 any technical knowledge of statistics? Can he, without 
 such special knowledge, analyze his data and get the 
 really significant things out of them? Or, without such 
 knowledge, can he present these results effectively to the 
 public ? 
 
 The Conservative's View. Attempts to answer these 
 questions have brought forth much nonsense and fruit- 
 less effort. For example, one group of school men take 
 the position that a superintendent needs no special knowl- 
 edge of statistics, simply because there is in their opinion 
 no virtue in statistics. They quote the old statement of 
 Bagehot : " There are three kinds of lies — lies, damned 
 lies, and statistics." Or they reiterate : " Figures don't 
 lie, but liars do figure." Or else they would at least 
 
 90 
 
Technical Methods in School Statistics 91 
 
 agree with the author of a recent article that '' whatever 
 the causes, the fact is that any one who presents his 
 arguments in the form of tables, and his conclusions in 
 dogmatic statements presumably based on the tables, 
 is sure to convince nine tenths of his readers." ^ Con- 
 sequently, superintendents having this viewpoint see no 
 need of any special knowledge of statistics. They think 
 that any effort to do anything unusual with statistical 
 data is simply a waste of time. The absurdity of such an 
 opinion should be evident to any one who has even glanced 
 through the preceding chapters of this book. Professor 
 King states the whole point very strikingly thus : ''To 
 attempt to handle statistics properly without a knowledge 
 of statistical method is only a little less absurd, though 
 vastly more common, than to attempt to build a great 
 steel bridge without a knowledge of trigonometry." ^ . 
 
 The Specialist's View. Another group, composed of 
 scientific educators and statistical experts, advocate 
 either a very thorough course in statistical method or 
 none whatever. They quote Pope on " A little learning 
 is a dangerous thing," etc. Or they say that giving a 
 school man only a little, or very superficial, knowledge of 
 statistics is like putting a razor in the hands of a baby. 
 Or they compare the results of such a procedure with those 
 ensuing when very delicate and expensive machinery is put 
 in the hands of a novice. Such machinery, which would 
 produce wonders if run by a competent man, is, of course, 
 soon ruined by a bungler, and the result produced is very 
 inferior or altogether lacking. Even Professor King says : 
 " The science of statistics, then, is a most useful servant, 
 
 1 "Lies, Damned Lies and Statistics," Unpopular Review, 1915^ 
 Vol. II, 352-353 
 
 2 King, W. I. : Elements of Statistical Method, pp. 37-38 
 
92 School Statistics and Publicity 
 
 but only of great value to those who understand its 
 proper use." ^ 
 
 The Golden Mean. The truth probably lies between 
 these extremes. Even Professor Thorndike, the pioneer 
 in the application of modern statistical method to edu- 
 cational problems, favors this moderate view when he 
 says : '' There is, happily, nothing in the great principles 
 of modern statistical theory but refined common sense, 
 and little in technique resulting from them that general 
 intelligence cannot readily master." ^ Of course, he 
 later says that mathematical gifts and training will be 
 very useful to students of quantitative mental science, 
 but such things are not absolutely necessary for learning 
 the elements of statistical method. 
 
 Observations in other fields also support this " golden 
 mean " view. Often the successful politician, minister, 
 or business man has better practical ways of controlling 
 people and reading human nature than has the expert in 
 the psychology of such matters. It is not an uncommon 
 thing for an experienced public speaker to influence an 
 audience more than a teacher of public speaking could 
 hope to do. We are only saying in other words that 
 common sense and first-hand experience in reading and 
 controlling human minds are as powerful factors in in- 
 fluencing the public as expert knowledge in the mechanics 
 of any art having the same end in view. Is it not reason- 
 able, then, to expect the experienced superintendent 
 with a small amount of statistical theory to outdistance, 
 in practical statistics with the public, the best-trained 
 experts in statistical theory only? 
 
 1 King: oj)- cit., p. 33 
 
 ^ Thorndike, E. L. : Mental and Social Measurements, p. 2 
 
Technical Methods in School Statistics 93 
 
 II. STATISTICAL KNOWLEDGE NEEDED FOR SCHOOL 
 
 SURVEYS 
 
 But another phase of this question arises. The modern 
 superintendent must be able to read and understand 
 school surveys and apply the statistical methods used to 
 his own school problems. How much statistical work 
 does he need for this sort of thing? 
 
 A study of practically every school survey thus far 
 published shows that to meet this requirement the 
 superintendent should, in addition to things previously 
 mentioned, know : 
 
 1. The meaning of these terms : median, average, quartile, range, 
 central tendency, variability, overlapping, and coefficient of varia- 
 bility. 
 
 2. The methods of computing medians, quartiles, averages, and 
 variability so that any fallacies or mistakes arising from poor or false 
 methods may be detected. 
 
 3. How to read and understand tables and graphs. 
 
 4. The principles underlying the theory of good and bad units, and 
 how the unit in the particular study was derived. 
 
 5. The principles underlying the construction of graphs, so that 
 he will not be misled by a badly constructed graph. 
 
 6. How scales are derived, so that he will not be misled by the 
 interpretations made from data measured on these scales. 
 
 However, the citation of survey material as evidence 
 of the relatively small amount of statistical knowledge 
 needed by the superintendent is open to at least one 
 criticism, that the surveys are written for the public, and 
 to be effective must contain little or no technical material. 
 But the proper translation of such facts into popular 
 language presupposes accurate statistical work. And 
 few superintendents could hope to do more than to give 
 the best possible statistical treatment to such problems 
 as are so treated in all our best school surveys to date. 
 
94 School Statistics and Publicity 
 
 III. STATISTICAL KNOWLEDGE NEEDED FOR READING 
 EDUCATIONAL INVESTIGATIONS 
 
 Moreover, for publicity work a good superintendent 
 needs to keep up with recent investigations in education 
 and psychology, as carried on by investigators in the 
 various schools of education and by educational founda- 
 tions. Some of the most valuable of these are not written 
 especially for laymen and require some knowledge of 
 statistical terms and methods to be understood at all. 
 Good abstracts or news items of such studies often employ 
 these terms. The superintendent needs to understand 
 the terms and general processes used, but he need not 
 know how to calculate or employ these terms accurately 
 himself. It would be better if he could so use them, but 
 it is not absolutely necessary. The situation is similar 
 to that in reading for all of us. We must know how to 
 employ, use, and spell correctly certain words. We need 
 to be able to read with sufficient understanding a much 
 larger list of words. But it is not necessary for us to 
 know how to spell these latter words, or how to use 
 them correctly in our speech or writing. 
 
 The superintendent needs to understand the meaning, 
 for reading purposes, of at least the following : 
 
 1. Such terms as : average deviation, standard deviation, mode, 
 probable error, inter-percentile range, correlation, skewness, dis- 
 persion. 
 
 2. The reliability of the various measures of variability. 
 
 3. The effect of different methods of grouping data, on the con- 
 clusions reached. 
 
 4. Some of the common methods of making allowance for the un- 
 reliability of data. 
 
Technical Methods in School Statistics 95 
 
 IV. ILLUSTRATION OF VALUE OF STATISTICAL METHOD 
 TO THE SUPERINTENDENT 
 
 The value of statistical method and presentation to the 
 superintendent may be most clearly presented by a con- 
 crete illustration. Suppose a superintendent wishes to 
 know whether his high school classes are too large or 
 too small for good work. He may take as his standards 
 the pronouncements of colleges or universities, or the 
 actual classes found in high schools that he or some 
 competent person rates as good ones. 
 
 He would, of course, get the enrollments of all the 
 classes in his own high school, say fifty or more. He 
 would get similar figures for twenty other high schools, 
 or more if possible, say for at least a thousand classes. 
 Then he would be confronted with the problem of handling 
 this enormous mass of data so as to bring any clear idea 
 out of it. His data would cover pages and pages. He 
 would have an unwieldy mass of facts that needed sim- 
 plifying. Without proper treatment, his ideas of the 
 whole would be '' decidedly vague and indefinite." He 
 would need some procedure that would enable him to 
 " give clear-cut form to this hazy conception " and to 
 " set objects in their proper perspectives and relation- 
 ship." ' 
 
 Now if he knew the elements of statistical method, he 
 could very shortly summarize his data on a half page, as 
 Professor Bobbitt does in the School Review for October, 
 1915.2 In this article, to which we have previously 
 referred, Professor Bobbitt is making studies of the cost 
 
 1 King, W. I. : Elements of Statistical Method, p. 28 
 
 2 Bobbitt, J. F. : " High School Costs," School Review, 23 : 505- 
 534 
 
96 
 
 School Statistics and Publicity 
 
 of instruction per one thousand student hours in a number 
 of high schools. First, each study, as English, mathe- 
 matics, etc., is worked out separately as shown in Table 
 1, on page 18 of this book. Table 12 is the summarizing 
 table. 
 
 Table 12. Bobbitt Table Showing Sizes of High School 
 Classes by Subjects 
 
 1 
 
 Median 
 No. Pupils 
 
 " Zone of S a 
 
 
 Pupils 
 
 58 
 
 42-88 
 
 32 
 
 28-55 
 
 22 
 
 20-24 
 
 21 
 
 18-24 
 
 21 
 
 17-23 
 
 20 
 
 16-22 
 
 19 
 
 18-25 
 
 19 
 
 15-23 
 
 18 
 
 14-24 
 
 17 
 
 15-20 
 
 17 
 
 14-19 
 
 17 
 
 13-23 
 
 15 
 
 10-21 
 
 14 
 
 12-18 
 
 Music 
 
 Physical Training . . 
 
 English 
 
 Mathematics . . . . 
 History ...... 
 
 Science 
 
 Agriculture . . . . 
 Commercial . . . . 
 
 Drawing 
 
 Modern Languages . . 
 
 Latin 
 
 Household Occupations 
 Normal Training . . 
 Shopwork 
 
 To any one who understood the simplest things about 
 statistics, this table would at a glance disclose such facts 
 as these: In music, half the schools have more than 58 
 pupils in a class, and half have less; half of them have 
 between 42 and 88 pupils; a fourth have less than 42 
 pupils; a fourth have more than 88 pupils. Similar 
 statements would hold for the other subjects down the 
 table. The table would also disclose that the " average 
 classes " are in agriculture and commercial subjects. 
 
I 
 
 Technical Methods in School Statistics 97 
 
 with 19 pupils each as a rule. Half the classes in other 
 subjects have more than this number and half of them 
 have less. The table would show which had more and 
 which less, ranging from music with 58 down to shopwork 
 with 14. A little more inspection would disclose which 
 classes ranged more in their variations from the ''typical" 
 or '' average " class in that subject. All these things, if 
 told in words, would occupy pages and pages of description 
 that would be about as clear and interesting as a real- 
 estate deed to the lot on which the superintendent's home 
 stood. 
 
 The Bobbitt table,^ of course, is infinitely clearer and 
 more forcible than the great masses of data or the long 
 and tedious description could ever be. It has indeed 
 fulfilled " one of the prime objects of statistics." This, 
 according to Professor King,^ is " to give us a bird's-eye 
 view of a large mass of facts, to simplify this extensive 
 and complex array of isolated instances and reduce it to a 
 form which will be comprehensible to the ordinary mind." 
 
 V. STATISTICAL METHOD AS A FORM OF EXPRESSION 
 
 Finally, the superintendent needs statistical method 
 just as he does any other method of effective presentation 
 or expression. A good description of scenery, of an 
 object, a face, etc., always proceeds by giving first a 
 bird's-eye view, or very brief comprehensive sketch, 
 called usually the " fundamental image," or in exposition, 
 " the topic sentence." Then the details are later filled 
 in. The success of the description or explanations depends 
 upon the clearness, brevity, and vividness of the funda- 
 
 1 The writer uses this name because such tables appear to have 
 been first used by Professor J. F. Bobbitt of the University c:f Chicago. 
 
 2 King : op. cit., p. 22 
 
98 School Statistics and Publicity 
 
 mental image, and then upon the extent to which approxi- 
 mately all the important details place themselves clearly 
 under it. Of course, the whole process must not be so 
 mechanical and obvious as to disgust the reader. But 
 without the fundamental image and this procedure, the 
 reader would soon be utterly lost in the details, or he 
 would get so few of them in mind, or in such a disorganized 
 manner, that he would get no clear idea of the whole. 
 And he would not waste any more time in trying to do so. 
 In the same way statistical method forces a mass of 
 numerical data into a form which describes the whole 
 by giving a good fundamental image or picture. At the 
 same time it leaves all the data so grouped and classified 
 that significant points stand out, but both points and 
 minor details of any consequence may all be easily located 
 under the fundamental image. 
 
 If suitable graphic presentations of the statistical results 
 are made, they will have the same advantage that a 
 line drawing has over a photograph. The superintendent 
 without a knowledge of statistics might give in words 
 only a picture that would correspond to a photograph 
 taken without proper focus. This, of course, gives all 
 the details, but blurred. A suitable graph would make 
 the essential facts of the whole and the essential details 
 stand out clearly. It has been found repeatedly by text- 
 book writers for beginners that line drawings emphasizing 
 the essential elements in apparatus, pictures, etc., are 
 preferable to actual photographs of the objects, because 
 the photographs give too many details and so obscure 
 the big things. The superintendent with no knowledge of 
 statistical method could, under the most favorable cir- 
 cumstances, give his reader only a sort of blurred photo- 
 graph of his ideas. And, indeed, he would probably have 
 
Technical Methods in School Statistics 99 
 
 only a blurred picture of them in his own mind. With 
 statistical method he could have in mind a sharp line 
 drawing and give his readers the same kind of picture. 
 The essentials of this method that are of value to him 
 will be taken up in the next chapter. 
 
 EXERCISES 
 
 1. Which of the statistical terms mentioned on page 93 do you 
 think you understand fully? 
 
 2. Which of the statistical processes given on the same page do 
 you think you know how to do ? 
 
 Note : The best way to know whether you understand a term fully 
 is to see whether you can quickly write a clear explanation of it. 
 Similarly, to know whether you can tell how to do a process, see if 
 you can quickly write directions for doing it. 
 
 3. Which of the statistical terms mentioned on page 94 have you 
 come across in your reading, and just what do they mean to you at 
 present ? 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 King, W. I. Elements of Statistical Method, Chapters I-III. 
 Rugg, H. 0. Statistical Methods Applied to Education, Chapter I. 
 Thorndike, E. L. Mental and Social Measurements, Introduction 
 and pages 36-41. 
 
 I 
 
 I 
 
CHAPTER IV 
 
 SCALES, DISTRIBUTION TABLES, AND SURFACES 
 
 OF FREQUENCY 
 
 Thus far we have discussed only the most elementary 
 statistical matters. But we have seen the need of some 
 technical knowledge of statistics, which we shall now 
 proceed to develop. The treatment will include the 
 meanings of the various statistical terms, methods of 
 calculating them, cautions as to their use, and devices for 
 showing them graphically. 
 
 I. SCALES 
 
 Review of Scales. The first essential in all statistical 
 work is to determine the units and scales to be used. It is 
 impossible to collect data profitably until this has been 
 done. For this reason, these terms were discussed at 
 length in connection with the collection of data. If the 
 reader is not familiar with the treatment given there,i he 
 should read it before proceeding with this section. 
 
 For our purposes here it is necessary to understand : 
 
 1. That whenever possible all the measures in a group should be 
 expressed in terms of a definite common measure called a "unit." 
 
 2. That all measures in any group, when arranged in order of size, 
 make a scale going from high to low. 
 
 1 See pp. 43-57 
 100 
 
Scales and Distribution Tables 
 
 101 
 
 $170 
 160 
 150 
 140 
 130 
 120 
 110 
 
 Qi47 
 
 20 
 
 10 
 
 University High 169 
 
 Mishawaka 112 
 
 Elgin »00 Maple Lake, Minn. 100 
 
 Norfolk, Neb. 42 
 
 Bonner Springs 38 
 Junction City, Kaa 33 ^^^^^^^^^^3^ 
 
 Washington, Mo. 41 
 
 Russell, Kan. 34 
 
 Fig. 6. — Device for Representing a Discrete Scale Graphically. 
 
 This graph shows the cost per 1000 student hours in mathematics in certain high 
 schools. (From J. F. Bobbitt, School Review, 23 : 509.) 
 
102 
 
 School Statistics and Publicity 
 
 3. That we must know what a given measure on a scale means, 
 i.e., whether 6 extends from 5.5 to 6.5 or from 6.0 to 6.99 or from 5.95 
 to 6.05, etc. 
 
 4. That we must know whether the scale is discrete (all measures 
 separate or with gaps between) or continuous (measures running into 
 one another and spread out all over the scale). 
 
 It is worth noting that many of the scales the 
 superintendent uses, especially those he makes up for him- 
 self, are discrete in appear- 
 ance but are really used as 
 though they were continu- 
 ous. That is, each separate 
 item is regarded as extend- 
 ing half the distance to 
 the nearest items above and 
 below. 
 
 When the measures are 
 discrete and there is a rela- 
 tively small number of cases, 
 say twenty to thirty, they 
 may be shown in a Bobbitt 
 table. 1 In this kind of table 
 the name of each measure 
 is written in the left-hand 
 column and the size of the 
 measure in the right-hand 
 column. The measures begin 
 with the highest and run 
 to the lowest, thus giving 
 an idea of a scale like 
 that of a thermometer, 
 which goes up from the 
 bottom. 
 18, 96 
 
 -REVERE- 23.50 
 QUINCY- 22.00 
 CHEL3EA-21.40 
 -ARLlNGTON-20.80 
 ICAMBRIDGE- 20.40 
 
 iMELROSE - 20. 40 
 EVERETT- 19.70 
 
 IMALDEN -19.20 
 WINTHROP- 19.20 
 lOMERVILLE- 18.80 
 ELMONT- 18.30 
 WATERTOWN- 18.20 
 MEDFORD- 18.00 
 WINCHESTER- 18.00 
 NEWT ON -\7. 40 
 DEDHAM-17.40 
 
 ^BOSTON- 16-4.0. 
 ^WALTHAM- 15.90 
 BROOKLINE- 12.00 
 MILTON- 11. 50 
 
 Fig. 7. — Thermometer Chart for 
 Presenting a Discrete Scale. 
 
 It shows the total tax on $1000, for the 
 year 1912, in all the cities and towns of 
 the metropolitan district, Boston, Massa- 
 chusetts. Note omission of zero line. 
 
 (From 1912, Newtoji, Massachusetts, 
 School Report, page 113.) 
 
 1 See pp. 
 
Scales and Distribution Tables 
 
 103 
 
 Graphic Presentation of Discrete Scales. A discrete 
 scale is easily presented graphically by various methods. 
 Four of these methods are given here, three of them using 
 the data from the Bobbitt table on page 17. 
 
 1. By vertical scale on left with names of items to the right. 
 Figure 6 shows this. 
 
 Cost per 1000 5. Hours $0 30 6o so iso iso leo 
 
 Name of school. 
 Uo'verai+y High 
 M'Shawaka, Ind. 
 Elgin, III. 
 Maple Lake,Minn 
 Grani+eCity, III. 
 Ea3+ Chicago, Ind. 
 OeKolb.IlL 
 San Antonio, Tex. 
 Harvey, 111. 
 Waukegqn.TII. 
 South Bencl,Ind. 
 East Aurora, III. 
 Rockford, III. 
 Booneville.Mo 
 Brazi(,Ind. 
 Leavenworth, Kans 
 Greens burg. Ind 
 Morgan Park, II I. 
 Noblesville, Ind 
 Norfolk Neb.- 
 Washington, Mo. 
 Bonner Springs.Kans 
 Russell,Kans. 
 Junction Cify, Kans 
 M+Corroll,III 
 
 Fig. 8. — Graphic«Representation of a Bobbitt Table, Histogram Form. 
 The data are from Table 1, page 18. 
 
 2. By thermometer device. 
 
 Superintendent Spaulding in his 1912 report for the schools of 
 Newton, Massachusetts, showed a similar table as a scale on a 
 thermometer. As this is now out of print, the device is here re- 
 produced. It is a very excellent one, save that the zero line is not 
 properly shown. See Figure 7. 
 
 
 
 ! 
 
 
 
 
 $169 
 
 
 
 
 
 
 1 
 
 11^ 
 
 
 
 
 1 
 
 
 
 100 
 
 
 
 
 
 
 100 
 
 
 
 
 
 88 
 
 
 
 1 
 
 
 
 82 
 
 
 
 1 
 
 
 74 
 
 
 
 
 1 
 
 b-y 
 
 
 
 
 
 69 
 
 
 
 
 63 
 
 
 
 J 
 
 
 <62 
 
 
 
 1 
 
 61 
 
 
 
 
 59 
 
 
 J 
 
 
 58 
 
 
 1 
 
 56 
 
 
 
 56 
 
 
 
 54 
 
 
 J 
 
 53 
 
 
 
 52 
 
 
 f 
 
 42 
 
 
 
 41 
 
 
 1 
 
 s 38 
 
 
 1 
 
 34 
 
 
 
 . 33 
 
 
 
 30 
 
 
 
 
 
 
 
 
104 
 
 School Statistics and Publicity 
 
 3. By bars to represent magnitvde. 
 
 The procedure for this is : First, the names of the items (schools 
 in this instance) are placed on the left, the highest at the top. In 
 the next column is placed the corresponding magnitude (cost per 
 1000 student hours in this case). At the top and running out to the 
 
 Cost per 1000 S hrs. ^0 
 
 Name of scbool. 
 UniYersi+y HigW ^169 
 
 AAishawaKQ, Ind- I '2 
 
 l\Q,\r\.W. 100 
 
 Map\e Loke, H\ nn. lOO 
 
 Grani+eCi-Vy,IU. 88 
 
 East ChicQCjclnd. 82 
 
 DeKQ|b,Ill. 74 
 
 5a n An^on\o,Tex. 69 
 
 Harvey. l\l. 69 
 
 Waukec^anAVV. 63 
 
 South Bend, \nd. 62 
 
 South Avjrora,W- 61 
 
 RocKtord.lll- 59 
 
 Boonev'\lle,lAo. 58 
 
 Braz.il, \nd. 56 
 
 Leavenworth, Kanv 56 
 
 6reer>\3or«j,\nd. 5^ 
 
 Morgan ParKUl. 53 
 
 Nob\es\'(\\e, \Y\d. 52 
 
 Norto\W,Neto. 42 
 
 Woshincjton.Mo. 41 
 Bonner Springs, Kan S. dQ 
 
 Rus3e\\,V^ans. 34 
 
 Junction City, Kans. 33 
 
 Mt.Carro\|,\\\. 30 
 
 30 
 
 60 
 
 90 
 
 120 
 
 150 
 
 180 
 
 ^-••^-^—a * ' 
 
 Fig. 9. — Graphic Representation of a Bobbitt Table, Smoothed Curve Form. 
 
 The data are from Table 1, page 18. This is simply the smoothed form of Fig. 8. 
 
 right, the scale is placed (at $10 intervals here). Each distance of 
 five small squares on the scale represents $10 and one small square 
 will thus mean $2. Therefore, to get the bar for the first school, go 
 out on the scale to 170, drop back half a square to get 169, and then 
 construct the bar from the base or zero line to this point. In the 
 
Scales and Distribution Tables 105 
 
 same way the bar for each other school may be constructed. The 
 significance of the bar in such a graph lies in its length only, not in 
 its width. Hence, all bars in the same diagram must be uniform in 
 width. The bars are here drawn adjacent to each other, but there 
 can be spaces between if desired. See Figure 8. 
 
 4. By a '^ curve "drawn with the magnitudes on the vertical 
 axis and the names of the cases running in order from high 
 to low {or vice versa) on the horizontal axis. 
 
 See Figure 9. 
 
 This diagram or graph has been constructed in exactly the same 
 way as the preceding one, except that dots have been placed at the 
 middle of the spaces where the ends of the bars came in the preced- 
 ing diagram, and then joined with a line. The dots were put in 
 faintly and have been covered up by the line. Obviously, then, the 
 "curve" is nothing more than the smoothing down of the corners 
 of the bars. 
 
 EXERCISES 
 
 1. The salaries of school superintendents in Missouri cities be- 
 tween 2500 and 5000 population in 1914-15, for all cities on which 
 data could be obtained, were as follows : 
 
 Booneville, $1,650 — Butler, $1,320 — Cameron, $1,400 — Carter- 
 ville, $1,200 — Caruthersville, $1,200 — Charleston, $1,500 — Clin- 
 ton, $1,920 — De Soto, $1,400 — Excelsior Springs, $1,500 — Far- 
 mington, $1,400 — Fayette, $1,500 — Festus, $1,140 — Frederick- 
 town, $1,450 — Kennett, $1,500 — Kirkwood, $2,400 — Liberty, 
 $1,800 — Louisiana, $1,350 — Macon, $1,700 — Marceline, $1,100 
 
 — Marshall, $2,100 — Maryville, $1,500 — Monette, $1,500 — Rich 
 Hill, $1,200 — Richmond, $1,500 — Sikeston, $1,500 — Slater, $1,200 
 
 — Warrensburg, $1,600 — Washington, $1,170 — West Plains, $1,500. 
 Make up a Bobbitt table to show the status of these salaries, and graph 
 this table in as many ways as you can. 
 
 2. Make up a similar table on superintendents' salaries in a group 
 of cities in some other state that may be legitimately compared, 
 choosing cities of some other size if preferred. 
 
 Statistics on population may be gotten from census reports or a 
 
106 
 
 School Statistics and Publicity 
 
 good almanac like the World Almanac. Figures for salaries may be 
 gotten from directories issued by book companies, from reports of 
 the state superintendent of education, or from " A Comparative Study 
 of the Salaries of Teachers and School Officers" (Bulletin U. S. Bureau 
 of Education, 1915, No. 31). Volume II of the Annual Report of the 
 U. S. Commissioner of Education will furnish figures for the total ex- 
 pense of the superintendent's office but sometimes this contains other 
 items in addition to his salary. In the smaller cities it very closely 
 approximates the superintendent's salary. 
 
 n. DISTRIBUTION TABLES 
 
 In a continuous scale or distribution, it is customary 
 to group measures of magnitudes that are almost the I 
 same in value, and call them by a group name. Thus, 
 in the Ayres spelling scale, all words spelled by from 70 
 to 76 per cent inclusive of children in a given grade 
 are lumped into one group and called 73 in difficulty for 
 that grade. 
 
 It is also customary in such a distribution to make up 
 a table called a " distribution table " or '' table of fre- 
 quency." The table is made up with magnitudes in 
 order of size in the left-hand column and the corresponding 
 numbers of cases or frequencies in a parallel column on 
 the right, the smaller measures preferably being at the 
 bottom. The grouping is for the purpose of condensation 
 and clearness, but the cases can always be kept individually 
 if necessary. Professor Dearborn in a study of grades 
 has kept all the cases of each magnitude separate. For 
 example, he presents the distribution of marks given in 
 English to 69 eighth grade children as shown in Table 
 13.1 
 
 1 Dearborn, W. F. : ** School and University Grades," Bulletin, 
 University of Wisconsin, No. 368, H. S. Series, No. 9, 1910. From 
 Figure 9, p. 25 
 
Scales and Distribution Tables 
 
 107 
 
 Table 13. Distribution Table of Marks of Eighth Grade 
 Children (from Dearborn) 
 
 Mark 
 
 No. Making 
 
 Mark 
 
 No. Making 
 
 100 
 
 
 
 80 
 
 3 
 
 99 
 
 
 
 79 
 
 3 
 
 98 
 
 
 
 78 
 
 3 
 
 97 
 
 2 
 
 77 
 
 2 
 
 96 
 
 
 
 76 
 
 1 
 
 95 
 
 2 
 
 75 
 
 2 
 
 94 
 
 1 
 
 74 
 
 
 
 93 
 
 3 
 
 73 
 
 3 
 
 92 
 
 4 
 
 72 
 
 2 
 
 91 
 
 6 
 
 71 
 
 2 
 
 90 
 
 2 
 
 70 
 
 
 
 89 
 
 3 
 
 69 
 
 
 
 88 
 
 3 
 
 68 
 
 2 
 
 87 
 
 3 
 
 67 
 
 
 
 86 
 
 1 
 
 66 
 
 
 
 85 
 
 2 
 
 65 
 
 1 
 
 84 
 
 3 
 
 64 
 
 1 
 
 83 
 
 2 
 
 63 
 
 
 
 82 
 
 3 
 
 62 
 
 1 
 
 81 
 
 3 
 
 61 
 
 
 
 
 
 60 
 
 
 
 The magnitudes in this table run from 60 to 97. This 
 table is exactly like a Bobbitt table, except that in this 
 instance there are several cases of the same magnitude. 
 But even here these cases are supposed or assumed to run 
 from the lower limit of the magnitude to the lower limit 
 of the next magnitude. That is, the six 91's are not all 
 exactly 91 but are spread from barely 91 up to not 
 quite 92, or from just 90.5 to almost 91.5, depending 
 upon which system of definition of the measure 91 is 
 used. 
 
108 School Statistics and Publicity 
 
 But we know that such grouping as Professor Dearborn 
 has used in this instance is rather finer than we are 
 accustomed to use in examining teachers' marks. 
 
 Let us now make coarser groupings. The first 
 possibiUty that occurs to the reader is probably that of 
 making the groupings cover supposedly equal parts of 
 the scale, say five units. Thus, we may make the groups 
 cover 60-64, 65-69, etc., setting the limits very definitely 
 and getting these groups : 
 
 Marks 
 
 No. Making 
 
 95-100 
 
 4 
 
 90-94 
 
 16 
 
 .85-89 
 
 12 
 
 80-84 
 
 14 
 
 75-79 
 
 11 
 
 70-74 
 
 7 
 
 65-69 
 
 3 
 
 60-64 
 
 2 
 
 
 69 
 
 Or we may make them cover 60-69, 70-79, etc., and 
 
 have: 
 
 Marks No. Making 
 
 90-100 20 
 
 80-89 26 
 
 70-79 18 
 
 60-69 _5 
 
 69 
 
 Or we may group as many school systems do, and have : 
 
 Magnitude 
 
 No. Cases 
 
 95-100 
 
 4 
 
 90-94 
 
 16 
 
 80-89 
 
 26 
 
 70-79 
 
 18 
 
 Below 70 
 
 5 
 
 
 69 
 
Scales and Distribution Tables 109 
 
 For grouping, this from Professor Thorndike should be 
 kept in mind : 
 
 In general, in mental and social measurements, in the calculation 
 of averages, average deviations, and mean square deviations, when 
 the face value of the series gives a grouping of 40 to 60 steps, it is al- 
 lowable to group by double steps, and when the face value of a series 
 gives a grouping of 60 to 80 steps, to group by triple steps. But it 
 should be observed that coarse grouping saves little time except in 
 the calculation of the average, average deviation, and mean square 
 deviation. In the case of the calculation of the median, 25 percentile, 
 75 percentile, and median deviation, it is the author's opinion that 
 the gain in precision from the finer scale is greater than the loss in 
 time, if one economizes time in recording measures in the finer group- 
 ing.i 
 
 The superintendent, however, may not understand the 
 finer points of this without considerable statistical theory 
 and experience. The best simple rule for him to follow 
 is not to divide into small groups where the cases seem to 
 bunch more closely than usual, and not to include in the 
 same group cases that are manifestly far apart. In the 
 example of the marks given above, it will be noted that 
 the cases bunch together very closely when grouped in 
 the third form. Nor are the cases to be found in one 
 group too far apart from one another to be in that group. 
 For instance, 81 and 89 are to be found in the same group. 
 But experience with marks shows us that when one teacher 
 marks one boy 81 and another teacher marks another 
 boy 89 in the same subject, there may be little appreciable 
 difference in the achievements of the boys. 
 
 In most cases common sense and experience must be 
 utilized in considering grouping. Salaries of superintend- 
 ents may be safely grouped by hundreds (1200-1299, 
 1300-1399, etc.) because their salary increases usually 
 
 1 Mental and Social Measurements, p. 50 
 
110 School Statistics and Publicity 
 
 come by hundreds. But the salaries of grade teachers 
 are more profitably grouped by fifties or twenty-fives, 
 thus : 400-449, 450-499, or 400-424, 425^49, 450-474, 
 474-499, etc., because their usual salary increases are 
 covered by the smaller steps. A grouping for training 
 in weeks, of teachers that covered summer school work, 
 would be 1-6, 7-12, 13-18, etc., or 1-3, 4-6, 7-9, 10-12 
 and not 1-4, 5-8, 9-12, because summer schools usually 
 run either 6, 9, or 12 weeks. 
 
 EXERCISES 
 
 1. The salaries of superintendents in cities of 2500-5000 for 1914- 
 15, for all obtainable, were in the states given, as follows : 
 
 Alabama. $1,500 — 1,800 — 1,250 — 1,680 — 1,900 — 1,600 — 
 
 1,800 — 1,500. 
 Arkansas. $2,000 — 1,620 — 1,000 — 1,500 — 1,600 — 1,600 — 
 
 1,500 — 1,100 — 1,500 — 1,200 — 1,500 — 1,500 — 1,600 — 
 
 1,350. 
 Florida. $1,500 — 1,200 — 1,500 — 1,650. 
 Georgia. $1,800 — 1,800 — 1,800 — 2,000 — 1,500 — 1,200 — 
 
 1,500 — 1,500 — 1,200 — 1,600 — 1,800 — 2,000 — 2,000 — 1,650. 
 Kentucky. $1,000 — 1,500 — 1,800 — 1,350 — 1,800 — 1,600 — 
 
 1,800 — 1,400 — 1,200 — 1,400 — 1,650 — 1,200 — 1,400 — 
 
 1,500. 
 Louisiana. $1,500 — 1,800 — 1,500 — 1,800 — 1,500. 
 Maryland. $1,400 — 1,450. 
 
 Mississippi. $1,800 — 2,200 — 1,125 — 1,700 — 1,650 — 1,800. 
 Missouri. $1,650 — 1,320 — 1,400 — 1,200 — 1,200 — 1,500 — 
 
 1,920 — 1,400 — 1,500 — 1,400 — 1,500 — 1,140 — 1,450 — 
 
 1,500 — 2,400 — 1,800 — 1,350 — 1,700 — 1,100 — 2,100 — 
 
 1,500 — 1,200 — 1,500 — 1,500 — 1,200 — 1,600 — 1,170 — 
 
 1,500. 
 North Carolina. $1,500 — 1,500 — 1,200 — 1,500 — 1,200. 
 Oklahoma. $1,500 — 1,500 — 1,400 — 1,800 — 1,800 — 1,500 — 
 
 900 — 2,000 — 1,300 — 1,800 — 1,500 — 1,800 — 1,200 — 1,800 
 
 — 1,300 — 1,500 — 1,800 — 1,800. 
 
Scales and Distribution Tables 111 
 
 South Carolina. $1,200 — 1,500 — 1,215 — 1,500 — 1,350 — 
 1,800 — 1,200 — 1,250 — 1,500 — 2,000. 
 
 Tennessee. $2,000 — 1,000 — 1,200 — 1,080 — 1,200 — 1,600 — 
 1,600 — 1,500 — 1,000 — 1,800. 
 
 Texas. $1,960 — 2,000 — 1,500 — 1,800 — 2,100 — 2,200 — 
 2,000 — 1,800 — 1,500 — 1,800 — 2,000 — 1,200 — 1,500 — 
 1^560 — 1,800 — 1,500 — 1,800 — 1,300 — 1,800 — 2,300 — 
 1,500 — 1,675 — 2,200 — 1,200 — 2,000 — 1,400 — 1,800 — 
 1,500 — 1,500 — 1,500 — 1,800 — 1,800 — 1,800. 
 
 Virginia. $1,750 — 1,200 — 1,200. 
 
 West Virginia. $1,500 — 1,400 — 1,350 — 1,500 — 1,500 — 1,800 
 — 1,500 — 1,380 — 1,600 — 1,550 — 2080. 
 
 Ignore the matter of sampling and arrange these salaries in a distri- 
 bution table, being careful to justify the step chosen in your grouping. 
 
 2. Make a similar distribution table of superintendents' salaries 
 in cities of this size for any other section of the United States, getting 
 your data from the sources found in Exercise 2, page 105, and telling 
 just why you use each step in the process of making the table. 
 
 3. Make a similar distribution table for the following figures on 
 the number of hours required by individual pupils to complete one 
 half-grade in grammar : ^ 
 
 7 — 10 — 11 — 11 — 11—12 — 12 — 13 — 13 — 15 — 16 — 16 
 _ 16 — 17 — 18 — 18 — 19 — 19 — 20 — 20 — 21 — 21 — 22 — 
 22 — 22 — 23 — 23 — 25 — 27 — 29 — 33 — 33 — 33 — 34 — 34 
 _ 36 _ 37 _ 38 — 39 — 40 — 43 — 44 — 44 — 48 — 49 — 49. 
 
 III. SURFACE OF FREQUENCY 
 
 Graphing Distribution Tables. For the presentation of 
 grouped distributions by graphs, three simple devices 
 are available, the histogram or rectangular graph, the 
 smoothed graph, and the check form of the histogram. 
 
 The procedure for this histogram is as follows : 
 
 1. Lay off on cross-section paper a horizontal scale, on which the 
 magnitude scale runs by groups from the lowest magnitude at the left 
 to the highest magnitude at the right. 
 
 1 From Monograph C, Individual Instruction, San Francisco State 
 Normal School, p. 28 
 
112 
 
 School Statistics and Publicity 
 
 2. From the same zero point erect a perpendicular scale which 
 is to represent the number of cases. 
 
 
 
 
 
 
 r— 
 
 
 
 
 
 
 
 
 
 
 < 
 
 
 
 
 
 
 
 25 
 
 
 
 
 
 
 
 
 
 
 
 
 
 20 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 
 y 
 
 
 
 
 
 15 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 : 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 1 
 1 
 
 1 
 
 1 
 
 
 
 
 
 \ 
 
 
 
 
 
 ' 
 
 
 
 
 1 
 
 1 
 1 
 
 
 
 
 
 1 
 1 
 
 1 
 
 
 "No. Below 
 Cas65 70 
 
 70 t079 80 to 89 SO to 84 QSfoWO 
 
 Fig. 10. — Histogram Showing Data of Table 13, but Using Grouping as 
 
 Given on Page 108. 
 
 3. Then find on the horizontal scale the point marking the mag- 
 nitude of any given case, and count up to find the proper point to 
 denote the number of cases in that group. 
 
Scales and Distribution Tables 
 
 113 
 
 Do the same for each group. In so doing, one will get a number of 
 points at different heights strung out above the horizontal scale. 
 
 4. Then proceed to draw a line through these points coming down 
 to the base line on the right, and either coming down to the base line 
 
 
 
 
 
 
 
 
 
 
 
 
 25 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 70 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (5 
 
 
 
 / 
 
 
 
 
 > 
 
 ^ 
 
 
 
 
 
 
 / 
 
 
 
 
 
 \ 
 
 
 
 /O 
 
 
 / 
 
 ' 
 
 
 
 
 
 \ 
 
 
 
 
 / 
 
 
 
 
 
 
 
 \ 
 
 
 5 
 
 
 ^ 
 
 
 
 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 / 
 
 
 
 
 
 
 
 
 
 \ 
 
 Cases 70 
 
 Fig. 11. — Smoothed Form of Graph Shown in Figure 10. 
 
 70 to 79 ao f o 69 00 to 94 9 5 f o 100 
 
 on the left or going to the vertical scale on that side. There will thus 
 be inclosed an area which is called the "surface of frequency." 
 
 This surface may be made by making each point located after the 
 
114 School Statistics and Publicity 
 
 manner described above, the upper left-hand corner of a rectangle 
 which is as wide as the length of the space occupied by that group on 
 the horizontal scale. Thus the plotting of the Dearborn data as a 
 histogram is shown in Figure 10. 
 
 It is not customary to draw those parts of each rectangle 
 shared in common with other rectangles. Common por- 
 tions in the diagram are shown by dotted lines. 
 
 In the smoothed graph (Figure 11), the points located 
 to determine the rectangles may represent the middles 
 of the tops of such rectangles instead of the upper left- 
 hand corners. Then these points may be joined by straight 
 lines, giving a surface with apexes. This is somewhat 
 " smoothed," it will be noticed. 
 
 The check form of the histogram simply uses dots on 
 cross-section paper, one for each item, thus keeping the 
 columns the same width. There is no line drawn above, 
 the columns showing roughly the shape of the surface. 
 It is a very valuable form for tabulating data and at 
 the same time showing the shape of the surface of fre- 
 quency. That is, it may be made up before the dis- 
 tribution table. Thus, if cross-section paper had been 
 used for the. Dearborn data at the outset, a surface of 
 frequency like that in Figure 12 could have been obtained, 
 and from this surface, the distribution table given on page 
 107 could have been easily made up. 
 
 In graphing a distribution table, the scale in which the 
 items come regularly is always put on the base line. 
 The scale in which the items come more or less irregularly 
 is put on the vertical line. In other words, this means 
 that magnitude is measured on the horizontal line of the 
 graph and the number of cases is shown on the vertical 
 line. There is no bullet-proof reason for this ; it is simply 
 the conventional way of doing the thing, just as the order 
 
Scales and Distribution Tables 
 
 115 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 t3 
 O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 QO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 K« 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 lO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 ~ 
 
 — 
 
 «^ 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 CM 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 ~" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 "~" 
 
 
 <n 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 t- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 <0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 «o 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 "t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 ro 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 rvJ 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 ""* 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 O 
 flO 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 O) 
 
 
 
 
 
 
 
 
 
 
 
 
 «. 
 
 •— 
 
 — 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — . 
 
 f- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 <0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 ~ 
 
 ir» 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 CJ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 "* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 h- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <d 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 tn 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 CM 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a 
 
 o 
 
 
 o . 
 
 
 OS (H 
 
 
 V o 
 
 
 
 
 ^S 
 
 
 03 •-< 
 
 
 -t-j 
 
 
 O ^ 
 
 
 3co 
 
 CO 
 i-H 
 
 ^03 t}2 
 
 Eh ^ 
 
 3 
 
 _. »-* 
 
 Sj 
 
 HO 
 
 H 
 
 •43 fl 
 
 a 
 
 P o 
 
 o 
 
 -Q w 
 
 y-i 
 
 •S ^ 
 
 a> 
 
 4^ W 
 
 ^ 
 
 03 (S3 
 
 c4 
 
 «^ 
 
 1^ 
 
 +3 
 
 bD 
 
 rt 
 
 M.S 
 
 ■73 
 
 
 1 
 
 (D r£l 
 
 
 CO 
 
 
 (H >i 
 
 
 o-o 
 
 
 to 
 
 f*< 
 
116 School Statistics and Publicity 
 
 from left to right across the page and from top to bottom 
 in reading and writing is the proper and customary one 
 for European people and their descendants. One should 
 draw pictures or graphs of data to be read with as much 
 care as he would exercise in preparing the manuscript of 
 an article to be printed. ^ 
 
 Characteristics of Surface of Frequency. The dis- 
 tribution table is a great economy over a miscellaneous 
 mass of unassorted data. But it is too cumbersome to 
 be kept in mind in all its details. We need to apply here 
 our idea of the fundamental image or bird's-eye view of 
 the whole. This can best be done through the use of 
 two special qualifications, characteristics, or earmarks of 
 the distribution. The first of these qualities is that 
 measure which indicates the typical, average, or central 
 size of the group. The second is that number which 
 indicates how far the other members of the group on the 
 average vary, spread, or deviate from the first-named 
 quality. These two characteristics of the distribution 
 table serve to make it full of meaning to any person who 
 understands statistics; and with a little care the lay 
 reader may readily acquire the ideas back of these 
 devices. 
 
 Both of these characteristics may be expressed as magni- 
 tudes or as so many multiples of whatever unit may be 
 used in the distribution table. They may also be shown 
 graphically on the surface of frequency. The particular 
 kind of central tendency to be used and the measure from 
 it depend altogether upon the shape of this surface of 
 frequency. The question at once comes up, then: 
 How many variations in the surface of frequency are of 
 
 1 Paraphrased from W. C. Brinton : Graphic Methods of Present- 
 ing Facts 
 
Scales and Distribution Tables 117 
 
 significance to the superintendent, and how may he know 
 them? 
 
 Normal Surface of Frequency. If the cases in any 
 distribution are taken by chance or by a combination of 
 causes that amount to chance, the shape of the surface 
 of frequency invariably becomes bell-shaped with symmet- 
 
 Percenf 
 inaKirig 
 given score 
 
 Lowesf Highest 
 
 score score 
 
 Fig. 13. — Example of Normal Surface of Frequency. This graph shows 
 the per cent of pupils attaining given scores in Stone Reasoning Tests, all 
 pupils being tested. (Adapted from Butte Survey, page 95.) 
 
 rical sides. Figures 13, 14, and 15 are good examples 
 because all the children were taken, and the variations 
 then arise only from chance. 
 
 Notice that in one of these graphs or diagrams the cases 
 composing it are bunched much more closely around the 
 highest point or apex of the distribution than in the 
 second diagram. But it might sometimes happen that 
 two different sets of children, when tested on the same 
 thing, might give distribution tables which, when graphed, 
 would make graphs or surfaces of frequency varying as 
 widely as these two. For example two sixth grades 
 might make the same average or central tendency on the 
 Courtis Tests. But in one case the achievements of all 
 
118 
 
 School Statistics and Publicity 
 
 the children might be close to the average, while in the 
 second case some of the children might make very high 
 records and others very low. This bell-shaped surface is 
 called the " normal " or " probability " surface because 
 it is the one found in natural and mental phenomena of 
 all kinds, when the distributions are made up from un- 
 
 5043 
 
 4462 
 
 3536 
 
 2878 
 
 1373 
 
 452 
 
 2 59 
 
 3536 
 
 1884 
 
 1225 
 
 527 
 
 399 
 
 10 20 30 40 50 60 70 80 90 100 110 120 
 to to to to to to to to to to to to to 
 9 19 29 39 49 59 69 79 89 99 109 119 129 
 
 Fig. 14. — Example of Normal Surface of Frequency. This figure shows 
 the number of pupils writing at each speed from to 9 letters per minute 
 to 120 to 129 letters per minute. Data for 25,387 pupils in four upper 
 grades of Cleveland. (From Measuring the Work of Public Schools, Cleve- 
 land Survey, page 67, by permission.) 
 
 selected cases or from those picked at random. In other 
 words, it is the normal or probable thing to expect from 
 such data. 
 
 Skew Surface of Frequency. But there are some 
 distributions in which the cases bunch much more on one 
 side of the apex than on the other, for the reason that a 
 certain cause or combination of causes operates on some 
 of the cases in the distribution but not on all. The 
 
Scales and Distribution Tables 119 
 
 distribution or surface is said to be " skewed " toward 
 the thin, drawn-out side and is called a '' skew '' dis- 
 tribution or '' skew " surface. ^ A good example of such 
 a distribution is one made up of the number of children 
 of different ages in school. As the children grow older, 
 some die, and consequently the group of a given age is 
 
 5000- - 
 
 Fig. 15. — Smoothed Form of Figure 14. 
 
 slightly smaller in size than any group of lower age. But 
 at the age of fourteen the compulsory education laws 
 usually cease to operate, and many children immediately 
 drop out of school. Others want to go to work. Such 
 forces tend to decrease the group rapidly. The follow- 
 ing table of frequency for 1908 for Nashville, Tennessee, 
 as reported by Stray er in his " Age and Grade Census of 
 
 1 From the verb "skew," meaning to put askew or twist to one side. 
 While there is perfect agreement among educational writers as to what 
 constitutes a skew distribution, there is no agreement as to which is 
 the skew end. By this term some writers mean the blunt end and 
 some the sharp end. The author uses it to describe the sharp end, 
 following Professor Thorndike. 
 
120 
 
 School Statistics and Publicity 
 
 Schools and Colleges," ^ well illustrates the point under 
 discussion : 
 
 Table 14. Distribution by Ages of Pupils in the Nashville 
 
 Schools, 1908 
 
 \ 
 
 Age 
 
 Number in group 
 
 
 72 
 
 1493 
 
 
 8 
 
 1733 
 
 
 9 
 
 1584 
 
 
 10 
 
 1712 
 
 
 11 
 
 1595 
 
 
 12 
 
 1626 
 
 
 13 
 
 1490 
 
 
 14 
 
 1198 
 
 
 15 
 
 885 
 
 
 16 
 
 528 
 
 
 17 and over 
 
 390 
 
 Notice that there is practically no increase or decrease 
 of importance in these groups until the age of fourteen 
 is reached. From this point on, the decrease is very 
 rapid. If the ages beyond seventeen had been given sep- 
 arately, the extension would narrow down to a very 
 slender one. Figure 16 shows the facts in the histogram 
 or unsmoothed form. 
 
 Smoothed out, as in Figure 17, the graph shows the 
 " skew " even better. 
 
 A good example of a skew surface is shown in the results 
 of the spelling test in Figure 18. 
 
 1 Bulletin U. S. Bureau of Education, 1911, No. 5, p. 34 
 ' Children are not permitted to enter the Nashville schools until 
 they are seven years of age. 
 
Scales and Distribution Tables 
 
 121 
 
 < 1 
 
 Mo, of 
 
 pupils 
 
 1800 
 1600 
 
 1400 
 /200 
 1000 
 600 
 600 
 400 
 
 ^00 
 
 A36S 6 7 8 10 II 12 13 14 15 \Q 17 and over 
 
 Fig. 16. — Skewed Histogram Representing Distribution of Pupils by 
 Ages in Nashville Public Schools, 1908. (From data in Table 14.) 
 
 No. of 
 pupils 
 1800 
 
 1600 
 
 1400 
 1200 
 
 (000 
 800 
 600 
 400 
 200 
 
 Ages 6 7 8 S 10 II 12 13 14 15 16 17 and over 
 Fig. 17. — Smoothed Form of Graph Given in Figure 16. 
 
 -V 
 
 1 — I ^ 
 
 i I I I I I I I I I 
 
122 
 
 fo Of children 
 40 A 
 
 School Statistics and Publicity 
 
 M91 
 
 30- 
 
 20- 
 
 10- 
 
 ENTIRE CITY 
 3988 cinildren 
 
 I 
 
 10 20 30 40 50 60 -70 80 90 100 
 
 Fig. 18. — Skewed Histogram Showing the Percentage of Children 
 Attaining Each of the Possible Scores on the Spelling Test in Salt Lake City 
 as a Whole. (Adapted from Salt Lake City Survey, page 135.) 
 
 EXERCISES 
 
 1. Draw a surface of frequency for each of the distribution tables 
 used or gotten up by you in the previous exercises. 
 
 2. Draw a surface of frequency for each of the distributions given 
 in the table on page 123. 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 King, W. I. Elements of Statistical Method, Chapters V, XI. 
 Rugg, H. 0. Statistical Methods Applied to Education, Chapters 
 
 IV, VII, VIII. 
 Thorndike, E. L. Mental and Social Measurements, Chapter II and 
 
 pages 28-36. 
 
Scales and Distribution Tables 
 
 123 
 
 'Frequency of the Different Percentages of Boys and 
 
 Girls Retarded Two Years in Certain Cities of 25,000 
 
 Population and Over, 1908 ^ 
 
 Per cent of total 
 no. of hoys 
 
 No. cities 
 
 Per cent of total 
 no. of girls 
 
 No. cities 
 
 2 
 
 1 
 
 2 
 
 3 
 
 3 
 
 5 
 
 3 
 
 6 
 
 4 
 
 3 
 
 4 
 
 9 
 
 5 
 
 7 
 
 5 
 
 9 
 
 6 
 
 7 
 
 6 
 
 9 
 
 7 
 
 4 
 
 7 
 
 12 
 
 8 
 
 9 
 
 8 
 
 18 
 
 9 
 
 16 
 
 9 
 
 15 
 
 10 
 
 12 64 
 
 10 
 
 13 
 
 ! 11 
 
 12 
 
 11 
 
 11 
 
 12 
 
 19 56 
 
 12 
 
 7 
 
 13 
 
 11 
 
 13 
 
 3 
 
 14 
 
 9 
 
 14 
 
 2 
 
 15 
 
 2 
 
 15 
 
 5 
 
 16 
 
 2 
 
 16 
 
 3 
 
 17 
 
 1 
 
 17 
 
 3 
 
 18 
 
 7 
 
 18 
 
 2 
 
 19 
 
 1 
 
 19 
 
 2 
 
 20 
 
 2 
 
 21 
 
 1 
 
 21 
 
 1 
 
 
 
 22 
 
 1 
 
 
 
 1 From Strayer, G. D. : "Age and Grade Census of Schools and 
 Colleges," Bulletin U. S. Bureau of Education, 1911, No. 5, pp. 86-87 
 
CHAPTER V 
 
 MEASURES OF TYPE 
 
 There are three measures of the " type " or central 
 tendency of importance in school work — the mode, the 
 median, and the average — which will now be discussed 
 in order. 
 
 I. THE MODE 
 
 Definition. The mode is that number which represents 
 the size of the most numerous item or items in a group. 
 That is, it is the vogue or fashion in the cases, because 
 there are more of this size than of any other. The mode 
 is precisely what the ordinary man usually has in mind 
 when he speaks of the '' average.'' He is referring to that 
 measure which includes the greatest number of cases. If 
 he says that teachers instruct forty children on the 
 average, he means that more teachers instruct just about 
 forty children than teach thirty, fifty, or any number 
 far removed from forty. 
 
 Graphic Representation. Graphically, the mode is 
 the magnitude represented by the point on the scale 
 above which the surface of frequency is highest. It may 
 be marked by a perpendicular erected from this point 
 to the apex of the surface. But note that the mode is a 
 measure, not a number of cases. 
 
 Calculation. All that is necessary here is to pick out 
 the group containing the largest number of cases and 
 
 124 
 
Measures of Type 
 
 125 
 
 see what its magnitude is. If several adjoining groups 
 have about the same number of cases in them, they 
 should be run together into larger groups. Usually this 
 procedure will give a more pronounced mode, which is 
 its purpose. In case two widely separated groups are 
 larger than the others, the distribution is said to be " bi- 
 
 Number receiving 
 
 1251 
 
 iOO 
 
 15 
 
 50 
 
 25 
 
 n 
 
 ^ 
 
 
 ^^ 
 
 ,0 (O 
 
 o 
 
 " 0) — > >- V. 
 
 
 o O 
 
 ^ <x> 
 
 o 
 
 0; 
 
 Salary 
 
 C\J 
 
 C 
 
 Q> 
 
 CO 
 
 
 O 
 
 ^ooo o ooO(io oo ooSoo 
 
 Fig. 19. — Example of Multi-modal Surface of Frequency. 
 
 This shows the distribution of salaries paid elementary school teachers, Salt Lake 
 City, 1914-15. Note that the printing on this graph cannot be read easily from one 
 position. (Adapted from Salt Lake City Survey, page 51.) 
 
 modal, ^' that is, it has two modes, and probably is 
 composed of two rather different classes which have been 
 lumped together but which possibly should not be so 
 considered. A frequency table showing the number of 
 teachers getting different salaries will often show more 
 than one mode. For example, the surface of frequency 
 
126 School Statistics and Publicity 
 
 for the salaries of elementary teachers in the Salt Lake 
 City Survey has three well-defined modes at approximately 
 $650, $850, and $1020. This means that, as regards 
 salaries, there are really three distinct classes of teachers 
 within the whole group. The graph is here shown in 
 Figure 19. 
 
 Advantages of the Mode for School Statistics. 1. The 
 mode is useful where it is desirable to eliminate extreme 
 variations. 
 
 For example, the amount of work a given group of children can do 
 in a school year is determined by the modal attendance of the group, 
 not by that of the few who are absent almost continuously, or that 
 of the small number who never miss a day. 
 
 2. In finding the mode, it is unnecessary to know 
 anything about extreme cases except that they are few 
 in number. 
 
 In comparing his school with other schools, the superintendent 
 need not worry about the one or two schools that are higher than any 
 of the others, if his own school falls in the mode or close to it. The 
 extreme cases may not be measured accurately, and they may or may 
 not really come legitimately into the distribution. But whether 
 they do or not, they cannot affect the mode. 
 
 3. It is very easy to determine with considerable 
 accuracy from well-selected data. 
 
 4. It is the best measure of type to the ordinary mind. 
 
 As before indicated, this is what the ordinary man often means by 
 "average." 
 
 5. It is unambiguous. 
 
 No one ever thinks from it that all the measures in the group are 
 practically on it. 
 
 6. The mode is often the most typical measure of a 
 skew distribution. 
 
Measures of Type 127 
 
 Probably the most significant thing about a frequency table of 
 teachers' salaries is that largest group which get the same salary or 
 a salary within certain limits. The extreme salaries, the median 
 salary, or the average salary might be of no especial significance. But 
 the modal salary would point out the significant group at once. The 
 three modal salaries for the Salt Lake City elementary teachers, as 
 shown in the graph on page 125, indicate the significant salaries at a 
 glance. 
 
 Disadvantages of the Mode for School Statistics. 1. In 
 
 many groups, no single, well-defined type actually exists. 
 
 There is no such thing as a modal age for children or a modal num- 
 ber of children in a grade in school. All the age groups up to 14 are 
 about the same in size, and all the grades up to about the sixth or 
 seventh keep about the same size. Of course children drop out but 
 usually enough are held over to make the grades approximately the 
 same size. 
 
 When all cases are kept separate as in Bobbitt tables, there is, of 
 course, no mode unless several cases happen to be of exactly the same 
 size or are considered to be of the same size. 
 
 2. The mode is of no value if weight is to be given to 
 extreme cases. 
 
 It would take no special account of the high per capita cost of a 
 city at the upper end of a per capita group, so far as the size of the 
 item was concerned, although such city might admittedly have the 
 best schools in the group. Similarly it would take no special note of 
 the lowest city in the group although it might admittedly have the 
 worst schools in the group. 
 
 3. The mode cannot be determined by any simple 
 arithmetical process and is sometimes difficult to get by 
 any method. 
 
 4. The product of the mode by the number of items 
 does not give the correct total. 
 
 For example, take Table 15. 
 
128 School Statistics and Publicity 
 
 Table 15. Distribution Table Showing Penmanship 
 Records of Second Grade at Butte ^ 
 
 Score ■ Number pupils making 
 
 4 5 
 
 5 22 
 
 6 21 
 
 7 29 
 
 8 28 
 
 9 42- 
 
 10 7 
 
 11 29 
 
 12 5 
 
 13 7 
 16 _1 
 
 196 
 
 The mode in this example is 9. 196 X 9 = 1764. The sum of 
 the products of the cases in each group by its score, however, is only 
 1617. Such a total sometimes proves very useful for checking other 
 steps. 
 
 5. The mode may be determined by a very few items 
 in case none of the groups contains more than a few 
 items. 
 
 This, of course, may be offset by wider grouping. 
 
 EXERCISES 
 
 1. What is the mode in each of the distribution tables used in pre- 
 vious exercises ? 
 
 2. Draw the line to represent the position of the mode on each of 
 the surfaces of frequency used in previous exercises. 
 
 II. THE MEDIAN 
 
 Definition. The median is the magnitude represented 
 by the mid-point on a scale or distribution. Obviously, 
 half the cases fall below this mid-point and half above it. 
 
 1 Butte Survey, p. 80 
 
Measures of Type 129 
 
 Note that the median is a magnitude or size of a case, 
 not the number of the case. 
 
 Calculation. Various devices and formulas have been 
 given for calculating the median, according as there is an 
 odd number of cases, an even number of cases, a gap 
 between two groups that are equal in size, etc. But the 
 simplest and surest plan hy far is to regard the distribution 
 as a scale and always to find the magnitude of the mid-point 
 on it. 
 
 If there is an odd number of cases, the median is, of 
 course, where the mid-point of the middle case lies. If 
 the median falls in a gap, the mid-point in the gap must 
 be taken. If the median falls in a group distributed 
 over part of the scale, one must run up the part covered 
 by this group until he finds the point that will exactly 
 place half the cases in the whole distribution below it 
 and half above it, splitting a case into halves if it is 
 necessary. 
 
 The essential thing is to find the mid-point. The mag- 
 nitude denoted by this mid-point will be the exact median. 
 The magnitude corresponding to the group containing the 
 mid-point will be the approximate median. This mid- 
 point method of calculation will now be illustrated with 
 various examples, starting with an even number of cases. 
 
 If the distribution is a discrete one and contains an 
 even number of cases, the median falls between the two 
 middle cases. The place for it to fall is found by dividing 
 the number of cases by 2, which gives the number of 
 cases to have on one side. Then count in from one end 
 till this number of cases has been checked off. 
 
 For instance, suppose we make up a Bobbitt table with an even 
 number of cases by taking only the first twelve cases from the table 
 on page 18, as in Table 16. 
 
130 
 
 School Statistics and Publicity 
 
 Table 16. Bobbitt Table Showing Cost of Instruction per 
 1000 Student Hours (Mathematics) 
 
 Name of school 
 
 University High 
 Mishawaka, Ind. . 
 
 Elgin, 111 
 
 Maple Lake, Minn. 
 Granite City, 111. . 
 East Chicago, Ind. 
 
 De Kalb, 111. ^ 
 San Antonio, Tex. . 
 Harvey, 111. . . . 
 Waukegan, 111. . . 
 South Bend, Ind. . 
 East Aurora, 111. 
 
 Cost per 1000 
 student hours 
 
 $169 
 112 
 100 
 100 
 
 88 
 
 82 
 
 74 
 69 
 69 
 63 
 62 
 61 
 
 The mid-point of these twelve cases will obviously have to throw six 
 cases on each side of it, that is, it must come between cases six and 
 seven. For those who desire a formula, this will be found by dividing 
 the number of cases by 2. The mid-point, then, is at the magnitude 
 halfway between $82 and $74 or at $78. ($82 - $74 = $8. h of $8 
 = $4. $74 + $4 = $78.) The same result, of course, could be 
 obtained by merely taking the average of the two middle cases. 
 ($74 + $82 = $156. $156 ^ 2 = $78.) 
 
 If the two cases between which the median in a Bobbitt table falls 
 are the same size, as in the Bobbitt table of size of classes on page 96, 
 the median, of course, is represented by the size of either case, — 19 in 
 this instance. 
 
 For an example of a continuous series and even number 
 of cases, take the achievements of the eighth grade in 
 composition at Butte. ^ 
 
 These may be adapted for our purposes as follows, paying attention 
 for the present to the two left-hand columns only : 
 
 1 Butte Survey, p. 74 
 
Measures of Type 131 
 
 Rated at 
 
 Number of ; 
 
 papers 
 
 7 
 
 2 
 
 
 6 
 
 6 
 
 (8) Adding down 
 
 5 
 
 22 
 
 (30) 
 
 i4 
 
 43 
 
 (73) 
 
 ^^3 
 
 39 
 
 
 ^ 
 
 32 
 
 (42) 
 
 1 
 
 9 
 
 (10) Adding up 
 
 
 
 1 
 
 
 2 )154 
 
 77 
 
 There are 154 cases in all. The median, then, must be at the point 
 which will throw 77 cases on one side and 77 cases on the other. This 
 point will obviously be at the end of the 77th case or the beginning of 
 the 78th case, as one prefers to call it. Say that this point will be 
 located at the end of the 77th case. Counting up we find 42 cases in 
 steps 0, 1, and 2. If the 39 cases in step 3 are added, we find that we 
 have more than the required 77 cases. Subtracting 42 from 77 we 
 find that we must have 35 cases more. That is, we must add 35 cases 
 from group 3 to the other 42 so as to reach the end of the 77th case. 
 This means that the median is located |f of the distance up the scale 
 represented by the step 3. (ff = .90) 
 
 But does step 3 extend from 3 to 4, or from 2.5 to 3.5? If it ex- 
 tends from 3 to 4, the median is obviously 3.90. (3 + .90 = 3.90) 
 In the Butte scoring, however, the latter method was used, and the 
 actual values on the Hillegas scale are as follows : 
 
 is 3 is 3.69 
 
 1 is 1.83 4 is 4.74 
 
 2 is 2.60 5 is 5.85, etc. 
 
 With this in mind, we must use the true measures on the Hillegas 
 scale. Step 3 would extend from the halfway point between 2.60 
 and 3.69 (or 3.145) and the mid-point between 3.69 and 4.74 (or 
 4.215). The distance between 3.145 and 4.215 is 1.07. f| of 1.07 
 is .96. Adding this .96 to 3.145 we get the median or mid-point, 4.105. 
 The median may also be figured from the other end. The pro- 
 cedure is the same, the figures this time being as follows. Counting 
 down we find that from quality 4 up we have 73 cases. We need to 
 take 4 cases from the upper part of the group rated 3. That is, we 
 must go down /^ of the part covered by that group, (/j = .10) If 
 
132 School Statistics and Publicity 
 
 the step meant 3 to 4, the median would then be 4 — .10 or 3.9. But 
 as we saw before, this group covers 1.07 and extends up to 4.215. 
 3% of 1.07 is .11. Then 4.215 - .11 gives 4.105 for the median. 
 
 If the distribution is a discrete one and contains an odd 
 number of cases, the magnitude of the middle case is the 
 median. This is easily found by counting in, usually 
 adding 1 to the number of cases and dividing by 2 to 
 get the number of the middle case. 
 
 Thus, in the discrete series represented by the first two columns 
 of the table on page 51, the median real wealth behind each $1 for 
 schools is $234, because it is the eighteenth case on the scale. There 
 are 17 cases below and 17 cases above. Note that 18, the number 
 of case wanted, is found by adding 1 to the total number of cases (35) 
 and dividing by 2. (1 + 35 -^ 2 = 18.) In the Bobbitt table on page 
 18, the median is represented by the thirteenth case. (13 = 1+25 -^2.) 
 Note the horizontal lines inclosing the median. 
 
 If, however, the series is a continuous one and has an 
 odd number of cases, the median is manifestly located at 
 the mid-point of the middle case. In the discrete series, 
 we took the whole middle case for the median. In the 
 continuous series, the middle case is itself supposed to be 
 spread out along the scale, and consequently we have to 
 find its mid-point. 
 
 For example, let us take the median for the fifth grade composi- 
 tion scores at Butte.^ These were as follows : 
 
 Rated at 
 5 
 
 Number making 
 
 1 
 
 4 
 3 
 2 
 
 18 
 49 
 86 
 
 (19) Adding down 
 (68) 
 
 1 
 
 
 
 46 
 
 1 
 
 2)201 
 
 (47) Adding up 
 
 100.5 
 Butte Survey, page 74 
 
Measures of Type 
 
 This time we have an odd number of cases. The median will fall 
 on that point where 100| cases come on either side, that is, in the 
 middle of the 101st case. In steps and 1, we have 47 cases. We 
 need 53 § cases out of the 86 in step 2 to find our halfway place. Chang- 
 ing step 2 to the Hillegas value as before, we find that it extends from 
 the mid-point between 1.83 and 2.60 to the mid-point between 2.60 
 
 and 3.69, that is, from 2.215 to 3.145, the distance being .93. ^^ 
 
 86 
 of .93 is .58. 2.215 plus .58 makes 2.795, the median. 
 
 Coming down, we find that we have 68 cases from 3 up. To get 
 
 100| cases, we must take 32^ cases from the upper end of group 2. 
 
 Figuring as before, ^ of 93 = .35. 3.145 - .35 gives us 2.795 
 86 
 
 for the median, the same result as before. 
 
 Some books give rules for finding the median which 
 involve finding the middle case. The middle case does 
 represent the median in that its magnitude is the median. 
 But there is danger in using the formula of adding 1 to 
 the total number of cases and dividing by 2 to get the 
 middle case. The danger lies in the tendency to add the 
 whole of the middle case to the part taken from the 
 group. Thus in the example preceding, a beginner is apt 
 to take fl of step 2 to add to the lower limit of that 
 step, or to take || from the upper limit if he is coming 
 down. The former procedure really puts 101 cases below 
 and only 100 cases above the median calculated. The 
 other procedure puts 101 cases above and only 100 cases be- 
 low the median calculated. Obviously, two different re- 
 sults would come from these two calculations. That is, 
 the median calculated coming down would not agree with 
 the median calculated coming up. The mid-point plan, 
 however, will give the same result from either end and 
 is consequently safer. 
 
 In order to emphasize the fact that the median is best 
 located by taking the mid-point on the scale or dis- 
 
134 School Statistics and Publicity 
 
 tribution, irrespective of whether the number of cases is 
 odd or even, let us take several other examples. 
 
 In 1916, the Courtis Tests were given in certain schools in a western 
 city, with the following results for the number of problems attempted 
 by each eighth grade child in one process : 
 
 No. 
 
 problems attempted 
 
 No. attempting 
 
 24 
 
 5 
 
 
 23 
 
 2 
 
 (7) Adding down 
 
 22 
 
 2 
 
 (9) 
 
 21 
 
 1 
 
 (10) 
 
 20 
 
 5 
 
 (15) 
 
 19 
 
 5 
 
 (20) 
 
 18 
 
 5 
 
 (25) 
 
 17 
 
 7 
 
 (32) 
 
 16 
 
 13 
 
 (45) 
 
 15 
 
 15 
 
 (60) 
 
 14 
 
 10 
 
 (70) 
 
 13 
 
 15 
 
 (85) 
 
 12 
 
 24 
 
 (109) 
 
 11 
 
 22 
 
 
 10 
 
 26 
 
 (88) 
 
 9 
 
 21 
 
 (62) 
 
 8 
 
 19 
 
 (41) 
 
 7 
 
 9 
 
 (22) 
 
 6 
 
 8 
 
 (13) 
 
 5 
 
 3 
 
 (5) 
 
 4 
 
 1 
 
 (2) Adding up 
 
 3 
 
 1 
 2)219 
 109.5 
 
 
 There are 219 cases. Therefore, the median will fall in the middle of 
 the 110th case, since there must be 109 § cases on either side of it. 
 Steps 3 to 10 inclusive take in 88 cases. Subtracting 88 from 109.5, 
 we find that we must have 21.5 cases out of the next step. There are 
 22 cases in the step. Therefore, ^ of the distance of this step, 
 which is 1, is .98. By the nature of the Courtis Tests, however, step 
 
Measures of Type 135 
 
 11 extends from 11 to 12o Therefore, the median is 11 + .98 
 = 11.98. Figuring down, we would have ^ of 1 = .02 to be taken 
 
 from the upper limit of step 11, which is 12. 12 — .02 = 11.98 for 
 our median, the same result as before. 
 
 Suppose, now, that we manipulate the data in the above problem 
 so that step 11 contains only 1 case instead of 22, the remaining 21 
 being given to step 10, making its total 47. The total of all the 
 cases is unchanged. We find, however, that steps 3 to 10 inclusive 
 now contain 109 cases, and the median must fall in the 11th step, 
 which has only one case. It will, therefore, fall in the middle of the 
 step, since the one case must be considered as extending over the whole 
 space in the step. The median, therefore, in this supposed case would 
 be 11.5. 
 
 In order to show the working of the same principle in the case of 
 the fifth grade composition results at Butte, given above, let us ma- 
 nipulate the data until they appear as follows : 
 
 Rated at 
 
 No. Papers 
 
 5 
 
 23 
 
 4 
 
 28 (51) Adding down 
 
 3 
 
 49 (100) 
 
 2 
 
 1 
 
 1 
 
 46 (100) Adding up 
 
 
 
 54 
 
 
 2)201 
 
 
 100.5 
 
 The total number of cases remains the same, and the median would 
 fall as above, namely, at the middle of the 101st case, or at the 100| 
 case. Steps and 1 now together contain 100 cases. Therefore, h 
 case must be taken from the next step. As there is only one case in 
 this step, the median will fall at the mid-point of this step, for the same 
 reason as stated in the solution of the problem just above this one. 
 The distance covered by step 2 on the Hillegas scale, as shown before, 
 is .93. One-half of this is .465. Add .465 to 2.215 (the lower limit 
 of the step) and we have for the median, 2.68. 
 
 Suppose, however, that we have an even number of cases, and the 
 median falls in a step containing cases, that is, in a gap in the dis- 
 tribution. For example, let the data used in the last problem be ma- 
 nipulated so as to appear thus : 
 
136 School Statistics and Publicity 
 
 ed at 
 5 
 
 No. papers 
 23 
 
 4 
 3 
 2 
 
 28 (51) Adding down 
 49 (100) 
 
 
 1 
 
 
 46 (100) Adding up 
 54 
 
 
 2)200 
 100 
 
 In this example, the median would come at the end of the 100th case. 
 We find that the end of this case coincides with the end of step 1. 
 But coming down, the end of the 100th case would be at the begin- 
 ning of step 3. Obviously any point in step 2 could be taken and 
 there would still be 100 cases on each side. In a problem of this 
 kind, the best procedure is to divide the vacant space on the scale 
 so that the two nearest actual cases on it will be equidistant. Then 
 the median will fall in the middle of step 2 or at 2.68 for the real value, 
 and will be the same as if one case had been in the step with the total 
 number of cases odd. 
 
 If the data given on page 134 from the tests in arithmetic are ma- 
 nipulated so that there are only 218 cases in all and one-half of these 
 are above 11 and one-half below, with none in step 11, but actual 
 cases in steps 10 to 12, precisely the same procedure would be followed. 
 That is, in this case, the median would be 11.5. 
 
 Sometimes the median is not calculated exactly but 
 only approximately. Professor Monroe, for example, 
 gives some devices for obtaining an approximate median 
 and then correcting it to get the true median.^ But in 
 general, about the only time that an approximate median 
 may be used to advantage is when one takes for it the 
 magnitude of the lower limit of the group that contains 
 the real median. That is, 11 would be taken for the 
 approximate median for the problem on page 134, since 
 the median falls within the 11-11.99 group. The straight 
 
 * Monroe, W. S. : Educational Tests and Measurements, pp. 242- 
 247 
 
Measures of Type 
 
 137 
 
 mid-point method of calculating the median will get it 
 with less confusion, greater accuracy, and more speed for 
 the average person. 
 
 Graphic Representation of the Median. In a Bobbitt 
 table having an odd number of items, the median is the 
 No. attempting 
 
 ow 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 25 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 zo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \D 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ( 
 
 No. Pr 
 
 Dbli 
 
 sm 
 
 s 
 
 
 1 
 
 & 
 
 
 G 
 
 9. 
 
 juar 
 
 '0 II. 
 
 Medi 
 
 65 
 
 tile 1 
 
 98 
 an 
 
 2G 
 
 15 
 
 15. 
 
 Quar 
 ^ 
 
 35 
 
 file 
 
 3 
 
 
 2 
 
 
 
 
 
 
 2 
 
 5 
 
 Fig. 20. — Graphic Representation of the Median on a Surface of Frequency. 
 The corresponding distribution table is given on page 134. 
 
138 School Statistics and Publicity 
 
 size of the middle item, and it is usually inclosed in parallel 
 horizontal lines to indicate that it is the median. (See 
 page 18.) If the table has an even number of items, a 
 single horizontal line is drawn across between the two 
 middle cases, thus throwing half the cases above and half 
 below. (See page 130.) The line then represents the 
 median, but the exact magnitude is not shown in 
 figures. 
 
 To represent the median graphically on a surface of 
 frequency, find the point on the base line of the surface 
 of frequency represented by the calculated median and 
 through it erect a perpendicular. This perpendicular is 
 sometimes erroneously called the median, but really the 
 median is the size of the magnitude on the horizontal 
 scale at the foot of the perpendicular. (See Figure 20.) 
 This perpendicular divides the surface of frequency into 
 two equal parts. 
 
 Through the point on the base representing the median (11.98) 
 draw a perpendicular (represented by a dotted line here) which will 
 cut the surface into two equal areas. The small squares may be 
 counted to prove this. There are 88 small squares to the left of the 
 11-11.99 group and 109 to the right of it. The 11-11.99 group 
 contains 22 small squares, of which .98 must go with the ones to the 
 left, and .02 with the ones to the right. .98 of 22 = 21.56. Adding 
 to 88 we get 109.56. .02 of 22 = .44. Adding to 109, we get 109.44, 
 or approximately the same area. The slight discrepancy is due to 
 carrying the decimals out to only two places. 
 
 Advantages of the Median for School Work. 1. The 
 
 median can usually be located exactly without much 
 trouble. 
 
 This is of great service where the mode cannot be exactly deter- 
 mined, as in distributions where practically any legitimate groupings 
 will give several groups of about the same size. 
 
Measures of Type 139 
 
 2. Extreme cases influence it little. In this it resembles 
 the mode. 
 
 For example, dropping off a number of cases at either end of the 
 distribution table on page 134 would only shift the median a part of 
 one step. As long as the median stays within the 22 cases of step 11, 
 for instance, one case dropped at either end could shift the median 
 only jV or .045 of one step. 
 
 3. Its location can never depend upon a small number 
 of items. 
 
 It falls at the midpoint of the distribution irrespective of how 
 many cases are in a group or where the groups are. 
 
 4. If the number of extreme cases is known, or known 
 approximately, we do not have to know their size. 
 
 Thus, if we wish to know the median or typical salary of Latin 
 teachers in a state, it is not necessary to get the salaries of teachers in 
 the large high schools that will not report to the state superintendent, 
 nor in the small unorganized high schools, provided we know about 
 how many teachers are in each class of schools. If we know these 
 numbers approximately, we can still get the median salary of Latin 
 teachers for the whole state. 
 
 5. The median is of special value for data where the 
 items cannot be measured in definite units. 
 
 Thus, we may get the median child on any particular ability for a 
 room, or the median performer in a debating or oratorical contest, 
 without ever being able to measure in definite units the performance 
 of a single contestant. An arithmetical average cannot be calculated 
 here with any useful accuracy, but the median can be found and com- 
 pared with similar medians. 
 
 Disadvantages of the Median for School Statistics. 
 1. The median is not so easily calculated as the average. 
 
 The average is computed by an arithmetical process familiar to 
 most children in the grammar grades, and it may be computed without 
 
140 School Statistics and Publicity 
 
 1 
 
 rearranging the items. The median cannot be calculated until the 
 data are rearranged in order of size, and while the process of calcula- 
 tion is simple, the previous examples show that considerable care 
 must be exercised in getting it. 
 
 2. The total cannot be gotten by multiplying the 
 median by the number of items. In this respect it is 
 like the mode. 
 
 3. It is not useful in those cases where it is desirable 
 to give large weight to extreme variations. 
 
 Thus, the average daily attendance of a school is much affected by 
 those who are absent a large part of the time. The median attendance 
 would not be so affected. But what we wish in this particular in- 
 stance is that the great effect of the few extreme cases shall exert its 
 full influence. Consequently, the average daily attendance is used. 
 
 4. Unlike the mode but like the average, the median 
 may be located in the distribution where the actual cases 
 are few. 
 
 5. In a discrete series there may be so many cases the 
 same size as the median that it will become almost 
 meaningless. 
 
 It cannot mean much here unless there is some reasonable basis 
 for regarding the given measures as spread out, that is, for regarding 
 the distribution as in some ways continuous. 
 
 EXERCISES 
 
 1. Calculate the median from both ends for each of the distribu- 
 tion tables used in previous exercises. 
 
 2. Draw the line to represent the position of the median in each 
 of the surfaces of frequency used in previous exercises. 
 
 f 
 
Measures of Type 141 
 
 III. THE AVERAGE 
 
 Definition. The average is a measure much used in 
 ordinary life without being defined. Indeed, it is not 
 capable of easy definition. The ordinary person, if 
 pinned down long enough, will define it as a measure 
 which will give '* the general run " of a group by taking 
 into account both the number of cases and the size of 
 each one. 
 
 But actually he means one of two things, which differ 
 widely from each other and neither of which really 
 corresponds to this definition. Most of the time the 
 " average " means to him the most frequent measure in 
 the group, i.e., the mode. But in some instances, it 
 means to him a very unreal and unjust thing obtained 
 by statistical sleight of hand. Thus he will say that 
 there is no such thing as an average boy, nobody ever 
 saw such a boy, etc. This comes nearer his definition 
 than does his other use of the term. But even here he 
 probably is not conscious of the fact that the average 
 is really the size of the balancing point or center of 
 gravity in the distribution. 
 
 Calculation. 1. Ordinary method. Ordinarily, the 
 average is calculated by dividing the sum of all the 
 measures (or cases) by the number of measures. By 
 formula it is : 
 
 . /AN Sum of all measures 
 
 Average (Av.) = — ;jr^ 5 
 
 No. of measures 
 
 Thus, for the results on the Courtis Tests, page 134, we could find 
 the average as follows : 
 
142 School Statistics and Publicity 
 
 No. 
 
 
 
 • 
 
 
 Problems 
 
 
 No. 
 
 
 All 
 
 Attempted 
 
 
 Attempting 
 
 
 Measures 
 
 24 
 
 X 
 
 5 
 
 = 
 
 120 
 
 23 
 
 X 
 
 2 
 
 = 
 
 46 
 
 22 
 
 X 
 
 2 
 
 = 
 
 44 
 
 21 
 
 X 
 
 1 
 
 = 
 
 21 
 
 20 
 
 X 
 
 5 
 
 = 
 
 100 
 
 19 
 
 X 
 
 5 
 
 = 
 
 95 
 
 18 
 
 X 
 
 5 
 
 = 
 
 90 
 
 17 
 
 X 
 
 7 
 
 = 
 
 119 
 
 16 
 
 X 
 
 13 
 
 = 
 
 208 
 
 15 
 
 X 
 
 15 
 
 = 
 
 225 
 
 14 
 
 X 
 
 10 
 
 = 
 
 140 
 
 13 
 
 X 
 
 15 
 
 = 
 
 195 
 
 12 
 
 X 
 
 24 
 
 = 
 
 288 
 
 11 
 
 X 
 
 22 
 
 = 
 
 242 
 
 10 
 
 X 
 
 26 
 
 = 
 
 260 
 
 9 
 
 X 
 
 21 
 
 = 
 
 189 
 
 8 
 
 X 
 
 19 
 
 = 
 
 152 
 
 7 
 
 X 
 
 9 
 
 = 
 
 63 
 
 6 
 
 X 
 
 8 
 
 = 
 
 48 
 
 5 
 
 X 
 
 3 
 
 = 
 
 15 
 
 4 
 
 X 
 
 1 
 
 = 
 
 4 
 
 3 
 
 X 
 
 1 
 219 
 
 2] 
 
 3 
 
 .9)2667 
 
 
 12.18 Average No. 
 
 
 
 
 
 Problems Attempted. 
 
 2. Short method. By experienced workers in statistics 
 averages are often computed by the following method of 
 guessing the average and then correcting it. 
 
 First arrange figures in a distribution table. Then 
 guess the average by inspection, usually taking the 
 approximate median. To be absolutely correct, this 
 must be guessed within one step of the true average. 
 
 Then correct the guessed average by the average of all 
 the deviations from it. 
 
Measures of Type 
 
 143 
 
 In the Courtis Test problem just worked, the procedure is as fol- 
 lows: 
 
 Guessed average is 11. 
 
 Deviations above or + deviations are : 
 
 
 + 
 
 + 
 
 24 deviations of 
 
 1 or 
 
 24 
 
 15 
 
 2 . 
 
 30 
 
 10 
 
 3 . 
 
 30 
 
 15 
 
 4 . 
 
 60 
 
 13 
 
 5 . 
 
 65 
 
 7 
 
 6 . 
 
 42 
 
 5 
 
 7 . 
 
 35 
 
 5 
 
 8 . 
 
 40 
 
 5 
 
 9 . 
 
 45 
 
 1 
 
 10 . 
 
 10 
 
 2 
 
 11 . 
 
 22 
 
 2 
 
 12 . 
 
 24 
 
 5 
 
 13 . 
 
 65 
 
 +492 
 
 Deviations below or — deviations are : 
 
 26 deviations of 1 or 26 
 
 21 ... . 
 
 . 2 . 
 
 . 42 
 
 19 ... . 
 
 . 3 . 
 
 . 57 
 
 9 . . . . 
 
 . 4 . 
 
 . 36 
 
 8 . . . . 
 
 . 5 . 
 
 . 40 
 
 3 . . . . 
 
 . 6 . 
 
 . 18 
 
 r-l 
 
 . 7 . 
 
 . 7 
 
 1 . . . . 
 
 . 8 . 
 
 . 8 
 -234 
 
 + Deviations 492 
 
 
 
 234 
 
 
 
 Excess of + deviations =258 
 
 
 
 This means that we got the guessed average too low because there 
 are more deviations above than below it. The 258 is the excess of 
 deviations above. As there are 219 cases, the average excess devia- 
 tion to satisfy every case is |f| or 1.18 of a step. As the guessed aver- 
 
144 School Statistics and Publicity 
 
 age of 11 was too low, we add this 1.18 average deviation and get' 
 12.18 for our corrected average. 
 
 Had the minus deviations been in excess, it would have meant 
 that the guessed average was too high, and we should have sub- 
 tracted the correction. 
 
 The rule for computing the average by the short-cut method 
 is: " Arrange the numbers in the order of their magni- 
 tude ; choose any number hkely to be nearest the average ; 
 add together, regarding signs, the deviations from it of all 
 the numbers; divide this result by the number of the 
 measures of the average which you are obtaining; add 
 the quotient to the chosen number." ^ 
 
 In using the short-cut method, three cautions must be 
 kept in mind : 
 
 1. To get much accuracy, the guessed average must be within one 
 step of the true average. 
 
 2. The correction must be added to the guessed average if the 
 plus deviations are in the majority; it must be subtracted if the 
 minus deviations are in the majority. 
 
 3. If the guessed average is in a group, the deviations for all that 
 group are 0. The number of cases, however, must be used in dividing 
 to get the average deviation. 
 
 The short method is not always short by itself ; but it 
 often saves time on certain calculations on deviations. 
 
 Graphic Representation. The average can be rep- 
 resented graphically in the same way as the median. 
 (See page 137.) That is, the point representing the 
 calculated average is found on the base line and a per- 
 pendicular is erected to call attention to the size of the 
 average. However, there is in general little value in 
 representing the average this way because the line drawn 
 has no appreciable relation to the surface of frequency. 
 
 1 Thorndike, E. L. : Mental and Social Measurements, p. 3 
 
Measures of Type 145 
 
 A line for the mode runs through the highest part of 
 the surface, a fact the eye easily grasps. A line for the 
 median cuts the surface into two equal areas, which the 
 eye will readily compare. But the average has no such 
 relationship to show. 
 
 Advantages of the Average for School Statistics. 
 1. " Unlike the median or mode, it may be definitely 
 located by a simple process of addition and division, and 
 it is unnecessary to draw diagrams or arrange the data in 
 any set form or series." ^ 
 
 The great ease with which the average can be calculated from 
 figures in almost any form is doubtless the main reason it has been so 
 often used. It is not even necessary to throw figures into a distribu- 
 tion table or table of frequency, as is the case in finding the median 
 or the mode. 
 
 2. It weighs extreme cases, which is a desirable thing 
 in certain instances. 
 
 3. " Unlike the mode, it is affected by every item in the 
 group, and its location can never be due to a small class 
 of items." 2 
 
 Thus Superintendent Spaulding in a recent bulletin gets the aver- 
 age expenditure for Minneapolis by taking the average for five years. 
 No one year is any more important than another and so gets counted 
 no more and no less than another year. The median in such a small 
 number of cases, with an odd number especially, would emphasize 
 one year unduly. 
 
 4. The method of calculating it is familiar to every one. 
 
 5. It may be determined when the aggregate and the 
 number of items are the only things known. 
 
 Thus, in determining the typical number of days' attendance for 
 a child in a certain grade, we may have the total number of days' 
 
 1 King, W. I. : Elements of Statistical Method, p. 136 
 
 2 Ibid., p. 136 
 
146 School Statistics and Publicity 
 
 ■ 
 
 attendance and the total number of children. We might not have the 
 actual record of a single child given us. But from the other two items 
 we could calculate the average number of days' attendance for a child. 
 Sometimes these are the only two items that can be secured from 
 reports. Thus we may have only the total for teachers' salaries and 
 the number of teachers ; the total sum spent on repairs and the num- 
 ber of buildings; the total expenditures for a certain kind of school 
 and the number of such schools, etc. 
 
 Disadvantages of the Average for School Statistics. 
 
 1. It cannot be located on a surface of frequency from 
 looking at the surface alone. 
 
 The median or mode can be quickly located by inspecting the sur- 
 face of frequency. The average can be put on it only after the aver- 
 age has been calculated, and the point counted out on the base scale. 
 
 2. It cannot be located accurately if the extremes are 
 missing or in any way doubtful. 
 
 This is particularly troublesome when we recall that we are often 
 inclined to expect extreme cases of expenditures and such things 
 in school work to be doubtful or inaccurate. 
 
 3. It lays too much stress on extreme variations. 
 
 This is as bad a thing for school statistics as judging a patent 
 medicine by the enthusiastic few who write testimonials about it for 
 publication, or a school by what is done the first and last week in each 
 session. The trouble can, however, be eliminated to some extent 
 by dropping very extreme cases and calculating the average from the 
 remaining ones. 
 
 4. It cannot be used where we cannot accurately 
 measure the quantities studied. 
 
 The so-called averaging of points on contestants in a debate is often 
 futile because they are not measured by the different judges on the 
 same scale. 
 
 5. It may fall where no data actually exist. 
 
 That is, it may fall in a gap (the median does this sometimes as 
 well) and be almost as absurd as the case of the duck reported by a 
 
Measures of Type 147 
 
 western professor. The duck was shot at with a double-barreled 
 shotgun, one shot going two feet to the right of the duck and the other 
 two feet to the left. The average performance was, of course, zero 
 and centered on the duck. Statistically the duck was dead ; actually 
 it flew away. 
 
 6. The average often means to the ordinary man 
 something different from what the calculated thing means. 
 
 The process of calculation is undoubtedly familiar to all. But 
 what the ordinary man often means by the "average boy" is a boy 
 like the majority of boys, not like the few above or below this majority. 
 The man in all probability has seldom thought through the fact that 
 his method of calculation does not always give him this "majority" 
 measure. 
 
 EXERCISES 
 
 1. Calculate by both the long and short method the average for 
 each of the distribution tables used in previous exercises. 
 
 2. Draw the line to represent the position of the average in each 
 of the surfaces of frequency used in previous exercises. 
 
 IV. WHICH MEASURE OF TYPE TO USE IN A GIVEN 
 
 DISTRIBUTION 
 
 If the distribution is symmetrical or approximately so, 
 the average, median, or mode are, of course, the same or 
 approximately so. Consequently, as regards magnitude, 
 it is a matter of indifference which of the three is used as 
 a measure of central tendency. But it is generally im- 
 possible to tell whether the distribution is symmetrical 
 or not till a frequency table has been made. Once this 
 table has been made, it is much easier to determine the 
 median than the average. In a Bobbitt table or similar 
 distribution where all cases are separate, the median 
 can be found much more quickly than the average. 
 
 The mode is the measure to use in all skew distributions 
 or bi-modal ones. 
 
148 School Statistics and Publicity 
 
 The matter of symmetry or skewness can be seen at a 
 glance from the surface of frequency. With a little 
 experience it can be recognized directly from the table of 
 frequency. 
 
 For school statistics, this is probably the safest rule : 
 
 Use the mode for skew and hi-modal distributions; 
 the average, for cases where every item must he counted 
 as much as any other item {say in finding the average 
 expenditures for five or less years); and the median for 
 all others. 
 
 Of the three, the median is by far the best " all pur- 
 pose " measure of central tendency for school statistics. 
 
 EXERCISE 
 
 In each of the distribution tables for which you have calculated 
 the mode, median, and average, which is the best measure of central 
 tendency ? Why ? 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 King, W. I. Elements of Statistical Method, Chapter XII. 
 
 Rugg, H. 0. Statistical Methods Applied to Education, Chapter V. 
 
 Thorndike, E. L. Mental and Social Measurements, pp. 36-39. 
 
CHAPTER VI 
 
 MEASURES OF DEVIATION OR DISPERSION 
 
 The second element necessary in giving the bird's-eye 
 view, it will be recalled, is some measure of how much 
 the cases range from the central tendency. This is 
 variously called '' spread," " range," " deviation," or 
 " dispersion." It is of just as much importance as the 
 central tendency. For example, it makes a vast difference 
 whether a teacher is called upon to teach children that 
 are all very close to the typical or " average " child, or 
 ones that vary enormously both above and below. 
 Supervision means one thing for a superintendent when 
 all his teachers are very close in ability, experience, 
 spirit, etc., to his typical teacher; it means an entirely 
 different thing when many of his teachers range widely 
 either side of this typical teacher. 
 
 There are several ways of measuring this range, the 
 most useful of which will now be given. 
 
 I. EXTREME RANGE VARIATION 
 
 The dispersion may be shown by giving the extreme 
 cases and thus showing how far it is between them. This 
 measure may be useful in showing the difference of ability 
 in achievements of a grade in school. 
 
 For example, the eighth grade in a western city as reported on 
 some Courtis Tests varied in attempts on the addition problems 
 from 4 to 24, or a range of 20 out of a possible range of 24. This 
 
 149 
 
150 
 
 School Statistics and Publicity 
 
 measure shows at once that there is a wide range of ability of eighth 
 grade children in that city as regards attempts on addition problems. 
 
 But as a general rule, the extreme range variation is not 
 a good measure of dispersion because the extreme cases 
 are likely to be very unreliable, especially if there are 
 only one or two that are isolated from the rest of the 
 measures in the distribution. 
 
 The range of teachers' salaries in the typical school system is 
 not from that of the superintendent down to the salary of the poorest 
 paid teacher, because there is only one superintendent and his salary 
 is far removed on the scale from that of the highest paid teacher under 
 him. His salary must be eliminated before the extreme range for 
 the others has any value. 
 
 The influence of extreme variations upon this measure is shown 
 by Table 17, taken from a study in costs of instruction in home 
 economics in fifteen southern normal schools.^ 
 
 Table 17. Variations in Cost of Instruction in Home 
 Economics in 15 Southern Normal Schools, per 1000 Stu- 
 dent Hours, 1916 
 
 Key number of 
 
 normal school 
 
 III 
 
 VII 
 
 XIII 
 
 II 
 
 VIII 
 
 IV 
 
 XI 
 
 XIV 
 
 XV 
 
 I 
 
 VI 
 
 X 
 
 V 
 
 IX 
 
 XII 
 
 Cost of home economics 
 
 instruction per 1000 
 
 student hours 
 
 $194 
 
 115 
 
 89 
 
 88 
 
 85 
 
 82 
 
 79 
 
 73 
 
 70 
 
 64 
 
 60 
 
 58 
 
 53 
 
 52 
 
 15 
 
 1 Alexander, Carter: "Costs of Instruction in Normal Schools," 
 Elementary School Journal, XVII, 653 
 
Measures of Deviation or Dispersion 151 
 
 The extreme range variation between 194 and 15 is 179. But it 
 is very evident that both extremes are the results of some very unusual 
 factor. If the top case is cut off, the range is reduced to 100. If the 
 bottom case is cut off, but the top one retained, the range is 142. 
 If both are cut off, the range is reduced to 63. If the two top cases 
 and the bottom one, all of which seem unusual, are eliminated, the 
 range is reduced to 37, which is probably a fairly reliable figure. 
 
 The uncertainty as to just how many cases to leave 
 out in any one distribution when computing the spread 
 emphasizes the need of having some standard proportion 
 cut off, say a fourth at each end. Then the average 
 deviation or spread from the central tendency of the two 
 intersections or points thus obtained can be computed. 
 Let us now discuss some of the devices for getting devia- 
 tions from such points. 
 
 II. QUARTILE DEVIATION (SEMI-INTER-QUARTILE RANGE 
 
 OR " Q ") 
 
 This is by far the easiest and most important measure 
 of dispersion for school statistics. But before it can be 
 easily understood, it is necessary to get the quartiles. 
 These are the magnitudes of the points on the scale which 
 divide the distribution into four equal groups of cases (or 
 quarters) . 
 
 Quartiles. Obviously, it takes three such points to 
 divide the whole into four groups with equal numbers of 
 cases, — the first quartile or 25 percentile, the second 
 quartile (which, of course, is the median), and the third 
 quartile or 75 percentile. Obviously, also, the first 
 quartile is the median of the lower half of the distribution, 
 and the third quartile is the median of the upper half. 
 Below the first quartile come one-fourth of the cases 
 and above it, three-fourths. Above the third quartile 
 
152 School Statistics and Publicity 
 
 come one-fourth of the cases, and below it, three-fourths. 
 Between the first and third quartiles come one-half or 
 50 per cent of the cases. 
 
 The problem of locating the first quartile is statistically 
 the same as that of locating the median, except that cases 
 are counted in from one end so as to get only one-fourth 
 of them on one side. That is, the object this time is to 
 find the " quarter point " from one end of the distribution. 
 
 In a Bobbitt tableit is generally best to fix the number of cases so 
 as to know in advance whether the quartiles will be shown as separate 
 cases or fall between separate cases. In this way, there will be no 
 trouble with splitting cases. Thus if the number of cases is some 
 multiple of 4, plus 1, the median will appear as a separate case and 
 the quartiles will fall between cases., For the Bobbitt table on page 
 18, the quartiles are .respectively 47 for Q 1 (|^ of 42 + 52) and 78 for 
 Q 3 (I of 74 + 82). If the total number of cases is some multiple of 
 4, the median and both quartiles will fall between cases. If the total 
 number of cases is some multiple of 4, plus 3, the median and both 
 quartiles will appear as separate cases. 
 
 The calculation of the quartiles in the distribution given on page 
 134 is as follows. Since there are 219 cases, the first quartile must 
 come at the point that will throw | of 219 or 54 1 cases below it. 
 Counting up, we find that this will require 13 f cases of the 9-10 group, 
 
 there being only 41 cases below. This will make it come -^ — or 
 
 21 
 .65 of the step up. Q 1 then is 9 + .65 = 9.65. Similarly Q 3 is 
 
 m the 15 group commg down. 54| — 45 = 9|. ^^-^ = .65. 
 
 Q 3 then is 16 - .65 = 15.35. ^ 
 
 Graphically, the first quartile is the magnitude on the 
 base at the foot of a perpendicular so located that it will 
 cut off one-fourth of the area of the surface of frequency 
 to the left of it and three-fourths to the right. The third 
 quartile has one-fourth of the area to the right and three- 
 fourths to the left. The median has one-half the area 
 on either side of it. (See Figure 20, page 137.) 
 
Measures of Deviation or Dispersion 153 
 
 Quartile Deviation. The Q is found by taking half 
 the difference between the first and third quartiles. This 
 average will give the average distance of the quartiles 
 from the median of central tendency selected. The aver- 
 age distance is used because sometimes the median is not 
 exactly halfway between the quartiles (it cannot be if the 
 distribution is not perfectly symmetrical). 
 
 By formula, then, 
 
 ^ Quartile 3 — Quartile 1 
 Q = 2 
 
 For the Bobbitt table on page 18, the calculation is : 
 
 Q = $Z§-iiML=i?l= $15.50 
 
 For the distribution on page 134, the calculation is: 
 _ 15.35 - 9.65 ^ 5.70 ^ 2 85 
 
 Li Li 
 
 Graphically, Q is one-half the distance from Quartile 3 
 to Quartile 1 on the base of the surface of frequency. But 
 as it is difficult to show only one-half of this, the whole of 
 this distance is usually represented as 2 Q . (See Figure 20 . ) 
 
 III. OTHER PERCENTILE DEVIATIONS 
 
 In some of the educational investigations, especially 
 those issued by workers at the University of Chicago, 
 dispersion is indicated by the distance from the median 
 of some point marking off a convenient number of the 
 cases, other than the quartile. The number of cases is 
 usually a common percentage; hence the point is called 
 a *' percentile." In some studies the cases are divided 
 into thirds by " tertiles." The median here is, of course, 
 in the middle of the second group. The tertile deviation, 
 then, is half the difference of the first and second tertiles 
 
154 
 
 School Statistics and Publicity 
 
 (only two tertiles are needed to make three groups). 
 Similarly four '' quintiles " will divide the distribution 
 into five groups with the same number of cases, the median 
 being at the middle of the third group. The quintile 
 deviation then would be one-half the difference of the 
 first and fourth quintiles. 
 
 Table 18. Table to Show Variation by Percentage Groups, 
 Using Distribution of Annual Salaries of Regular Teach- 
 ers IN Elementary Schools in Cleveland and in 13 Other 
 Cities of More Than 250,000 Inhabitants ^ 
 
 
 Salaries not exceeding the amounts specified were 
 
 
 earned by teachers bearing to the aggregate num- 
 
 City 
 
 ber employed in each city the proportion of: 
 
 
 10 per 
 
 30 per 
 
 50 per 
 
 70 per 
 
 90 per 
 
 
 cent 
 
 cent 
 
 cent 
 
 cent 
 
 cent 
 
 Baltimore , . . 
 
 $600 
 
 $ 700 
 
 $ 700 
 
 $ 750 
 
 $ 800 
 
 Boston .... 
 
 648 
 
 840 
 
 1,176 
 
 1,176 
 
 1,224 
 
 Chicago . . . 
 
 675 
 
 975 
 
 1,175 
 
 1,175 
 
 1,200 
 
 Cincinnati . . . 
 
 700 
 
 900 
 
 1,000 
 
 1,000 
 
 1,000 
 
 Cleveland . . . 
 
 600 
 
 750 
 
 900 
 
 1,000 
 
 1,000 
 
 Indianapolis . . 
 
 475 
 
 625 
 
 875 
 
 925 
 
 925 
 
 Milwaukee . . 
 
 876 
 
 876 
 
 876 
 
 876 
 
 876 
 
 Minneapolis . . 
 
 750 
 
 950 
 
 1,000 
 
 1,000 
 
 1,000 
 
 Newark .... 
 
 630 
 
 780 
 
 1,000 
 
 1,000 
 
 1,300 
 
 New Orleans . . 
 
 500 
 
 600 
 
 700 
 
 750 
 
 800 
 
 Philadelphia . . 
 
 630 
 
 780 
 
 900 
 
 940 
 
 1,000 
 
 San Francisco . . 
 
 840 
 
 1,164 
 
 1,200 
 
 1,224 
 
 1,224 
 
 St. Louis . . . 
 
 700 
 
 1,032 
 
 1,032 
 
 1,032 
 
 1,120 
 
 Washington . . 
 
 625 
 
 700 
 
 750 
 
 890 
 
 980 
 
 Average . . . 
 
 $661 
 
 $ 834 
 
 $ 949 
 
 $ 988 
 
 $1,032 
 
 ^ Data for Cleveland from payroll for 1914-15; data for other 
 cities for 1913-14, from "Tangible Rewards of Teaching," U. S, 
 Bureau of Education 
 
Measures of Deviation or Dispersion 155 
 
 The calculations and graphic representations for these 
 various percentile deviations are precisely similar to those 
 for the Q or quartile deviation. 
 
 These deviations are found as yet in few educational 
 investigations, but should be understood for reading 
 purposes. 
 
 A simple but effective device for popular consumption 
 indicates the variability indirectly by showing the magni- 
 tudes which equal or exceed certain fixed percentages of 
 the cases. This device has been found especially valu- 
 able for making comparisons between different distribu- 
 tions. 
 
 A good example occurs in the Cleveland Survey ^ and 
 is reproduced in Table 18. 
 
 IV. MEDIAN DEVIATION (MED. DEV. OR P. E.) 
 
 This is the median of all the deviations arranged in 
 order of size and irrespective of whether they are above or 
 below the central tendency. In other words, it is cal- 
 culated by arranging the deviations in order of size and 
 then finding the median of these deviations in precisely 
 the same way as the median of anything else would be 
 found. 
 
 Within the median deviation of the central tendency 
 used, the middle 50 per cent of the cases come. Within 
 the Q of the central tendency approximately 50 per cent 
 of the cases come, but they are not necessarily the middle 
 50 per cent, because the central tendency does not 
 lie exactly halfway between the quartiles, except in a 
 sjnumetrical distribution. In such a distribution the 
 median deviation is of course the same as Q. 
 
 1 Volume on ** Financing the Schools," p. 57 
 
156 School Statistics and Publicity 
 
 The median deviation is little used in school statistics, 
 but the superintendent may come across it in his reading 
 of educational investigations and in attempts to indicate 
 variability. It is sometimes called the '' probable error," 
 and this is where it gets the letters '* P. E." However, 
 this name is not a good one for, as Professor Thorndike 
 suggests, the median deviation '' is not specially probable 
 and not an error at all." ^ A statement of central 
 tendency of 30 with a P. E. of 4 means that half the cases 
 deviate more than 4 and half of them less than 4 from this 
 30. It also means that for any given case chosen at 
 random, the chances are 50 to 50 that it will deviate more 
 than 4 from this 30, that is, be below 26 or above 34; 
 the chances are likewise 50 to 50 that it will deviate less 
 than 4 from this 30, that is, be between 26 and 34. In 
 other words, it is a toss-up as to whether any given case 
 will be likely to deviate more or less than 4 from this 
 central tendency of 30 ; that is, any given case is as likely 
 to be between 26 and 34 as it is to be below 26 or 
 above 34. 
 
 V. AVERAGE DEVIATION (A. D.) 
 
 This is simply the average of all the deviations from 
 whatever central tendency is selected. It may be figured 
 from any measure of central tendency. But it is best 
 calculated from the average or approximate average, 
 next from the median or approximate median, and seldom 
 if ever from the mode, unless the average spread for each 
 side is given separately. 
 
 The A. D. for the Bobbitt table on page 18 is figured 
 thus : 
 
 » Mental and Social Measurements, p. 40, footnote 
 
Measures of Deviation or Dispersion 157 
 
 Deviation from median (59) of 61 is 2 
 
 62 
 
 3 
 
 63 
 
 4 
 
 69 
 
 10 
 
 69 
 
 10 
 
 74 
 
 15 
 
 82 
 
 23 
 
 88 
 
 29 
 
 100 
 
 41 
 
 100 
 
 41 
 
 112 
 
 53 
 
 169 
 
 110 
 
 59 
 
 
 
 58 
 
 1 
 
 56 
 
 3 
 
 56 
 
 3 
 
 54 
 
 5 
 
 53 
 
 6 
 
 52 
 
 7 
 
 42 
 
 17 
 
 41 
 
 18 
 
 38 
 
 21 
 
 34 
 
 25 
 
 33 
 
 26 
 
 30 
 
 29 
 
 !es, 25)502 sum 
 
 
 20.1 A. D 
 
 The A. D. for a distribution in groups is usually figured 
 from the approximate average or the approximate median, 
 so as to have the deviations in whole steps. The result 
 is close enough for all practical purposes. 
 
 The A. D. for the distribution given on page 134 is figured thus 
 from the guessed average : 
 
 492 + deviations ] ^ . ^ 
 
 234- deviations l^^^P^S^ 143 
 
 No. of cases, 219 )726 sum of all deviations 
 
 3.31 A. D. 
 
158 School Statistics and Publicity 
 
 VI. STANDARD DEVIATION (MEAN SQUARE DEVIATION, S. D.) 
 
 This is found by taking the square root of the average 
 of the squares of the deviations, counting zero deviations. 
 
 ^^ -p. _ / Sum of squares of deviations 
 ^ Number of deviations 
 
 It is of no particular value in school statistics except 
 that the superintendent should understand it so that he 
 may read intelligently educational or economic treat- 
 ments that use it. It saves time to use this as the measure 
 of dispersion if the Pearson Coefficient of Correlation is 
 later to be used. It is better than the A. D. if the de- 
 viations of the extreme cases need to be weighted. 
 
 The calculation of the S. D. from the guessed average in the dis- 
 tribution on page 134 is figured as follows, using the deviations as 
 given on page 143 : 
 
 24 X 12 
 
 = 
 
 24 
 
 15 X 22 
 
 = 
 
 60 
 
 10 X 32 
 
 = 
 
 90 
 
 15 X 42 
 
 = 
 
 240 
 
 13 X 52 
 
 = 
 
 325 
 
 7 X 62 
 
 = 
 
 252 
 
 5 X 72 
 
 = 
 
 245 
 
 5 X 82 
 
 = 
 
 320 
 
 5 X 92 
 
 = 
 
 405 
 
 1 X 102 
 
 = 
 
 100 
 
 2 X 11^ 
 
 = 
 
 242 
 
 2 X 122 
 
 = 
 
 288 
 
 5 X 132 
 
 =: 
 
 845 
 
 26 X 12 
 
 = 
 
 26 
 
 21 X 22 
 
 = 
 
 84 
 
 19 X 32 
 
 = 
 
 171 
 
 9 X 42 
 
 = 
 
 144 
 
 8 X 52 
 
 = 
 
 200 
 
 3 X 62 
 
 = 
 
 108 
 
 1 X 72 
 
 = 
 
 49 
 
 1 X 82 
 
 = 
 
 64 
 
 ^^ 
 
 S.D. =j4-282=4.42 
 219 
 
 4282 
 
Measures of Deviation or Dispersion 159 
 
 VII. DEVIATIONS FOR SKEW DISTRIBUTIONS 
 
 The measures for deviations so far given are serviceable 
 only for symmetrical, or approximately symmetrical, 
 distributions. If the distribution is a skew one, it is 
 evident that no average deviation for the whole can be 
 of any service, because the deviation on one side of the 
 central tendency will be markedly different from the 
 deviation on the other side. This average or median 
 deviation for the whole would be a deviation which did 
 not actually exist. 
 
 Hence in a skew distribution it is customary to take 
 the mode or median (preferably the mode) for the central 
 tendency, and then to give the average or median deviation 
 for the cases above this central tendency, and the same 
 for the cases below it. 
 
 This procedure may be illustrated from the salaries paid the 
 Cleveland elementary school teachers in 1914-15.1 (See page 160.) 
 
 The approximate median is sufficient here and inasmuch as only 
 an average deviation is to be figured, it is unnecessary to make the 
 groupings by even steps. 
 
 There are in all 2204 teachers, making 1102 above the median and 
 1102 below. The A. D. above is • 
 
 $118810 (MAO 
 
 The A. D. below is 
 
 $200920 »ia, . 
 1101"^^^^^ + 
 The median of the part above the group containing the median 
 is at the end of the 525th case, evidently in the $1000 group, making 
 the quartile deviation above roughly $100. The median of the part 
 below the group containing the median is evidently in the $700 
 group, making the quartile deviation below roughly $200. The 
 exact median and the exact quartile deviation could be figured if 
 desired, counting 1102 cases above the median and 551 cases above 
 the quartile. 
 
 1 Adapted from Cleveland Survey, Summary Volume, p. 98 
 
160 
 
 School Statistics and Publicity 
 
 Salary paid 
 
 Number getting 
 salary 
 
 Deviation from 
 median 
 
 Sum of 
 deviations 
 
 
 $1650 
 
 1 
 
 750 
 
 + 750 
 
 
 1540 
 
 2 
 
 640 
 
 1280 
 
 
 1500 
 
 4 
 
 600 
 
 2400 
 
 
 1430 
 
 1 
 
 530 
 
 530 
 
 
 1400 
 
 1 
 
 500 
 
 500 
 
 
 1300 
 
 5 
 
 400 
 
 2000 
 
 
 1210 
 
 1 
 
 310 
 
 310 
 
 
 1200 
 
 7 
 
 300 
 
 2100 
 
 
 1155 
 
 1 
 
 255 
 
 255 
 
 
 1100 
 
 71 
 
 200 
 
 14200 
 
 
 1050 
 
 83 Total 
 
 150 
 
 12450 
 
 
 1045 
 
 3 teachers 
 
 145 
 
 435 
 
 Approxi- 
 
 1000 
 
 762 above = 
 
 100 
 
 76200 
 
 mate 
 
 950 
 
 108 1050 
 
 50 
 
 
 
 50 
 
 5400 118810 
 
 Median = 
 
 = 900 
 
 196 196 
 
 
 
 
 850 
 
 112 
 
 -5600 
 
 
 825 
 
 2 
 
 75 
 
 150 
 
 
 800 
 
 130 
 
 100 
 
 13000 
 
 
 770 
 
 1 
 
 130 
 
 130 
 
 
 750 
 
 133 
 
 150 
 
 19950 
 
 
 715 
 
 4 
 
 185 
 
 740 
 
 * 
 
 700 
 
 164 
 
 200 
 
 32800 
 
 
 650 
 
 145 
 
 250 
 
 36250 
 
 
 600 
 
 136 Total 
 
 300 
 
 40800 
 
 
 550 
 
 20 teachers 
 
 350 
 
 7000 
 
 
 500 
 
 110 below - 
 
 400 
 
 44000 
 
 
 400 
 
 1 958 
 
 2)2204 
 
 1102 
 
 500 
 
 500 200920 
 
 The deviations have been calculated with reference to the median 
 in this particular example because this was the central tendency 
 used in the survey. But it is apparent that if the modal salary of 
 $1000 as denoted by the largest group of 762 cases were taken, the 
 discrepancy between the deviation measures on the two sides would 
 be much greater. 
 
Measures of Deviation or Dispersion 161 
 
 "1200- 
 1160 4- 
 1 120-1- 
 1060- 
 1040- 
 1000- 
 960 r 
 
 920 - 
 
 660 
 
 640- 
 
 eoo- 
 
 Hobak en 
 
 760 -- 
 
 720 -- 
 
 M 
 
 680 
 
 640- 
 
 560- 
 520- 
 460-U 
 440 -- 
 
 Springfield 
 
 Bayonnc 
 
 Youngstown 
 Fort Wayne 
 
 D«i Moines 
 
 Passaic -^omervilk— Springfield—Lawrence — New Bedford^Evanay/He-Doiuth 
 
 ■ Utica- 
 
 -Canton- 
 
 — Wafer bury f-lolyoke — : Lynn^ 
 
 Pa^ focket^j^^^J^^nton Tcrre ^aute^.^^.^ /Kansas Cy. 
 
 £. St. Louis 
 
 Schenectady Saginaw Sf.Josepii Wi/kes-Barre - 
 
 M , . Harrisbu/^ 
 nanchcater 
 
 Altoona Souffj Bend 
 
 Reading 
 
 Fig. 21. — Graphed Bobbitt Table of Mean Annual Salaries Paid Ele- 
 mentary Teachers in Certain Cities. 
 (J. F. Bobbitt : Elementary School Journal, 15 : 45.) 
 
162 School Statistics and Publicity 
 
 In this connection it should be noticed that a Bobbitt 
 table may be really a skew distribution. This skewness 
 will not be apparent at a glance because the table will 
 show just as many lines or cases between one quartile 
 and the median as between the other quartile and the 
 median. It needs inspection to see whether cases listed 
 on separate lines are really the same in size or approxi- 
 mately so. Professor Bobbitt removes this difficulty by 
 graphing his results. (See page 101.) The crowding in 
 of the cases below the median shows that the devia- 
 tion is on the average much less on that side than on 
 the upper. The same is true of the graph given in Fig- 
 ure 21. 
 
 VIII. WHICH MEASURE OF DEVIATION TO USE IN A GIVEN 
 
 DISTRIBUTION 
 
 The choice of the measure of deviation depends upon 
 the measure of central tendency selected. If the latter is 
 the average, then the deviation should be expressed by 
 the average deviation. If the median is used, deviation 
 should be expressed by the Q, although the average de- 
 viation may be employed. If for any reason extreme 
 variations are to be emphasized, the extreme range 
 measure or the standard deviation should be employed. 
 In a skew distribution, if the median is used, the quartile 
 deviation or the average deviation for each side should be 
 given separately; if the mode is used, the average 
 deviation or the median deviation for each side should be 
 given separately. 
 
Measures of Deviation or Dispersion 163 
 
 EXERCISES 
 
 1. Calculate all the measures of deviation or dispersion you can 
 for each of several of the distribution tables used in previous exercises. 
 For this use the measures of central tendency chosen in the exercise 
 on page 148. 
 
 2. For each of the distributions used in the exercise above, which 
 measures of deviation are preferable? Why? 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 King, W. I. Elements of Statistical Method, Chapter XIII. 
 
 Rugg, H. O. Statistical Methods Applied to Education, pp. 149- 
 
 173 and 178-180. 
 Thorndike, E. L. Mental and Social Measurements, pp. 46-50 and 
 
 Chapter VI. 
 
CHAPTER VII 
 
 MEASURES OF RELATIONSHIPS 
 
 So far we have for the most part considered distribu- 
 tions of measures as wholes and a single distribution at 
 a time. But one of the greatest values of statistical method 
 is the ease with which it makes possible the study of 
 relationships, including those between separate distribu- 
 tions. Since things have no meaning except through 
 their connections with other things, statistical method 
 must bring out such relationships very clearly. It was, 
 of course, impossible to cover the previous topics in 
 this book without dealing with relationships. But it 
 is now advisable to give these connections a special and 
 separate treatment. 
 
 I. RELATIONSHIPS INSIDE OF ONE GROUP 
 
 Discrete Series. As soon as a Bobbitt table or cen- 
 tral tendency and the deviation for any distribution 
 are given, by any of the various ways for calculating 
 these measures, a relationship is, of course, indicated. 
 But in general, inside of one distribution, relationship is 
 most easily shown by some form of the bar graph. 
 
 If every item is kept separate, each should be repre- 
 sented by a separate bar. All bars should be the same 
 width and differ only in length. This length indicates 
 the measure of the case. 
 
 164 
 
Measures of Relationships 165 
 
 The graphing of the Bobbitt table in Figures 8 and 9 is an example 
 of this. The bars in Figure 8, of course, may run vertically if pre- 
 ferred. 
 
 In such a table the cases are arranged from high to low, and the 
 tops of the bars or the curve for the graphic presentation will have a 
 general downward or upward direction, according to the end from which 
 it is viewed. But there are some distributions in which the cases come 
 in a time sequence or in an alphabetical order so that the tops of the 
 bars or the curve jog up and down. This would be the case with the 
 number of children in school by years or months in a city of fluctuat- 
 ing population, if the base scale represented the calendar months or 
 years in succession. In this instance, the central tendency and meas- 
 ure of deviation would have to be calculated and given separately, 
 or else shown with the same data rearranged as a Bobbitt table or 
 ordinary grouped distribution. 
 
 Continuous Series. If there is more than one case to 
 an item, that is, if the items are grouped, the length of 
 each bar will represent the number of cases. This 
 amounts then merely to drawing the surface of frequency 
 with bars representing groups which may or may not be 
 adjacent to each other. ^ The tops of the ends of these 
 bars form the broken line or ''curve" which, united with 
 the base line, makes up the surface of frequency. 
 
 If the distribution table is to be written or printed 
 without graphing, the relative sizes of the groups may 
 frequently be brought out more forcibly by turning the 
 numbers into percentages of the whole number of cases. 
 Of course, the ratio between 47 cases of magnitude 6 in a 
 total of 470 cases is exactly the same as the ratio of 10 
 per cent to 100 per cent. But the ordinary man is used 
 to thinking in terms of percentages and can grasp the 
 relationship much more quickly that way. Graphing 
 will show this relationship as easily from the original 
 figiires as from percentages, so there is no need for the 
 
 1 See p. Ill 
 
166 
 
 School Statistics and Publicity 
 
 change to percentages where graphing is to be done, in 
 the case of one distribution taken by itself. 
 
 II. SIMPLE RELATIONSHIPS BETWEEN DIFFERENT DIS- 
 TRIBUTIONS 
 
 If two different distribu- 
 tions have been made up on 
 the same scale, they can be 
 conveniently shown in bar 
 graphs, as in Figure 22. 
 
 It may be noted there 
 that the tops of the two dif- 
 ferent sets of bars give two 
 " curves." This device will 
 not be of service in compar- 
 ing more than two distribu- 
 tions at a time, but several 
 more " curves " may be 
 drawn provided they do not 
 overlap too much. It is cus- 
 tomary to represent curves 
 alone by different conven- 
 tional signs as : 
 
 No. IV 
 
 No. V -.-.-.-.-.-.-.-.-.-.-. 
 
 No. Yl fill II II 11 II II 1 1 II II II II II II II II II 
 
 Fig. 22. — Device for Comparing 
 Two Different Distributions Made 
 up on the Same Scale with Bar 
 Graphs. 
 
 The whole bara represent the total en- 
 rollment, while the shaded portions repre- 
 sent retarded children in the grades at 
 Memphis, about 1908. (From Laggards in 
 Our Schools by L. P. AjTes, page 39, by 
 permission of Russell SageFoundation) 
 
 No. I 
 
 No. II - 
 
 No. Ill rrrrrr....^.^ 
 
 Figure 23 is a good example. 
 
 However, it is often very difficult to grasp comparisons 
 between two different distributions using the same scale, 
 unless the number of cases for each magnitude is put 
 on the same basis as regards the rest of its distribution. 
 
Measures of Relationships 
 
 167 
 
 RESULTS OF SPELLING TESTS 
 PERCENTAGE OF WORDS SPELLED COR- 
 RECTLY BY GRADES 
 
 Percent 
 100 
 
 90 
 
 80 
 
 70 L 
 
 60 
 
 Highest 1 
 Average UuTTE 
 
 Lowest J 
 Average for 
 schools in 22 
 cities 
 
 This is most easily done by changing the number of cases 
 in each step to the proper percentage of the whole. That 
 is, one would not give the achievements in composition 
 for several school grades 
 by merely citing the 
 number of children in 
 each grade making the 
 various scores. Instead, 
 he would give tables 
 presenting the percent- 
 ages of children making 
 each score in each grade. 
 
 Table 18 on page 154 
 employs percentages for 
 making comparisons be- 
 tween cities on salaries 
 of teachers, by groups. 
 
 A good way to compare 
 several distributions is to 
 place the surfaces of fre- 
 quency made up from 
 the percentage tables one 
 above the other. This, 
 of course, may be used 
 only when they are all 
 applied to the same 
 scales. In this case, 
 they must be centered 
 exactly (the centers in 
 the same vertical line), or else the differences between the 
 central tendencies must be drawn to scale accurately. 
 Figure 24 is a good example. 
 
 Professor Bobbitt in the San Antonio Survey used 
 
 50 
 
 ■40 
 
 30 
 
 20 
 
 10 
 
 
 
 
 
 
 
 i 
 
 
 ^ 
 
 ^ 
 
 
 A 
 
 i 
 
 ^ 
 
 ^ 
 
 1 
 
 p 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 
 
 2 3 4 5 6 7 8 Grades 
 
 23. — Use of Curves for Comparing 
 More Than Two Distributions. 
 
 This chart represents the range from the 
 poorest to the best room tested in each grade 
 in spelling, Butte, Montana. For example, 
 the poorest second-grade room averaged 73. 
 The average for the whole city is repre- 
 sented by the dotted line, while the average 
 for twenty-two cities is represented by the 
 heavy line at 70. (From Butte, Montana, Sur- 
 vey, page 72) 
 
Scores 
 
 2 3 4 5 6 789 
 
 Median Scores 
 
 Scores: 01 23456789 
 
 Fig. 24. — Graph for Comparing Related Distributions, Using the Com- 
 position Scores in Salt Lake City. 
 
 (From Salt Lake City Survey, page 141) 
 
 168 
 
SPELLING ABILITY BY GRADES 
 
 100 
 
 eo 
 
 QO 
 70 
 60 
 
 Q^ 
 
 50 
 
 M 
 AO 
 
 Or 
 
 30 
 10 
 
 Cira de W 
 
 3 26 
 
 14 
 
 19 15 
 
 3 
 
 27 
 
 7 
 
 ^^4 ,2^ 
 
 17 
 2? 6 ?4 
 
 8 
 
 n 
 
 Q, 
 
 GrnrioW 
 
 19 
 
 15 
 
 S 3 
 
 13 7 
 
 20 (5 J8 
 
 % /^ 2 ^ '4 
 
 22 
 24 
 
 33 
 9 
 
 -22. 
 
 -21- 
 
 Gmdfl 7 
 
 /6 
 
 Ji. 
 
 "IW24 T2 
 
 4 _ 5 :2r 
 
 ^4. 
 
 Fig. 25. — Device for Graphing Bobbitt Tables from Different Distribu- 
 tions on Related Scales. 
 
 Each wide column represents the achievement in spelling of the ward schools for 
 that grade, on the Ayres scale. Each number represents a ward school, and the height 
 on the scale shows the achievement of that grade in that school. The quartile and 
 median lines are shown. (Adapted from the San Antonio Survey, page 105) 
 
 169 
 
170 School Statistics and Publicity 
 
 another device for making comparisons between different 
 distributions that had been made up as Bobbitt tables. 
 He drew their quartiles and medians as horizontal lines 
 across vertical columns, allowing each distribution one 
 column and adjusting the horizontal lines on the vertical 
 ones as scales. Figure 25 gives part of his device for 
 showing achievements of different grades in spelling. 
 
 This is a very convenient graph, for it shows a good 
 many relationships at a glance, such as the variations in 
 central tendencies, dispersions, etc. By observing only the 
 lines belonging to the medians, a curve may be read 
 across the page. That is, a little practice will enable 
 one to see three curves on the chart, the upper rep- 
 resenting quartile 3; the middle, the median; and the 
 lower, quartile 1. 
 
 III. COEFFICIENT OF VARIABILITY OR DISPERSION 
 
 A serious difficulty arises when we try to compare 
 two distributions as regards the amounts of dispersion 
 unless they have about the same central tendencies. If 
 the central tendencies are widely different and the abso- 
 lute number of units of dispersion is the same, the real 
 or relative amounts of dispersion are widely different. 
 
 For example, suppose a superintendent is studying the way his 
 high school teachers mark pupils. He gets 500 or more marks given 
 by each teacher. He finds that two teachers have the same median 
 in their marks, 80 on a numerical scale. But one teacher has a Q of 
 5 and the other a Q of 10. In the first case, the typical dispersion is 
 6| per cent of the median; in the other it is 12| per cent. Again, 
 a median of 90 with a Q of 5 would be markedly different from a 
 median of 80 with a Q of 5. In the former, the variability would 
 be 5f per cent and in the latter, 6i per cent, although they varied 
 10 points in the central tendency. 
 
 Consequently, in comparing distributions or groups 
 
Measures of Relationships 171 
 
 on variability, experienced persons compare them through 
 percentages of variabiUty or dispersion. The percentage 
 of dispersion is usually called the coefficient of variability. 
 By formula it is simply : 
 
 r^ ra ' ^ £ ^T • i„-T4- Mcasure of Devlatlou . , , 
 Coefficient of Variability = — :^ jj^ , pointed 
 
 off as per cent. 
 
 Thus, for the distribution on page 134, the calcula- 
 tion is : 
 
 Coefficientof Variability=§ = ^-g (from p 153) ^ 
 
 M 11.98 (from p. 135) 
 
 Professor Haggerty in his study of arithmetic in 
 twenty Indiana cities ^ has an interesting graph for com- 
 paring variation in two distributions. He had figures 
 for each school grade on the Courtis Tests in both attempts 
 and rights, thus getting two distributions of twenty cases 
 each. So he made a scale for the attempts and one for 
 the rights on each grade, with the medians centered and 
 opposite. Thus a line joining these two centers was 
 horizontal. A given city could be indicated by finding 
 its position on each of the two scales and joining these 
 points by a line. If the given city varied as did the whole 
 group of cities, its line was parallel to the original line 
 joining the medians. If it did not vary that way, its 
 line would slant, the end farthest from the median line 
 indicating that it varied more in that respect than in the 
 other. 
 
 Thus in Figure 26, for the fifth grade in addition, the upper heavy 
 line represents the work of the city of Bloomington and shows at a 
 glance that the fifth grade in Bloomington attempted 9 problems in 
 addition and got about 5| of these right ; that in both attempts and 
 
 1 Haggerty, M. E. : "Arithmetic: A Cooperative Study in Edu- 
 cational Measurements," Indiana University Studies, No. 27 
 
14 
 
 ■ 
 
 
 13 
 
 
 7 
 
 12 i 
 
 ■ 
 
 
 1) 
 
 
 6 
 
 10 
 9 
 
 / 
 
 B/oomington. 
 
 5 
 
 8 
 
 
 
 7 
 
 6 
 
 : 
 
 4 
 
 Median 
 2.0 Ind Cities 
 
 
 
 3 
 
 5' 
 
 
 
 4 
 
 
 
 
 
 2 
 
 3' 
 
 
 
 a 
 
 ' 
 
 1 
 
 1 
 
 
 
 , 
 
 
 
 A R 
 
 Fifth grade addition Bloomington Ind. compared 
 with standard for 20 Ind. cities, Haggerty. 
 
 Fig. 26. — Graphic Device for Comparing Variation in Two Different 
 
 Distributions. 
 
 The scale for attempts in the Courtis Tests from twenty cities is shown on the left, 
 centered with the scale for rights on the right. The slanting line represents the achieve- 
 ment of Bloomington, the slant upward indicating that it is relatively higher in rights 
 than in attempts. 
 
 172 
 
Measures of Relationships 173 
 
 rights it exceeded the standards set by the twenty Indiana cities as 
 a group (these standards are 6.5 and 3.6 respectively) ; also that its 
 achievement in rights was better than its achievement in attempts. 
 
 EXERCISE 
 
 Which of the two distributions given in Exercise 2, page 122, 
 is the more variable and just how much? Precisely how do you reach 
 your conclusion? For this, use the central tendencies and measures 
 of deviation previously calculated. 
 
 IV. CORRELATION 
 
 Meaning of Correlation. Sometimes it is very advan- 
 tageous to be able to show accurately and briefly the 
 general relation between two distributions that have 
 some common element or have been tested on the same 
 thing. Thus it may be desired to get the relation between 
 the two distributions obtained by testing the same group 
 of cases on two different tests. For example, one may 
 wish to know if a group of cities rank the same way on 
 excellence of schools that they do on per capita school 
 costs. 
 
 Now it is evident in all such cases that the chances are 
 very much against any situation where the cities would 
 rank exactly the same or exactly opposite in both lists. 
 Some will fall in exactly the same places, others exactly 
 opposite, and still others will change indiscriminately. 
 Consequently it is desirable to have some way of showing 
 the extent to which the individual cases generally keep the 
 same relative positions in the two distributions (that is, 
 first in each, second in each, last in each, etc.), or, putting 
 it another way, the extent to which the two distributions 
 are correlated with each other. 
 
 For a concrete example, let us take the following data 
 on Cleveland ward schools, accumulated during the sur- 
 
174 
 
 School Statistics and Publicity 
 
 vey there. 1 The records of eighteen schools for the same 
 grade on two qualities, which for our purposes we may call 
 the A test and the E test, are given in Table 19. 
 
 Table 19. Correlation Table Using Data from Eighteen 
 Cleveland Schools 
 
 School 
 
 Brownell . . 
 
 Clark . . . 
 
 Marion . . . 
 
 Detroit . . . 
 
 Fullerton . , . 
 
 Sackett . . . 
 North Doan 
 
 Bolton . . , 
 East Boulevard 
 
 Gilbert . . . 
 
 Rosedale . . 
 
 Landon . . . 
 
 Lawn . . . 
 
 Walton . . . 
 
 Gordon . . 
 
 Sibley . . . 
 
 Waverly . . 
 
 Halle . . . 
 
 Record in 
 A test 
 
 32.8 
 28.3 
 28.0 
 26.3 
 26.1 
 25.5 
 25.0 
 24.6 
 24.0 
 23.0 
 22.9 
 22.6 
 21.9 
 21.9 
 21.7 
 21.3 
 20.8 
 19.4 
 
 Record in 
 Etest 
 
 10.0 
 7.5 
 6.8 
 6.5 
 7.5 
 7.0 
 7.0 
 6.9 
 5.9 
 6.7 
 8.5 
 6.2 
 5.3 
 7.1 
 6.6 
 7.1 
 5.2 
 
 For our purposes it is not necessary to know which grades 
 were used, what the tests mean, or how they were figured. 
 With the records as given, the main question is : How did 
 the rankings of the schools on the two tests correspond ? 
 That is, did each school (or the majority of the schools) 
 
 1 From some material turned over to the author to be used as 
 practice work in his class on Statistical Methods Applied to Educa- 
 tion at the University of Chicago, summer quarter of 1915. 
 
Measures of Relationships 
 
 175 
 
 Rating 
 05 
 
 so 
 
 85 
 
 20 
 
 15 
 
 fO 
 
 Rating 
 35 
 
 
 30 
 
 25 
 
 20 
 
 A Test 
 
 IS 
 
 E Te4+ 
 
 z 
 o 
 
 a 
 (D 
 
 o Q 
 
 <J5z 
 
 d fc 
 
 kj o 
 
 -I (0 J 
 CD lJ CD 
 
 i! 
 
 o — 
 
 X 
 
 Fig. 27. — Graph to Show Correlation Based on Data of Table 19. 
 
176 School Statistics and Publicity 
 
 occupy the same, or approximately the same, relative 
 position from the best on one test that it did on the 
 other ? 
 
 Graphic Methods of Showing Correlations. On look- 
 ing at the figures we note that the poorest school on the 
 A test is also poorest on the E test. But the best on the 
 A test is only second on the E test. The second best on 
 the A test is best on the E test. Similar observations 
 might be continued without arriving at any idea of the 
 relation existing between the distributions as wholes. 
 
 But let us make two horizontal scales, one on each side 
 of the page, running from to 35, and plot the two tables 
 with the names of the schools on the base line, as in 
 Figure 27. 
 
 From this, it is apparent that in general the schools 
 have about the same relative positions on the two tests, 
 only there are some jogs in the lower curve showing that 
 the relationship does not hold for every case. If it did 
 hold absolutely for every case, the second curve would 
 be parallel to the upper. 
 
 Another way to study the relationship is to draw a 
 scale for each test on opposite sides of the paper, prefer- 
 ably making the two scales about the same height, and 
 drawing a straight line across from the correct place 
 on each scale to represent the performance of each school. 
 This is merely an extension of the device used by Pro- 
 fessor Haggerty.i If a school did as well on one test as 
 on another, its line, of course, is parallel to the lines of 
 most of the other schools. If it did not, the line will 
 slant, depending upon which test it made the best record 
 on. The Cleveland data are shown by this device in 
 Figure 28. Observe that many of the lines are roughly 
 
 iSeep. 172 
 
Measures of Relationships 
 
 177 
 
 parallel but that a fair number of them cut across each 
 other and do not follow the general trend of relationship. 
 
 Showing Correlation by Shifts of Cases within Parts 
 of the Distribution. One very rough way to show 
 correlation is to divide each . 
 group by its median. Then 
 look at each case to see if it ^^ 
 is in the same half (upper or 
 lower) in the second distri- 
 bution as in the first. Call ^^ 
 the upper half " plus " and 
 the lower half " minus " in 
 each distribution. Then get 
 the percentage of like signs. 
 This single figure which 
 expresses the general rela- 
 tionship between the two 
 distributions is called the 
 *' coefficient of correlation.' ' 
 
 Graphically this relation- 
 ship may be represented 15 
 easily by writing the cases 
 in one distribution in two ^ 
 
 - „ 1 1 li? Fig. 28. — Graphic Representation 
 
 colors, one tor each halt, say of Correlation between the Results 
 
 black for the upper and red ^^ ^^^ a Test and those in the 
 
 n ,1 1 rr\i 1 E Test for Eighteen Cleveland 
 
 tor the lower. then keep schools. 
 
 these same colors for each Each Une represents a school. (From 
 • 1 1 T X •!_ data in Table 19, page 174.) 
 
 case m the second distribu- 
 tion, no matter where the case falls. Suppose for example 
 that all cases in the upper half in the first distribution 
 appear black and all in the lower half appear red. In 
 the second distribution, if the cases fall exactly opposite, 
 the upper half will be red and the lower black. If there 
 
178 
 
 School Statistics and Publicity 
 
 is a good deal of shifting, there will be some red and some 
 black in each half. 
 
 For the Cleveland data, this device is shown in Table 
 20, the upper half of the first distribution being denoted 
 by all capital letters, and the lower half by small letters, 
 each case retaining its own type when it appears in the 
 second distribution. 
 
 Table 20. Like Signs Table for Correlation of Stand- 
 ings OF 18 Cleveland Schools on Two Tests, Form I 
 
 Median 
 
 Rank and name of school 
 
 Rank and name of school 
 
 
 for A test 
 
 
 for E test 
 
 1 
 
 BROWNELL 
 
 1 
 
 CLARK 
 
 2 
 
 CLARK 
 
 2 
 
 BROWNELL 
 
 3 
 
 MARION 
 
 3 
 
 London 
 
 4 
 
 DETROIT 
 
 4.5 
 
 MARION 
 
 5 
 
 FULLERTON 
 
 4.5 
 
 SACKETT 
 
 6 
 
 SACKETT 
 
 6.5 
 
 Gordon 
 
 7 
 
 NORTH DOAN 
 
 6.5 
 
 Waverly 
 
 8 
 
 BOLTON 
 
 8.5 
 
 NORTH DOAN 
 
 9 
 
 EAST BOULEVARD 
 
 8.5 
 
 BOLTON 
 
 10 
 
 Gilbert 
 
 10 
 
 EAST BOULEVARD 
 
 11 
 
 Rosedale 
 
 11 
 
 DETROIT 
 
 12 
 
 London 
 
 12 
 
 Rosedale 
 
 13 
 
 Lawn 
 
 13 
 
 Sibley 
 
 14 
 
 Walton 
 
 14 
 
 FULLERTON 
 
 15 
 
 Gordon 
 
 15 
 
 Lawn 
 
 16 
 
 Sibley 
 
 16 
 
 Gilbert 
 
 17 
 
 Waverly 
 
 17 
 
 Walton 
 
 18 
 
 Halle 
 
 18 
 
 Halle 
 
 Twelve of the cases are in the corresponding half in the 
 two distributions, so the coefficient of correlation is 
 roughly i| or 66| per cent.^ 
 
 1 This is not the real figure for this coefficient nor is the procedure 
 used a strictly accurate one. But both are sufficiently accurate for 
 many of the correlations that the superintendent will need until he 
 
Measures of Relationships 
 
 179 
 
 In using this method there must be as many ranks in 
 each distribution as there are pairs of cases, — in this 
 instance 18. If two schools tie, they get the average 
 rank of the cases occupied by them. Note that Marion 
 and Sackett on the E test are given 4.5 each because they 
 tied for the 4th place and thus occupy the 4th and 5th 
 ranks, or average 4.5 each. 
 
 Another form of such a table keeps every case on the 
 same line for the two distributions, as in Table 21. 
 
 Table 21. Like Signs Table for Correlation of Standings 
 OF 18 Cleveland Schools on Two Tests, Form II 
 
 Median 
 
 Rank and name of school 
 
 Rank and name of school 
 
 
 for A test 
 
 
 for E test 
 
 1 
 
 BROWNELL 
 
 2 
 
 BROWNELL 
 
 2 
 
 CLARK 
 
 1 
 
 CLARK 
 
 3 
 
 MARION 
 
 4.5 
 
 MARION 
 
 4 
 
 DETROIT 
 
 11 
 
 Detroit 
 
 5 
 
 FULLERTON 
 
 14 
 
 Fullerton 
 
 6 
 
 SACKETT 
 
 4.5 
 
 SACKETT 
 
 7 
 
 NORTH DOAN 
 
 8.5 
 
 NORTH DOAN 
 
 8 
 
 BOLTON 
 
 8.5 
 
 BOLTON 
 
 9 
 
 EAST BOULEVARD 
 
 10 
 16 
 
 East Boulevard 
 
 10 
 
 Gilbert 
 
 Gilbert 
 
 11 
 
 Rosedale 
 
 12 
 
 Rosedale 
 
 12 
 
 London 
 
 3 
 
 LONDON 
 
 13 
 
 Lawn 
 
 15 
 
 Lawn 
 
 14 
 
 Walton 
 
 17 
 
 Walton 
 
 15 
 
 Gordon 
 
 6.5 
 
 GORDON 
 
 16 
 
 Sibley 
 
 13 
 
 Sibley 
 
 17 
 
 Waverly 
 
 6.5 
 
 WAVERLY 
 
 18 
 
 Halle 
 
 18 
 
 Halle 
 
 By looking at the right hand or E test column, it is 
 seen at once that there are three cases out of place in the 
 
 has studied technical statistics much more thoroughly than is here 
 attempted. 
 
180 School Statistics and Publicity 
 
 upper part and three more in the lower. That is, 12 
 cases out of 18 or 66f per cent of the cases are ahke 
 in going into the same half of the two distributions. 
 
 A much more elaborate device of the same nature, but 
 covering two grouped distributions, was used by Pro- 
 fessor Dearborn to show the relation between the achieve- 
 ments of a group of students in high school mathematics 
 and their achievements in college mathematics.^ He 
 gave each student a number and placed this number in 
 the proper column on a base scale carrying the grades 
 from 60 to 100, so as to make a surface of frequency. In 
 the high school mathematics distribution, all numbers 
 below the first quartile were black ; all between the first 
 quartile and the median, green ; all between the median 
 and the third quartile, purple ; all above the third 
 quartile, red. Thus there were four equal areas, each 
 represented by a solid color (all numbers in it having the 
 same color). The lowest fourth appeared as a black 
 mass, the second fourth as a green one, the third 
 fourth as a purple one, and the upper fourth as a 
 red one. 
 
 With the same number and color to represent the same 
 student, the numbers were put in their proper places 
 so as to make up the surface of frequency for achievement 
 in university mathematics. This gave the second dis- 
 tribution. All the students did not keep exactly the 
 same positions that they had in the first distribution. 
 Their switching around showed at once on the second 
 distribution because the colors in one mass were not all 
 the same but appeared variegated. The switching 
 around in colors then gave a rough indication of the 
 
 1 Dearborn, W. F. : " School and University Grades," in Bulletin 
 of University of Wisconsin, No. 368, H. S. Series No. 9, p. 48 
 
Measures of Relationships 181 
 
 relation or opposition (correlation) of the order of cases 
 in the two distributions. 
 
 Coefficient of Correlation. The foregoing ways of 
 showing correlation are not exact enough and take too 
 much space to show the desired relation with the best 
 results in all cases. Consequently there have been 
 devised various ways of showing this relation by the 
 single figure called the coefficient of correlation. The 
 calculation of this coefficient except by the very rough 
 "like signs method " is generally very laborious, and the 
 knowledge of what the results mean in the given circum- 
 stances is very hard to acquire. 
 
 The superintendent should not, in general, enter upon 
 the calculation of this coefficient without much more 
 extensive study of technical statistics than can be given 
 here. Fortunately, there are few cases where the su- 
 perintendent really needs to calculate this coefficient 
 unless he is working on a thesis in some university, where 
 he will be shown how to make the calculations. However, 
 he does in his reading need to understand what the co- 
 efficient of correlation means. Accordingly, we shall 
 now explain clearly what it is, give some practical 
 examples, and then merely append two ways of cal- 
 culating it. 
 
 Briefly, the coefficient of correlation is a number ex- 
 pressing relationship between two distributions, based 
 upon the changes in order or ranks of the different cases 
 in the two groups. It ranges from + 1 (or + 100 per 
 cent) expressing perfect agi^eement in order, through 
 zero expressing only accidental or no relation, to - 1 (or 
 - 100 per cent) expressing perfect opposition in order or 
 ranking. In perfect relationship, the first case in distri- 
 bution one will be the first case in distribution two ; the 
 
182 School Statistics and Publicity 
 
 second case in distribution one will be the second case in 
 distribution two, etc. In perfect opposition, the first case 
 in distribution one is the last case in distribution two ; the 
 second case in distribution one is the next to the last 
 case in distribution two, etc. 
 
 There is marked difference of opinion among statisticians 
 and investigators as to the significance of the size of a 
 coefficient of correlation. Any reader of educational articles 
 in the last ten years which employ coefficients of correlation, 
 will have noted the shifts and controversies on this point. 
 Coefficients once thought large enough to be of importance, 
 are now considered negligible. The size depends often 
 upon the particular method of calculation. The importance 
 of the size depends upon the particular method used for the 
 particular distribution, a matter about which the authorities 
 often differ. Probably the following from Professor Rugg 
 is as safe as anything for the superintendent in his 
 reading : 
 
 This definition of limits (of high and low correlation) depends 
 largely on the personal experience of the person making the interpre- 
 tation. For example, it has been common for certain educational 
 investigators to interpret arbitrarily a coefficient of .25 as an indi- 
 cation of "high" positive 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 "negligible" 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 educational 
 testing, few correlations in testing will run above .70, and it is safe 
 to regard this as a very high coefficient.^ 
 
 1 Rugg, H. 0. : Statistical Methods Applied to Education, p. 256 
 
Measures of Relationships 
 
 183 
 
 Examples of Coefficients of Correlation. The following 
 examples of coefficients of correlation will be of signifi- 
 cance to superintendents : 
 
 Between what 
 items 
 
 Coefficient of 
 correlation 
 
 Where found 
 
 Total cost per pupil 
 
 
 Strayer, G. D. : City 
 
 with 
 
 + .93 
 
 School Expenditures, 
 
 Teaching and supervision 
 
 
 p. 95 
 
 (49 city systems, 1902-3) 
 
 
 
 Supervision 
 
 
 
 with 
 
 + .15 
 
 Same 
 
 Eepairs (same) 
 
 
 
 Payments for schools 
 
 
 Elliott, E. C: Some 
 
 with 
 
 - .541 
 
 Fiscal Aspects of Public 
 
 Payments for interest 
 
 
 Education, p. 85 
 
 on city debt 
 
 
 
 General merit of teachers 
 
 
 Boyce, A. C.: "Methods 
 
 with 
 
 + .54 
 
 for Measuring Teachers' 
 
 Neatness of room 
 
 
 Efficiency," in 1 Uth Year- 
 hook, National Society 
 for Study of Education, 
 Part II, p. 68 
 
 General merit of teachers 
 
 
 
 with 
 
 + .88 
 
 Same 
 
 General development of 
 
 
 
 pupils 
 
 
 
 General merit of teachers 
 
 
 
 with 
 
 + .79 
 
 Same 
 
 Discipline 
 
 
 
 Ability in reasoning 
 
 
 Stone, C. W. : Arithmeti- 
 
 with 
 
 + .73 
 
 cal Abilities, p. 37 
 
 Ability in fundamentals 
 
 
 
 in arithmetic 
 
 
 
 Ability in reasoning 
 
 
 
 with 
 
 + .32 
 
 Same 
 
 Ability in addition 
 
 
 
184 School Statistics and Publicity 
 
 Calculation of Coefficient of Correlation. The original 
 treatment of correlation ended with the preceding exam- 
 ples, but several readers of the manuscript recommended 
 adding one or more simple calculations of a coefficient. 
 For reasons which will not be clear to the casual reader, 
 the coefficient of correlation calculated from the Cleveland 
 data would not be reliable enough to be worth the effort to 
 get it.^ These reasons center about the fact that the orig- 
 inal measures were averaged from a number of individuals 
 and not a simple measure for each school. But as it would 
 take too much space and time to introduce and explain 
 new sets of data, we shall assume that the data are suit- 
 able for the purpose and proceed to give the calculations. 
 The method of calculation is the only thing we care for here 
 and it can be shown as well with the Cleveland data as with 
 any other. The reader who has had no other train- 
 ing in statistics is particularly cautioned against attempt- 
 ing to calculate coefficients. The examples are given 
 solely to clear up a reading knowledge of correlation. 
 
 1. Spearman Rank-Order Method ^ 
 
 6SD2 
 
 P =1 ~~:; 77 
 
 n(n2 - 1) 
 
 S = sum of 
 
 D2 = squares of differences in ranks (must be same number of ranks 
 
 in both distributions) 
 
 n = number of pairs in distributions 
 
 P = coefficient of correlation 
 
 1 See J. F. Kelley : Teachers' Marks, p. 88 
 
 2 The formula here given is only one factor in the correct formula, 
 but it is sufficiently accurate for administrative problems. See Thorn- 
 dike : Mental and Social Measurements, p. 167 
 
Measures of Relationships 
 
 185 
 
 From page 178 we get D^ as follows : 
 
 
 D 
 
 D' 
 
 Brownell ; 
 
 1 
 
 1.00 
 
 Clark 
 
 1 
 
 1.00 
 
 Marion 
 
 1.5 
 
 2.25 
 
 Detroit 
 
 7 
 
 49.00 
 
 Fullerton 
 
 9 
 
 81.00 
 
 Sackett 
 
 1.5 
 
 2.25 
 
 North Doan 
 
 1.5 
 
 2.25 
 
 Bolton 
 
 .5 
 
 .25 
 
 East Boulevard 
 
 1 
 
 1.00 
 
 Gilbert 
 
 6 
 
 36.00 
 
 Rosedale 
 
 1 
 
 1.00 
 
 London 
 
 9 
 
 81.00 
 
 Lawn 
 
 2 
 
 4.00 
 
 Walton 
 
 3 
 
 9.00 
 
 Gordon 
 
 8.5 
 
 72.25 
 
 
 3 
 
 9.00 
 
 Waverly. 
 
 10.5 
 
 110.25 
 
 Halle. 
 
 
 
 0.00 
 
 
 
 462.50 
 
 p = l- 
 
 6 X 462.50 
 
 = +.52 
 
 18(324 - 1) 
 
 2. The Pearson Method 
 
 The Pearson coefficient of correlation is calculated by the following 
 formula : 
 
 Coefficient of Correlation or r= (^ • V) — 
 
 Vsx2 VS2/2 
 
 S = algebraic sum 
 X = deviations in one distribution 
 y = deviations in other distribution 
 X ' y = product of deviations for corresponding case. 
 
186 
 
 School Statistics and Publicity 
 
 The calculation on the Cleveland data is as follows, the deviations 
 being taken from the median : 
 
 School 
 
 Score 
 
 X 
 
 x' 
 
 Score 
 
 y 
 
 y' 
 
 -\-xy 
 
 —xy 
 
 Brownell . 
 
 32.8 
 
 +9.3 
 
 86.49 
 
 8.8 
 
 + 1.85 
 
 3.42 
 
 + 17.21 
 
 
 Clark . . 
 
 28.3 
 
 +4.8 
 
 23.04 
 
 10.0 
 
 +3.05 
 
 9.30 
 
 + 14.64 
 
 
 Marion 
 
 28.0 
 
 +4.5 
 
 20.25 
 
 7.5 
 
 + .55 
 
 .30 
 
 + 2.48 
 
 
 Detroit 
 
 26.3 
 
 +2.8 
 
 7.84 
 
 6.8 
 
 - .15 
 
 .02 
 
 
 - .42 
 
 FuUerton . 
 
 26.1 
 
 +2.6 
 
 6.76 
 
 6.5 
 
 - .45 
 
 .20 
 
 
 -1.17 
 
 Sackett 
 
 25.5 
 
 +2.0 
 
 4.00 
 
 7.5 
 
 + .55 
 
 .30 
 
 + 1.10 
 
 
 N. Doan . 
 
 25.0 
 
 +1.5 
 
 2.25 
 
 7.0 
 
 + .05 
 
 .003 
 
 + .80 
 
 
 Bolton . . 
 
 24.6 
 
 + 1.1 
 
 1.21 
 
 7.0 
 
 + .05 
 
 .003 
 
 + .06 
 
 
 E. Blvd. . 
 
 24.0 
 
 + .5 
 
 .25 
 
 6.9 
 
 - .05 
 
 .003 
 
 
 .03 
 
 GUbert . 
 
 23.0 
 
 - .5 
 
 .25 
 
 5.9 
 
 -1.05 
 
 1.10 
 
 + .53 
 
 
 Rosedale . 
 
 22.9 
 
 - .6 
 
 .36 
 
 6.7 
 
 - .25 
 
 .06 
 
 + .15 
 
 
 London . 
 
 22.6 
 
 - .9 
 
 .81 
 
 8.5 
 
 + 1.55 
 
 2.40 
 
 
 1.40 
 
 Lawn . . 
 
 21.9 
 
 -1.6 
 
 2.56 
 
 6.2 
 
 - .75 
 
 .56 
 
 + 1.20 
 
 
 Walton . 
 
 21.9 
 
 -1.6 
 
 2.56 
 
 5.3 
 
 -1.65 
 
 2.72 
 
 + 2.64 
 
 
 Gordon 
 
 21.7 
 
 -1.8 
 
 3.24 
 
 7.1 
 
 + .15 
 
 .02 
 
 
 - .27 
 
 Sibley . . 
 
 21.3 
 
 -2.2 
 
 4.84 
 
 6.6 
 
 - .35 
 
 .12 
 
 + .77 
 
 
 Waverly . 
 
 20.8 
 
 -2.7 
 
 7.29 
 
 7.1 
 
 + .15 
 
 .02 
 
 
 - .41 
 
 Halle . . 
 
 19.4 
 
 -4.1 
 
 16.81 
 
 5.2 
 
 -1.75 
 
 3.06 
 
 + 7.18 
 
 
 
 Med. 23.50 
 
 190.81 
 
 Med. 6.95 
 
 23.61 
 
 +48.04 
 
 -3.70 
 
 I 
 
 S xy 
 
 44.34 
 
 V 
 
 2;x-^ 
 
 Vs 
 
 = + .66 Ans. 
 
 V190.81 V23.61 
 
 (13.81) (4.86) 
 
 The Pearson coefficient in actual work is often figured from the 
 guessed average, the result being corrected by a formula given in this 
 Thorndike and Rugg references. This saves time because the devia- 
 tions will thus always be integers or steps denoted by integers, and the 
 handling of these will be much easier than when deviations with deci- 
 mals are used as in the example just given. 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 King, W. I. Elements of Statistical Method, Chapters XIV, XVI, 
 
 XVII, XVIII. 
 Rugg, H. O. Statistical Methods Applied to Education, Chapters 
 
 VII, VIII, IX. 
 Thorndike, E. L. Mental and Social Measurements, Chapters X, XI. 
 
CHAPTER VIII 
 SUPPLEMENT ON STATISTICAL TREATMENT 
 
 The preceding chapters cover most of the problems 
 which the superintendent will encounter in working up 
 his school statistics, until he reaches the actual presenta- 
 tion to the public. But there are three other practical 
 problems, not directly connected with any of the fore- 
 going problems nor with one another, which will be of 
 interest to him. These are : 
 
 1. How may one insure reliability in the statistical results or at 
 least know about how accurate they are? 
 
 2. What special economies may be employed to reduce the in- 
 evitably large labor involved in statistical calculations? 
 
 3. How may data given in ranks only (for example, the decisions 
 of judges in a contest) be easily combined? 
 
 These problems will now be discussed in order. 
 
 I. RELIABILITY OF STATISTICAL RESULTS 
 
 The aim throughout this book has, of course, been to 
 produce reliable results, and cautions for this purpose 
 have been included in many places. But in spite of 
 these, the reader will probably wish to know how to 
 avoid the numerous errors in adding, multiplying, omit- 
 ting figures, etc., that all of us understand are liable to 
 creep into any numerical work. 
 
 Variable Errors. The simplest way to regard these is 
 to distinguish between variable and constant errors. A 
 
 187 
 
188 School Statistics and Publicity 
 
 variable error is one that is as likely to occur one way as 
 the other and so the various results will offset each other. 
 For example, in adding, the average person is as apt to get 
 results too large about as often as he gets them too small 
 and in about the same amounts. In any problem in- 
 volving many additions, these errors will balance each 
 other and so the result may be assumed to be fairly correct. 
 The principle is exactly that involved in the common 
 practice of discarding the last decimal place by adding 1 
 to the preceding place if this decimal is 5 or more, and 
 throwing it away altogether if it is less than 5. Or it is 
 the same principle as that behind the old business prac- 
 tice of settling a bill to the nearest five cents, — paying 
 $1.25 if the bill comes to $1.23 or only $1.20 if it comes to 
 $1.22. In both of these, it is assumed that in the long 
 run things will even up and no essential injustice or in- 
 accuracy will result. 
 
 Most of the errors in ordinary statistical calculations 
 are variable ones and, if reasonable care is given the work, 
 they may then be ignored. But they may be still further 
 removed by the devices of utilizing students or assistants 
 to check each other's work, or of going through each pro- 
 cess at least twice. If the result comes out the same each 
 time, it may be safely assumed to be correct. If, after 
 repeated trials, the results all vary slightly, the average 
 of them is safe enough for practical purposes. 
 
 Constant Errors. A constant error, on the other hand, 
 is one that always tends in the same direction. The 
 marks in deportment given healthy boys by a well- 
 poised, healthy teacher would in the long run be sub- 
 stantially correct. If he were ill one month and cut 
 them very low, the result would probably be offset by 
 correspondingly high marks in some month when he was 
 
Supplement on Statistical Treatment 189 
 
 feeling well. That is, his variable errors would offset 
 each other. But if these same boys were under a nervous 
 or irritable teacher, his errors would probably all tend 
 in the direction of giving them very low marks. That is, 
 he would have a constant error. Similar constant errors 
 may be expected in such cases as these : reports on 
 number of tardies, because teachers easily forget to mark 
 tardies but are very unlikely to mark present pupils as 
 absent at the time of noting tardies; reports on training 
 of teachers, because teachers are apt to try to make their 
 training seem as extensive as possible and very unlikely 
 to understate it ; itemized and verified expense accounts, 
 which usually tend to be less than the actual expenditure 
 because of the general tendency to forget to put down 
 items, and the absence of intent to be dishonest. 
 
 There is no general rule for avoiding constant errors, 
 except possibly to watch for fallacies in sampling. Con- 
 stant errors are never , negligible but " skill in avoiding 
 them is due to capacity and watchfulness far more than 
 to knowledge of any formal rules." ^ 
 
 Weighting Results. Sometimes the attempt is made 
 to avoid errors by weighting certain factors to give them 
 more influence in determining the results. This should 
 never be attempted in the belief that it will give more 
 accurate results, except by one especially trained in 
 statistics. " Bowley gives a rule that is satisfactory 
 for most cases that occur in practice; namely, to give 
 your attention to eliminating constant errors and not to 
 manipulating weights." ^ 
 
 Estimating Reliability of Results. All complete treat- 
 ments of statistical methods include formulas for estimat- 
 
 1 Thorndike, E. L. : Mental and Social Measurements, p. 209 
 
 2 Ibid., p. 212 
 
I 
 
 190 School Statistics and Publicity 
 
 ing the reliability of the various averages, measures of 
 dispersion, or coefficient of correlation. But in practi- 
 cal school statistics there is no need for these. The 
 superintendent had far better employ his time in seeing 
 that he avoids the errors in collecting and manipulating 
 his data that common sense and reasonable care would 
 indicate. Beyond this, he needs to know only that, if I 
 constant errors are avoided, accuracy is increased by 
 taking more measures of an item, or samples from a 
 distribution. From a random sampling of any material 
 number of cases, accuracy increases as the square root 
 of the number of samples. That is, 400 samples would | 
 double the accuracy obtained by taking only 100 samples. 
 
 II. SPECIAL ECONOMIES IN CALCULATION 
 
 Many economies have been indicated in connection 
 with the particular processes to which they pertain. 
 But there are several others that merit special notice. 
 
 Use of Cross-Section Paper. Plain cross-section paper 
 is very valuable for saving time in getting figures lined 
 up in columns and in keeping horizontal lines straight. 
 It also lessens eyestrain. The kind with a little square 
 just large enough for one digit is best. This paper is 
 especially valuable for saving copying, because a part 
 of a calculation that is correct or that needs to be shifted 
 can easily be cut out, removed to where it is needed, and 
 quickly pasted into line there. 
 
 Checking in from Both Ends. The old device of 
 checking addition by adding the column up and then down 
 until the results agree is very helpful for insuring accuracy. 
 The device of calculating the median by going up, and 
 then by coming down, until the two results agree, is 
 equally serviceable. 
 
Supplement on Statistical Treatment 191 
 
 Use of Steps in Distributions Involving Large Numbers. 
 If the distribution involves large numbers, time can 
 usually be saved by employing steps instead of the actual 
 numbers for getting the average from the guessed average, 
 for getting the deviation measures, etc. Thus, if one is 
 studying superintendents' salaries grouped by hundreds, 
 and the guessed average is 1500, the deviations do not 
 have to be given in hundreds. The 1600 group could be 
 counted as one step above ; the 1400 group, one step 
 below, etc. The deviation measure finally calculated would 
 then be in steps ; if multiplied by $100, the value of a step, 
 the real deviation measure would be at once secured. 
 This method obviously saves much labor in copying zeros. 
 It would save much more labor in multiplication, if the 
 value of a step was some such number as 25, 5.5, etc. 
 
 One Step All the Way Through. It is much easier men- 
 tally and insures accuracy as well, to perform the same 
 process all the way through at one time. Thus, all data 
 may be copied to the end ; all multiplications may be 
 made at one step ; all additions made at another ; all 
 squares found at another, etc. 
 
 Calculating Devices. Various tables may be utilized 
 to save time in multiplying, dividing, squaring, or ex- 
 tracting roots. These are mentioned in the bibliography. 
 The writer has used Crelle for years, but this is not now 
 obtainable in this country. The short tables given in 
 Thorndike's Mental and Social Measurements can often 
 be utilized by employing the step method as given 
 above. These tables are not at all difficult for even school 
 children to use. Some years ago, the twelve-year-old 
 son of a friend of the writer regularly used the Thorndike 
 tables in computing the batting averages of his favorite 
 baseball stars. 
 
 I 
 
192 
 
 School Statistics and Publicity 
 
 Most superintendents will be able to get the use of an 
 adding machine at the city hall, the county courthouse, 
 or some bank, if the school system does not have one. 
 
 Utilizing Students in Calculations. What was said 
 about the value to students of collecting data (see page 
 86), applies even more to practice on the calculations 
 involved in working them up. Such data afford the 
 finest sort of laboratory material for the upper arithmetic 
 and high school mathematics classes. And much of the 
 work is very valuable for clerical practice. 
 
 m. COMBINING DATA GIVEN IN RANK ORDER ONLY 
 
 The school administrator often faces a problem which 
 involves a summary for the relative merits of a number 
 of items that have been graded on different phases with 
 widely varying standards, or by different judges with 
 varying standards. Thus, he may wish to find the vale- 
 dictorian in the high school from the grades given by 
 many teachers. He may wish to find the best contestant 
 in a debating contest from decisions given by several 
 judges. Or he may wish to find which ward school is 
 on the whole the best, as judged by results from standard 
 tests in arithmetic, composition, spelling, and handwrit- 
 ing, all given on very different scales. In all of these 
 he must find the relative standing or rank of several 
 people or schools. How can he combine the rankings 
 given from one judge or one test with those from another 
 judge or another test ? 
 
 These problems may be conveniently classified for 
 practical purposes as being of five types : 
 
 1. How may rankings from different distributions, having the same 
 number of ranks and being considered as of equal value, be combined ? 
 For example, the decisions of judges in a contest. 
 
 I 
 
Supplement on Statistical Treatment 193 
 
 2. How are tie rankings within one or more distributions to be 
 treated when being combined with rankings from the other distribu- 
 tion? 
 
 For example, one judge in the contest gives a tie vote on two 
 contestants. 
 
 3. How are rankings from one distribution to be combined with 
 rankings from other distributions which have a different number of 
 ranks ? 
 
 For example, a rank from a class of 12 in chemistry is to be 
 combined with a rank from a class of 20 in English ; or rankings 
 on daily grades from a class of 20 with rankings for only 18 
 who took the examination. 
 
 4. How are rankings from different distributions to be combined 
 when it is desired to give special weight to one or more of the dis- 
 tributions ? 
 
 For example, it is desired to count rankings on daily grades 
 three times as much as rankings on examination, to secure the 
 rankings for final grades. 
 
 5. How is absolute unfairness on the combinations to be avoided? 
 
 For example, in the author's class in statistical methods at one 
 time, tests were being given on which the class were ranked. 
 The rankings were obtained from two separate distributions, 
 one of rankings on speed, the other of rankings on accuracy. 
 Speed and accuracy were counted equal. On one test, a stu- 
 dent at the beginning handed in a blank paper, claiming first 
 rank on speed and being willing to take lowest rank on accuracy. 
 With thirty in the class, he counted on a final or average ranking 
 
 of — X_ , or 15.5, or about the middle of the class. 
 2 
 
 These will be discussed in order. 
 
 1. Distributions Having Same Number of Ranks 
 
 This is managed by adding the various ranks ; then 
 ranking the items in order of the sums, giving the one 
 with the smallest sum a final rank of 1, the one with the 
 next smallest sum a rank of 2, etc. 
 
194 
 
 School Statistics and Publicity 
 
 For example, take the illustration of the judges in a contest, given 
 on page 12, changing some of the figures to avoid any ties. This 
 would keep the number of ranks the same for each judge. The orig- 
 inal marks would be : 
 
 Changing these to ranks we would have : 
 
 Contestant 
 
 Ranked by 
 
 Ranked by 
 
 Ranked by 
 
 Sum of 
 
 Final 
 
 Judge A 
 
 Judge B 
 
 Judge C 
 
 ranks 
 
 rank 
 
 1 
 
 10 
 
 6 
 
 10 
 
 . 26 
 
 9 
 
 2 
 
 1 
 
 8 
 
 5 
 
 14 
 
 5 
 
 3 
 
 8 
 
 5 
 
 7 
 
 20 
 
 7 
 
 4 
 
 2 
 
 1 
 
 6 
 
 9 
 
 lor 21 
 
 5 
 
 7 
 
 7 
 
 8 
 
 22 
 
 8 
 
 6 
 
 3 
 
 3 
 
 4 
 
 10 
 
 3 
 
 7 
 
 4 
 
 2 
 
 3 
 
 9 
 
 lor2i 
 
 8 
 
 6 
 
 4 
 
 2 
 
 12 
 
 4 
 
 9 
 
 5 
 
 10 
 
 1 
 
 16 
 
 6 
 
 10 
 
 9 
 
 9 
 
 9 
 
 27 
 
 10 
 
 If the judges had rated the contestants by rankings in the first 
 place, the result could, of course, have been obtained much more 
 quickly. 
 
 1 These tie, and average 1.5 each. See page 195. 
 
Supplement on Statistical Treatment 195 
 
 2. Tie Rankings 
 
 This is handled by changing the tie rankings so as to 
 make the same number of ranks in each distribution, 
 then proceeding as in 1. 
 
 For example, the rankings from the marks of the judges as given 
 on page 12 would be : 
 
 
 Ranked by 
 
 Ranked by 
 
 Ranked by 
 
 
 Judge A 
 
 Judge B 
 
 Judge C 
 
 1 
 
 8 
 
 4 
 
 5 
 
 2 
 
 1 
 
 5 
 
 2 
 
 3 
 
 6 
 
 4 
 
 3 
 
 4 
 
 2 
 
 1 
 
 2 
 
 5 
 
 6 
 
 4 
 
 4 • 
 
 6 
 
 3 
 
 2 
 
 1 
 
 7 
 
 4 
 
 2 
 
 1 
 
 8 
 
 5 
 
 3 
 
 1 
 
 9 
 
 5 
 
 5 
 
 1 
 
 10 
 
 7 
 
 4 
 
 3 
 
 Obviously, these could not be combined as they now are, for two 
 judges used only five ranks and the other one had eight, so that rank 
 1 does not mean the same in the three distributions. 
 
 The trouble will be removed by changing the rankings so as to 
 make each judge cover the entire ten ranks. Thus, in the original 
 table on page 12, Judge A has given Contestants 8 and 9 each the 
 rank of 5. This means he has really made them take up ranks 5 and 
 6, or average 5.5 in rank. This leaves Contestants 3 and 5 as ranked 
 by him to occupy ranks 7 and 8 or average rank 7.5 each. Proceeding 
 thus, the real rankings from the figures on page 12 would be those 
 given on the following page. 
 
 It would, of course, have been easier to require the judges to give 
 no tie rankings, but if they should insist on turning in tie ranks, the 
 only safe procedure is that given. 
 
196 
 
 School Statistics and Publicity 
 
 
 Real rank 
 
 Real rank 
 
 Real rank 
 
 Sum 
 
 Final 
 rank 
 
 Contestant 
 
 by 
 
 by 
 
 by 
 
 of 
 
 
 Judge A 
 
 Judge B 
 
 Judge C 
 
 ranks 
 
 1 
 
 10 
 
 6.5 
 
 10 
 
 26.5 
 
 10 
 
 2 
 
 1 
 
 9.5 
 
 5.5 
 
 16 
 
 5 
 
 3 
 
 7.5 
 
 6.5 
 
 7.5 
 
 21.5 
 
 7 
 
 4 
 
 2 
 
 1 
 
 5.5 
 
 8.5 
 
 2 
 
 5 
 
 7.5 
 
 6.5 
 
 9 
 
 23 
 
 8.5 
 
 6 
 
 3 
 
 2.5 
 
 2.5 
 
 8 
 
 1 
 
 7 
 
 4 
 
 2.5 
 
 2.5 
 
 9 
 
 3 
 
 8 
 
 5.5 
 
 4 
 
 2.5 
 
 12 
 
 4 
 
 9 
 
 5.5 
 
 9.5 
 
 2.5 
 
 17.5 
 
 6 
 
 10 
 
 9 
 
 6.5 
 
 7.5 
 
 23 
 
 8.5 
 
 3. Different Number of Ranks in the Distributions 
 
 This is practically the same as the problem of a tie 
 vote, except that here, instead of taking the average of a 
 set of items and getting the average rank for each, one may 
 have to take the average range of an item to get its rank. 
 
 For example, suppose that one wishes to combine 
 ranks of a class of 12 in chemistry with those from a class 
 of 20 in English. The ranks in chemistry should cover 
 20. This will make one rank in chemistry cover 1.6 
 ranks in English. If the second and third pupils in 
 chemistry tied, they would cover ranks 3.2 and 4.8, or 
 average rank 4 each. 
 
 In the case where 20 pupils were ranked on daily grades, 
 but only 18 took the examination, a similar procedure 
 could be followed. But a much shorter approximate 
 result might be obtained by making the 18 pupils get 
 ranks 1 to 18 on examination, giving 19.5 each on examina- 
 tion to the two absent pupils. The question of passing 
 pupils absent from examinations would have to be 
 decided entirely apart from the matter of ranks. 
 
Supplement on Statistical Treatment 197 
 
 4. Weighting Factors Given in Ranks from Different 
 
 Distributions 
 
 The way to do this is to arrange the factors by ranks, 
 with the same number of ranks in each distribution. Then 
 multiply the factor to be weighted by the number of 
 times it is desired to weight it. The ranks can then be 
 added as in previous instances. 
 
 For instance, suppose pupils have been rated as follows on exam- 
 ination and daily grades : 
 
 Pupil 
 
 Ranks on 
 
 Rank on 
 
 
 daily grade 
 
 examination 
 
 
 A 
 
 1 
 
 5 
 
 
 B 
 
 2 
 
 2 
 
 
 C 
 
 3 
 
 4 
 
 
 D 
 
 4 
 
 3 
 
 
 ■ E 
 
 5 
 
 1 
 
 
 If it is desired to count the daily grades three parts and the examina- 
 tion grade one part in determining the final rating, the original ratings 
 for daily grades should be multiplied by 3 ; thus : 
 
 
 Weighted 
 
 Rank in 
 
 Sum of 
 ranks 
 
 Final 
 
 Pupil 
 
 daily grade 
 
 examination 
 
 rank 
 
 
 rank 
 
 grade 
 
 
 
 A 
 
 3 
 
 5 
 
 8 
 
 1.5 
 
 B 
 
 6 
 
 2 
 
 8 
 
 1.5 
 
 C 
 
 9 
 
 4 
 
 13 
 
 3 
 
 D 
 
 12 
 
 3 
 
 15 
 
 4 
 
 E 
 
 ■ 15 
 
 1 
 
 16 
 
 5 
 
 5. Avoiding Absurdities in Combining Ranks 
 
 No mechanical device or general advice will take the 
 place of common sense in avoiding absurdities such as 
 
198 School Statistics and Publicity 
 
 that of the student mentioned on page 193, who handed 
 in a blank paper. In this particular instance, the com- 
 mittee of students appointed to consider the case rightly 
 decided that common sense and fairness demanded that 
 the student was not entitled to any consideration what- 
 ever until all those who had really tried to do something 
 had been assigned ranks. Similarly, the matter of giving 
 any ranks whatever to the two students who failed to 
 take the examination in the instance given on page 196 
 would have to be decided entirely apart from the mechan- 
 ical problem of adjusting the ranks. The whole matter 
 of ranking might be left until these two had taken the 
 examination. But as such a delay on reporting grades 
 or ranks is hardly practicable in most places, it would be 
 simpler to rank all the others and later, after the two had 
 taken the examination, to give them tie ranks that would 
 place them approximately where they belonged. Much 
 energy is sometimes wasted in trying to combine rankings 
 where the same subjects did go wholly through two differ- 
 ent sets. Thus a student was worried over what to do 
 with an experiment with fourteen white rats where one died 
 in training so that he had to train a new fourteenth rat. 
 A junior high school principal studying twenty pupils had 
 only nineteen of these in both distributions and wished to 
 take a new twentieth one for the second test. The simplest 
 and safest procedure is to take only the cases that appear 
 all the way through. 
 
 EXERCISE 
 
 Find the relative standing of each of the twenty cities for rights 
 in arithmetic, on the four processes combined, from the data on the 
 following page. 
 
Supplement on Statistical Treatment 199 
 
 Rankings of Twenty Indiana Cities on Rights in Arithmetic 
 FOR THE Courtis Tests in Fifth Grade Only ^ 
 
 
 
 Ranks 
 
 
 City 
 
 
 
 
 
 
 
 
 
 Addition 
 
 Subtraction 
 
 Multiplication 
 
 Division 
 
 1 
 
 8 
 
 6.5 
 
 13 
 
 14.5 
 
 2 
 
 19.5 
 
 17.5 
 
 20 
 
 20 
 
 3 
 
 3 
 
 4 
 
 3 
 
 6 
 
 4 
 
 12.5 
 
 16 
 
 15 
 
 7.5 
 
 5 
 
 19.5 
 
 19 
 
 19 
 
 19 
 
 6 
 
 5.5 
 
 5 
 
 9 
 
 14.5 
 
 7 
 
 16.5 
 
 12.5 
 
 10.5 
 
 9 
 
 8 
 
 16.5 
 
 14.5 
 
 7.5 
 
 3 
 
 9 
 
 15 
 
 20 
 
 18 
 
 17.5 
 
 10 
 
 7 
 
 3 
 
 4.5 
 
 13 
 
 11 
 
 5.5 
 
 9.5 
 
 13 
 
 16 
 
 12 
 
 14 
 
 9.5 
 
 6 
 
 11 
 
 13 
 
 2 
 
 1.5 
 
 1 
 
 1 
 
 14 
 
 12.5 
 
 9.5 
 
 10.5 
 
 10 
 
 15 
 
 9.5 
 
 14.5 
 
 16.5 
 
 12 
 
 16 
 
 18 
 
 17.5 
 
 16.5 
 
 17.5 
 
 17 
 
 1 
 
 1.5 
 
 2 
 
 2 
 
 18 
 
 9.5 
 
 -9.5 
 
 7.5 
 
 4.5 
 
 19 
 
 11 
 
 6.5 
 
 13 
 
 7.5 
 
 20 
 
 4 
 
 12.5 
 
 4.5 
 
 4.5 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 Rugg, H. O. Statistical Methods Applied to Education, pp. 134-147. 
 Thorndike, E. L. Mental and Social Measurements, pp. 51-59, 
 Chapters XII, XIV. 
 
 1 Adapted from M. E. Haggerty's "Arithmetic: A Cooperative 
 Study in Educational Measurements," Indiana University Bulletin, 
 Vol. XII, No. 18, p. 443. 
 
CHAPTER IX 
 
 USELESSNESS OF STATISTICS IN CURRENT 
 SCHOOL REPORTS 
 
 I. THE SITUATION 
 
 Thoughtful school men will agree with Professor 
 Hanus in his statement that many of the ordinary presen- 
 tations of statistical material on schools to the public are 
 useless. 1 In his investigation, which included a random 
 selection covering the entire country, he found that the 
 reports studied by him were '' vague in purpose, miscel- 
 laneous in subject matter, and hence ineffective/' ^ jjg 
 found that many of the tables were printed just as they 
 had been tabulated by the superintendent; that exactly 
 50 per cent of the reports studied had no adequate inter- 
 pretations of tables; that only 42 per cent contained 
 any comparative statistics whatever. 
 
 1 Compare, for example, the statement of Superintendent Giles 
 of Richmond, Ind. : "Much of the labor expended on school reports 
 is lost because they are not in a form useful for comparison." This 
 is contained in an article in Educational Administration and Super- 
 vision, Vol. II, p. 305, where he attempts "to make available for ad- 
 ministrative purposes a part of the annual reports of city superin- 
 tendents in Indiana to the State Department of Public Instruc- 
 tion." 
 
 2 Hanus, Paul H: "Town and City Reports" (more particularly 
 superintendents' reports), School and Society, Vol. Ill, p. 196 
 
 200 
 
statistics in Current School Reports 201 
 
 n. CAUSES OF THE USELESSNESS 
 
 Hanus makes the point that '' the useless tables of 
 statistics are often quite useless because they are mere 
 collections of working data unrelated and uninterpreted, 
 and also because they pertain only to the year under re- 
 view." 1 Or as he puts it in another place, '' without inter- 
 pretation or discussion, many statistics are meaningless 
 to most readers, whether lay or professional." ^ 
 
 Hanus believes that a part of this confusion exists 
 because one report is gotten out for several classes of 
 people who really need separate reports. He says that 
 the superintendents must give information concerning 
 schools to three classes : (1) the school board ; (2) the 
 teaching staff; (3) the public. The so-called annual 
 report is usually gotten up for the three classes with one 
 or two more or less consciously in mind, but it often 
 fails to be serviceable to any of the three. '' Reports to 
 the board or staff should always contain both the statistical 
 summaries and the details from which they are derived. 
 . . . Reports to the people should contain only sum- 
 maries, by items, of course, and for successive years or 
 periods; and special pains should be taken to interpret 
 them so clearly that he who runs may read." ^ The 
 report to the people should be ''an abstract or digest of 
 the reports to the board, brief but comprehensive and 
 clear." In addition to the three classes noted by Hanus 
 in his thoroughly representative reports, some super- 
 intendents apparently write for two other classes : 
 (1) the state or national educational officials, (2) students 
 of education, as indicated by Snedden and Allen. * But 
 
 1 Ibid., p. 196 2 j})ifi^^ p 195 3 /^^^^.^ p, 195 
 
 ^ School Reports and School Efficiency, pp. 4, 5 
 
202 
 
 School Statistics and Publicity 
 
 this, of course, only makes the reports all the worse for 
 the public. 
 
 Whatever the relative values of the different reports, 
 the superintendent should consider very carefully the 
 preparation of the school statistics for the public. Ifl 
 this is to be done effectively, he must first master the 
 statistical material in hand and get hold of the significant ■ 
 points to be found in the data. For help on this, hei 
 may consult previous chapters. But his time spent in 
 preparing the special report for the public will not be| 
 lost for either himself or his teachers. Honest statistics 
 gotten up to influence the public will influence school 
 men and teachers all the more strongly because the 
 striking points set forth so vividly in the popular presen- 
 tation will make more impression on teachers. The old 
 saying that we never completely know a thing until we 
 can tell it effectively to some one else applies here. The 
 superintendent never fully understands his own statistics 
 and their relations until he can present the significant 
 points effectively to the public. In short, to paraphrase 
 Snedden and Allen, in the main, the methods that will 
 give the maximum of publicity on school statistics will 
 probably result also in providing the most effective statis- 
 tical basis for school administration. ^ 
 
 I 
 
 III. DEVICES FOR EFFECTIVE PRESENTATION 
 
 But granted that the superintendent realizes the in- 
 effectiveness of his school statistics and is determined 
 to issue a good report specifically for the public, what 
 devices are available for him? There are only three 
 main devices for this purpose. He may graph his statis- 
 ^ School Reports and School Efficiency, p. 8 
 
Statistics in Current School Reports 203 
 
 tics; he may translate them into words; or he may 
 tabulate them. Each device is best suited for certain 
 conditions. Each has its strengths and weaknesses. 
 
 i 1. Graphic Presentations 
 
 A graph or drawing can be so constructed as to show 
 at a glance the significant things in a mass of technical 
 data. Then, too, it is so very simple that, if properly 
 made up, it can be readily grasped by the average man 
 who is totally unversed in the intricacies of statistical 
 manipulation. In general, however, it cannot be 
 accurately remembered or reproduced without consider- 
 able effort. 
 
 2. Translation 
 
 By translation is meant the transferring or inter- 
 pretation into ordinary language of statistical ideas, 
 relationships, and results. These need to be translated 
 for the ordinary man before they can be clear to him, 
 much less affect him to any appreciable extent. Such 
 translation is difficult because sometimes it is as hard to 
 give statistical results through mere words as it is to 
 describe a painting in words only. But the superintendent 
 in employing translation has only the same problem as 
 has any able editorial writer, any successful campaign 
 orator, any writer on agricultural or food topics, or any 
 writer for a society engaged in propaganda on a large 
 scale. Some of the best effects produced by such persons 
 are managed through translating statistical material in 
 word language only. A good translation can be easily 
 remembered and used in a speech or article without any 
 special preparation. 
 
204 School Statistics and Publicity 
 
 3. Tabulation 
 
 Probably on the whole the least serviceable device is 
 tabulation. The ordinary tabulations and masses of 
 statistics in most school reports are of very little value 
 for the public. The significant facts do not stand out in 
 tables of this sort. It requires much labor for any 
 person other than the maker of an ineffective table to 
 get anything out of it. Certainly it would be unreasonable 
 to expect the average man to waste any time on tables 
 of this nature, even if he can be persuaded to read the 
 rest of the report. In such tables, comparisons with 
 similar data are usually impossible. If facts about a 
 number of ward schools in the system are given, the 
 arrangement may be alphabetical, and the reader cannot 
 find whether School A is better than School B without 
 some labor. Norms or standards with which any item 
 in the table might be compared may be left out. There 
 may be no units of measurement. Often great masses 
 of facts related slightly or not at all to each other are 
 thrown together in the same table. 
 
 Even if helpful devices are employed in tabulation, 
 or if the table is arranged like a scale with the largest 
 magnitude at the top and constantly decreasing, or if 
 heavy type is used for the home city or special school, or 
 if the table is simplified by breaking it up into parts and 
 so on, still the graphs and translation are probably best 
 for the public' In either of the latter, the significant 
 facts are outstanding and may be grasped by the reader 
 with less labor than he will have to use in understanding 
 even good tabulations. The impression that the graphs 
 and translation make is more striking and lasting. 
 
 However, tabulations are necessary in the statistical 
 
Statistics in Current School Reports 205 
 
 process because they must precede the graph or trans- 
 lation. The graph is only a pictorial representation of 
 the significant facts in a tabulation. The translation is 
 only a word representation of the same thing. Often- 
 times we do not know what the things we wish to graph 
 or translate for the public are, until we have made the 
 proper tabulation. Consequently we shall take up in 
 detail the matter of tabulating school statistics, following 
 it with treatments of graphs and translations. 
 
CHAPTER X 
 
 PRESENTING TABULATED STATISTICS TO THE 
 
 PUBLIC 
 
 I. POSSIBILITIES OF USING TABULATIONS OF STATISTICS 
 TO INFLUENCE THE PUBLIC 
 
 As previously shown, one of the chief reasons for the 
 many useless tables of statistics in school reports is that 
 the average school superintendent tends to publish 
 without modification the tables he gets up for his own 
 use or for state and national requirements. ^ But the 
 mere publishing of such tables is about the worst thing 
 he can do if he is trying to influence his public. It is 
 practically certain that the majority of people who read 
 school reports at all tend to skip conscientiously most of 
 the tabulated matter of this nature. 2 
 
 What, then, is the object of preparing a tabulation for 
 
 1 See p. 201 
 
 2 This does not in any way militate against the contentions of 
 Professor Thorndike and others that educational investigators should 
 get into the habit of publishing in full the data secured in their studies. 
 Such complete publications are highly desirable, but not for the 
 general public. They should appear in educational periodicals, mono- 
 graphs, dissertations, and similar places. If included in reports to 
 the public, they should at least be in appendices where they will not 
 prevent the citizen from reading the parts of the report that will in- 
 fluence him. But even thus they will probably weaken the effect 
 of the other parts upon him. 
 
 206 
 
Presenting Statistics to the Public 207 
 
 the public? The original intention undoubtedly is to 
 give the public quickly and easily a bird's-eye view of a 
 great mass of data which are entirely unintelligible 
 until collected and systematized. Again, tabulation 
 achieves a great economy of space through combining 
 many items which had been isolated up to that time. 
 This saving of space will not be achieved without some 
 care. For example, it is possible to make numerous 
 separate tables with all the headings repeated for each, 
 as in reporting facts about all the grades, so that the 
 total occupies more pages than straight reading matter 
 giving the same facts would take up. But in general, the 
 argument is that, if the ordinary reader can understand 
 the tabulation and will read it, he is more likely to be 
 influenced by it than by the pages of print necessary to 
 give the same idea. There is, however, a great danger in 
 undue condensation, especially in getting too many 
 items into one table. For example, the 1914-15 report 
 for Des Moines has one table with ninety different 
 columns.^ The ordinary reader can hold in mind only 
 a few different items at one time. It is, therefore, out 
 of the question to count on such a complicated table's 
 influencing the public. And it is probably unwise to 
 insert a table of this kind in a report intended even 
 partially for public consumption. A third object in 
 tabulation is the aid to logical thinking and presentation 
 which it affords for all work capable of expression in 
 numbers. Most of us recall how our science teachers 
 insisted that the extent to which we mastered scientific 
 thinking would be shown by our laboratory reports. 
 Probably these reports first acquainted us with tabulation. 
 
 ^Annual Report of the Des Moines Public Schools, for the year 
 ending July 1, 1915, opposite p. 82 
 
208 
 
 School Statistics and Publicity 
 
 But the distinction between tabulations intelligible to 
 school men or experts and those easily understood by the 
 public cannot be overemphasized. Tabulation, although 
 a very useful device for work, does not come from 
 instinct. It must be acquired, a truth often forgotten 
 by the expert tabulator. Most of us probably recall 
 some good mathematician who makes a very poor in- 
 structor because he has forgotten how painfully slow was 
 his own mastery of some of the work he is teaching. He 
 has consequently no patience with his students. Similarly 
 a superintendent familiar with tabulation is very apt to 
 forget that the public is not so familiar. It is true that 
 many a business man makes much use of tabulation in 
 his work. He has been trained to do this, however, for 
 no man can successfully conduct a business involving 
 many details unless he 'knows how to collect and con- 
 solidate data concerning his complex enterprise, in what 
 corresponds to tabular form. 
 
 But even an expert, trained to see data thrown into 
 certain conventional or familiar forms, is easily upset 
 if confronted with a tabulation differing from the form 
 to which he is accustomed. If a table is arranged with 
 magnitudes running from low to high instead of from 
 high to low, or if the order of years runs from the most 
 recent to the most remote instead of vice versa, or if 
 any similar change is made, the expert accustomed to 
 his own different tabulations will often hesitate and 
 become lost for a while. It may take him some time to 
 understand the new forms. The ordinary man would 
 not give even that much time to trying to understand 
 them. He would pass by the table the instant that he 
 could not easily grasp its meaning. 
 
 It should be evident by now that the making of tab- 
 
Presenting Statistics to the Public 209 
 
 ulations to influence the public is no easy task. It is 
 indeed of so difficult a nature as to require study. We 
 shall now proceed to make this study, taking up first the 
 problem of how to give a bird's-eye view through tab- 
 ulation, and following it with a consideration of how to 
 get up a series of tables. The reader should clearly 
 understand that all this presupposes an acquaintance 
 with the preliminary treatment of blanks and tabulating, 
 pages 71-81. 
 
 II. HOW TO GIVE A BIRD'S-EYE VIEW THROUGH 
 
 TABULATION 
 
 Many points are to be considered in making a table for 
 the public. But it should before all else strive to give 
 the reader quickly a bird's-eye view of the data. Among 
 the things required for this are : 
 
 1. A Very Carefully Worded and Specific Heading 
 
 The heading should be so well chosen as to give the 
 key to the understanding of the table, even if no other 
 explanatory material follows. 
 
 As examples of incomplete headings, the following from the South 
 Bend Survey are pertinent: 
 
 (a) Table headed "Value of School Property" (page 13). 
 
 It contains the dates of the erections of the various build- 
 ings, sizes of the lots, value of buildings and equipment. 
 
 (6) Table headed "Registration" (page 15). 
 
 It contains not only the registration by years but the in- 
 crease in registration and average daily attendance. 
 
 (c) Table headed "Cost of Fuel, 1912-13" (page 21). 
 
 This is by ward schools and contains the number of tons 
 each school used, the total cost in a lump, and the cost per 
 pupil. 
 
 In none of these does the title really specify what is in the table. 
 
210 School Statistics and Publicity 
 
 2. Comparatively Few Classes of Facts in One Table 
 
 Of course, it is hard to say just how many classes of 
 facts should go into one table. But it is safe to say that a 
 table containing more than four such classes will not be 
 read by the average man. By far the greater portion of 
 the tables in the best school surveys are concerned with 
 only one class of facts, and those including as many as 
 three classes are very rare. No one will question the 
 statement that the best methods of presenting school 
 statistics to the public have so far appeared in these 
 surveys or in annual reports that are virtually surveys. 
 
 3. Careful Grouping of the Classes That are Used 
 
 A complete running list of items is of little value. It 
 corresponds to the term of derision used toward a person 
 who puts too many items into a story or explanation, 
 '' total recall." 
 
 Thus in the Birmingham, Alabama, report for 1914-15 there is a 
 bird's-eye tabulation of five city school systems, covering twelve 
 items (pages 18 and 19). In the Des Moines report for 1914-15 
 there appears a table of ninety columns (grouped by sixes, however), 
 which contains the judgments of ninety judges on some specimens 
 of handwriting (page 82). This table will never be read by the public. 
 The Birmingham report for 1914-15 contains a table of "Distribu- 
 tion of Expenditures" (page 8) which has twenty-six columns not 
 broken in any way. Many similar examples might be cited. In all 
 probability none of these tables, so far as the average man is con- 
 cerned, is worth printing. 
 
 Once in a while a long table of receipts and expenditures 
 is justified on the ground that the citizens demand it. 
 They may demand it even though they will never read it, 
 because they think it will secure greater honesty and 
 efficiency in handling school moneys. But such a table 
 
Presenting Statistics to the Public 211 
 
 may be more tolerable by using double entry as in Table 
 22. 
 
 Table 22. Double Entry Table Showing Receipts and Pay- 
 ments, Springfield, Illinois, Public Schools, 1912-13 ^ 
 
 Receipts 
 
 State school fund . . . . 
 General property taxes . . 
 Rent and interest . . . . 
 
 Tuition fees 
 
 Sales of property . . . . 
 Sales of equipment . . . . 
 Balance at beginning of year 
 
 Total 
 
 Payments 
 
 Board of Education office 
 School elections .... 
 Finance accounts . . . 
 Legal services .... 
 Operation of office building 
 Office of superintendent . 
 Compulsory education . . 
 
 Instruction 
 
 Salaries of supervisors . . 
 Salaries of principals . . 
 Salaries of teachers 
 
 Textbooks 
 
 Stationery and supplies 
 Other expenses .... 
 
 Operation of plant 
 
 Wages of janitors . . . 
 
 Fuel 
 
 Water 
 
 Light and power .... 
 Janitors' supplies . . . 
 
 
 $17,414.19 
 
 — 
 
 314,788.96 
 
 — 
 
 4,176.84 
 
 — 
 
 2,538.50 
 
 — 
 
 2,807.40 
 
 — 
 
 297.28 
 
 — 
 
 264,597.87 
 
 — 
 
 606,621.04 
 
 $2,353.68 
 
 
 40.00 
 
 — 
 
 720.00 
 
 — 
 
 1,295.50 
 
 — 
 
 730.00 
 
 . — 
 
 3,000.00 
 
 — 
 
 1,000.00 
 
 — 
 
 6,550.00 
 
 
 
 30,177.25 
 
 — 
 
 169,012.99 
 
 — 
 
 325.00 
 
 — 
 
 8,914.90 
 
 — 
 
 3,807.25 
 
 — 
 
 16,420.91 
 
 
 
 6,519.37 
 
 — 
 
 2,046.08 
 
 ■ — 
 
 1,268.05 
 
 — 
 
 1,050.00 
 
 — 
 
 Springfield, Illinois, Survey, p. 99 
 
212 
 
 School Statistics and Publicity 
 
 Maintenance of plant 
 
 Repair of buildings and care of grounds 
 
 Repair and replacement of equipment . 
 
 Insurance 
 
 Auxiliary agencies 
 
 Libraries 
 
 Promotion of health 
 
 Miscellaneous 
 
 Rent 
 
 Permanent outlays 
 
 Land . . 
 
 New buildings 
 
 Alteration of old buildings .... 
 
 New equipment of new and old build- 
 ings 
 
 Other payments 
 
 Redemption of bonds 
 
 Payments of interest 
 
 Total 
 
 Balance at end of year 
 
 Grand Total 
 
 $16,276.98 
 
 
 5,346.11 
 
 — 
 
 533.10 
 
 — 
 
 850.00 
 
 
 
 1,434.00 
 
 — - 
 
 120.00 
 
 — 
 
 9,249.80 
 
 — 
 
 85,527.91 
 
 — 
 
 20,000.00 
 
 — 
 
 2,963.61 
 
 — 
 
 10,500.00 
 
 
 
 6,663.75 
 
 — 
 
 414,696.24 
 
 
 
 191,924.80 
 
 — 
 
 — 
 
 $606,621.04 
 
 However, although Table 22 gives the classified feature, 
 it does not appear on one page and could not be gotten on 
 one page in print large enough to be read easily. It would 
 make a better showing if printed on adjacent pages, a thing 
 easily possible in reports but not feasible here. 
 
 Long tables are also broken into smaller units through 
 the use of subdivisions of the main classes of items. The 
 question, however, arises as to how far such subdivision 
 may extend without making the table too complicated. 
 Certainly the process of subdivision should not go farther 
 than the description of school buildings in Table 23. 
 
CO 
 
 o 
 
 O 
 1^ 
 
 >^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^^ 
 
 
 
 
 
 
 
 
 CO 
 
 o 
 
 O 
 
 o 
 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 o 
 
 ■♦J. 
 ^ ■ 
 
 
 
 
 
 
 
 
 CO 
 CO 
 
 
 
 
 
 
 
 
 CO 
 O 
 
 
 
 
 
 
 
 
 Brick or 
 frame 
 
 
 
 
 
 
 
 
 O 
 
 
 
 
 
 
 
 
 
 O 
 
 o 
 
 t/2 
 
 
 
 
 
 
 
 
 
 
 
 o 
 o 
 
 CO 
 
 o 
 
 ^ 
 ^ 
 
 02 
 
 
 213 
 
214 
 
 School Statistics and Publicity 
 
 Long tables may also be broken up by grouping the items. 
 
 Thus, contrast the tables of Cincinnati and Cleveland for com- 
 parison of expenditures for different years, as given by Snedden and 
 Allen, pages 35-36, here shown as Tables 24 and 25 : 
 
 Table 24. Example of Effect of Unclassified Items. Com- 
 parison OF School Expenditures for the Years 
 1895-1905 (Cincinnati) 
 
 
 Year ending 
 
 Year ending 
 
 Year ending 
 
 
 Aug. 31, 1895 
 
 Aug. 31, 1900 
 
 Aug. 31, 1905 
 
 Teachers, day schools 
 
 $669,752 
 
 $799,286 
 
 $815,719 
 
 Teachers, night schools . 
 
 9,606 
 
 6,612 
 
 8,321 
 
 Officers and examiners . 
 
 15,143 
 
 16,646 
 
 17,792 
 
 Librarians 
 
 
 
 
 Janitors 
 
 
 
 
 New buildings .... 
 
 
 
 
 Repairs 
 
 
 
 
 Lots 
 
 
 
 
 Furniture 
 
 
 
 
 Heating fixtures . . . 
 
 
 
 
 Rent 
 
 
 
 
 Fuel 
 
 
 
 
 Supplies 
 
 
 
 
 Printing 
 
 
 
 
 Advertising 
 
 
 
 
 Gas 
 
 
 
 
 Census 
 
 
 
 
 Textbooks and supple- 
 
 
 
 
 mentary readers . . 
 
 1,641 
 
 3,502 
 
 13,448 
 
 Incidentals, etc. . . . 
 
 5,678 
 
 4,643 
 
 3,284 
 
 Teachers' Institute . . 
 
 
 
 
 Interest and redemption 
 
 
 
 
 of bonds 
 
 
 
 
 Public library .... 
 
 
 
 
 Deaf-mute taxes . . . 
 
 
 
 
 Transfer of funds . . . 
 
 
 
 
 Apparatus 
 
 
 
 
 Totals 
 
 
 
 
Presenting Statistics to the Public 215 
 
 Table 25. Example of Effect of Classified Items. Com- 
 parison OF School Expenditures (Cleveland) 
 
 August 31 
 
 1900 
 
 1905 
 
 Tuition 
 
 Supervisors' salaries 
 
 Teachers' salaries 
 
 Maintenance 
 
 Officers' and employees' salaries .... 
 
 Fuel and light 
 
 $37,406 
 897,190 
 
 118,664 
 1,295 
 
 $50,964 
 1,314,660 
 
 184,144 
 
 Repairs 
 
 
 Stationery and supplies 
 
 Contingent 
 
 "Water 
 
 10,335 
 
 Fixed charges 
 
 Interest 
 
 
 Bonds 
 
 
 Rent and insurance 
 
 Furniture and fixtures 
 
 Permanent improvements 
 
 Land 
 
 
 Buildings 
 
 
 Grading, paving, etc 
 
 Improvement on existing buildings . . . 
 
 Miscellaneous 
 
 School books 
 
 
 St. Louis Exposition 
 
 Glenville annexation 
 
 
 Total 
 
 
 
 
 
216 School Statistics and Publicity 
 
 4. Arrangement of Totals So That They Can Be Quickly 
 
 Grasped 
 
 This will aid the superintendent to discover significant 
 relations, as well. The most common devices are the 
 use of the word '' total," the employment of black-face 
 type or italics, and the outset in another right-hand 
 column ; thus : 
 
 $ 673 $ 673 $ 673 
 
 1240 1240 1240 
 
 890 890 890 $2803 
 
 Total. $ 2803 $ 2803 
 
 5. Printing of Headings So They Can All Be Read from 
 
 One Way 
 
 It is not pleasant to have to twist one's head or turn 
 the book around to read vertical headings. The average 
 reader will skip such rather than do the turning. For 
 the same reason, tables that are printed lengthwise on 
 the page should be avoided if at all possible. In general, 
 the headings may, by the use of syllables and abbre- 
 viations, be horizontally printed in just as small a space 
 as the wrong way takes. See pages 72, 213. 
 
 6. Filling in Tables Only Where Data Actually Exist 
 
 If vacant places are filled with zeros, the labor involved 
 in reading the zeros is as heavy as that in reading the 
 actual numbers.^ However, if the paper is not cross- 
 lined, it is better to fill the vacant spaces with dotted 
 lines, as the reader may get lost if several vacancies occur 
 together. Compare for example Tables 26 and 27 2. 
 
 1 This does not in any way discount what was said on page 84 about 
 making an entry for every case. That referred to a blank for the 
 superintendent to use in collecting data, not for the public. 
 
 2 Snedden and Allen : School Reports and School Efficiency, p. 78 
 
Presenting Statistics to the Public 217 
 
 Table 26. Showing Use of Zeros — Detroit Central 
 
 High School 
 
 Total number of " first year " pupils who have left since Sept., 1904 
 
 Cause 
 
 Illness 
 
 Illness in family 
 Failing eyesight 
 
 Work 
 
 Transferred . . . 
 Left city .... 
 Indifference to work 
 
 Music 
 
 Unknown . . . 
 
 Total .... 
 
 Ages 
 
 13 
 
 Bi G T 
 
 13 
 
 B G T 
 
 14 
 
 B G T 
 
 15 
 
 B G T 
 
 Totals 
 
 B G T 
 
 65 
 
 23 
 4 
 2 
 
 41 
 6 
 
 10 
 9 
 1 
 
 23 
 
 119 
 
 Table 27. Omitting Zeros — Detroit Central High 
 
 School 
 
 Total number of "first 
 
 year 
 
 ' pupils who have left 
 
 since 
 
 Sept. 
 
 , 1904 
 
 
 
 Ages 
 
 Totals 
 
 Cause 
 
 13 
 
 13 
 
 14 
 
 15 
 
 
 
 B 
 
 G 
 
 T 
 
 B 
 
 G 
 
 T 
 
 B 
 
 G 
 
 T 
 
 B 
 
 G 
 
 T 
 
 B 
 
 G 
 
 18 
 4 
 1 
 7 
 2 
 7 
 1 
 1 
 13 
 54 
 
 T 
 
 Illness 
 
 1 
 
 
 1 
 
 2 
 
 1 
 
 1 
 
 2 
 
 1 
 
 1 
 
 5 
 
 1 
 
 1 
 
 1 
 1 
 
 1 
 
 2 
 
 1 
 6 
 1 
 1 
 
 1 
 
 1 
 
 10 
 2 
 
 2 
 
 4 
 
 4 
 1 
 1 
 2 
 1 
 3 
 
 1 
 6 
 
 5 
 1 
 1 
 
 12 
 3 
 3 
 2 
 1 
 
 10 
 
 5 
 
 1 
 
 34 
 
 4 
 
 3 
 
 8 
 
 10 
 65 
 
 23 
 
 Illness in family . . . 
 Failing eyesight . . . 
 
 Work • . . 
 
 Transferred 
 
 Left city 
 
 4 
 2 
 
 41 
 6 
 
 10 
 
 Indifference to work . . 
 
 Music . 
 
 Unknown 
 
 Total 
 
 9 
 
 1 
 
 23 
 
 119 
 
 
 
 ^ In order to include all the columns within the limits of the page, it has 
 been necessary to use the abbreviation B for Boys, G for Girls, and T for 
 Total. 
 
218 School Statistics and Publicity 
 
 7. Care to Avoid Eyestrain 
 
 All the type in a table should be large enough to be 
 read easily and quickly. It is a false economy to compress 
 tables into a very small space through the use of fine print. 
 This may save money on printing bills. But it loses 
 more money, because the fine print will not be read by the 
 very persons for whom the table was prepared. 
 
 Dotted lines are very helpful to aid the eye in covering 
 long horizontal stretches between data that are to be 
 connected. They should, however, go as far as they are 
 needed. 
 
 In Table 28 dotted lines start out, but do not go far enough to be 
 of real service. 
 
 Table 28. Example of Bad Use of Dotted Lines. Cost for 
 Overhead Administrative Control in Western Cities ^ 
 
 City Percent of total mainte- 
 
 nance cost spent for ad- 
 ministrative control 
 
 Sacramento, Cal 1.8 
 
 Spokane, Wash 2.2 
 
 Pasadena, Cal 2.4 
 
 Seattle, Wash 2.6 
 
 Oakland, Cal 2.7 
 
 Denver, Colo 2.7 
 
 etc. etc. 
 
 8. Avoidance of the Alternate Column Scheme Unless 
 the Types and Arrangement in the Columns Stand Out 
 
 Sharply 
 
 For example, it is usually unwise to use alternate lines 
 for statistics on boys and girls, for public consumption, 
 unless one set is in red type, black-face type, or italics. 
 Italics as used by Ayres in his ** Laggards in Our Schools " 
 
 1 Oakland Survey, p. 17 
 
 I 
 
Presenting Statistics to the Public 219 
 
 are generally less satisfactory than black-face type. As 
 red, for practical purposes, can be used only on individual 
 copies and not in printed reports, it is often better to 
 make a separate table for boys and one for girls if they 
 are to be printed. This simplifies the matter. Both 
 tables may be later summarized into one, as on page 217, 
 for purposes of comparison which will be clear to any one 
 who has looked at the previous tables. 
 
 In the Des Moines report for 1914-15, a table of this kind appears, 
 opposite page 98. It is an age-progress table. Children who have 
 always been in school in Des Moines are shown in the upper left-hand 
 column of each rectangle; those coming in, in italics in the upper 
 right-hand column ; and the total below in heavy type : 
 
 20 SU 
 54 
 
 The differentiation in lines within divisions of a table 
 must also be carefully attended to. Note in Tables 26 
 and 27, page 217, how lighter lines are used to subdivide 
 the larger divisions. Dotted lines would have done as 
 well for the interior lines, but the use of colons for this 
 purpose is not advisable. 
 
 The Salt Lake City Survey has a table ^ in which colons have been 
 used to indicate interior lines. But as set up, they appear at first 
 to indicate ratios between boys per seat and the percentage of suffi- 
 ciency, and so on. It takes the average reader some time to realize 
 that these colons are really intended to form dotted dividing lines. 
 
 9. Clear Headings and Subheadings 
 
 The subheadings must be clear without much explana- 
 tion. The use of only key numbers or letters to head 
 columns is bad enough for the person doing the work.^ 
 
 1 Page 249 2 gee page 74 
 
220 School Statistics and Publicity 
 
 Every effort should be made to avoid it in a tabulation 
 intended for the average man. 
 
 On page 50 of the Hammond Survey appears a table employing key- 
 letters which actually occupy almost as much space for explanation 
 as the tabulation itself. 
 
 10. Neatness and Artistic Features 
 
 In addition to the foregoing, it is highly desirable to 
 have all printed tables as neat and artistic as possible. 
 Tables that are pleasing to the eye will by their very 
 form and convenience attract unconscious attention, 
 which may then easily be diverted to a consideration of 
 the ideas or conclusions embodied, in the tabulated 
 material. Or they will cause the reader so little strain 
 that whatever attention he has to give them will be 
 concentrated on the thought alone. A competent printer 
 will be able to secure such results by himself but may 
 need to be held down to setting up all type so that it can 
 be read from one position. Where the superintendent 
 has to give complete directions for printing, he should be 
 especially careful about uniform headings, uniform spac- 
 ing, large print, inclosing each table in a border, or at 
 least using distinctive ruling or open spaces above and 
 below each table to set it off properly. The tables in 
 this book have been planned to serve as models on these 
 matters. 
 
 III. HOW TO MAKE UP A SERIES OF TABLES OF THE 
 SAME GENERAL NATURE 
 
 Often it is desired to have a series of tables of the 
 same general nature or at least closely related. This 
 necessitates : 
 
 I 
 
Presenting Statistics to the Public 221 
 
 1. A Summary Table at the Start and Minor Tables in 
 the Same Sequence as Items in the Summary Table 
 
 For example, if it is desired to give a series of tables 
 on various school costs per pupil, the procedure might 
 be thus : First, have a summary table with various items, 
 as cost of superintendent's office, cost of instruction, 
 cost of supervision, etc., all appearing in the same table 
 for the city as a whole. This could be followed by a 
 series of tables, each covering all the ward schools on one 
 item. These tables would follow the sequence of the 
 items in the summary table. 
 
 2. The Same Sequence of Items within Similar Tables 
 
 If, in these various tables, the lump sum, the average 
 number of pupils belonging, and the cost per pupil are 
 given in adjacent columns from left to right, it is advisable 
 to follow the same order of presentation and the same form 
 of table all the way through. In this way the " mind 
 set " of the reader may be utilized. 
 
 3. Keeping in Mind That the Main Purpose of Any 
 
 Tabulation Is the Showing of Relationships 
 
 (a) The purely alphabetical order of items in many tables 
 destroys or greatly handicaps the showing of relationships. 
 
 Thus in the report of the city superintendent of South Bend for 
 1913-14, all the tables involving the different wards or schools are 
 practically worthless for influencing the public because of this alpha- 
 betical arrangement. For example, take the table on comparative 
 costs of instruction and supervision by buildings for 1913-14, page 30. 
 If the last column of this table had been arranged in order of magni- 
 tude from high to low, the comparison would stand out. See Tables 
 29 and 30 for this. 
 
 The original table might do if the central tendency and quartiles 
 for the city, or some other standards, were printed in bold-faced type 
 
222 
 
 School Statistics and Publicity 
 
 at the top of the table, so that comparison might be made with them 
 for each ward school. Or it would not be so unsatisfactory if another 
 column had been added at the right giving the rank of each school 
 on cost per pupil, from highest to lowest. Grammar would be rank 
 2; Colfax, rank 9; and so on to Warren, which would be rank 3. 
 But the ranks would not show up so well as in the arrangement on 
 page 223. In any event, it is only just to note that the alphabetical 
 arrangement is of service in making easy the work of checking the 
 names of the schools so that no one will be omitted. This has its 
 value during the period of working up data inside the school system. 
 But this value disappears as soon as the table reaches the citizen. He 
 assumes that the work is correct and desires only to get at the meaning 
 of the whole and its parts as quickly as possible. 
 
 ORIGINAL FORM 
 Table 29. Example of Alphabetical Arrangement of Items 
 Showing Comparative Cost of Instruction and Super- 
 vision BY Buildings for 1912-13 (South Bend) 
 
 
 Total 
 
 Average 
 
 Cost 
 
 Schools 
 
 cost of 
 
 daily 
 
 per 
 
 
 instruction 
 
 attendance 
 
 pupil 
 
 Grammar 
 
 $10,054.05 
 
 332 
 
 $30.28 
 
 Colfax 
 
 etc. 
 
 etc. 
 
 24.53 
 
 Coquillard 
 
 
 
 25.36 
 
 Elder 
 
 
 
 22.19 
 
 Franklin 
 
 
 
 21.04 
 
 Jefferson 
 
 
 
 32.35 
 
 Kaley 
 
 
 
 22.26 
 
 Lafayette 
 
 
 
 27.97 
 
 Laurel 
 
 
 
 20.14 
 
 Lincoln 
 
 
 
 24.07 
 
 Linden 
 
 
 
 25.10 
 
 Madison 
 
 
 
 26.12 
 
 Muessel 
 
 
 
 27.94 
 
 Oliver 
 
 
 
 19.90 
 
 Perley 
 
 
 
 23.61 
 
 River Park ....... 
 
 
 
 19.25 
 
 Studebaker 
 
 
 
 23.89 
 
 Warren 
 
 
 
 29.16 
 
Presenting Statistics to the Public 223 
 
 REVISED FORM 
 
 Table 30. Bobbitt Table Arrangement of Data in 
 
 Table 29 
 
 Schools Cost per pupil 
 
 Jefferson $32.35 
 
 Grammar 30.28 
 
 Warren 29.16 
 
 Lafayette . . . . . 27.97 
 
 Muessel 27.94 
 
 Madison 26.12 
 
 Coquillard 25.36 The full figures from which 
 
 Linden 25.10 this table is derived are on 
 
 Colfax ....... 24.53 file in the superintendent's office 
 
 Lincoln 24.07 and may be inspected by any 
 
 Studebaker 23.89 interested person. 
 
 Perley ...... 23.61 
 
 Kaley 22.26 
 
 Elder 22.19 
 
 Franklin 21.04 
 
 Laurel 20.14 
 
 Oliver 19.90 
 
 River Park 19.25 
 
 (6) It is generally best to arrange the table in the form 
 of a scale running from high to low. 
 
 There are some exceptions. Perhaps a better rule 
 would be to arrange items in the table so that the city or 
 school having the desired trait in the greatest abundance 
 should be at the top. 
 
 For example, a table showing the number of dollars behind each $1 
 spent for schools should be placed with the smallest number at the 
 top. The smaller number is the more desirable, since a small number 
 of dollars behind each dollar spent for schools indicates a high school 
 tax and presumably good schools. Accordingly, Table 31 should 
 have been reversed. 
 
224 School Statistics and Publicity 
 
 Table 31. Real Wealth Behind Each Dollar Spent for 
 School Maintenance ' 
 
 1. Atlanta, Ga $559.00 
 
 2. Los Angeles, Cal 538.00 
 
 3. Richmond, Va 536.00 
 
 4. Birmingham, Ala 479.00 
 
 5. Portland, Ore 456.00 
 
 6. Memphis, Tenn 449.00 
 
 7. Indianapolis, Ind 408.00 
 
 and so on to 
 
 35. Toledo, Ohio 184.00 
 
 36. Worcester, Mass 180.00 
 
 37. Newark, N. J 165.00 
 
 Note that the ranks in this are really reversed. Atlanta in this 
 showing is doing less for schools than any of the other cities and should 
 have rank 37. Newark should have rank 1. 
 
 Table 32, which gives the same idea in another way, is correct in 
 putting the largest number at the top, because a high tax rate on real 
 wealth for schools is desirable. It is given this way : 
 
 Table 32. Comparative Rates of Tax Required for School 
 Maintenance (in Mills) 2 Based on Real Wealth of Cities ^ 
 
 1. Newark, N. J 00606 
 
 2. Toledo, Ohio 00543 
 
 3. New Haven, Conn 00541 
 
 4. Paterson, N. J. 00541 
 
 5. Lowell, Mass 00515 
 
 and so on to 
 
 31. St. Paul, Minn 00244 
 
 32. Memphis, Tenn 00244 
 
 33. Portland, Ore 00219 
 
 34. Birmingham, Ala 00209 
 
 35. Richmond, Va 00186 
 
 36. Los Angeles, Cal 00184 
 
 37. Atlanta, Ga 00180 
 
 1 Portland Survey, p. 310 
 
 2 In some parts of the country, this would be more easily under- 
 stood if given as cents on the one hundred dollars ; as : 
 
 1. Newark, N. J $.606 
 
 ^Portland Survey, p. 311 
 
Presenting Statistics to the Public 225 
 
 In printing Bobbitt tables, the most effective results 
 on the casual reader will undoubtedly be obtained by 
 giving each item a separate line. 
 
 Thus in Tables 33-35 from pages 62 and 63 of the San Antonio 
 Survey, Form 1 is best, Form 2, the next best. Form 3 should not be 
 used. A trained reader can understand one form as easily as the 
 other. But the average man will understand the first form much 
 more quickly than he will the others. 
 
 Table 33. Bobbitt Table, Form 1. Annual Per Capita 
 Expenditures for Street Maintenance, 1912 
 
 Nashville $2.79 
 
 Augusta 2.76 
 
 Tampa 2.10 
 
 Memphis 2.04 
 
 Houston 2.04 
 
 Savannah 1.71 
 
 Atlanta , 1.63 
 
 Dallas 1.55 
 
 Galveston 1.54 
 
 Jacksonville 1.53 
 
 Austin 1.51 
 
 New Orleans 1.50 
 
 Macon 1.43 
 
 Shreveport ........ 1.36 
 
 Montgomery 1.36 
 
 Mobile 1.33 
 
 Fort Worth 1.17 
 
 El Paso 1.10 
 
 Muskogee 1.07 
 
 Birmingham 1.02 
 
 San Antonio 99 
 
 Charleston 85 
 
 Little Rock ........ .63 
 
 Oklahoma City ..,.,.. .63 
 
226 
 
 School Statistics and Publicity 
 
 Table 34. Bobbitt Table, Form 2. Annual Per Capita 
 Expenditures for Street Maintenance, 1912 
 
 Nashville $2.79 
 
 Augusta 2.76 
 
 Tampa 2.10 
 
 Memphis 2.04 
 
 Houston 2.04 
 
 Savannah 1.71 
 
 Atlanta 1.63 
 
 Dallas 1.55 
 
 Galveston . . . . . 1.54 
 
 Jacksonville 1.53 
 
 Austin 1.51 
 
 New Orleans . . . . 1.50 
 
 Macon . . . 
 
 
 
 $1.43 
 
 Shreveport . . 
 
 
 
 1.36 
 
 Montgomery . 
 
 
 
 1.36 
 
 Mobile . . . 
 
 
 
 1.33 
 
 Fort Worth . . 
 
 
 
 1.17 
 
 El Paso . . . 
 
 
 
 1.10 
 
 Muskogee . . 
 
 
 
 1.07 
 
 Birmingham 
 
 
 
 1.02 
 
 San Antonio 
 
 
 
 .99 
 
 Charleston . . 
 
 
 
 .85 
 
 Little Rock . . 
 
 
 
 .63 
 
 Oklahoma City 
 
 
 
 .63 
 
 Table 35. Bobbitt Table, Form 3 (Practically Never De- 
 sirable). Annual Per Capita Expenditures for Street 
 Maintenance, 1912 
 
 Nashville 
 
 Tampa 
 
 Houston 
 
 $2.79 
 2.10 
 2.04 
 
 Augusta . 
 Memphis 
 Savannah 
 
 $2.76 
 2.04 
 1.71 
 
 Atlanta . 
 Galveston 
 Austin . . 
 
 1.63 
 1.54 
 1.51 
 
 Dallas . . 
 Jacksonville 
 New Orleans 
 
 1.55 
 1.53 
 1.50 
 
 Macon . . 
 Montgomery 
 Fort Worth . 
 
 1.43 
 1.36 
 1.17 
 
 Shreveport 
 Mobile . 
 El Paso . 
 
 1.36 
 1.33 
 1.10 
 
 Muskogee 
 San Antonio 
 Little Rock 
 
 1.07 
 .99 
 .63 
 
 Birmingham 
 Charleston . . 
 Oklahoma City 
 
 1.02 
 .85 
 .63 
 
 The use of Forms 2 and 3 probably grows out of a desire to utilize 
 part of a page for the table, or through a mistaken idea that it econ- 
 omizes space. But in most places where it is desired to use such a 
 table, there would be many such tables to present. It is readily 
 apparent that two tables of Form 1 placed side by side on a page 
 would take up practically no more space than if they were printed 
 either as Form 2 or Form 3. 
 
Presenting Statistics to the Public 227 
 
 (c) If many horizontal lines appear in the table, ease 
 in reading them is facilitated by running a heavy horizontal 
 line or leaving a space every five lines or less. 
 
 Examples of this are familiar to most readers. The lines drawn 
 across, or the gaps left in the Bobbitt tables to show the medians and 
 quartiles serve the same purpose, if there are not too many cases in 
 each section of the distribution. A third device for this purpose is 
 the numbering of the items from top to bottom on both margins. 
 Thus the data for item 3 on the left may be traced across to the data 
 on the line numbered 3 on the right. This is the device used by the 
 United States Bureau of Education on many of the tables that cover 
 two pages, with continuous horizontal lines. 
 
 (d) In most tables where the cases are kept separate, 
 it is advisable to make the name of the city, school, etc., 
 to be compared with the others, stand out prominently. 
 
 It will be recalled that this was done in the tables from the Port- 
 land Survey. The device is the familiar one used by newspapers to 
 call attention to the home city in a table giving the standing of the 
 baseball clubs of a league. Sometimes the emphasis is given with 
 capital letters as in Portland, or with black-faced type as in Salt Lake 
 City. On paper charts not to be printed, the particular case can be 
 marked in red or some bright color. It is hardly possible to make the 
 one case and the data with it stand out too prominently. 
 
 In the age-grade tables, it is customary to make the children of 
 normal age stand out prominently by marking off these numbers. 
 Heavy stairstep lines may be drawn to inclose the numbers for nor- 
 mal children; or heavy lines may be placed above and below these 
 numbers ; or they may be boxed in ; or they may be printed in bold- 
 faced type. Samples are given in Figure 29. 
 
 Where the printer can handle it, the boxed-in form is preferable, 
 because it enables one to separate the normal from both the retarded 
 and the accelerated children very easily. 
 
 In another form of age-grade table, the facts for one grade at a 
 time are presented. This is usually done by running the ages along 
 the top and the number of years the child has been in school down the 
 table. Then two heavy lines are drawn vertically through the table 
 to inclose the children of normal age in this grade. Two other heavy 
 
228 
 
 School Statistics and Publicity 
 
 
 
 
 Ag 
 
 as 
 
 
 
 Grades 
 
 under 
 6 
 
 6 
 
 to 
 
 6k 
 
 TO 
 7 
 
 7 
 
 to 
 
 7i. 
 
 to 
 
 8 
 
 
 IB 
 
 22 1 54-0 
 
 )59 1 105 
 
 38 
 
 
 1 A 
 
 
 30 
 
 98 
 
 il8 
 
 76 
 
 
 HB 
 
 
 17 
 
 6^ 1 133 
 
 102 
 
 
 HA 
 
 
 
 Z 
 
 £7 
 
 58 
 
 e.. 
 
 HIB 
 
 
 
 
 
 
 
 HEA 
 
 
 
 
 
 
 
 Ages 
 
 Grades 
 
 I B 
 lA 
 
 EB 
 BA 
 IHB 
 HIA 
 
 under 
 6 
 
 6 
 to 
 
 to 
 
 7 
 
 7 
 
 to 
 
 7i 
 
 7i 
 
 to 
 
 8 
 
 
 22 
 
 540 
 
 159 
 
 105 
 
 38 
 
 
 
 30 
 
 98 
 
 M8 
 
 76 
 
 
 
 17 
 
 65 
 
 133 
 
 102 
 
 
 
 
 a 
 
 Z7 
 
 58 
 
 84 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ages 
 
 under 
 6 
 
 6 
 
 to 
 
 6i 
 
 to 
 
 7 
 
 7 . 
 
 7i 
 
 7i 
 
 to 
 
 8 
 
 
 ZZ 
 
 540 
 
 159 
 
 105 
 
 38 
 
 
 
 30 
 
 98 
 
 lie 
 
 76 
 
 
 
 17 
 
 65 
 
 133 
 
 lOZ 
 
 
 
 
 a 
 
 27 
 
 58 
 
 e4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 29. 
 
 Grades 
 
 IB 
 
 I A 
 
 HB 
 
 HA 
 niB 
 
 HLK 
 
 Devices for Making the Number of Normal Children Stand 
 Out in Age-Grade Tables. 
 
Presenting Statistics to the Public 229 
 
 lines are drawn horizontally through to inclose the children of normal 
 progress irrespective of how old they were at entrance. The area 
 in the center, where the two sets of lines cross, incloses children who 
 are both normal for age and progress. Various other combinations 
 for the other parts of the table are easily worked out. 
 
 The following table form is similar to one used in the Bridgeport, 
 Connecticut, Survey, page 35. 
 
 Years 
 
 in 
 School. 
 
 Table 36. Age-Progress Table. 5B Grade 
 
 Ages. 
 
 Total. 
 
 
 6 
 
 6i 
 
 9 
 
 3k 
 
 10 
 
 I0| 
 
 II 
 
 Mi 
 
 12 
 
 »2i 
 
 13 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Total. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (e) It is often advisable to present only percentage 
 derivations instead of the original figures. 
 
230 School Statistics and Publicity 
 
 The reason is that the original figures may be large and not easily 
 comparable, whereas the percentage equivalents are small and easily 
 compared. Distribution tables of the results on standard tests, how- 
 ever valuable for educational investigations, are not at all suited for 
 the general public. Round numbers or the nearest whole units are often 
 better than exact figures, as they can be held in mind more easily. 
 If all the parts of 100 per cent are shown, care should be taken to as- 
 sign values to the parts that will total exactly 100 per cent. Approxi- 
 mations are satisfactory for the superintendent, but a discrepancy be- 
 tween the sum of the parts and 100 per cent would afford some readers 
 an excuse for attacking the accuracy of the report. These suggestions 
 must, however, be used with caution. For in some instances the use 
 of approximate figures may lead readers to suspect that the figures 
 have been " doctored." Especially is this true as regards statistical 
 presentations designed to increase the school tax. 
 
 (/) Special devices in tabulating are sometimes of great 
 value. 
 
 For example, a tabulation showing the part of each dollar of school 
 money that goes to various school expenses is effective with the average 
 man. Insurance companies take great pains to show where every part 
 of the dollar from premiums goes. Tables 37-39 show some of the 
 possibilities. Similar devices could show the part of a year or years 
 spent on each subject. 
 
 Table 37. How Portland Spends Its Dollar ^ 
 
 Interest 20.7 cents 
 
 General expenses of city government 6.0 cents 
 
 PoHce department 9-0 cents 
 
 Fire department 12.0 cents 
 
 Inspection 0.9 cents 
 
 Health 0.7 cents 
 
 Street cleaning and sanitation 6.2 cents 
 
 Care of streets and bridges 9.0 cents 
 
 Education 30.8 cents 
 
 Libraries 1.0 cents 
 
 Parks and playgrounds 2.6 cents 
 
 Damages 1-1 cents 
 
 Total lOO.O cents 
 
 1 Adapted from the Portland Survey Graph, p. 84 
 
Presenting Statistics to the Public 231 
 
 Table 38. How the Trolley Nickel is Divided ^ 
 
 2.02 cents for wages 
 
 .78 cents for expenses 
 
 .55 cents for taxes 
 1.06 cents for interest 
 
 .59 cents for dividend 
 5.00 cents — your nickel 
 
 Table 39. How Rockford Spent Its School Dollar, 1915 ^ 
 
 Elementary Schools 
 
 Teachers' salaries 41.67 cents 
 
 Building maintenance and upkeep . . 12.3 
 
 New buildings ........ 9.9 
 
 Educational supplies 3.9 
 
 Department of hygiene .97 68.74 cents 
 
 General 
 
 Interest on school fund 2.81 
 
 Executive employees (Educational) . 1.39 
 Executive employees (Board) ... .86 
 Evening schools and gymnasium . . .4 5.46 
 
 High School 
 
 Teachers' salaries 15.81 
 
 Upkeep of building 6.7 
 
 Educational supplies 1.43 
 
 New building 1.41 
 
 Educational employees .45 25.80 
 
 100.00 cents 
 
 Sometimes the variations in sizes of different items may be in- 
 dicated by difference in sizes of the type used in printing. Table 40 
 is a good example of this, but the variations in size are only very 
 rough approximations. 
 
 1 Poster used by Louisville, Ky., Railway Company 
 
 2 Adapted from the chart on p. 120 of the Review of Rockford Public 
 Schools, 1915-16. Some of the figures were changed slightly so as to 
 make the total exactly 100 cents. 
 
232 
 
 School Statistics and Publicity 
 
 Table 40. Variations in Type to Indicate Relative Size 
 
 "Watch the Central Association Grow!" 
 
 Year 
 
 i^ew memoers auucu 
 during year 
 
 1 1 uuti paiu up 
 membership 
 
 Net increase 
 
 1908 
 
 1909 
 
 1910 ' 
 
 1911 
 1912 
 1913 
 1914 
 
 102 
 
 161 
 
 132 
 
 117 
 
 115 
 
 146 
 
 177 
 
 824 
 445 
 
 484 
 486 
 
 659 
 
 619 
 
 680 
 
 121 
 
 39 
 
 2 
 
 73 
 60 
 
 61 
 
 1915 
 
 223 
 
 768 
 
 88 
 
 1916 
 
 320 
 
 973 
 
 205 
 
 917 
 
 " Every member 
 
 get one I " 
 
 Are you one 
 of these ? 
 
 "There is no 
 going back." 
 
 Do you need more convincing proof of the worth of membership in 
 
 The Central Association of SCIENCE and MATHEiVIATICS TEACHERS? 
 
 Table 41 uses very little space, for data classified in several ways. 
 
 Table 41. Example of Presenting in a Small Space Data 
 Classified in Several Ways. The Standing of Salt Lake 
 City in the Fundamentals of Arithmetic as Compared with 
 Other Cities, Judged by the Median Score Attained by 
 Each Grade ^ 
 
 
 Addition 
 
 
 
 
 Multiplication 
 
 V 
 
 VI 
 
 VII 
 
 VIII 
 
 
 V 
 
 VI 
 
 VII 
 
 VIII 
 
 3.9 
 
 4.6 
 
 5.4 
 
 6.7 
 
 Detroit 
 
 3.8 
 
 4.8 
 
 6.0 
 
 7.5 
 
 3.7 
 
 4.9 
 
 5.6 
 
 7.8 
 
 Boston 
 
 3.3 
 
 4.8 
 
 5.1 
 
 6.5 
 
 3.9 
 
 4.4 
 
 4.7 
 
 5.6 
 
 Other Cities 
 
 2.6 
 
 4.5 
 
 5.2 
 
 6.4 
 
 2.9 
 
 3.4 
 
 3.8 
 
 5.3 
 
 Butte 
 
 4.1 
 
 5.0 
 
 6.5 
 
 8.1 
 
 4.1 
 
 6.4 
 
 6.9 
 
 8.6 
 
 Salt Lake City 
 
 4.3 
 
 6.3 
 
 7.1 
 
 8.3 
 
 
 Subtraction 
 
 
 
 
 Division 
 
 
 5.5 
 
 6.2 
 
 7.3 
 
 9.5 
 
 Detroit 
 
 2.7 
 
 4.4 
 
 7.1 
 
 8.8 
 
 4.9 
 
 6.3 
 
 6.9 
 
 8.6 
 
 Boston 
 
 2.0 
 
 3.3 
 
 5.1 
 
 6.9 
 
 4.5 
 
 6.1 
 
 7.8 
 
 8.4 
 
 Other Cities 
 
 2.3 
 
 4.3 
 
 5.8 
 
 6.3 
 
 2.9 
 
 3.4 
 
 3.8 
 
 5.3 
 
 Butte 
 
 3.6 
 
 4.3 
 
 7.2 
 
 10.2 
 
 6.2 
 
 7.8 
 
 8.8 
 
 9.8 
 
 Salt Lake City 
 
 3.0 
 
 6.6 
 
 7.7 
 
 9.6 
 
 ^ Salt Lake City Survey, p. 174 
 
Presenting Statistics to the Public 233 
 
 (g) There are times when it is necessary to use con- 
 ventional tables. In so doing, the following points may 
 be of use: 
 
 1. Use double distribution tables and forms recommended by the 
 National Education Association committee.^ 
 
 2. Place dates old to new down the page or left to right. 
 
 3. Place magnitude so that the most desirable showing is at the 
 top or to the left. (This sometimes reverses the order of dates ad- 
 vocated in 2.) 
 
 4. Use Roman numerals for one classification, e.g., the numbers 
 for the school grades, and Arabic numerals for the data on pupils 
 within the grades. 
 
 5. It is best where the sense is not destroyed and the effect de- 
 sired will not be lost, to get up tables in the conventional forms. 
 
 Bizarre effects in tabulation are no more to be desired than is 
 writing from right to left or up the page. 
 
 EXERCISE 
 
 Take some annual school report or school survey in which you 
 are interested. Write out a detailed criticism of the tables or lack of 
 them in it, from the standpoint of their effectiveness with the public, 
 showing just why they are good or liable to be unsuccessful. In the 
 cases of the unsuccessful ones, or failure to employ tabulations where 
 desirable, draw up forms that would present the same data properly. 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 Report of the Committee on Uniform Records and Reports. U. S. 
 
 Bureau of Education Bulletin, 1912, No. 3. 
 Rugg, H. 0. Statistical Methods Applied to Education, Chapter X. 
 Snedden, David S., and Allen, William H. School Reports and School 
 
 Efficiency. 
 
 1 Report of Committee on Uniform Records and Reports, Bureau 
 of Education Bulletin, No. 2, 1912, p. 20 
 
CHAPTER XI 
 
 GRAPHIC PRESENTATIONS OF SCHOOL STATIS- 
 TICS, ESPECIALLY FOR THE PUBLIC 
 
 I. OBJECT OF GRAPHIC PRESENTATIONS 
 
 The object in making a graphic presentation of statis- 
 tical matter is to give as quickly as possible through the 
 eye a faithful and forceful bird's-eye view of the mass of 
 statistics, the significant parts, their relationships, etc. 
 Graphic presentation in statistics is simply a develop- 
 ment of a growing general tendency to make desired 
 impressions by pictures rather than by word descriptions. 
 This tendency is shown most clearly in the pictorial 
 supplements and cartoons, and in the constantly in- 
 creasing proportion of illustrations in printed material 
 appearing in all our leading newspapers and magazines 
 at the present time. 
 
 The graph, however, is more closely akin to the line 
 drawing than it is to the photograph. It is found to be 
 of great advantage in textbooks for the following reasons : 
 
 1. It presents the significant points in a clear and unmistakable 
 way. 
 
 These points are also presented apart from the great mass 
 of subsidiary data on which they rest. 
 
 2. It makes the presentation concrete by appealing to the eye. 
 
 As many people are unable to understand things they 
 cannot image, the graph will drive the significant points home 
 to those who could not be reached otherwise. 
 
 234 
 
Graphic Presentations of School Statistics 235 
 
 3. It often economizes time and space, for it will take up less room 
 than the description it displaces. 
 
 4. It gives to most persons a more accurate basis of comparison 
 than they could get with the same effort from word descriptions or 
 tabulations. 
 
 Textbooks must make things clear very quickly to 
 beginners in the subject. School reports must present 
 school facts very rapidly and clearly to citizens who 
 know little of them. Consequently, the graph, if prop- 
 erly used, should be about as valuable for school reports 
 as it has been for textbooks. 
 
 The graph, however, is open to dangers of mis- 
 representation and exaggeration. These things are just 
 as harmful in school graphs as they are in demagogic 
 politics, patent medicine advertisements, etc. The ideal 
 graph would probably have the forcefulness of the most 
 powerful motor car, department store, or patent medicine 
 advertisements, with something like the truthfulness 
 and accuracy on essential points of a first-rate scientist. 
 Another trouble is that a graph leaves with most persons 
 only a general impression ; it is very difficult for them to 
 recall it accurately, much less to reproduce it from memory 
 later for any one else. 
 
 II. HOW TO MAKE GRAPHS FOR THE PUBLIC FROM 
 
 STATISTICAL DATA 
 
 1. Component Parts 
 
 Circle Graph. It is often desirable to show in a graph 
 the relative size of the component parts of a statistical 
 whole. A popular device for doing this is the circle, 
 each sector by its area representing one proportional part 
 of the whole. 
 
236 School Statistics and Publicity 
 
 Figure 30 is a graph of this nature showing the distribution of time 
 through the eight grades for the various common school subjects. 
 Figure 31 gives a similar graph with fewer parts. 
 
 Fig. 30. — Component Part Circle Graph Showing Distribution of Time 
 to Various Subjects Throughout the Eight Elementary Grades. 
 
 (Adapted from the 1915-16 Review of the Rockford, Illinois, Schools, page 52.) 
 
 This kind of graph is famihar to the public and of 
 course makes the comparisons through the sizes of the 
 angles. But it has these disadvantages : 
 
 1. If more than a few parts are to be represented, there is trouble 
 in reading the names. 
 
i 
 
 r 
 
 I 
 
 Graphic Presentations of School Statistics 237 
 
 When there are only three parts and they are all large, the 
 names may all be printed horizontally as in Figure 31. But 
 if there are many parts and some of them are small, this is im- 
 possible. Then the names 
 must be printed as in Figure 
 30 or else at the side with 
 dotted lines leading into the 
 parts indicated. If the 
 former device is used, the 
 printing must be reversed 
 as the eye proceeds around 
 the circle, with consequent 
 delay. Note in Figure 30 
 how one has to read up 
 for "music" and down for 
 " geography." If the latter 
 device is used, much more 
 time is consumed in asso- 
 ciating the names with the 
 parts concerned, than when 
 the names are printed on 
 the parts. 
 
 Fig. 31. — Component Part Circle 
 Graph Showing Relative Propor- 
 tions of Normal, Retarded, and 
 Accelerated Pupils. 
 
 (From Salt Lake City Survey, page 190.) 
 
 2. The figures denoting the various parts cannot be placed in such 
 positions that they can be easily compared or added. This is espe- 
 cially bad when the parts are given in percentages. 
 
 3. It is extremely difficult to compare the same factors in different 
 wholes. 
 
 For example, suppose a circle similar to this one for Rockford 
 " had been drawn for Joliet; it would be very hard to com- 
 
 pare the sector on "reading" in the Rockford circle with the 
 ' corresponding sector in the Joliet circle. 
 
 Bar Graph. The data shown in this circle graph for 
 Rockford may be presented in a bar graph which permits 
 of placing the figures so they can be added. For this 
 the parts within each of the three main divisions should 
 be arranged from high to low. Figure 32 shows this 
 arrangement. 
 
 I 
 
 I 
 
238 
 
 School Statistics and Publicity 
 
 Reading 
 
 Arithmetic 
 
 Note that practically all the defects of the circle graph have been 
 remedied here. Note also that the component parts in a bar graph 
 are shown proportionately by their lengths only. Their widths have 
 
 nothing to do with it and their 
 areas are in exactly the same 
 proportion as their lengths. 
 
 Note, too, that the figures arc 
 given for any reader who cares 
 for them and in such shape that 
 they may be added easily. 
 
 Sometimes for economy in 
 printing, the bar may be a hori- 
 zontal one, in which case the 
 lines of printing and figures may 
 run vertically so that the reader 
 will have to turn the page to 
 read them. 
 
 English 12.86% . i i ^. • ., 
 
 An elaboration of the 
 
 component bar graph is 
 
 the form with two different 
 
 but related scales, one on 
 
 either side. 
 
 For example, the waste 
 through repeaters in a school 
 system might be shown as in 
 Figure 33, using the average 
 annual cost per elementary pupil, 
 say $30. 
 
 The bar graph idea with 
 component parts may be 
 100 7o carried out for popular 
 
 Fig. 32. — Bar Graph Showing by Presentation with cartoon 
 
 Component Parts with Subdivisions effccts. 
 the Distribution of Time for Com- 
 
 mon School Subjects. Po, example, the proportion of 
 
 (Adapted from Figure 30.) white and negro children out of 
 
 Geoqrophy 
 
 Hisloi 
 Music 
 
 7 
 
 Drawing 
 
 Manual Jr. 
 Physical Jr. 
 
 Gomes 
 Recess 
 
 Opening Ex. 
 
 t.b% 
 
 
 14.75 % 
 
 
 12.867. 
 
 
 7.64% 
 
 
 3.09% 
 
 
 5.85% 
 
 
 6.65% 
 
 
 2.85% 
 
 
 3.267. 
 
 
 1.09 7o 
 
 
 706% 
 
 
 2.94 7o 
 
 
Graphic Presentations of School Statistics 239 
 
 No. of 
 repeaters 
 
 50 
 Z5 
 50 
 
 75 
 
 50 
 50 
 
 50 
 
 75 
 425 
 
 8th Grad^ 
 
 7th 
 
 M 
 
 6th 
 
 II 
 
 5th 
 
 • 1 
 
 4 th 
 
 II 
 
 3rd 
 
 II 
 
 znd 
 
 •t 
 
 1st 
 
 M 
 
 
 « 
 
 Cost of repeat-' 
 ing work 
 
 H500 
 
 750 
 
 1500 
 
 a200 
 
 1500 
 1500 
 
 1500 
 
 2200 
 ,^2750 
 
 Fig. 33. — Bar Graph to Show Component Parts, with Two Different but 
 
 Related Scales. 
 
 each twelve may be shown as in Figure 34. The black children com- 
 ing in from the right on the row serve the same purpose as putting 
 the right end of a bar in black. But the idea of the children's figures 
 
 Fig. 34. — Bar Graph with Cartoon Effect Showing Proportion of White 
 and Colored Children out of Every Twelve in Alabama. 
 
 (From An Educational Survey of Three Counties in Alabama, page IS.) 
 
240 School Statistics and Publicity 
 
 here gives an interesting touch in addition. While this cartoon 
 graph is not accurate, strictly speaking, a little care will make it 
 accurate enough for its purpose. For example, by choosing twelve 
 rather than ten figures for the whole line, fractions of two-thirds 
 and one-third have been shown with whole figures. This device 
 is extremely effective in a popular presentation. 
 
 2. Simple Comparisons 
 
 The simplest graphic comparison is usually made 
 through some form of the bar graph. The latter is used 
 a great deal for comparisons in presentations intended 
 for the public, but its possibihties are not generally 
 recognized. Many graphic comparisons for pubhc con- 
 sumption so far have involved one or both of two 
 errors : 
 
 (a) The comparison is made by using similar areas. 
 
 (6) The comparison is made through a cartoon effect, when the 
 latter is needed only to attract attention to the graph, in which bars 
 would show the relationship much better. 
 
 We shall now examine the possibihties of the bar 
 graph for comparisons, and after that take up typical 
 graphs of other kinds for this purpose. 
 
 Comparisons with Bar Graphs. Some of the possi- 
 bihties of the bar graph in making comparisons have 
 been indicated heretofore, particularly in connection 
 with a distribution arranged as a Bobbitt table. ^ If the 
 component parts of any bar graph on pages 238 and 239 
 were taken out, arranged in order of size and lined up at 
 the left, they would give a bar graph effect running from 
 high to low. 
 
 In making comparisons with bars, it is generally advis- 
 
 1 See p. 103 
 
Graphic Presentations of School Statistics 241 
 
 able to follow this order, with the bars lined up at the left 
 ends, as in Figure 35. 
 
 Name of item Size Bar 
 
 $703 
 5,52 
 
 Newark 
 Jersey City 
 
 Fig. 35. — Illustration of Correct Order of Items in a Bar Graph. 
 
 By doing this the magnitudes will come in a column where 
 they can be seen by those who like to have the figures, 
 and these can easily be added, with the sum at the bottom 
 of the middle column. Very seldom should the numbers 
 be placed at the immediate right-hand ends of the bars. 
 They tend to make the bars seem disproportionately 
 larger and they cannot be easily compared or added. 
 
 lo 20 30 40 50 60 70 SO 90 <00 
 
 1.- Vermont 
 etCy to 
 ^QrLoutsiana 
 
 m. 
 
 Fig. 36. — Device Used by Dr. L. P. Ayres to Show by States Percentages 
 of the School Population Enrolled in Public Schools, in Private Schools, 
 and Not in Any School, in 1910. 
 
 White portion indicates children in public schools ; shaded, those in private schools ; 
 and black, those not in any school. (Adapted from A Comparative Study of the Public 
 School Systems in the Forty-eight States and reproduced by permission of the Russell 
 Sage Foundation.) 
 
 To make the bars stand out clearly, the distance between 
 them must be markedly different from the width of a 
 bar. Otherwise the lines will tend to run together or 
 sink into the background. 
 
 •Sometimes, to economize space, some of the largest 
 items may be shown with double bars. A familiar example 
 is the graph used by the American Book Company on its 
 
242 
 
 School Statistics and Publicity 
 
 calendar. On this a long double bar represents the 
 expenditures in liquor and a short single bar, the expendi- 
 tures in textbooks. This is very doubtful practice, as it 
 greatly lessens the difference between the magnitudes for 
 the untrained reader. 
 
 The horizontal bar graph is also very valuable where 
 it is desirable to make comparisons between component 
 parts in similar wholes. For this purpose, the wholes 
 are represented by bars of the same width and length. 
 
 Republican V//A Progressive ES3 Democrat I I 
 
 AO 50 60 70 80 90 100 
 
 10 20 30 
 
 Alabama J^yyAwww"^ 
 
 to 
 
 New York 
 
 Fig. 37. — Device to Show Parts of a Total and Also to Indicate Relative 
 
 Sizes of the Totals. 
 
 Adapted from Professor Irving Fisher's chart showing parts of the total vote for 
 president in 1912, at the same time indicating the relative voting strength of each state. 
 
 Then the component parts for the same item are lined up on the 
 left margin, and those for another item will appear lined up on the 
 right margin. A fine example is shown in Figure 36. 
 
 It is to be noted that this will not work well for more than two 
 component parts. Observe how difficult it is to get an idea of the 
 size of the middle item representing the number of children enrolled 
 in private schools when this item for different states is compared. 
 
 If, for any reason, it is desirable to make comparisons 
 between the relative sizes of the wholes, the widths of 
 the bar may be adjusted accordingly. 
 
 Mr. W. C. Brinton borrows such a device for giving an 
 analysis of the total vote for president in 1912 in the 
 forty-eight states from Professor Irving Fisher.^ Each 
 
 1 Brinton, W. C. : Graphic Methods cf Presenting Facts, p. 10 
 
Graphic Presentations of School Statistics 243 
 
 state is represented by the same length, and the percentage 
 given to each party varies, while the number of voters 
 also varies in the states. (See Figure 37.) 
 
 The following are some of the places in school work 
 where a graph of this kind would be helpful : 
 
 1. It could be used in the Ayres graph given before for comparing 
 the total number of school children in the states. 
 
 2. In a chart comparing the number of retarded, normal, and ac- 
 celerated children in a number of cities, the relative totals of children 
 enrolled could be shown by the width of the bars. 
 
 100 100 
 V 
 
 21 
 
 82 
 
 68 
 
 r: 
 
 ^ 
 
 55 
 
 35 
 
 
 
 
 38 
 
 
 
 
 
 
 
 ! 
 
 
 
 23 
 
 1 
 
 
 
 19 
 
 i 
 
 28 
 
 
 12 12 
 
 
 1 
 
 
 r^y^~^ 
 
 13 14 15 16 17 18 19 
 
 Fig. 38. — Bar Graph for Comparing Two Things Whose Proportions Are 
 
 Constantly Varying. 
 
 Columns represent number of boys and girls among each hundred beginners who 
 remain in school at each age from 13 to 19. Shaded columns represent boys and white 
 columns girls. (From Springfield Survey, page 52, by permission of the Russell Sage 
 Foundation.) 
 
244 
 
 School Statistics and Publicity 
 
 3. If the distribution of the cost of educating one child were being 
 shown for various items such as instruction, supplies, etc., for several 
 cities, the width of the bar might represent the total amount spent 
 by each city to educate one of its children, etc. 
 
 The bar graph may be used to compare two or more 
 things whose proportions are constantly varying, by using 
 a different shading for each separate kind of item. 
 
 Thus Dr. Ayres uses Figure 38 to show the number of beginners 
 that remain in school at various ages at Springfield, Illinois, using 
 shading for boys and white for girls. The reader not only gets a 
 comparison between the number of boys and girls remaining in school 
 at any given age, but also a comparison between the different age groups. 
 
 Other examples are : 
 
 1. Enrollment of pupils in different grades, white columns for boys 
 and black columns for girls [Cleveland Summary Volume, page 85). 
 
 2. Contrast in percentage of retardation of white and colored 
 pupils by grades, shaded bars for white, black bars for colored pupils, 
 all on horizontal basis {Louisville Report for 1911^-15, page 27). 
 
 School. 
 
 year 
 
 Daily 
 cost 
 
 Hi<3h 
 schools 
 
 Sal 
 
 ones 
 
 1 
 
 '////y 
 
 y//A y////////, y////////.w^^ 
 
 // /// 
 
 y///, mrn^M, '/////////. yrn^M, 
 
 ^^^ WMM/M ^vy/////A 
 
 
 I WASH I NQ TON 
 12 VtRMONT 
 24 KANSAS 
 36 MARYLAND 
 48 ALABAMA 
 
 Fig. 39. — Device for Showing Relative Standing of Several Cases on 
 
 Each of Several Items. 
 Adapted from Ayres's graph showing the standing of the forty-eight states, by per- 
 mission of the Russell Sage Foundation. The highest quarter is represented by white, 
 the second quarter by light shading, the third quarter by dark shading, and the lowest 
 quarter by black. 
 
 An elaborate yet easily understood chart, making com- 
 parisons for the forty-eight states on ten different items, 
 is found in Ayres's bulletin on the state school systems. ^ 
 
 1 Page 32 
 
I a a o 
 
 rt _ fe ^ o 
 
 .2 .2 f^ a oj 
 
 . .~ fl tn o) , 
 
 o ^ c3 fl, a 
 
 '^^^ ^ 
 
 o c3 ,, ej 1^ 
 •r -Q ti 
 
 ^ S 3 •-' 
 
 g 73 
 5 O fl 
 
 <u 2 >> >> 
 M g -a .1i 5 
 
 O) " <D 4) 0) 
 
 ^■5 S 2 != 
 
 j3 >^ <» S * 
 
 -Q a ^ 02 
 
 S o =* O 73 
 
 ■if ^ 03 QJ 
 
 c .s « 3 
 ^ ^ 2 d « 
 
 ^ tn « a; Its 
 ^ a 
 
 -w |_i IE S< =*-! 
 
 •ft « Oi « ° 
 
 a fl fe C! 2 
 
 -►^ ^ « fa 
 
 ro , ^ 4^ 
 
 -fj t; ■-' ■ vi a> 
 
 C O .ti T) ^> 
 
 2^ ^ S & ra 
 ft a> a >< "" 
 
 57 , X « 4) 
 
 t-i ^ o _fl 
 
 a> '^ tc - ^^ 
 
 c3 S 9 O O 
 
 ;i (- -u cc j3 
 ~ 3 tc o -t^ 
 
 05 -^ W "^ -g 
 
 «« a:S o -c 
 
 0) 8:3:2 ^ 
 -^ fl > S 
 
 IK ir: ^j o 
 
 ■+^ •« -rH M 
 
 o S 
 
 OQ d 
 
 245 
 
246 School Statistics and Publicity 
 
 This is made up by starting at the top with the best state, Washing- 
 ton, and going down to the lowest, Alabama. Each state is repre- 
 sented by a separate bar, and each bar is divided into ten parts. 
 Each of these ten parts represents an item on which the state is graded. 
 For each item, the states in the lowest quarter have that part black ; 
 the states in the next quarter have dark shading; the states in the 
 next to the highest quarter have light shading ; and the states in the 
 highest quarter have white. (See Figure 39.) 
 
 This form of chart is excellent for graphing the numerous items of a 
 summary table. The use of the different forms of shading uniformly 
 through the table enables the reader quickly to locate any state's 
 rank on any item. The use of the darker shades for the worst ranks 
 on any item is also a good device, for it would have a tendency to 
 sting the p^ide of the average citizen. Note also that an idea of the 
 general standing of any state on the ten items can be gotten quickly. 
 For example, Washington appears with practically white spaces and 
 so is manifestly very high on the whole. Alabama has a black space 
 on every item and so is very low. These things can be seen at a glance. 
 
 Such a graph would be of use in comparing the per- 
 formances of several pupils, of several teachers, of several 
 classes, of several rooms, or of several schools where each 
 individual case had been ranked on a number of qualities 
 or achievements. 
 
 For example, the showing on standard tests for several eighth 
 grades might be made thus, using a horizontal bar for each grade, and 
 a vertical column for each test used. 
 
 The Boston Survey has a graph to show how Boston's 
 expenditures compare with those of the average city, 
 using component parts of a bar graph that is a square. 
 It is reproduced here as Figure 40, but experiments show 
 that while it is not readily understood at first by the 
 average school man, it is exceedingly effective once it is 
 grasped. 
 
 Sometimes it is desirable to compare two distributions 
 that have been grouped by similar steps, for the typical 
 
Graphic Presentations of School Statistics 247 
 
 amount of the same quality in each step. In this case, 
 a set of bar graphs running to the right may be used for 
 one distribution, abutting on a similar set of bars running 
 
 HIGH SCHOOL EDUCATION. PAYS 
 YEARLY INCOME 
 
 HIGH SCHOOL 
 TRAINING 
 
 NO K S. 
 
 TRAINING 
 
 N HIGH SCHOOL 
 
 IN HIGH SCHOOL 
 
 9 500 
 
 1000 
 
 I. 150 
 
 ^Z337 
 
 H SCHOOL TRAINED BOYS-WAGES -53.50 PER DAY 
 
 NO H. SCHOOL TRAINING -WAGES^ 1.50 PER DAY 
 
 Fig. 41. — Example of Right and Left Device for Comparing Distributions 
 
 with Bar Graphs. 
 
 It shows average yearly income of high school graduates as compared with that of 
 persons not having high school training. (From one of the 1917 folders of the Agri- 
 cultural Extension Division of the International Harvester Company.) 
 
 to the left for the other distribution. It is, however, 
 difficult to compare magnitudes on the right with those 
 on the left. 
 
248 
 
 School Statistics and Publicity 
 
 An example is the graph used by the Bureau of Edu- 
 cation, shown in Figure 41. 
 
 From this, it is only a short step to the graph which 
 exhibits the facts about several qualities in a city or 
 
 DECREASES INCREASES 
 
 70 
 
 6.0 
 
 5.0 
 
 • 4.0 
 
 3.0 
 
 2.0 
 
 1.0 
 
 
 1.0 
 
 I II III 1 - 
 
 
 ' I^^Grade 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 3rd 
 
 
 
 
 
 
 
 
 
 4+h 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -,th 
 
 
 
 
 
 
 
 
 gth 
 
 
 
 
 
 
 
 
 oth 
 
 
 
 
 
 
 
 ■ 
 
 
 - 10^^ 
 
 
 
 
 
 
 
 
 
 
 „tK 
 
 
 
 
 
 
 
 
 ^^ 
 
 ,2th 
 
 . Fig. 42. — Example of Right and Left Device for Comparing Distributions 
 
 with Bar Graphs. 
 
 It shows changes in distribution of enrollments by grades in Baltimore between 1899 
 and 1909. (From Baltimore Survey, page 98.) 
 
 school, by showing increases on bars extending to the 
 right of a vertical line, and decreases by bars extending 
 to the left of it. 
 
 A good example is Figure 42 from the Baltimore Survey, showing 
 tha change in enrollments in the various grades by percentages. The 
 
Graphic Presentations of School Statistics 249 
 
 decrease in the first grade was due to special efforts to move up re- 
 tarded children. 
 
 This form is preferable to that in Figure 41 because the bars to 
 be compared are closer to each other. 
 
 The " Monument " Graph. Areas are not at all easy 
 to compare. Such comparisons would be far more 
 effective if made by bar graphs lined up at one end so 
 that the comparison would merely be a matter of com- 
 paring the lengths of the various bars. 
 
 For example, take the "monument" graph which is frequently 
 found in school reports. A good illustration is found in one repre- 
 senting the number of pupils enrolled in each grade in the Alabama 
 Survey of Three Counties, shown in Figure 43. 
 
 Here each stone represents the enrollment for one grade, beginning 
 with the first grade for 5423 pupils and topping off with a little stone 
 for the 60 pupils in the last year of the high school. 
 
 The errors in using this sort of graph are as follows: 
 The areas of the " stones " may be taken into account 
 for the relationship by the reader, when it really is shown 
 by their lengths only. As there is no common point 
 from which to measure either the length or area of the 
 stones, no adequate idea of relative sizes can be obtained 
 from such a graph. The effect of this form of comparison 
 is to make the difference between the larger and the 
 smaller numbers seem less than it is. This may be 
 shown by rearranging the data in the graph in a regular 
 bar graph lined up at the left. (See Figure 44.) Notice 
 how much accentuated the differences appear. ' It may 
 be contended that the monument graph is only one form 
 of the graph given in Figures 41 and 42. That is, it 
 really has two halves formed by an imaginary line down 
 the center, either half of which gives a simple bar graph 
 effect. But if that is the case, why not use the simple 
 
250 School Statistics and Publicity 
 
 bar graph alone in the first place? When children are 
 lined up in '' stair steps " to get their varying heights, 
 
 IV H.S. 60 
 JHH.S. I\9 
 n H.S. 198 
 I H.S. 455 
 
 7^^(5r.Q\b 
 
 ^^ 6KI485 
 
 b^^Qtr. 1800 
 
 4^^6^2080 
 
 3^Qr. EI32 
 
 Z^Qr 2687 
 
 |9* Gr. 54 Z3 
 
 Fig. 43. — Monument Graph Showing Number of Pupils Enrolled in 
 
 Each Grade. 
 (From An Educational Survey of Three Counties in Alabama, page 63.) 
 
 they are all placed on the floor. No one would ever think 
 of putting their waist lines on the same level and then 
 of taking account only of variations above the waist. 
 
 
 
 ET H.S. 
 
 60 
 
 4ttj qrade 20bO 
 
 1 
 
 \ 1st grade M23" 
 
 Fig. 44. — Graph Showing Apparent Size of Certain "Stones" from the 
 Monument Graph Lined up at One Side. 
 
 Comparisons with Circle Graphs. Errors frequently 
 arise in making comparisons with circles. It makes all 
 the difference whether the comparison is made through 
 the diameters or through their areas. The ordinary 
 reader tends to make the comparison on the diameter 
 basis. If the circles are drawn on the area basis, how- 
 ever, it is apparent that the comparison will not be so 
 striking as the maker of the graph intended. If, on the 
 
Graphic Presentations of School Statistics 251 
 
 other hand, circles are drawn on the diameter basis, some 
 readers will tend to overestimate the facts. 
 
 The graph used on page 91 of the Cleveland Survey, Summary 
 Volume, is a good example of the difficulty in using circles on the area 
 
 Under age and 
 rapid progress 
 
 © 
 
 Normal age and 
 rapid progress 
 
 © 
 
 Over age and 
 rapid progress 
 
 Under age a\nd 
 normal progress 
 
 Normal age and 
 norma\ progress 
 
 Over age and 
 normal progress 
 
 ® 
 
 Under age and 
 slow progress 
 
 Normal age and 
 slow progress 
 
 
 
 Over age and 
 slow progress 
 
 Fig. 45. — Comparison of Circles by Areas Using the Percentage of 
 Children in Each Age and Progress Group in Elementary Schools of 
 Cleveland at Close of Year 1914-15. 
 
 (From Cleveland Survey, Summary Volume, page 91. By permission of the Siirvey 
 Committee of the Cleveland Foundation.) 
 
 basis. It is reproduced in Figure 45. Here nine circles are employed 
 to show the percentages of children in the under-age, normal-age, and 
 over-age subdivisions of the rapid, normal, and slow groups. This 
 
252 
 
 School Statistics and Publicity 
 
 requires nine circles and it is almost impossible to compare them ac- 
 curately. One circle is marked 6 and another 30, but the former 
 appears to be about one-fourth of the latter, or larger than in reality. 
 
 A Larger Proportion of Children are Going to Nigh School. 
 
 Fig. 46. — Concentric Circle Graph to Show Relative Increase in High 
 
 School Enrollment. 
 (From the 1915-16 Review of the Bockford, Illinois, Schools, page 107.) 
 
 The use of concentric circles as a means of comparison 
 is even worse than that of the circles apart from each 
 other. 
 
Graphic Presentations of School Statistics 253 
 
 Figure 46 is a concentric circle graph showing the enrollment of 
 the Rockford schools for the years 1895, 1900, 1905, 1910, and 
 1915, with a comparison of the high school enrollment with the total 
 enrollment. This graph is worth little for the public, because it is 
 hard to understand, and because it is almost impossible to get a cor- 
 rect notion of the areas of the different circles. One tends to look 
 only at the rings and not at the circles, and the whole effect is some- 
 thing like the advertisements showing the various layers in an auto- 
 mobile tire. 
 
 The bar graph would be very much better for this 
 comparison, as in Figure 47. 
 
 Comparisons made by using the areas of segments of 
 the same circle are questionable, because people have not 
 
 Hig h School Grades 
 
 1895 - 5141 
 
 to 
 1915 - e932 
 
 Fig. 47. — Component Bar Graph Comparison of Data Shown in Figure 46, 
 
 been trained to estimate the areas of segments. Even 
 as simple a graph as " How Portland Spends Its Dollar " 
 (see Figure 48) is hard to interpret correctly. But 
 when the graph becomes as complicated as the one 
 in Figure 49, it is probably useless for the average 
 reader. 
 
 It is even more difficult to compare sectors in different 
 circles than in the same circle. 
 
 For example, take the graph on the percentage of home-trained and 
 non-trained teachers and principals in the Cleveland Survey, shown in 
 Figure 50. The sectors here serve but little to emphasize the different 
 percentages given. In addition, the white labels on the black parts 
 probably cut down the apparent size of the black parts very mate- 
 rially. 
 
254 
 
 School Statistics and Publicity 
 
 Triangle Graphs. A few surveys make comparison 
 by the heights or areas of overlapping isosceles trianglet 
 having equal bases, as in Figures 51 and 52. 
 
 Interest 
 
 207* 
 
 Libra K[g5 1.01 
 
 ,1/ ^-^- 
 
 .0'' 
 
 A 
 
 .^ 
 
 <J^' 
 
 /qa 
 
 00 
 
 Education 
 30.7i 
 
 
 
 % 
 
 ^<>?/> 
 
 
 'Sf^ 
 
 fo. 
 
 \ 
 
 ■op_. 
 
 Fig. 48. — Example of Difficulty of Comparing Component Parts in a 
 Circle Graph When the Angles Are Not Clearly Shown. 
 
 This figure gives an itemized statement of "How Portland Spends Its Dollar." 
 (From Portland Survey, page 84.) 
 
 Since two triangles with equal bases are to each other as 
 their altitudes, the comparison is perfectly accurate from 
 either a height or area standpoint. But as heights alone 
 are really wanted for determining the areas, plain bars 
 
1/ 
 
 y0(. 
 
 \ 
 
 \ 
 
 ^ 
 
 X 
 
 / 
 
 1 
 
 (0 
 
 
 \/\/X 
 
 ^ 
 
 \ 
 
 ^ 
 
 > 
 
 ^(O 
 
 CM 
 
 
 \/V^ 
 
 I ^ 
 
 V 
 
 \ 
 
 / 
 
 CO 
 
 W 
 
 
 V/\ 
 
 \ 
 
 \ 
 
 \ 
 
 /5 
 
 1 
 
 (U 
 
 
 \\ 
 
 \ 
 
 \ 
 
 y 
 
 r 
 
 (0 
 
 o 
 
 c 
 
 
 lU 
 
 
 o 
 
 1 
 
 
 "B* 
 
 D 
 
 c 
 
 
 ^ ^ 
 
 lO 
 
 IS 
 
 03 
 
 a> 
 
 u. 
 
 
 0> 
 
 •5^ 
 
 2 
 o 
 
 (0 
 
 o 
 o 
 
 o 
 
 5 
 
 
 X.* 
 
 52 
 
 CO 
 
 tJ 
 
 CO 
 
 
 
 Q> 
 
 D 
 
 3 
 
 
 cd 
 
 
 
 
 03 (1| 
 
 "ft § 
 
 ^ <0 
 
 CO 
 
 a 
 
 ^1 
 
 OS 
 
 6 
 
 M 
 
 "Diagram III. Surface of circle represents total per capita expenditure in the 
 average city. Sectors are proportional to amount spent for each of the twelve main 
 purposes for which funds are expended. Shaded portion represents expenditure in 
 Bridgeport. Under each heading the first figure gives in dollars and cents the 
 amount spent per child per year in the average city and the second figure the corre- 
 sponding amount for Bridgeport." 
 
 255 
 
256 
 
 School Statistics and Publicity 
 
 Fig. 50. — Graph Showing 
 DiflSculty in Comparing Sec- 
 tors from Different Circles. 
 
 (From Cleveland Survey, Sum- 
 mary Volume, page 107. By per- 
 mission of the Survey Committee 
 of the Cleveland Foundation.) 
 
 would show the relative lengths 
 of the altitudes with much less 
 work. Besides, since the aver- 
 age reader has had no experience 
 in comparing heights or areas of 
 triangles, about the only justifi- 
 cation for them is the one matter 
 of adding variety. 
 
 Comparisons with Cartoon 
 Effects. Another bad use of 
 comparison through areas is 
 sometimes found in the employ- 
 ment of cartoons in which the 
 data are represented by the areas 
 of persons or objects. For ex- 
 ample, take Figure 53. The 
 trouble with such a chart is 
 that the area grows much faster 
 than the height, so that the 
 expenditure for 1914-15, instead 
 of appearing only about 35 per 
 cent greater than for 1910-11 as 
 it should, really appears to be 
 several hundred per cent greater. 
 In this particular chart, the hori- 
 zontal lines at the back help to 
 reduce the exaggeration by em- 
 phasizing the height factor. 
 
 Even when the figures are 
 given with the chart, as in this 
 instance, the visual inaccuracy 
 is serious enough to cause a dis- 
 trust of the whole thing. How- 
 
Graphic Presentations of School Statistics 257 
 
 ever, the cartoon effect can be secured in a graph that, 
 by use of units or separate figures, allows no chance for 
 error in making comparisons . 
 
 )( 
 
 A very effective chart of this kind is the one from the Des Moines 
 ^Report for 1914-15.^ It is too large to be shown effectively in this 
 book, so will be described in words only. It aims to show in a car- 
 toon the relative numbers of retarded children, normal children, and 
 
 SALARIES OP 
 
 PRIMCiPALS 
 
 WAQES OF 
 JANITORS 
 
 Fig. 51. — Comparison by Triangles 
 between the Salaries of Principals in 
 Springfield and the Average for Ten 
 Other Cities in 1911-12. 
 
 Shaded triangle represents average annual 
 per capita expense for principals' salaries for 
 each child in average attendance in the day 
 schools of Springfield, and triangle in outline 
 represents corresponding expenditures for 
 the average of ten other cities. (From 
 Springfield Survey, page 98, by permission of 
 the Russell Sage Foundation.) 
 
 Fig. 52. — Comparison by 
 Triangles of the Cost of Janitor 
 Service in the Average City 
 and in Bridgeport. 
 
 Triangle in outline represents 
 portion of each thousand dollars 
 spent for janitors' wages in the 
 average city ; shaded triangle repre- 
 sents the amount spent in Bridge- 
 port. (From Bridgeport Survey, 
 page 27.) 
 
 accelerated children, boys and girls separately. To the left, 27 boys 
 and 20 girls are represented as climbing a hill, book in hand and read- 
 ing. In the middle, 18 boys and 23 girls are walking along on a level 
 with books tucked away under their arms. At the right, 5 boys and 
 7 girls are going down hill with no books at all. This is similar to 
 the Alabama illustration in that it gives relative proportions, but note 
 how much better this relationship is shown by separate children than 
 it would be by a few children varying in size. The graph would have 
 been a little more effective if each group of children had been in one 
 
 1 Page 99 
 
258 School Statistics and Publicity 
 
 line so that the length of the line might also have entered into the 
 comparison. This could have been easily managed by adding a few 
 more lines to indicate a portion of a hill for each child not on the level. 
 
 See how the child in the grades is growing 
 
 1910-11 1911-12 1912-13 1913-14 1914-15 
 
 t=rli 
 
 $23^ $25^ $27^ 
 
 Fig. 53. — Example of the Difficulty in Making Comparisons with Cartoon 
 
 Effects. 
 
 This shows what is spent in Louisville on each child in the grades. The comparison 
 is really made by heights only, but the reader tends to take it by areas. (From 1914-15 
 Louisville Report, page 35.) 
 
 This would take no more room on the whole, and it would not seriously 
 weaken the effect, since the hill conveys practically the same idea as 
 the use of the books. 
 
 Another example of a cartoon applying a good idea but in a very 
 unreal way, and also using circles for comparison, is given in Figure 54. 
 
 FOR EVERY DOLLAR THAT 
 
 The average Bridofeporl 
 
 cit y sp ends spends 
 
 A 
 
 BOARD OF EDUCATION OFFICE ^^^^H ^1 ^^ 22t 
 
 Fig. 54. — Cartoon Graph Using a Sector of a Circle to Represent Part 
 
 of a Dollar. 
 (From Bridgeport Survey, page 22.) 
 
 But the cartoon effect could have been kept and a much more accurate 
 comparison made by using cents in a bar effect, thus : 
 
 Average city $1.00 to 100 
 
 Bridgeport .22 . . . to 22 
 
Graphic Presentations of School Statistics 259 
 
 Ayres uses such a device to show the cost of schooling per child per 
 day in the various states, in his bulletin on comparing the school 
 systems of forty-eight states.^ 
 
 When the cartoon effect is very necessary to show 
 relative parts of a dollar, and accuracy is not essential, a 
 cartoon similar to Figure 55 may be advisable. 
 
 ADVENTURES OF MR. TAXPAYER. 
 
 With a municipal budget and without. 
 
 How one city now How a city 
 
 SUPPOSES KNOWS 
 
 the money is spent. where the money ^oes. 
 
 Fig. 55. — Cartoon Effect to Show Parts of a Dollar. 
 (From Newburgh Survey, page 93, by permission of the Russell Sage Foundation.) 
 
 Figure 56 is a cartoon effect to show the lack of suffi- 
 cient playground space. 
 
 Figure 57 is a skillful use of a bar graph effect. 
 
 Other good examples of cartoon effects may be found in : 
 
 a. The Ohio Survey, page 65. 
 
 Here there is a line of twelve teachers for each class of school, begin- 
 ning with one-room township schools and going on up to high schooL 
 Teachers without professional training are in black ; those with one 
 or more terms in summer schools are in gray ; those with one or 
 
 1 Page 18 
 
260 
 
 School Statistics and Publicity 
 
 more years in a professional school in white. A high degree of ac- 
 curacy is attained by using half a figure to represent one twenty- 
 fourth. Thus seven twenty-fourths of the twelve teachers in one 
 line are represented by three women clothed in black and another 
 with a black skirt and gray waist. 
 
 LAWN vs. PLAYGROUND 
 
 William Street 
 
 How One 
 Newbury 
 School Saves 
 the Grass at 
 the Expense 
 of the 
 Children 
 
 Fig. 56. — Cartoon Effect to Show Lack of Playground Space. 
 (From Newburgh Survey, page 63, by permission Of the Russell Sage Foundation.) 
 
 b. Ayres's A Comparative Study of the Public School Systems in the 
 Forty-eight States, page 6. 
 
 The value of school property in different states is represented by in- 
 dividual dollar marks, a line for each state, giving a bar effect ; thus : 
 
 Florida 
 Kentucky 
 Arkansas 
 Mississippi 
 
 IK (ft (J m (J (J* (J (P ^ (p (P <P ^ (P <P <p 
 
 XO ip tp Jp fp ip ip ip ip ip ip tp ip tj) *f) «p 
 
 "1 O (ft (ft <ft (ft (J (ft (ft (ft (P (P (P (J (P 
 
 XO tptptptptptptpytptptptptp 
 
 4 $$$$ 
 
Graphic Presentations of School Statistics 261 
 
 c. The 1912 Newton, Massachusetts, Report, page 86. 
 
 The enrollment of girls is represented proportionately for the dif- 
 ferent kinds of schools. The lower grades are shown with smaller 
 children and the high school with large ones. But the true relation- 
 
 15 THE HIGH TAX CRY JUSTIFIED ? 
 
 Totol Tax per Ccipifa ,\9\\ 
 
 New Rochelle 
 
 Hiag(^rQ Fa] Is Hi 
 
 ■■■■■■ 
 
 Average 10 Ci\ie5 b| 
 
 ■■■■■ 19. 
 
 Hiti^slon HH 
 
 HHHl Id./ 
 
 Pdu^hkeepsie HI 
 
 HHHI 
 
 Jamestown HI 
 
 WKKM 13.56 
 
 Hewbur^h |^| 
 
 HHI 13.6 a 
 
 Water town HI 
 
 HHI 
 
 AubuK/i HI 
 
 WKM 12.48 
 
 Amsferdam HI 
 
 I 7.02 
 
 25.10 
 
 Fig. 57. — Illustration of Use of a Bar Graph for Publicity Purposes. 
 
 Note that the figvires come at the right ends of the bars, but are much lighter in 
 effect and so do not make the bars appear longer. (From the Newburgh Survey, page 
 97, by permission of the Russell Sage Foundation.) 
 
 ships are shown by the numbers of figures in the various groups. 
 This gets all the value of the cartoon effect and avoids any trouble 
 about comparisons through areas. 
 
 Time Charts. A special device for comparing the use 
 of something for several purposes, or the time put in by 
 different persons, is the time chart. This is only one 
 
262 
 
 School Statistics and Publicity 
 
 form of a distribution table. The time element appears 
 on one scale and the items on the other. The data to 
 be entered are put in the rectangles determined by the 
 two scales. These areas may be shaded, or they may 
 simply be filled with words. Such a chart may be very 
 profitably used to show how teachers spend their time, 
 and it thus becomes simply the regulation school program ; 
 or it may be used to show how much of the time various 
 rooms are occupied ; or it may be utilized to show how 
 much of the time the school plant or grounds are used 
 during a year, etc. Table 42 is an example of this; 
 the rooms could be listed down the side and the hours 
 at the top if preferred. 
 
 Table 42. Time Chart by Rooms and Classes 
 
 Hours 
 
 Room I 
 
 Room II 
 
 Room III 
 
 Room I\ 
 
 Room V 
 
 8 
 
 Math. 1 
 Math. 1 
 
 Eng. 4 
 
 Hist. 2 
 
 Sci. 3 
 
 For. Lang. 1 
 
 9 
 
 Eng. 1 
 
 
 10 
 
 Math. 2 
 
 Eng. 2 
 
 Hist. 4 
 
 Sci. 1 
 
 For, Lang. 2 
 
 11 
 
 
 
 
 1 
 
 
 Eng. 1 
 Eng. 3 
 
 Hist. 3 
 
 
 
 2 
 
 Math. 3 
 
 Sci. 2 
 
 
 3 
 
 Math. 4 
 
 
 Hist. 1 
 
 
 For. Lang. 3 
 
 Comparisons with " Curves." It is probably unwise 
 to use curves often in presenting statistical material to 
 the public. It is well for the superintendent to know how 
 to employ curves of various kinds, for they are helpful 
 in his own analysis of his data. Such uses have been 
 shown on pages 166 ff. Engineers, draftsmen, economists, 
 social workers, and other statisticians, of course, use 
 
Graphic Presentations of School Statistics 263 
 
 them. They are to be seen in a few advertisements. 
 But they probably have little significance for the general 
 public as yet. 
 
 At present, curves can probably be used most effectively 
 for the public when the aim is to make comparisons be- 
 tween the changes in the 
 
 19 03-10. 1910-11. 1911- \Z 
 $95 
 
 TECH. HIGH 
 
 NEWTON HIGH 
 
 same classes of items on 
 different dates. For this 
 the same vertical scale 
 may be used on either side 
 of the diagram, with the 
 years going out from left 
 to right. 
 
 A good example is found in 
 the chart showing the decreas- 
 ing cost per pupil for various 
 groups, given in Figure 58. 
 The chart is very fine, except 
 that the omission of the zero 
 line exaggerates the decrease 
 altogether too much. 
 
 A still better example of the 
 use of curves for this purpose is 
 Figure 59. 
 
 Another example is a form 
 showing the amount of money 
 schools get as compared with 
 what the city as a whole 
 spends. 
 
 Thus Professor Cubberley gives two graphs to show such facts for 
 two cities. (See Figure 60.) 
 
 Curve Effects with Bar Graphs. At present, instead of 
 presenting curves to the public, it is probably better to 
 use bar graphs arranged to secure the same general effect. 
 
 ALL PUPILS 
 
 - SHADES 
 
 KINDERGARTEN 
 
 Fig. 58. — Use of Curves to Compare 
 Changes in the Same Classes of Items 
 on Different Dates. 
 
 The figure shows decreasing cost per pupil 
 in various schools for the years 1909-12. 
 Note omission of zero line. (From 1912 
 Newton, Mass., School Report, page 35.) 
 
$65 
 60 
 55 
 50 
 45 
 40 
 
 35 
 30 
 25 
 20 
 15 
 10 
 5 
 
 1907 '08 '09 '10 Ml '12 '13 '14 '15 
 
 Fig. 59. — Use of Cvirves to Compare Changes in the Same Classes of 
 Items on Different Dates. 
 
 It show3 cost per pupil in grades and high school, 1907-15, Rockford, Illinois. Note 
 zero line is given here. (From 1915 School Report, Rockford, Illinois, page 122.) 
 
 264 
 
 
 
 
 
 
 
 
 / 
 
 
 C05 
 
 r Per Pu| 
 
 )il h 
 
 igh ; 
 
 ichoc 
 
 ^. 
 
 / 
 
 
 
 
 y 
 
 
 
 
 
 
 
 / 
 
 ^^ 
 
 / 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Cos 
 
 ,t ?ey 
 
 Pup 
 
 i! Ele 
 
 ment 
 
 3ry C 
 
 Irade,! 
 
 ■■)y 
 
 
 
 
 y 
 
 
 
 
 f 
 
 y 
 
 
 
 — ■ ■ 
 
 y^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Cost 
 
 Per P 
 
 upil f 
 
 or Ge 
 
 neral 
 
 Expe, 
 
 ise.(Gr 
 
 ades9 
 
 H.S.) 
 
 
 ■ 
 
 
 
 
 
 
 
 
Graphic Presentations of School Statistics 265 
 
 Where two items are represented upon each of a number 
 of bars, and the part representing one of these items is 
 colored or shaded, the effect of two curves is obtained. 
 The graph as it stands satisfies the untrained reader, 
 while the trained reader can easily read off the curves. 
 The diagram showing the retarded children in the grades 
 
 Total City Tox Rate- 
 
 s' fr5 
 
 00 ^ 
 
 ^ CO t. 
 
 < o :^ 
 
 ^^^^TSToTTorSXh^gFI^^^^^ 
 
 Schenectady, fsf.Y 
 
 § § o ^ w CO 3: 
 ^ o^ o> ^ ijj 5^ O) 
 
 San franc \SCO,CQi 
 
 Fig. 60. — Example of Using Curves to Show Changes on Different 
 Dates in Two Items for Two Cities. 
 
 The two charts show the competition for city funds. (From Cubberley's Public 
 School Administration, page 414, by permission of Houghton Mifflin Company.) 
 
 of Memphis, Tennessee, public schools affords a fine 
 illustration. (See Figure 22, page 166.) 
 
 This same curve effect can be obtained by a trained man 
 from the component part bar graphs. Obviously, if only 
 two component parts are shown, the curve is read one 
 way for one set, and another for the other, that is, looked 
 at from the two sides separately, as in Figure 61. If 
 
266 
 
 School Statistics and Publicity 
 
 there are more than two component parts, the curves 
 show only for the two end parts, as in Figure 62. 
 
 c 
 
 Fig. 61. — Curve Effect on Bars with Two Component Parts. 
 
 The curve effect is also noticeable in a graph where 
 various items in one group are compared with similar 
 items in other groups. 
 
 A good example is the standing in the four fundamental operations 
 in the Courtis arithmetic tests for different grades. Letting A stand 
 for Addition, S for Subtraction, M for Multiplication, and D for 
 
 Fig. 62. — Curve Effect on Bars with Three Component Parts. 
 
 Division, the results might be shown as in Figure 63. A line connect- 
 ing the ends of all the bars of the same kind will give a curve. See 
 also Figure 38. 
 
 Standards for Drawing Curves. If, after all the pre- 
 ceding, it is still felt desirable to present school statistics 
 to the public with curves, the curves should be drawn 
 properly. Accordingly, the suggestions of the Joint 
 
Graphic Presentations of School Statistics 267 
 
 Committee on Standards for Graphic Presentation ^ 
 should be followed. The words alone are given here, but 
 
 No. 
 
 16 
 14 • 
 12 • 
 
 10 - 
 8 • 
 6 ■ 
 
 A ■ 
 
 probs. 
 
 GRADE E" 
 
 VI 
 
 q\c. 
 
 A 5 M D 
 
 A S M D 
 
 A S 
 
 M 
 
 Fig. 63. — Bar Graph Device with Curve Effect for Comparing Several 
 
 Groups on Several Items. 
 
 Each bar represents the standing in the Courtis Tests. A means addition, S, 
 subtraction, etc. 
 
 the full report contains graphs which make the text 
 much clearer: 
 
 1. The general arrangement of a diagram should proceed from 
 left to right. 
 
 2. Where possible, represent quantities by linear magnitudes, as 
 areas or volumes are more likely to be misinterpreted. 
 
 3. For a curve, the vertical scale, whenever practicable, should be 
 so selected that the zero line will appear on the diagram. 
 
 4. If the zero line of the vertical scale will not normally appear on 
 the curve diagram, the zero line should be shown by the use of a hori- 
 zontal break in the diagram. 
 
 5. The zero lines of the scales for a curve should be sharply dis- 
 tinguished from the other coordinate lines. 
 
 1 Copies may be obtained from the American Society of Mechani- 
 cal Engineers, 29 West 39th Street, New York, price 10 cents, discount 
 in quantities. 
 
268 School Statistics and Publicity 
 
 6. For curves having a scale representing percentages, it is usually 
 desirable to emphasize in some distinctive way the 100 per cent line 
 or other line used as a basis of comparison. 
 
 7. When the scale of a diagram refers to dates, and the period rep- 
 resented is not a complete unit, it is better not to emphasize the first 
 and last ordinates, since such a diagram does not represent the begin- 
 ning or end of time. 
 
 8. When curves are drawn on logarithmic coordinates, the limiting 
 lines of the diagram should each be at some power of ten on the log- 
 arithmic scales. 
 
 9. It is advisable not to show any more coordinate lines than 
 necessary to guide the eye in reading the diagram. 
 
 10. The curve lines of a diagram should be sharply distinguished 
 from the ruling. 
 
 11. In curves representing a series of observations, it is advisable, 
 whenever possible, to indicate clearly on the diagram all the points 
 representing the separate observations. 
 
 12. The horizontal scale for curves should usually read from left 
 to right and the vertical scale from bottom to top. 
 
 13. Figures for the scales of a diagram should be placed at the left 
 and at the bottom or along the respective axes. 
 
 14. It is often desirable to include in the diagram the numerical 
 data or formulae represented. 
 
 15. If numerical data are not included in the diagram, it is desirable 
 to give the data in tabular form accompanying the diagram. 
 
 16. All lettering and all figures on a diagram should be placed so 
 as to be easily read from the base as the bottom, or from the right- 
 hand edge of the diagram as the bottom. 
 
 17. The title of a diagram should be made as clear and complete 
 as possible. Sub-titles or descriptions should be added if necessary 
 to insure clearness. 
 
 3. Special Summarizing Graphs 
 
 Sometimes it is desired to give a graphic summary of 
 something that has been measured by relative position 
 on a number of items. The way of showing this roughly 
 by the bar graph was given on page 244, but this indicated 
 only the quarter of the distribution in which the case fell 
 
Graphic Presentations of School Statistics 269 
 
 3 
 
 4- 
 
 lO 
 
 1 1 
 
 ; 7. 
 
 !3 
 
 1 + 
 
 J5 
 
 I 7 
 
 I 8 
 
 19 
 
 20 
 
 21 
 
 7 
 
 I O 
 
 / 4 
 
 \5 
 
 t6 
 
 I 7 
 
 I 8 
 
 IS 
 
 20 
 
 21 
 
 lO 
 
 I I 
 
 12 
 
 15 
 
 IT 
 
 I 8 
 
 19 
 
 20 
 
 21 
 
 Expenditure Expenditure per Expenditure ^Qr 
 
 per $1,000 of tax- child in average 
 
 inhabitant able wealth daily attendance 
 
 The shaded rectangles represent Boston, 
 
 Fig. 64. — Graphic Device for Summarizing the Relative Position of a 
 Given Case in a Number of Different Distributions. 
 
 This figure shows the rank of Boston in a group of twenty-one cities in expenditiire 
 for operation and maintenance of schools. (From Boston Report, 1916, page 158.) 
 
270 
 
 School Statistics and Publicity 
 
 Teacher City 
 
 (Indicate sex) 
 
 EFFICIENCY RECORD 
 
 Grade taught. 
 
 (or building) (or subject) 
 
 Experience years. Salary per month. 
 
 Highest academic training '. 
 
 Extent of professional training 
 
 Detailed Ratixg V.P 
 
 II. 
 
 III. 
 
 IV. 
 
 General appearance 
 
 Health 
 
 Voice 
 
 Intellectual capacity 
 
 Initiative and self-reliance . . . . 
 Adaptability and resourcefulness . 
 
 Accuracy 
 
 Industry 
 
 Enthusiasm and optimism . . . . 
 
 Integrity and sincerity 
 
 Self-control 
 
 Promptness 
 
 Tact 
 
 Sense of justice 
 
 Academic preparation 
 
 Professional preparation 
 
 Grasp of subject-matter 
 
 Understanding of children . . . . 
 Interest in the life of the school . 
 Interest in the life of the community 
 Ability to meet and interest patroni 
 Interest in lives of pupils . . . . 
 Co-operation and loj'alty . . . . 
 Professional interest and growth . . 
 
 Daily preparation 
 
 Use of English 
 
 Care of light, heat, and ventilation . 
 
 Neatness of room 
 
 Care of routine 
 
 Discipline (governing skill) . . . . 
 Definiteness and clearness of aim . 
 
 Skill in habit formation 
 
 Skill in stimulating thought 
 Skill in teaching how to study . 
 
 Skill in questioning 
 
 Choice of subject-matter . . . . 
 Organization of subject-matter . . 
 Skill and care in assignment 
 Skill in motivating work . . . . 
 Attention to individual needs . 
 Attention and response of the class . 
 Growth of pupils in subject-matter . 
 General development of pupils 
 Stimulation of community . . . . 
 Moral influence 
 
 General Rating 
 
 Poor 
 
 Medium 
 
 Good 
 
 Ex. 
 
 Recorded by Position Date 
 
 Fig. 65. — Summarizing Graph to Show Efficiency Record of a Teacher, 
 Used by School of Education, University of Chicago. 
 
Graphic Presentations of School Statistics 271 
 
 on that quality. A refinement of this is found in Figure 
 64. 
 
 Another example is that used by the School of Education 
 at the University of Chicago to sum up the rating of a 
 
 Grade 
 
 Arithmetic 
 
 Addition 
 
 Spe&d 
 
 Accuracy 
 
 Subtraction 
 
 Speed 
 
 Accuracy 
 
 Multiplication 
 
 Speed 
 
 Accuracy 
 
 Division 
 
 Speed 
 
 Accuracy 
 
 Silent Reading 
 
 Handwriting 
 
 Speed 
 Qualify 
 
 Grade n m m sr la im: 2m. 
 
 Fig. 66. — Graphic Device for Summarizing the Achievements of One 
 Pupil or School in Several Fields as Related to Standards in Those Fields. 
 
 (From Educational Tests and Measurements of Monroe, De Vo33 and Kelly, by 
 permission of the authors and Houghton MiflBin Company.) 
 
 teacher on several different qualities. Part of this graph is 
 shown in Figure 65.^ 
 
 It should be noted that a general heading will loom 
 up on this graph in proportion to the number of sub- 
 heads it has. Consequently, the way to make any general 
 heading have weight will be to add sub-heads, without 
 caring particularly how important these sub-heads are. 
 
 1 The original blank has suitable main headings for each Roman numeral, 
 printed horizontally, but it was not feasible to show them on a page of this 
 size. 
 
272 School Statistics and Publicity 
 
 There appears to be no way of overcoming this defect 
 except by varying the widths of the horizontal spaces, 
 which would probably complicate the device beyond the 
 point of practical value. 
 
 A third summarizing device, and one capable of wide 
 adaptation, is shown in Figure 66. The same device 
 may be used to show the standing of a school or city on 
 a number of items, by placing the names of the standards 
 at the top in place of the Roman numerals for the grades. 
 
 Figure 24 on page 168, showing the surfaces of fre- 
 quency one above the other, is a fourth summarizing 
 graph that is of value to the public. 
 
 4. Brinton's Rules for Graphic Presentation 
 
 Brinton, in his standard book. Graphic Methods of 
 Presenting Facts, has a very convenient set of rules 
 followed by a set of check items. These were printed 
 before the suggestions of the Committee on Standards 
 for Graphic Presentation (of which he is chairman) were 
 published (see pages 267-268), and some of the suggestions 
 appear in both sets. But Mr. Brinton's original lists 
 are not confined to suggestions for curves, as is the report 
 of the committee. The practical school man will prefer 
 to have all the suggestions given in one place, so they are 
 here appended. 
 
 Helps on Graphic Presentations 
 
 (Selected from Brinton : Graphic Methods of Presenting Facts, 
 
 pages 360-362) 
 
 I. Rules for Graphic Presentation 
 
 1. Avoid using areas, or volumes, when representing quanti- 
 ties. Presentations read from only one dimension are the least 
 likely to be misinterpreted. 
 
Graphic Presentations of School Statistics 273 
 
 2. The general arrangement of a chart should proceed from left 
 to right. 
 
 3. Figures for the horizontal scale should always be placed at 
 the bottom of a chart. If needed, a scale may be placed at the 
 top also. 
 
 4. Figures for the vertical scale should always be placed at 
 the left of a chart. If needed, a scale may be placed at the right 
 also. 
 
 5. Whenever possible, include in the chart the numerical data 
 from which the chart was made. 
 
 6. If numerical data cannot be included in the chart, it is well 
 to show the numerical data in tabular form accompanying the chart. 
 
 7. All lettering and all figures on a chart should be placed so 
 as to be read from the base or from the right-hand edge of the 
 chart. 
 
 8. A column of figures relating to dates should be arranged with 
 the earliest date at the top. 
 
 9. Separate columns of figures, with each column relating to a 
 different date, should be arranged to show the column for the 
 earliest date at the left. 
 
 10. When charts are colored, the color green should be used to 
 indicate features which are desirable or which are commended, 
 and red for features which are undesirable or criticized adversely. 
 
 11. For most charts and for all curves, the independent variable 
 should be shown in the horizontal direction. 
 
 12. As a general rule, the horizontal scale for curves should 
 read from left to right and the vertical scale from bottom to top. 
 (See "special.") 
 
 13. For curves drawn on arithmetically ruled paper, the ver- 
 tical scale whenever possible should be so selected that the zero 
 line will be shown on the chart. 
 
 14. The zero line of the vertical scale for a curve should be a 
 much broader line than the average coordinate lines. 
 
 15. If the zero line of the vertical scale cannot be shown at the 
 bottom of a curve chart, the bottom line should be a slightly 
 wavy line indicating that the field has been broken off and does 
 not reach to zero. 
 
 16. When the scale of a curve chart refers to percentages, the 
 line at 100 per cent should be a broad line of the same width as a 
 zero line. 
 
274 School Statistics and Publicity 
 
 18. If the horizontal scale for a curve begins at zero, the verti- 
 cal line at zero (usually the left-hand edge of the field) should be 
 a broad line. 
 
 19. When the horizontal scale expresses time, the lines at the 
 left-hand and the right-hand edges of a curve chart should not 
 be made heavy, since a chart cannot be made to include the be- 
 ginning or the end of time. 
 
 20. When curves are to be printed, do not show any more 
 coordinate lines than are necessary for the data and to guide the 
 eye. Lines one-fourth inch apart are sufficient to guide the eye. 
 
 21. Make curves with much broader lines than the coordinate 
 ruling, so that the curves may be clearly distinguished from the 
 background. 
 
 22. Whenever possible, have a vertical line of the coordinate 
 ruling for each point plotted on a curve, so that the vertical lines 
 may show the frequency of the data observations. 
 
 23. If there are not too many curves drawn in one field, it is 
 desirable to show at the top of the chart the figures representing 
 the value of each point plotted in a curve. 
 
 24. When figures are given at the top of a chart for each point 
 in a curve, have the figures added if possible to show yearly totals 
 or other totals which may be useful in reading. 
 
 25. Make the title of a chart so complete and so clear that 
 misinterpretation will be impossible. 
 
 Special. 
 
 In showing deviations from a central tendency, on the vertical 
 scale, upwards is plus, and downwards, minus ; on the horizontal 
 scale, to the right is plus, and to the left, minus. 
 
 II. Checking List for Graphic Presentations. 
 
 1. Are the data of a chart correct? 
 
 2. Has the best method been used for showing the data? 
 
 3. Are the proportions of the chart the best possible to show 
 the data? 
 
 4. When the chart is reduced in size, will the proportions be 
 those best suited to the space in which it must be printed ? 
 
 5. Are the proportions such that there will be sufficient space for 
 the title of the chart when the chart has been reduced to final 
 printing size? 
 
Graphic Presentations of School Statistics 275 
 
 6. Are all scales in place? 
 
 7. Have the scales been selected and placed in the best pos- 
 sible manner? 
 
 8. Are the points accurately plotted? 
 
 9. Are the numerical figures for the data shown as a portion of 
 the chart? 
 
 10. Have the figures for the data been copied correctly? 
 
 11. Can the figures for the data be added and the total shown? 
 
 12. Are all the dates accurately shown? 
 
 13. Is the zero of the vertical scale shown on the chart? 
 
 14. Are all zero lines and the 100 per cent lines made broad 
 enough ? 
 
 15. Are all lines on the chart broad enough to stand the re- 
 duction to the size used in printing? 
 
 16. Does lettering appear large enough and black enough when 
 seen under a reducing glass in the size which will be used for print- 
 ing? 
 
 17. Is all the lettering placed on the chart in the proper direc- 
 tion for reading? 
 
 18. Is cross-hatching well made with lines evenly spaced? 
 
 22. Are dimension lines used wherever advantageous? 
 
 23. Is a key or legend necessary? 
 
 24. Does the key or legend correspond with the drawing? 
 
 25. Is there a complete title, clear and concise? 
 
 26. Is the drafting work of good quality? 
 
 27. Have all pencil lines which might show in the engraving 
 been erased? 
 
 28. Is there any portion of the illustration which should be 
 cropped off to save space? 
 
 29. Are the instructions for the final size of the plate so given 
 that the engraver cannot make a mistake? 
 
 30. Is the chart in every way ready to mark "0. K. "? 
 
 5. Presenting Statistical Data with Maps 
 
 Sometimes it is necessary to impress the public with the 
 way items vary in size or frequency in different geographic 
 areas. Thus, a state superintendent may wish to show 
 
276 School Statistics and Publicity 
 
 the legislature how high school facilities vary in different 
 counties of the state; a city superintendent may desire 
 to demonstrate how far pupils have to go to reach a 
 school, or the crowded conditions necessitating a new 
 building in a certain locality ; or a county superintendent 
 may need to show just how his county ought to be dis- 
 tricted for schools so that all children will be within a 
 reasonable distance of a school. These things may be 
 shown fairly well by variations in shading the maps for 
 different localities, a device much used by the United 
 States Bureau of Census and by geography, history, and 
 economics texts. However, all such maps require keys for 
 their interpretation, and it is difficult for any one, except 
 a person very familiar with such work, to understand 
 them quickly. 
 
 A much better way is to represent every case or every 
 certain number of cases (say ten) by a dot. Figure 67 
 is a map used by George Peabody College for Teachers 
 to show the communities from which students have 
 come to the college during the years 1914 to 1918. This 
 illustration shows admirably just how much territory is 
 coming under the direct influence of the school. It does 
 not give a correct notion of the number of students 
 that have attended the college during these years, as 
 some cities have contributed possibly a hundred or more 
 each and yet such a city would be represented by only 
 one dot. If a dot had been inserted for each student, 
 some portions of the map would have been solid black. 
 This could be remedied by using perpendicular wires 
 with a bead for each student, as recommended by Brinton,i 
 but it is very difficult to reproduce such a map by pho- 
 tography or by a drawing. In showing the widespread 
 1 Graphic Methods of Presenting Facts, p. 251 
 
^ 
 
 
 O lU 
 
 a 
 
 % 
 o 
 
 r/i 
 
 (-> 
 o 
 
 o 
 
 h o d 
 
 .2 -si 
 
 0) to 
 
 OS 
 
 a '" 
 
 2a 
 
 TO H 
 
 o o 
 
 0) 
 
 2 f=i 
 ;:; o 
 
 .2^ 
 
 CO 
 
 a; □ 
 -^ 2 
 
 "in 
 
 a s 
 
 .S eS 
 
 277 
 
278 
 
 School Statistics and Publicity 
 
 influence of the college, the map is more effective than 
 the one from the Report of the General Education Board 
 
 Fig. 68. — Device for Showing Distribution of Cases on a Map. 
 
 Each dot represents a student from the different counties, attending the University 
 of Georgia. Note the radiating lines to indicate that the university is exerting an 
 influence upon all the state. 
 
 for 1914, page 4. This is merely a map of the United 
 States in outline, with the number of students in each 
 
Graphic Presentations of School Statistics 279 
 
 state attending Vanderbilt University. It is thus only a 
 map with a table on it, and not even in good tabular 
 form, because there is no direct way to compare the 
 numbers by having them in one, column, and especially 
 running from low to high or vice versa. It is impossible 
 for one to visualize the number of students attending 
 Vanderbilt University, and it takes a little time to get a 
 correct notion of the sections of the country coming under 
 its direct influence. 
 
 The University of Georgia uses a good map to show its 
 attendance, heightening the effect by adding radiating 
 lines to indicate its influence in the state. (See Figure 68.) 
 
 The dot device on a map of the city was used by the 
 superintendent in the Rockford schools ^ to show the 
 residence of pre-tubercular pupils and also the residence 
 of students in the evening schools. In the one case, he 
 showed clearly that the pre-tubercular children were 
 scattered widely through the city and that the problem 
 of handling them was city- wide. He also showed that 
 the attendance at evening school was not restricted to a 
 small area of the city. In some of the Red Cross work 
 huge state maps are shown with the number of tubercular 
 soldiers for each county pasted in as so many paper doll sol- 
 diers, upright in ranks. It is a very effective presentation. 
 
 If the distances on a map are to be measured in some 
 time unit, a map similar to that in Figure 69 is useful. 
 
 The accredited schools of a state may be shown on an 
 outline map, using different-colored pins or tacks for the 
 different classes of schools. In the division of rural 
 education in the state department of education for 
 Missouri, there is an immense map of the state painted 
 on the wall. On this, each approved rural school is shown 
 1 Rockford Review for 1915-16, pp. 85, 104 
 
280 
 
 School Statistics and Publicity 
 
 by a small kodak picture of the building. This enables a 
 visitor to get almost immediately an idea of where the good 
 rural school work is being done, and closer examination 
 will show the kinds of school buildings being put up. Of 
 
 • Fig. 69. — Device for Showing Distance with a Time Element on a Map. 
 
 Map iised by the Nashville Commercial Club to show how accessible the city is. 
 Every city indicated is within twelve hours' travel of Nashville. 
 
 course, such a map cannot be easily reproduced in printing. 
 Still, something can be done with conventional drawings 
 for each item in a class. Most readers of this, for example, 
 probably recall the recent Y. M. C. A. propaganda with 
 
Graphic Presentations of School Statistics 281 
 
 the sketch of the Western Battle Front, each building of 
 the organization being represented by a tiny drawing of 
 the right kind. 
 
 III. HOW GRAPHS FOR THE PUBLIC DIFFER FROM THOSE 
 FOR THE ADMINISTRATOR 
 
 The difference has been referred to many times before, 
 especially in discussion of the superior value for the 
 public of bar graphs over curves and in Brinton's check 
 list.i Let us now analyze it further, restating certain 
 points for additional emphasis. 
 
 In general, the public will view graphs in much the 
 same way as they view any explanation or presentation. 
 The ordinary man cannot quickly get from a rough copy 
 of a chapter the meaning that an experienced writer can ; 
 he cannot extract from a confused and verbose mass of 
 evidence the essentials that a trained lawyer can; he 
 cannot grasp so quickly, nor in such large numbers, 
 many intricate graphic presentations that seem relatively 
 simple to a trained school man. School^graphs for the 
 public must be simple, with^jrelatively few" elements or 
 lines, and very forcible. The trained reader can extract 
 the significant things from most graphs, however poorly 
 constructed ; the average man cannot. Let tfe^now take 
 up some of the most significant aids for making graphs 
 clear to the public. 
 
 1. The title should give all the significant points to he 
 found in the graph, so that the graph would he quite in- 
 telligihle apart from the context where it is found. 
 
 This implies that there should not be many different points found 
 in one chart. If the chart can show only one thing, it is all the better. 
 Some examples of good titles are : 
 
 1 See pp. 274-5 
 
282 School Statistics and Publicity 
 
 Standing of the children of Salt Lake City in the fundamentals 
 of arithmetic, judged by the median score attained by each grade. 
 {Salt Lake City Survey, p. 175.) 
 
 Distribution of ages at which Salt Lake City children enter the 
 first school grade. (Same, p. 201.) 
 
 Columns represent number of pupils among each hundred begin- 
 ners who remain in school at each grade from the first elementary 
 to the fourth high. {Cleveland Survey, p. 88.) 
 
 Percentage of elementary teachers, high school teachers, and 
 elementary principals in Cleveland who are home trained and not 
 home trained. {Cleveland Survey, p. 107.) 
 
 How Portland spends its dollar. {Portland Survey, p. 84.) 
 
 Figure 7, representing the percentage of children in several 
 grades who make the given scores in composition. For ex- 
 ample, 1.7 per cent of the fourth-grade children wrote com- 
 positions scored at 0; 43.8 per cent of the fourth grade were 
 scored at 1; etc. By following the median lines, the overlap- 
 ping of ability from grade to grade is disclosed. {Butte Survey, 
 p. 75.) 
 
 In each of these instances, the title and the chart make a complete 
 unit. The last one is especially noteworthy. It gives a very brief 
 title, then expands this with a full but concise explanation of all points 
 in the diagram that may cause confusion. 
 
 2. A chart or graph too large to he seen without turning 
 the head is apt to he a poor chart for the puhlic, no matter 
 how simple it may he. 
 
 This is shown by some charts and folded graphs in some of the 
 surveys. It is very seldom that a chart or graph should occupy more 
 than one page of the publication in which it appears. 
 
 In reducing charts for publication, however, one must be careful 
 that the reduction is not carried to the point of making the differences 
 negligible or the lettering too small to be read easily. Advertisers 
 long ago discovered that the public will not read advertisements in 
 fine print, and school graphs are only a form of advertising the school 
 work. Often graphs are reduced greatly to economize space. If 
 this is pushed to the extent of making the graph hard to read, it is 
 clearly advisable to omit some of the graphs altogether, and make the 
 remainder large enough to be effective. 
 
Graphic Presentations of School Statistics 283 
 
 3. The background of a chart should not he made any 
 more prominent than necessary. 
 
 Many charts are plotted on coordinate paper heavily and finely 
 ruled, while the curves or bars are but a trifle heavier than the co- 
 ordinate ruling. Such charts do not stand out clearly from their 
 background. Only as many coordinate lines should appear on a 
 chart as are necessary to guide the eye of the reader and to permit of 
 easy reading of the curves. 
 
 The difficulty may be rather easily avoided by drawing on the 
 coordinate paper in very heavy lines with India ink all the lines and 
 figures which it is desired to reproduce, including the coordinate 
 ruling. The necessary lettering can be put in with a typewriter, using 
 a practically new black ribbon. The proper exposure in making the 
 cut will "take" all the desired lines and lettering, but not the others. 
 
 4. Exaggerations should he avoided as much as possible. 
 
 (a) Usually, a very forceful presentation may be had without any 
 great sacrifice of accuracy. 
 
 Of course, only complete scientific presentation can ever give 
 the whole truth. However, if only a phase of a problem at a 
 time is presented to the public (and often this seems necessary) , 
 some exaggeration is inevitable. Here the problem is to choose 
 between absolute accuracy and forcefulness of presentation. 
 
 (6) It is, in general, dangerous to leave the zero line off a chart 
 intended for the public, or even to send it out with the conventional 
 wave line at the bottom. 
 
 Many school men in making charts have found it convenient 
 to leave the zero line off. Sometimes, when all the quantities 
 used come high on the vertical scale, this is done to economize 
 space. 
 
 An excellent example of such a chart is Figure 70. The 
 upper chart conveys the idea that salaries have increased greatly 
 in Louisville during this period of five years. But the chart 
 begins at about $475 instead of zero. The chart below is drawn 
 in full. It is clearly seen from the complete chart that the salary 
 increase, while noticeable, does not appear anything like so large 
 as in the original and incorrectly drawn chart. 
 
284 
 
 School Statistics and Publicity 
 
 (c) Care should be taken that the scales chosen for the graph do 
 not exaggerate things unduly. 
 
 The novice in chart making will probably become confused by 
 the ever changing ratios between the perpendicular and the hori- 
 zontal scales. No definite rules can be laid down for guidance in 
 
 ^5 if was drawn 
 
 ^550 $600 -1650 itJOO 
 
 19*0-11 1 
 
 
 
 
 1 19/1-12 1 
 
 
 
 
 1 I9IZ-I3 1 1 
 
 
 
 
 [ igi3-i4 1 1 
 
 
 
 
 1 19(4-15 1 1 1 
 
 
 
 
 
 
 As 
 
 t should 
 
 hdve 
 
 been 
 
 
 
 
 
 
 
 IJO ~?|00 fl50 4200 ^250 ^300 4350 f400-#450 -?500 #550 ^600 ^650 ^700 
 
 I9l0|-ll 
 
 
 1 
 
 
 1911 
 
 -IZ 
 
 
 
 
 
 
 
 
 
 
 
 1912 
 
 -13 
 
 
 
 _ 
 
 
 
 
 
 
 
 
 i9(3 
 
 -14 
 
 
 
 
 
 
 
 
 
 
 
 1914 
 
 -15 
 
 
 ^ 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 Fig. 70. — Example of Danger of Leaving Zero Line Off a Chart. 
 
 The top set of bars was intended to show a "comparison of salaries paid to elementary 
 school teachers in Louisville for the years 1910-1915." The bottom set shows what the 
 comparison really was. (Adapted from the Louisville Report, 1914-1915, page 18.) 
 
 this matter. The only way to get facihty in adjusting these 
 . ratios in the proper way is through the trial and error method. 
 That a change in the ratio makes a great difference in the im- 
 pression produced by the graph is shown in Figure 71 
 
Graphic Presentations of School Statistics 285 
 
 The left-hand graph shows the results of the Kansas Silent 
 Reading Tests in the Rockford schools. The right-hand graph 
 shows the same data plotted with the horizontal scale increased 
 while the vertical scale is decreased. It will be seen at once 
 
 Scole Grade 
 
 345 6 7 8 
 20 
 
 10 
 
 5 
 
 Scale Grade 
 
 3 4 5 6 7 8 
 
 15 
 
 10 
 
 Fig. 71. 
 
 Example of Effect Produced by Changing Ratio of Horizontal to 
 Vertical Scale on a Graph. 
 
 Results of Kansas Silent Reading Tests. (Adapted from Review of Rockford, Illinois, 
 Schools, 1915-1916.) 
 
286 School Statistics and Publicity 
 
 that the difference in achievement between the grades does not 
 appear so marked in the second illustration as in the first. 
 
 (d) The superintendent must beware of graphs containing optical 
 illusions. 
 
 There is little danger of this in charts made small enough to 
 publish, but there may be danger in large wall charts. Brinton 
 calls attention to illusions caused by a row of perpendicular lines 
 compared with a row of horizontal ones the same distance apart. ^ 
 
 Fig. 72. — Cartoon Graph Representing Ratio of Lighting Space to Floor 
 
 Space. 
 
 (From Alabama Three-County Survey, page 92.) 
 
 The lines in the first row appear shorter than they really are and 
 spread farther apart ; those in the second seem to be longer than 
 they really are. Another illusion is caused when a white square 
 and a black one of exactly the same size are drawn adjacent. The 
 white one seems larger. This might affect slightly the ratio of 
 lighting space as shown by white windows against a dead black wall 
 space, in some surveys. (See Figure 72.) 
 
 5. Special effort should he made to introduce variety, 
 novelty, and various striking features to attract attention 
 to the statistical relations to he presented. 
 
 Variety in graphs is as necessary to keep the attention as is variety 
 anywhere else. No ordinary reader could stand it to wade through 
 a report of any length which had a great many bar graphs of one 
 pattern and no other illustrations. 
 
 Sometimes pictures or devices may be used simply to catch the 
 attention. The bulletin on illiteracy in Virginia, 2 for example, uses 
 a picture of a rural school to get the reader's attention for the state- 
 
 1 Graphic Methods of Presenting Facts, p. 358 
 
 2 Illiteracy in Virginia, published by State Department of Public 
 Instruction, p. 7 
 
Graphic Presentations of School Statistics 287 
 
 THE PROFIT FROM TWO HERDS FOR ONE YEAR 
 
 rr 
 
 
 
 $95.73 
 
 
 ._.^j l Ml ' I I I I I 
 
 '■"•' ' State Bank 
 
 WHY THIS DIFFERENCE? 
 
 Herd A 
 
 IT WAS NOT THE SIZE OP HERD .11 COWS 
 
 9 PURE Bred 
 
 It was mot the breed % grades 
 
 IT was mot the feed cost -♦sze.yo 
 
 (silos and cood buildings on each farm) 
 
 HERE IS THE ANSWER 
 
 AVCRAGIE PRODUCTION OP 
 
 BUTTER FAT I7I.I LBS. 
 
 PER COW 
 
 This Is A True Story As Told Dy 
 
 Herd B 
 
 II COWS 
 
 t NATIVE 
 10 GRADES 
 
 $569.96 
 
 386.9 LBS 
 PER COW 
 
 *^n 1 ■ I J ' ^ ■ * I t,IZI 
 
 ^ 
 
 H0RAL:-IT would have taken 93 POOR cows TO MAKE 
 THE PROFIT THE 11 GOOD COWS MADE; 
 
 DOES IT PAY TO KEEP RECORDS? 
 
 Fig. 73. — Example of Use of Special Devices to Attract Attention to the 
 
 Statistics Involved. 
 
 This figure compares the profit from two herds for one year and shows how many 
 dairymen are wasting time and money on low-producing cows. "Why not get rid of 
 your 'visitors'?" (From a pubUcation of the University of Wisconsin, Experiment 
 Division, by permission.) 
 
288 
 
 School Statistics and Publicity 
 
 ment that only six of these children are beyond the first reader and 
 none beyond the fifth reader. 
 
 Or devices similar to those in Figure 73 may be used to attract 
 attention to the statistics. 
 
 Then there are special touches which no one can tell precisely how 
 to go about acquiring. For example, in the Alabama three-county 
 
 A ^ ^ ^ 
 
 ftfl 
 
 One outof every ten white men must ask another 
 to mark his ballot for him. 
 
 ttMMIII 
 
 Foor outof every ten negroes must ask another 
 to ^ign their names for them. 
 
 Fig. 74. — Bar Graph with Cartoon Effect Showing Illiteracy in Alabama. 
 (From the Survey of Three Counties in Alabama, page 19.) 
 
 survey, illiteracy is shown among voters by having the literate voters 
 face the reader and the illiterate ones turn their backs. (See Figure 74.) 
 Probably this is about as forceful a showing of the shame of illiteracy 
 as could be devised. 
 
 IV. EXAMPLES OF GOOD GRAPHS ON SCHOOL STATISTICS 
 
 FOR THE PUBLIC 
 
 1. To Show Rise in School Costs 
 
 In the Newton, Massachusetts, Report for 1912,^ the 
 scale on tax for school maintenance per $1000 is shown 
 on a thermometer device which is reproduced on page 102 
 
 1 Page 113 
 
Graphic Presentations of School Statistics 289 
 
 of this book. This gives the idea that costs for school 
 maintenance should rise. The names of various cities 
 with which Newton is compared appear at one side of the 
 graph, with lines running from each name to the proper 
 degree on the thermometer where the mercury should 
 stand for that city. This graph shows very forcibly that 
 the mercury must rise many degrees for Newton before 
 it will equal the best record made by the other cities. 
 
 For some cities, probably a cartoon utilizing the weigh- 
 ing machine seen at fairs and carnivals would be equally 
 effective. 1 Instead of pounds, tax leyies or per capita 
 amounts of money could be shown on the scale of the 
 upright, with the highest amount reached or desired 
 at the top. Men representing the other cities could be 
 standing around, evidently having struck the machine, 
 and their records could be shown on a bulletin board in 
 the background. Another man, representing the home 
 city, could be shown as just getting ready to strike the 
 machine to see what he can do, in the midst of words of 
 encouragement or taunts from the other men. His old 
 record might appear on the bulletin board. Underneath 
 might be some such question as, " Can't he send it to the 
 top? '' " Who is the best man? " '' How much will he 
 beat his old record? " 
 
 2. To Show Relative Investments in School Property 
 
 In the Educational Survey of Three Counties of Alabama, 
 the number of dollars invested for each child of school 
 age by each state is given. ^ Each dollar is represented by 
 a dollar mark. Thus, Massachusetts has $115 invested 
 in school property for each child of school age, while 
 
 1 Suggested by Mr. F. C. Lowry 2 Page 212 
 
290 School Statistics and Publicity 
 
 Mississippi has only $4. The advantages of this graph 
 are : The sjmibol aids in calhng attention to the graph ; 
 the length of the row of dollar marks gives the effect of a 
 bar ; the data are accurately represented, — there is no 
 material exaggeration or anything in the device to mis- 
 lead. 
 
 3. To Show a Lack of Funds for Maintenance 
 
 In the Survey of Three Counties of Alabama,^ there 
 appear pictures of a schoolhouse and an automobile. 
 From suitable figures, we learn that the initial cost of a 
 cheap automobile is more than that of the average rural 
 schoolhouse; and that the upkeep of the machine is 
 more than that of the average rural school. This com- 
 parison depends for its power on contrast, not on accuracy, 
 for there is nothing particularly accurate about it. All 
 the same, it is a very powerful device in shaming rural 
 people into doing their duty by schools. If there is a 
 single automobile in such a district, it represents more 
 than the rural school expenses. This illustration will 
 be of service chiefly in suggesting similar comparisons. 
 
 The Columbus Dispatch some months ago had a very 
 effective cartoon to represent the disparity in wages of 
 women teachers and statehouse janitors in Ohio.^ It 
 depicts a gruff old man with hands in his pockets, labeled 
 " Old Man Ohio.'' Above him are two inserts. The 
 left insert represents a pitiful woman teacher in her 
 classroom, with the statement that the average salary 
 for the public school teacher in Ohio is $54 a month. The 
 right insert depicts a shuffling negro janitor in cap and 
 overalls, bearing broom, mop, and bucket, with the 
 
 1 Page 72 
 
 2 Reproduced in American. School Board Journal, Jan., 1917, p. 33 
 
Graphic Presentations of School Statistics 291 
 
 statement that Ohio pays the janitors in the statehouse 
 $60. On either side of Old Man Ohio is a hand with index 
 finger pointing at him to emphasize the title of the whole 
 — " For Shame ! " 
 
 4. To Show Length of School Term, Average Attend- 
 ance, Etc. 
 
 For this the Ayres bulletin on the forty-eight states has 
 a good graph. Each day is represented by a small square, 
 the whole representing a bar graph, with each bar two 
 squares wide to make the bars shorter. The total length 
 of the bar represents the average number of days in 
 which schools were open in that state. Beginning at the 
 left, a sufficient number of these little squares are shaded 
 
 40 60 60 
 
 49- NEW MEXICO, 
 
 
 Fig. 75. — Graph for Showing the Relation of the Average Number of Daj^s' 
 Attendance by Each Pupil to the Number of Days School Was Open. 
 
 (From Dr. Ayres's Comparative Study of the Public School Systems in the Forty-eight 
 States.) 
 
 to represent the average number of days attended by each 
 pupil enrolled in that state. The names of the states 
 appear on the left from high to low, beginning with Rhode 
 Island, which had her school open 193 days and kept 
 each pupil in 148.8 days. The lowest is New Mexico, 
 which had her schools open only 100 days and kept each 
 pupil in only 66.4 days. This is represented in Figure 75. 
 This chart shows clearly which states have the longest 
 school terms and which are making the best use of what 
 they have. It could be used for cities just as well as 
 for states. 
 
292 School Statistics and Publicity 
 
 Another way to show the attendance of different school 
 systems is suggested by the graph on page 10 of the same 
 bulletin. This shows the number of days of schooling 
 each child of school age would get per year if he got his. 
 share. Each day is represented by a small dot; the 
 dots are clustered in groups of five, thereby giving a 
 unit of the week as well as the day. The bar effect is 
 obtained, and the graph has every advantage mentioned 
 for those above. This graph could be used in any graph- 
 ing of attendance, probably. The copy used by Ayres 
 has the figures at the right end of the bars, which is bad, 
 because it makes the bars appear lengthened unequally. 
 This is corrected in the copy below. • 
 
 48. New Mexico 46 :•::•::.::•::• : etc. 
 
 5. To Show Per Capita Costs of Schooling 
 
 A graph for this, similar to the last two described, is 
 found on page 18 of the Ayres pamphlet. It sets forth 
 the cost of one day's schooling for one child in each state 
 in 1910. Each cent is represented by a black dot, and the 
 bar effect is obtained. This dot or the cent mark or the 
 dollar mark could be used in gi-aphing any data on costs, 
 the unit being chosen so as to keep the bar short enough.^ 
 
 South Carolina 7 :•::•::.::•::• : etc. 
 
 6. To Show a Disgraceful State of Affairs in Certain 
 
 Localities 
 
 On page 160 of the Alabama three-county survey is 
 shown a familiar map graph. This particular one shows 
 the map of the United States with all states having com- 
 
 ^ See page 260 of this book 
 
Graphic Presentations of School Statistics 293 
 
 pulsory education laws in white, and those not having 
 such laws in black. As black is usually associated with 
 shame and disgrace, the graph becomes a stinging accuser 
 against the sections that are backward in this respect. 
 The same idea, of course, has been used in religious maps. 
 This use of black was referred to in the description of the 
 chart on page 32 of the Ayres pamphlet. (See page 
 244 of this book.) The idea is capable of wide use in 
 cases where it is desirable to shame backward school 
 systems into doing something better. The objection that 
 it is difficult to show lettering on black areas is easily over- 
 come by using white ink for lettering. 
 
 7. To Show the Variability of Children in the Different 
 Grades in Their Achievement in Some Standard Test or 
 
 Similar Matter 
 
 A discrete distribution or Bobbitt table may be shown 
 with bar graphs as shown on page 103, with a curve as 
 shown on page 104, or with, a scale as shown on pages 101 
 and 102. 
 
 For a continuous distribution, some form of the block 
 graph is probably best. It is much used in the surveys of 
 Butte, Salt Lake City, and other places. (See pages 112 
 and 137 of this book.) 
 
 8. To Show the Relation between the Rank of Children 
 in Their Classes in the Elementary School and Their 
 
 Probable Entrance into High School 
 
 The pupils are divided into three classes, the upper, 
 middle, and lower thirds. Each third is represented by a 
 broad vertical bar, all bars the same length. The per- 
 
294 
 
 School Statistics and Publicity 
 
 centage of each bar representing those not going on should 
 be colored black or shaded, the rest being left unshaded. 
 The three bars should be placed side by side, with the 
 low third on the left and the high third on the right. (See 
 Figure 76.) 
 
 Low Middle High 
 third third third 
 
 Fig. 76. — Graph Showing Percentage of Eighth Grade Pupils Entering 
 High School from the Low Third, the Middle Third, and the High Third 
 of their Classes, Cleveland. 
 
 (From Cleveland Survey, Summary Volume, page 185, by permission.) 
 
 9. To Compare Achievements of the Various Grades in 
 Standard Tests with Similar Grades from Other Cities 
 
 The best graph for this purpose is probably a modifi- 
 cation of the bar given in Figure 63 on page 267. Each 
 grade could be represented by the one kind of shading 
 throughout. Each group could have the name of its 
 city beneath, the scale could appear on each margin, and, 
 if necessary, faint horizontal scale lines could be drawn 
 clear across the drawing. 
 
Graphic Presentations of School Statistics 295 
 
 10. To Show Ratio of Lighting to Floor Space 
 
 In the Springfield Survey, sl graph is used for this in 
 which the floor space is represented by a black square. ^ 
 In the midst of this is a white square representing the 
 lighting space. In the first diagram to the left appears 
 the standard ratio ; the second one gives the average ratio 
 for Springfield. The percentages appearing in the white 
 squares should be written below. White squares always 
 appear larger than black ones in such a graph, but this 
 one is too small for the feature to affect it materially. 
 The white square in the midst of the black gives a window- 
 like effect and so helps to call attention to the diagram. 
 This may be improved upon by making the whole the 
 shape of a side wall, with the white the shape of windows 
 in that wall. (See Figure 72, page 286.) 
 
 The preceding illustrations comprise only a few of the 
 best selections from school reports and surveys, in addition 
 to those given before. It will be noticed that many of 
 the problems the superintendent is sure to meet in 
 graphing his data have not been mentioned. The bar 
 graph, concerning which much has been said in the pre- 
 vious pages, is capable of wide adaptation, as is also the 
 Bobbitt table. One or the other of these two may be 
 pressed into service upon almost any occasion. It is 
 well, however, to introduce some of the special forms 
 mentioned above, for variety's sake if nothing else. 
 The efficient superintendent, of course, will always be on 
 the lookout for improving the devices we now have for 
 presenting statistical data to the public, or he may work 
 out some entirely new methods. But any graph he 
 
 1 Page 24 
 
296 School Statistics and Publicity 
 
 devises for the public should be relatively simple, very 
 clear, and, if possible, forceful. 
 
 V. ECONOMIES IN MAKING SCHOOL GRAPHS FOR THE 
 
 PUBLIC 
 
 Large Cross-Section Paper. If large charts are to be 
 made for display purposes, excellent results may be 
 obtained from the use of good wax or grease crayons of 
 assorted colors. The time required for getting accurate 
 measurements may be greatly reduced by the use of 
 large size cross-section paper, as the counting is then very 
 easily done for the two scales or for locating any particular 
 point. The paper for this should be heavy manila, 
 light enough in color and sufficiently free from spots to 
 make a good background for the colors, and rough enough 
 to take the colors easily. Sheets 36 by 40 inches ruled 
 faintly in one-inch squares give excellent results. 
 
 These may be obtained from any large printing house equipped 
 with ruling-pen machines, but are very expensive in small quantities. 
 The University of Chicago Press carries them in stock at from three 
 to five cents per sheet, depending upon the price of paper, transporta- 
 tion extra ; or they may be obtained from the Peabody College Book 
 Store on the same terms. 
 
 Making Cartoons. In some cases, good results may 
 be obtained by pasting pictures on charts, if the cartoon 
 effect is desired. 
 
 For example, the writer wished to reproduce in large form the 
 automobile and schoolhouse graph from the Alabama three-county 
 survey, referred to on page 290. He got a picture of an automobile 
 from a large-sized advertisement in the Saturday Evening Post and a 
 picture of a rural schoolhouse from the front cover of an American 
 School Board Journal. By putting on the title and the figures, the 
 chart was soon made. Students often employ this same device in 
 getting up posters for school entertainments. 
 
Graphic Presentations of School Statistics 297 
 
 Gummed Letters. Much time is saved and a beautiful 
 chart may be made at little cost of time and money by the 
 use of gummed paper letters and strips. These can be 
 gotten in several colors and any height from four inches 
 down. The superintendent can use letters from one to 
 one and a half inches high, costing from about $1.50 to 
 $2.00 per thousand. The letters may be spaced very 
 easily and quickly if the chart is made on a large sheet 
 of cross-section paper. 
 
 These letters and strips may be obtained from the Tablet and 
 Ticket Company of Chicago, which will send pictures of such charts. 
 The strips are very serviceable for making bar graphs. 
 
 Rubber Stamp Set. An advertising set such as is used 
 by merchants for display cards can be used advantageously 
 in making good charts. With a little practice, neat 
 charts can be quickly made with such a stamp set, 
 especially on the large cross-section sheets. 
 
 A satisfactory set may be obtained from the Milton Bradley 
 Company, 73 Fifth Avenue, New York City, or from Salisbury- 
 Schulz Company, 157 West Randolph Street, Chicago, or at any of 
 their various offices, for about $5, depending upon the price of 
 rubber. 
 
 Securing Clear Lines. In drawing charts for cuts, 
 the chart and all essential cross-section lines should be 
 reproduced. The error of reproducing the background 
 of numerous cross-section lines may be avoided by tracing 
 over in India ink the parts that should come out in the 
 cut. The photographic process will reproduce this easily 
 before the background with its fainter tones will '* take.'' 
 The too prominent background is usually due to making 
 the drawings with ordinary ink or faint typewriter letter- 
 
298 
 
 School Statistics and Publicity 
 
 ing. The latter may be easily traced in India ink and 
 give good results. (See page 76.) 
 
 Miscellaneous Aids. Numerous aids to making charts 
 quickly with various kinds of cross-section paper, special 
 scales, etc., may be obtained from the Educational 
 Exhibition Company, Providence, Rhode Island, or the 
 Tablet and Ticket Company of Chicago. 
 
 " Perpetual " Attendance Graph Device. An exceed- 
 ingly easily managed graph arrangement was observed 
 
 Fig. 77. — "Perpetual" Attendance Graph Device. 
 
 some years ago by the writer in the device used by 
 Principal R. L. Dimmitt of the Ensley High School at 
 Birmingham, Alabama, to compare the attendance 
 record of the classes in the high school. (See Figure 77.) 
 
Graphic Presentations of School Statistics 299 
 
 A large chart was made up on Bristol board, once for all, with the 
 percentage scale running up on the sides. Each class was repre- 
 sented by a paper ribbon that came through a slit on the base line. 
 By pulling the ribbons up and down each month and fastening the 
 ends with thumb tacks, the graph was quickly brought up to date. 
 The omission of the zero line exaggerated differences, but such ex- 
 aggeration was to some extent desired for emphasis. The only real 
 drawback seems to be that one cannot compare the records of the 
 classes by month with such a chart. The chart as operated, however, 
 does not need to show comparisons. The emphasis on attendance 
 is intended to keep up attendance all the time, and not to let children 
 slack up one month because of a good record the preceding month. 
 If, however, it is desirable to make comparisons, this can be easily 
 done by preserving kodak pictures of the chart at various times. 
 This cut is drawn from such a picture. 
 
 Graphs without Cuts. If the charts are to be set up 
 in type and not with cuts, the originals may be made 
 either by hand or on the typewriter. For all such work 
 the horizontal or vertical bar graph is especially useful. 
 By the aid of conventional signs, different lengths of 
 rules, etc., almost any chart containing a bar effect can 
 be made so simple that it can be set up in any ordinary 
 limitedly equipped printing office. The following are 
 examples : 
 
 XX 
 
 xxxx 
 
 j{)jj>;}>;j>Jt) xxxxxx 
 
 *mmm . xxxxxxxxx 
 
 XXXXXXX r^^^^-r^TTTTrTrrT^^^^^^^^^ XXXXXXXXXXXX 
 
 0000 xxxxxxxxxxxxxxxx 
 
 -^- ______-_-_-_-_-_------ xxxxxxxxxxxxxxxx 
 
 Utilizing Students. In most of the work on graphs for 
 the public, the superintendent need only furnish the idea 
 
300 
 
 School Statistics and Publicity 
 
 or design for the chart. The rest of the work can be 
 done by various upper grades or high school classes as 
 very profitable laboratory or practical exercises. The 
 mathematics classes, especially those in algebra, and the 
 drawing and art classes are the logical ones to call upon 
 to take charge of this work. 
 
 In Newton, Massachusetts, the boys in the high school printed 
 the title pages of the annual school reports and probably made the 
 graphs, although this latter is not so stated. In the World Book 
 Company's reprint of MacAndrew's The Public and Its School, the 
 drawings are all made by public school children. Children who could 
 do such drawing could make most of the graphs advocated in this 
 book. As it is, practically all the drawings in this book have been 
 
 (^^^W^^Es^iti: 
 
 rtSTS, BCTFt 5UR.VEY^ p. 93. [^fy\\ pupi I^ ICsteJ ) 
 
 Fig. 78. — Showing Sketch and Rough Notes from Which a Pupil Drew 
 
 Figure 13. 
 
 copied or drawn from the author's suggestions and statistical data 
 by the students of Mr. E. S. Maclin at the Atlanta Technological 
 High School, as a demonstration. Figure 78 shows the sketch and 
 notes furnished one of these pupils by the writer, from which Figure 
 13 was drawn. Figure 79 shows the cartoon drawn by a student 
 from notes given by Mr. Maclin. 
 
Graphic Presentations of School Statistics 301 
 
 ONLY TOO TRUE 
 
 cw>. I'm growing 
 
 |50 FAST THAT ILL 
 
 SOON be: in ThEl 
 
 ^TREET AND NO J 
 
 Fig. 79. — Cartoon Drawn by an Atlanta High School Student. 
 
 This cartoon represents the high school situation in Atlanta and was drawn by the 
 student with only these suggestions from his drawing instructor, Mr. E. S. Maclin : 
 
 Subject : The overcrowded conditions of the Atlanta High Schools. Represent a 
 good-natured boy who has outgrown his clothes at every point, trousers splitting, 
 feet running out of his shoes, shirt too small for him, etc. He is represented as 
 saying to his father, the City Fathers, "Dad, I'm growing in spite of you. I'll 
 soon be in the street and no place to go." His father is represented as being a 
 rich man with Atlanta's per capita bonded indebtedness of about $22 per in- 
 habitant. 
 
 It will be noted that the suggestions were not wholly followed, but the cartoon as 
 published in an Atlanta paper was sufficiently forceful. 
 
302 School Statistics and Publicity 
 
 EXERCISE 
 
 1 
 
 Take the school report or school survey used in "the exercise on 
 page 233. Write out a detailed criticism of the graphs or lack of 
 them in it from the standpoint of their effectiveness with the public, 
 showing just why they are good or liable to be unsuccessful. In the 
 cases of the unsuccessful ones, or failure to use graphs where desirable, 
 sketch graphs that would present the same data properly. 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 Brinton, W. C. Graphic Methods for Presenting School Facts. Prac- 
 tically all. 
 
 Ellis, A. Caswell. ** The Money Value of Education." U. S. Bureau 
 of Education Bulletin, 1917, No. 22. • 
 
 King, W. I. Elements of Statistical Method, Chapter X. 
 
 Rugg, H. 0. Statistical Methods Applied to Education, Chapter X. 
 
CHAPTER XII 
 
 TRANSLATING STATISTICAL MATERIAL ON 
 SCHOOLS FOR THE PUBLIC 
 
 I. THE NEED OF TRANSLATION 
 
 For the superintendent who wishes to present his school 
 statistics effectively to the public, three devices are 
 available. He may graph his material ; he may tabulate 
 it ; he may translate it into words. Each procedure has 
 its strengths and weaknesses. Each is best adapted to 
 certain conditions. 
 
 Translation is the most serviceable device in cases 
 where it is difficult to secure or to have printed good 
 tabulations or graphs. A translation of statistical 
 material into words with a few mere numbers can be 
 typewritten or set up in type anywhere with little effort. 
 Such a translation can be read or spoken at any public 
 meeting with no particular preparation. Furthermore, 
 it can be so neatly expressed that persons hearing it can 
 easily remember it and quickly pass it on to others, a 
 thing not possible with tabulations and graphs. It can 
 be so forcibly worded that it will arouse people to action. 
 
 For example, the New York Survey devoted many 
 pages to figures and graphs setting forth the results 
 attained in arithmetic. But it is very doubtful if the 
 whole or any part of this section could impress the 
 average man as does the simple translation of the summary 
 
 303 
 
304 School Statistics and Publicity 
 
 by Mc Andrew : "It takes us less time to get a thing 
 wrong here than it does in the average school system." ^ 
 
 Statistical material on schools is often presented in 
 words in such a way as to be clear to the trained school man 
 and yet be unintelligible to the average man. If it is to 
 be clear to the latter, it must be as definitely translated 
 for him as must material from the Latin or other foreign 
 languages, from a doctor's description in medical parlance 
 of a disease, from fundamental political theories, or from 
 scientific experiments in agriculture. We have various 
 writers to translate the classics ; we have Woods Hutchin- 
 son and other medical writers to translate medical 
 knowledge ; we had Miinsterberg and James to translate 
 psychology; we have various writers and speakers to 
 translate political theories into the party campaign 
 books; the agricultural colleges have numerous writers 
 of bulletins to translate agricultural knowledge for the 
 farmer and housewife. Have we not just as much need 
 for translating school statistics into the language of the 
 average man? 
 
 Beyond clearness, the matter of force is very important. 
 The translation must not only be clear to the man for 
 whom it is intended, but it must take hold of him in some 
 way. Force cannot be obtained by the mere repetition of 
 tabulations in straight sentences of reading matter, how- 
 ever much some state superintendents appear to think 
 it can. This is not translation. The problem is really f 
 the same problem as that of the life insurance compa- 
 nies, the corporation seeking to influence the public, and 
 advertisers in general, when they try to reach the public 
 with statistical material. They do far more than merely 
 express figures by words. 
 
 1 The Public and Its Schools, p. 8 
 
Translating Statistical Material 305 
 
 It cannot be too strongly stated that translation is 
 not the same as definition or explanation. In defining 
 statistical terms we simply aim to show precisely what 
 we mean by them, often in terms just as technical. In 
 explaining statistical terms we merely try to make our 
 particular use of them clear to persons who already have 
 the same definitions of them as ourselves, this too in 
 language often just as technical. Good translation of a 
 statistical term, of course, involves both definition and 
 explanation, but it is more. It puts the emphasis on the 
 fact that the meaning must be carried over into an entirely 
 different language or set of expressions. For example, the 
 median may be defined as the magnitude of the midpoint 
 in a distribution. In any given frequency table or sur- 
 face of frequency, it may be explained that the point 
 marked " m " signifies the median. But the idea of the 
 median in a set of superintendents' salaries may be 
 translated for the average man by telling him that half 
 the superintendents get more than that salary and half 
 of them get less. 
 
 The writer has never seen any discussion of this trans- 
 lation phase of school statistics. Consequently, the 
 treatment here is only a preliminary or tentative analysis, 
 to be supplemented with illustrations from any source. 
 For clearness and brevity, the points will be given rather 
 dogmatically. 
 
 II. SUGGESTIONS FOR GOOD TRANSLATIONS 
 
 The main things to be kept in mind in working out 
 good translations of school statistics are : 
 
 1. The illustrations and images used must be of an 
 elementary nature, or at least familiar to the people for 
 whom the translation is being made. 
 
306 School Statistics and Publicity 
 
 The announcement that retarded children in school are as thick as 
 Ford automobiles or men wearing Masonic pins is intelligible any- 
 where. On the other hand, the statement that retarded children are 
 as thick as negroes in southern cities would be a very effective way of 
 translating the facts to some southern audiences, but would hardly 
 be of much value elsewhere. It would not even work in all southern 
 cities, because the proportion of negroes in them varies from close to 
 50 per cent down to less than 5 per cent. 
 
 2. In some cases it may be necessary to use several 
 illustrations in order to be sure of reaching all classes of 
 people. 
 
 3. Instead of representing a total by imagining an 
 unreal extension of a familiar object, or by making up 
 from familiar units an aggregate so large as to be in- 
 comprehensible, it is usually better to employ some other 
 unit. Often this other unit is one of time. 
 
 It is of doubtful value to ask the average man to think of a line 
 of school children eight hundred miles long ; of a schoolhouse as large 
 as all the schoolhouses of the county put together; of a sheet of 
 writing paper large enough to cover a township; of a lump of coal 
 weighing as much as all the coal burned in one day in many schools ; 
 or of a total of any sort reaching into the hundreds of thousands. 
 Such translations are sometimes attempted. They might well be 
 called Jack-and-the-beanstalk translations, for they are about as far- 
 fetched. 
 
 Practically all statistical totals needing to be translated will, on 
 examination, be found to involve in some way units of length, area, 
 volume, weight, and time. The aim should be to translate the total 
 into another kind of unit that will keep it within the limits of com- 
 prehension or experience of the ordinary man.^ Often a relatively 
 larger unit of time that is forceful may be employed than in the case 
 of the other kinds of units, because most people comprehend long 
 stretches of time or the consumption of goods over long periods, 
 
 1 For elaborating this point the writer is indebted to Mr. H. A. 
 Webb, one of his students. 
 
Translating Statistical Material 307 
 
 fairly well. The periods of time, however, must in general be well 
 within the limits of the ordinary man's active life span. 
 
 Consider the following translation of the average daily absence 
 for the schools in Texas in 1911-13 : 
 
 "Placed twelve feet apart, these white pupils absent every day 
 from the schools of Texas would form a line extending across the 
 state from El Paso to Texarkana, a distance of over eight hundred 
 miles." 1 
 
 The average Texan, even though he is from an early age accustomed 
 to boasting of the size of the state, can have but a very hazy idea of 
 the distance mentioned. Even if he has traveled from one city to 
 the other, his idea is dependent chiefly upon his recollection of the 
 time it took to make the trip, and a good part of this time he may have 
 been asleep. In all likelihood, he has never seen children lined up 
 twelve feet apart this way. It would probably be better for most people 
 to consider these children as marching double file and say how many 
 days it would take them to pass a given point. The average man has 
 seen people marching double file in parades and has a good idea of 
 about how fast they would pass. 
 
 Instead of saying that, if the school costs of a certain city were 
 represented by silver dollars lying side by side, they would extend 
 the distance of ten blocks, it would be far better to say that at a cer- 
 tain sum, say $50 a month, it would take a man a certain number of 
 years to earn an equivalent amount of money. In this case, the 
 average man has a much better idea of how long it takes him to earn 
 money than he has of the distance dollars touching each other will 
 extend. He has never seen silver dollars lined up along a street, but 
 he has worked for wages and knows the value of his money. 
 
 A very effective example of this sort is found in the Negro Year- 
 Book for 1916-17? The compiler of this made a study of the average 
 number of days that each negro child of school age attended school 
 in each of the southern states. To make the smallness impressive, he 
 calculated the number of years it would take the average negro child 
 to complete the elementary school on the basis of eight grades and 
 nine months to the school year, thus : 
 
 1 White, E. v.: "A Study of Rural Schools in Texas," Bulletin 
 of University of Texas, No. 364, Oct. 10, 1914, p. 20 
 
 2 Page 233 
 
308 School Statistics and Publicity 
 
 No. of yrs. it would take 
 the average negro child to 
 complete the elementary 
 course in the public schools 
 State provided for him 
 
 Maryland 16 
 
 Texas 18 
 
 Virginia 18 
 
 Georgia 19 
 
 Florida 20 
 
 North Carolina 20 
 
 Alabama 22 
 
 Louisiana 25 
 
 South Carolina 33 
 
 This could have been made more forcible by changing the figures to 
 the age at which a negro entering school, when six years old, could 
 complete the grades, as 22 for Maryland, 24 for Texas, etc. 
 
 4. In cost statistics, it is sometimes advisable to 
 minimize the total by expressing it in amount per small 
 unit of time, usually a trivial sum. 
 
 Thus daily papers and many weekly periodicals advertise "10 cents 
 a week" and do not call attention to the total of $5.20 for the year. 
 
 A church calls for "30 cents a week" and does not emphasize the 
 fact that this reaches $15.60 for the year. 
 
 The Y. M. C. A. announces that membership with all privileges 
 costs "a nickel a day or the price of your daily cigar, " when the yearly 
 total is from $15 to $18, without mentioning these latter figures. 
 
 The Liberty Loan called for "a dollar a week" instead of stressing 
 the $50 that would be laid aside for the year. 
 
 The sum for the total is generally not stressed except in savings 
 bank advertisements and such appeals, where the aim is to surprise 
 the reader with the total saved, and not any amount that is expended. 
 
 5. Absolute accuracy frequently has to be sacrificed 
 to force and clearness in translation. 
 
 Thus, if the percentage of negroes in a city population was 40, and 
 the percentage of retarded children 35 or 45, the negro illustration 
 
Translating Statistical Material 309 
 
 mentioned before would be accurate enough for translating the 
 idea. 
 
 The "nickel a day" of the Y. M. C. A. is adequate for any an- 
 nual sum from $15 to $18. 
 
 6. Practically all totals have to be translated through 
 comparisons, using familiar objects or notions, before they 
 can be understood or have much force for the average man. 
 
 The statement that four per cent of all children of school age are 
 mentally defective and need special attention would mean little to 
 most men. But to say that every school system enrolling 500 chil- 
 dren had among that number 20 defective pupils, or enough to equal 
 the ordinary 6A grade taught by one teacher, would drive this fact 
 home to the average man. 
 
 The mere statement that there were on the average 64 children to 
 each grade teacher in a school system would mean little to the people 
 of the city and probably would receive no attention. However, if 
 the statement were changed to read that there were enough surplus 
 children over the standard number for the various classes to fill four 
 classrooms, the fact would certainly impress citizens. 
 
 The bare announcement of the amount spent for some item of 
 school expense is not nearly so forceful as the statement that it is 
 only a certain fraction of the ice cream and soda, liquor or tobacco 
 expenditures of the town, if such figures or approximate estimates 
 of them can be obtained from merchants. 
 
 A southern county superintendent recently translated his valua- 
 tion estimates as follows : 
 
 "The total value of all school property in the county outside 
 of the city is $34,420, which is $3080 less than half the value of 
 the county jail and site, and $3000 less than one fourth of the 
 value of the county courthouse, site, furniture and fixtures. 
 
 " Of the $34,420 invested in school property in the county, only 
 $15,665, or less than half, belongs to the state and county. In 
 other words, the value of all the school property of those schools 
 with titles vested in the state is $5335 less than the cost of three 
 of her best motor trucks used in constructing good roads. 
 
 "The total value of all supplies and equipment, including 
 musical instruments and libraries, is $5875, which is $1125 less 
 
310 School Statistics and Publicity 
 
 than the cost of one motor truck used in building the roads of 
 the county. 
 
 "All school equipment in the county outside the city of 
 
 is equal in value to less than one seventh of the value 
 
 of the machinery owned by the county and used in the making 
 of good roads. 'Seven to one' is the ratio of the county's in- 
 vestment in equipment for making roads as compared to her in- 
 vestment in equipment for making men and women." 
 
 The same superintendent translated the total area of all the school 
 grounds in the county, 73 acres, by comparing it with the 180 acres 
 of playgrounds in the property of two country clubs near the county- 
 seat. 
 
 In similar fashion, the increased enrollment of the public high schools 
 of the United States in 1914 over 1913 would make a city as large as 
 Chattanooga and Knoxville, Tennessee, combined. 
 
 Dr. L. P. Ayres did a great service for the school survey movement 
 when he used the following translation to show its results : 
 
 "About seven years ago this retardation became one of the 
 most widely studied problems of educational administration, and 
 in the past four it has been one of the prominent parts of the 
 school survey. During the entire period, hundreds of superin- 
 tendents throughout the country have been readjusting the 
 schools to better the conditions disclosed. 
 
 "In these seven years the number of children graduating each 
 year from the elementary schools of America has doubled. The 
 number now is three quarters of a million greater annually than 
 it was then. The only great organized industry in America that 
 has increased the output of its finished product as rapidly as the 
 public schools during the past seven years is the automobile 
 industry." ^ 
 
 The Warren County (Kentucky) Bulletin translates the losses 
 through retardation as amounting to "thousands of years." This 
 is probably more impressive than really comprehensible. 
 
 Suppose a teacher is repeating every answer after every pupil, 
 hour in and hour out, for days, as did some teachers observed by the 
 author's students. To tell a citizen that such a teacher is wasting 
 
 ^ Ayres, I-.. P.: "A Survey of Surveys," Indiana University Bul- 
 letin, Vol. 13, No. 11, p. 180 
 
Translating Statistical Material 311 
 
 time would mean little. But to show him that if the teacher had 
 forty pupils, she probably would have twenty of them reciting half 
 a day each day; that if she repeated every answer, she could only 
 cover about half as much as if she did not repeat ; that this meant 
 virtually wasting a fourth of a day for every pupil ; that a fourth of a 
 day for forty pupils amounted to ten days for one ; that this ten days 
 was two weeks in school for one pupil ; that this teacher by her repe- 
 titions was each day consuming or wasting the equivalent of one 
 pupil's time for two whole weeks in school, — all this would mean a 
 good deal to him. 
 
 7. Many questions involving value, and particularly 
 exhibits of loss or waste, can be profitably translated 
 into a money equivalent. This is particularly true of all 
 proposals involving an increase in school taxes, which 
 must, of course, be addressed to the taxpayers. 
 
 Thus, the need of health education and sanitation may be shown 
 by translating the loss in money through death and sickness into the 
 cost estimates furnished by Professor Irving Fisher of Yale, and 
 others. This is well done in the Warren County (Kentucky) Survey, 
 page 2. It was found there that in the year 1915 there were 155 
 preventable deaths, and the potential loss occasioned by these deaths, 
 according to Professor Fisher's estimate, was shown to be $263,500. 
 
 The well-known chart of the United States Bureau of Education, 
 which attempted to show that every day spent in school is worth 
 $9 to a child, is a good example. The fallacies in it are likely to 
 give actual pain to any one who knows much about statistical 
 method, or who will use his common sense effectively. But it does 
 translate the material into something that is intelligible and appealing 
 to most people. It has been of great value for thousands of high 
 school commencement addresses and campaigns for increasing school 
 levies. 
 
 In any propaganda for increasing school taxes, it is well to trans- 
 late the increase into the respective amounts of money which will be 
 due from men who already pay certain round sums for total taxes, 
 the number in each class, thus : 
 
 This increase in school taxes will require : 
 
 50 cents more per year from each of the 1500 men who now pay a 
 total yearly tax of $5 each. 
 
o 
 
 12 School Statistics and Publicity 
 
 $1 more per year from each of the 500 men who now pay a total 
 yearly tax of $10 each. 
 
 $2 more per year from each of the 250 men who now pay a total 
 yearly tax of $20 each, etc., etc. 
 
 III. EXAMPLES OF GOOD TRANSLATIONS OF SCHOOL 
 
 STATISTICS 
 
 In many of the school surveys which use modern statis- 
 tical method, the technical statistical terms and results 
 have not been translated so as to influence the average 
 man. To just this extent they are certain to fall short 
 of what a survey or review of a school claims to be. 
 True, there are to be found in isolated places some very- 
 successful translations of these terms But these are so 
 scattered as not to be generally accessible. The remainder 
 of this chapter aims to make a few of these available for 
 
 general use. 
 
 1. Sampling 
 
 The following translation of the process of sampling 
 was used by the author's students in preparing some 
 material for a survey of one of the western cities : 
 
 In giving the standard tests to the children of this city, it was found 
 to be too great a task to test every child, as labor in grading the papers 
 would be enormous. So certain schools and grades were chosen at 
 random, and tests were given in these places. 
 
 When a carload of wheat is being graded, the grader does not 
 look at all the grains, as every one knows that would take too long. 
 What he does is to take a few grains from each of several places well 
 distributed throughout the whole lot of wheat, and to make his rating 
 from these samples. This process has been found accurate enough so 
 that it is constantly used in business without complaint from either 
 the buyer or seller. The same thing is true in grading fruit. Not 
 every apple or peach is actually looked at, or even every box, but 
 only certain apples or peaches taken from certain boxes (determined 
 at random), and the quality of these determines the grade assigned 
 to the whole consignment. 
 
Translating Statistical Material 313 
 
 These examples are well known to the people of this 
 section, in all probability because of the amount of wheat 
 and fruit raised in the state. After they have read such 
 a translation, are they likely to doubt the validity of the 
 sampling done in the school survey? 
 
 Sampling may also be explained by comparing it with 
 the process of taking a straw vote. Most men understand 
 how this is done, how reliable it is, etc. 
 
 2. The Average 
 
 A familiar idea for translating the average is afforded 
 by the " see-saw " or by the lever. Every one knows that 
 the farther the person is from the object upon which the 
 lever is resting, the more weight it takes on the other 
 side to counterbalance him. That is, the center is what 
 the physicist means by " center of gravity." It is the 
 same with the average. The average is the balancing 
 point of all the cases in a distribution, with their distances 
 from it and their sizes taken into account. The farther 
 an item is from the average, the more weight it has. 
 
 Professor King uses the expression '' a type " for the 
 average. 
 
 3. The Median 
 
 The following translations are taken from surveys: 
 
 "Among the teachers in the elementary schools, the median or 
 midway age is twenty-nine years, half of the teachers being twenty- 
 nine years old or older, and the other half, twenty-nine years of age 
 or younger." ^ 
 
 "With teachers ranked in descending order according to size of 
 salaries, the median salary is the salary received by the teacher half 
 way down the line." ^ 
 
 1 "The PubHc Schools of Springfield, Illinois," Springfield Survey, 
 p. 59 2 "Financing the Schools," Cleveland Survey, p. 53 
 
314 School Statistics and Publicity 
 
 The idea of a median was translated with the aid of a picture by 
 Superintendent Womack in his Conway (Arkansas) Survey by taking 
 his ungraded class and lining them up for height from low to high. 
 The middle child was standing on a drain which ran out white in 
 front, and there were two gaps in the line to indicate the quartiles. 
 Any reader could look at the line of pupils and quickly get a clear 
 notion of what was meant by median height. 
 
 " The point above which and below which fifty per cent of the cases 
 fall." 1 
 
 ** The median is the case which was found in the investigation to 
 have as many cases below it as there are above it." ^ 
 
 A very effective translation of the median can be se- 
 cured by using a description of a typical person or school, 
 which will have the median amount in each of a number 
 of different qualities. The first example of this use of 
 the median ever noted by the author was the description 
 of the typical teacher in Professor Coffman's Social Com- 
 position of the Teaching Population, pages 79-80. A por- 
 tion of this description follows : 
 
 The typical American male public school teacher ... is twenty- 
 nine years of age, having begun teaching at almost twenty years of 
 age, after he had received but three or four years of training beyond 
 the elementary school. In the nine years elapsing between the age 
 he began teaching and his present age, he has had seven years of ex- 
 perience, and his salary at the present time is $489 a year. Both his 
 parents were living when he entered and both spoke the English 
 language. They had an annual income from their farm of $700, 
 which they were compelled to use to support themselves and their 
 four or five children. 
 
 His first experience as a teacher was secured in the rural schools, 
 where he remained for two years at a salary of $390 per year. He 
 
 1 Cubberley, E. P.: "Survey of the Organization, Scope and Fi- 
 nances of the Public School System of Oakland, California," Board of 
 Education Bulletin, No. 8, June, 1915. 
 
 2 Ellifif, J. D.: "A Study of the Rural Schools of Saline County, 
 Missouri," University of Missouri Bulletin, Vol. 16, No. 22, p. 8, 
 footnote 
 
Translating Statistical Material 315 
 
 found it customary for rural teachers to have only three years of 
 training beyond the elementary school, but in order for him to ad- 
 vance to a town school position, he had to get an additional year of 
 training. Etc., etc. 
 
 This has been imitated since in many cases. Thus, in 
 the American School Board Journal, for August, 1917, page 
 70, there is a description of the typical Iowa high school 
 principal, based on medians. The author has had this 
 device used by many of his students in writing up in- 
 vestigations. 
 
 4. The Mode 
 
 The mode may be translated as follows : A certain 
 article of clothing is said to be in " fashion " when more 
 people wear it than do without it. Likewise in a distribu- 
 tion, the mode is the *' fashion " in cases ; more appear 
 there than anywhere else. 
 
 5. Spread or Dispersion or Variability 
 
 There are wide differences in the wealth of the people 
 of this country. Wealth of individuals varies all the way 
 from Rockefeller and his millions to the poor street 
 beggar. This variation in statistics is called the spread 
 or dispersion. Now some agitators, if they had their 
 way, would eliminate this spread by making all people's 
 wealth equal. 
 
 In the same way there are all sorts of variations in 
 children in school work, for any particular line of work. 
 It would be just as great an error as that of the agitators 
 to reduce the variability in any one line so that all children 
 were considered equal in performance. 
 
 Professor J. F. Bobbitt, in the School Review for October, 
 1915, page 508, uses the translation '' zone of safety " to 
 
316 School Statistics and Publicity 
 
 indicate that on high school costs of instruction, it would 
 be well to try to get within the middle 50 per cent. 
 That is, he translates the spread between the quartiles by 
 '' zone of safety." 
 
 6. Correlation 
 
 The Biblical phrase, " the first shall be last and the last 
 shall be first," might be used to good advantage in trans- 
 lating a perfect negative correlation. ^ 
 
 The following is a good translation of a coefficient of 
 correlation of .48 between abilities in shop practice and 
 abilities in drawing: 
 
 There is marked evidence that abilities 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 specifically 
 in what way the two abilities are centrally connected, or to what ex- 
 tent the presence of either one is an indication of the other .2 
 
 EXERCISE 
 
 Take the school report or school survey used in the exercises on 
 pages 233 and 302. Write out a detailed criticism of the transla- 
 tions of statistics or lack of them in it from the standpoint of their 
 effectiveness with the public, showing just why they are good or liable 
 to be unsuccessful. In the cases of the unsuccessful ones, or failure 
 to use translations where desirable, make up translations that would 
 present the same data properly. 
 
 REFERENCES FOR SUPPLEMENTARY READING 
 
 Ellis, A. Caswell. " The Money Value of Education." United States 
 
 Bureau of Education Bulletin, 1917, No. 22. 
 McAndrew, William. The Public and Its School. 
 
 1 Suggested by one of the writer's students, Mr. L. A. Sharp 
 
 2 Rusg, H. O. : Statistical Method Applied to Education, p. 257 
 
SELECTED AND ANNOTATED BIBLIOGRAPHY 
 
 The aim in this is to give a minimum list of the simpler and more 
 easily accessible materials. 
 
 I. Statistical Method 
 
 Chapman and Rush. The Scientific Measurement of Classroom 
 Products. Silver, Burdett and Company, Boston, 1917. 
 
 Contains excellent brief chapters on the theory of scales, their 
 application in schools, and dangers incident to their use. Other 
 chapters present the more important scales for measuring work 
 in the formal subjects in the elementary school, describe processes 
 for getting results, and show how the results may be used to 
 better classroom Work. 
 
 Elderton, W. p. and E.- M. Primer of Statistics. A. & C. Black, 
 London, 1910. 
 
 A brief, very simple, and readable treatment, with no special 
 reference to education. 
 
 King, W. I. The Elements of Statistical Method. The Macmillan 
 Company, New York, 1915. 
 
 An elementary, concise, straightforward treatment, but adapted 
 more to economic or historical work than to school problems. 
 
 Monroe, W. S. Educational Tests and Measurements. Riverside 
 Press, Cambridge, Mass., 1917. 
 
 A treatment which works in many of the elements of sta- 
 tistics very simply and forcibly, under discussions of various tests 
 and scales. 
 
 RuGG, H. O. Statistical Methods Applied to Education. The River- 
 side Press, Cambridge, Mass., 1917. 
 
 An admirable book for its general purpose, emphasizing the 
 problems of the school administrator. The statistical part proper, 
 while written as much as possible in a non-technical style, is natu- 
 rally carried to a much greater refinement and intricacy than are 
 
 317 
 
318 School Statistics and Publicity 
 
 necessary in preparing school statistics for publicity. However, 
 the last chapter is very fine for publicity work. The bibli- 
 ography covers all the main problems which superintendents need 
 to study quantitatively, especially surveys, and is alone worth 
 the price of the book. 
 Thorndike, E. L. An Introduction to the Theory of Mental and 
 Social Measurements. Teachers College, Columbia University, 
 New York, 1913. 
 
 A complete exposition of things needed in the fields indi- 
 cated by its title. It has few direct applications for the admin- 
 istrator, and is extremely difficult for the beginner in statistics. 
 
 II. Calculating Tables 
 
 Crelle, a. L. Rechentafeln. G. Reiner, Berlin, new edition, 1907. 
 
 Gives products to 1000 by 1000. 
 Peters, J. Neue Rechentafeln fiir Multiplikation und Division. 
 
 G. Reiner, Berlin. 
 
 III. Exercises and Problems 
 
 Thorndike's Mental and Social Measurements and Rugg's Statistical 
 Methods Applied to Education have problems in various places, 
 some of which may be easily adapted for practice work. 
 
 RUGG, H. O. Illustrative Problems in Educational Statistics. Pub- 
 lished by the author. University of Chicago Press, 1917. 
 
 IV. Graphic Methods 
 
 Brinton, W. C. Graphic Methods of Presenting Facts. Engineering 
 Magazine Company, New York, 1914. 
 
 An excellent non-technical treatment, profusely illustrated. 
 While not written especially for school men, its conclusions and 
 suggestions are easily adapted to school problems. 
 Ellis, A. Caswell. The Money Value of Education. Bulletin of the 
 United States Bureau of Education, 1917, No. 22, Washington. 
 Contains numerous charts used in educational campaigns. 
 RuGG, H. O. Statistical Methods Applied to Education. The River- 
 side Press, Cambridge, Mass., 1917. 
 
 Chapter X presents numerous good examples. 
 
Selected Bibliography 319 
 
 V. School Reports, General Treatments 
 
 Bliss, D. C. Methods and Standards for Local School Surveys. D. C. 
 Heath and Company, Boston, 1918. 
 
 A simple but adequate treatment of the topics indicated, which 
 will be of great value to the superintendent. Has many good 
 tabulations and some graphs. 
 
 Giles, J, T. A Statistical Study of School Reports from the Twenty- 
 five Largest Cities of Indiana. Educational Administration and 
 Supervision, Vol. II, pp. 305-311. 
 
 Hanus, Paul H. School Efficiency. A Constructive Study. School 
 Efficiency Series. World Book Company, Yonkers-on-Hudson, 
 N.Y., 1913. 
 
 A study of twenty-six widely selected city reports in the United 
 States. 
 
 Snedden, David S. and Allen, William H. School Reports and 
 School Efficiency. The Macmillan Company, New York, 1908. 
 A pioneer book in this field, now useful chiefly for its sug- 
 gestions on what to include in a report and on using tabulations. 
 
 VI. School Reports and Surveys Especially Valuable from a 
 
 Publicity Standpoint 
 
 The publisher is indicated for each item given here. For a copy 
 of any other survey or report mentioned in the body of the text, 
 address the superintendent of the school system concerned. 
 Alabama. An Educational Survey of Three Counties in Alabama. 
 
 Department of Education, Montgomery, Ala., 1914. 
 Boston, Massachusetts. Report of a Study of Certain Phases of 
 
 the Public School System of Boston, Massachusetts. Teachers 
 
 College, Columbia University, New York City, 1916. 
 Butte, Montana. Report of a Survey of the School System of Butte, 
 
 Montana. By Strayer, G. D., and others. Board of School 
 
 Trustees, 1914. 
 Cleveland, Ohio. The Cleveland Education Survey. Ayres, L. P., 
 
 Director. Published in twenty-five separate monographs by 
 
 the Survey Committee of the Cleveland Foundation, Cleveland, 
 
 Ohio. The following are especially valuable: 
 
 Child Accounting in the Public Schools — Ayres 
 Financing the Public Schools — Clark 
 
320 School Statistics and Publicity 
 
 4 
 
 Measuring the Work of the Public Schools — Judd 
 The Cleveland School Survey (summary) — Ayres 
 
 Dansville, New York. A Study — The Dansville High School. 
 By Foster, J. M. F. A. Owen Publishing Co., Dansville, N. Y. 
 
 Denver, Colorado. Report of the School Survey of School District 
 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 Manage- 
 ment; Part V, The Building Situation and Medical Inspection. 
 The School Survey Committee, Denver, Colorado, 1916. 
 
 Des Moines, Iowa. Annual Report of the Des Moines Public Schools. 
 For the year ending July 1, 1915. Board of Education, Des 
 Moines, Iowa. 
 
 Ellis, A. Caswell. The Money Value of Education. Bulletin of 
 the United States Bureau of Education, 1917, No. 22. 
 
 Grand Rapids, Michigan. School Survey. By a large staff. 1916. 
 
 Janesville, Wisconsin. An Educational Survey. By Theisen, 
 W. W., and staff of state department. Published by State De- 
 partment of Public Instruction, Madison, Wisconsin. 
 
 McAndrew, William. The Public and Its School. World Book 
 Company, Yonkers-on-Hudson, N. Y., 1916. 
 
 Minneapolis, Minnesota. Three Monographs on School Finance 
 in Minneapolis. By Spaulding, F. E. Board of Education, 
 Minneapolis, Minn. : 
 
 A Million a Year 
 
 Financing the Minneapolis Schools 
 
 The Price of Progress 
 
 Newburgh, New York. The Newburgh Survey. Department of 
 Surveys and Exhibits, Russell Sage Foundation, 128 East 23d 
 Street, New York City, 1913. 
 
 Newton, Massachusetts. The Newton Public Schools. By Spauld- 
 ing, F. E. 1912 and 1913, Newton, Mass. (Out of print.) 
 
 New York City. Report of Committee on School Inquiry. By Hanus, 
 Paul H., and others. School Efficiency Series, World Book 
 Company, Yonkers-on-Hudson, N. Y. 
 
 Ohio. Report of the Ohio State School Survey Commission. By Camp- 
 bell, M. E., Allendorf, W. L., and Thatcher, C. J., 1914. 
 
 Portland, Oregon. The Portland Survey. By Cubberley, E. P., 
 and others, 1913. 
 
Selected Bibliography 321 
 
 Report of the Committee on Uniform Records and Reports. U. S. 
 
 Bureau of Education Bulletin, 1912, No. 3. 
 RocKFORD, Illinois. A Review of the Rockford Public Schools, 1915- 
 
 1916. Board of Education, Rockford, Illinois, 1916. 
 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. 
 San Antonio, Texas. The San Antonio Public School System. By 
 
 Bobbitt, J. F. The San Antonio School Board, 1915. 
 Springfield, Illinois. The Public Schools of Springfield, Illinois. 
 
 By Ayres, L. P. Division of Education, Russell Sage Founda- 
 tion, New York City, 1914. 
 St. Louis, Missouri. Report of Survey of St. Louis School System. 
 
 Board of Education, 1917. 
 Texas. A Study of Rural Schools in Texas. By White, E. V. and 
 
 Davis, E. E. University of Texas, Austin, Texas, 1914. 
 United States. A Comparative Study of Public School Systems in 
 
 the Forty-eight States. Division of Education, Russell Sage 
 
 Foundation, New. York City, 1912. 
 
 This is now out of print, but copies will doubtless be available 
 
 for the superintendent at his state department of education or 
 
 state university. Copies were sent to all the members of the 
 
 state legislatures at the time of its issue, 1912. 
 
INDEX 
 
 Note. — To save space, the words education, publicity, school, 
 statistics, and teachers are in the main omitted from this index. In 
 the numerous combinations in which they naturally occur, look for 
 the next most significant word. 
 
 Absence, translation for, 307 
 
 Absurdities in rank-order combi- 
 nations, 198 
 
 Accuracy, 187-190, 230; errors 
 in attempts at too great ac- 
 curacy, 16-19 ; in graphs, 283 ; 
 in translations, 308 
 
 Advertising, school, error in bas- 
 ing upon exceptional graduates, 
 23 
 
 Age-grade tables, emphasis in, 228 
 
 Age-progress, table form for, 229 ; 
 circle graphs for, 251 
 
 Age, school, error in determining, 
 5 
 
 Ages of pupils, distribution table 
 for, 120 ; graphs for, 121 
 
 Aikins, Professor, 47 
 
 Alabama, Survey of Three Counties, 
 239, 250, 288, 289, 290 
 
 Allen, W. H., 25, 27, 33, 72, 78, 
 201, 202, 213, 216 
 
 Alphabetical order in tables, 221- 
 222 
 
 Alternate columns in tables, 218 
 
 Amarillo, Tex., 68 
 
 American Book Company, graph, 
 241 
 
 Area graph for comparison on 
 component parts, 245 ; for 
 comparisons with circles, 251 
 
 Arithmetic tests, 46, 47, 49 ; sur- 
 face of frequency for, 117; 
 frequency table for, 134; table 
 for results, 232 ; variability in, 
 171 
 
 Arkadelphia, Ark., 8 
 
 Artistic features in tables, 220 
 
 Association of science and mathe- 
 matics teachers, table to show 
 growth, 232 
 
 Atlanta, 300, 301 
 
 Attendance, at teachers' associa- 
 tions, error in computing, 14 ; 
 school, blanks for showing, 72, 
 73 ; error in indefinite units, 
 4; perpetual graph device for 
 showing, 298 ; problems of, 34 
 
 Automobile graph, 290, 296 
 
 Average, 141-147 ; advantages 
 of, 145; computation of, long 
 method, 141-142 ; computa- 
 
 323 
 
324 
 
 Index 
 
 tion of, short method, 142 ; 
 definition of, 141 ; disadvan- 
 tages of, 146; errors in com- 
 puting, 19-21 ; graphic repre- 
 sentation of, 144; translation 
 for, 313 
 
 Average deviation, 156 
 
 Averages, deviations from, 21-22 
 
 Ayres, L. P., 5, 166, 241, 243, 
 244, 251, 259, 291, 292, 310 
 
 Ayres handwriting scale, 53 
 
 Ayres spelling scale, 106 
 
 Background for graphs, 241, 283 
 
 Bagehot, Walter, 90 
 
 Baltimore Survey, 248 
 
 Bar graph, 237; cartoon effect, 
 239, 260, 288; comparisons 
 with, 243, 244, 253; com- 
 ponent parts, use for, 239; 
 curve effects with, 263 ; order 
 of items for, 241 ; right and 
 left form, 247, 248 
 
 Bi-modal distribution, 147 
 
 Bird's-eye view through tabula- 
 tion, 209 
 
 Birmingham, Ala., 210, 298 
 
 Black, use of, on maps, 293 
 
 Blanks, 71-78; making, 78; 
 examples of good, 78; re- 
 vision of, 42; vs. card index, 
 79-81 
 
 Bloomington, Ind., 172 
 
 Bobbitt, J. F., 17, 22, 69, 70, 86, 
 95, 96, 101, 162, 167, 315 
 
 Bobbitt table, 18, 96, 103, 104, 
 127, 130, 147, 152, 153, 156, 
 161, 162, 165, 169, 223, 225, 
 226, 240, 293, 295 
 
 Bold-face type, use in tables, 219, 
 227 
 
 Boston, circulation of school re- 
 port, 27 
 
 Boston Report, 245, 246, 269 
 
 Bowley, A. L., 189 
 
 Bridgeport Survey, 229, 255, 257, 
 258 
 
 Brief, use of, in planning, 42 
 
 Brinton, W. C, 116, 242, 272, 
 276, 286; graphic methods of 
 presentation, 242 ; rules for 
 graphic presentation, 272-.275 
 
 Buffalo, 213 
 
 Buildings, lack of, cartoon effect 
 for, 301 ; problems on, 34 ; 
 table for, 213; units and 
 scales for, 53 
 
 Butte Survey, 117, 128, 130, 167, 
 293 
 
 By-products in collecting data, 
 41 
 
 Calculating devices, 191 
 Calculating tables, 191 
 Calculation, economies in, 190- 
 
 192 
 Card index vs. one blank, 79-81 
 Carelessness in securing data, 15 
 Cartoon effects in graphs, 256, 
 
 287, 301 
 Cartoons, economies in making, 
 
 296 
 Census, school, problems of, 34 
 Central tendency, measures of, 
 
 124-147 
 Charts, time, 261 
 Checking, in calculations, 190 ; 
 
 on b!?.nks, 84 
 
Index 
 
 325 
 
 Chicago, University of, statistical 
 method in School of Education, 
 29 ; blank for rating teachers, 
 270 
 
 Cincinnati, 214 
 
 Circle graph, 235, 259 ; for com- 
 parisons, 250, 254, 255, 256, 
 259; with cartoon effect, 258, 
 259 
 
 Cleveland, 5, 36, 174, 214 
 
 Cleveland Survey, 118, 154, 
 159, 251, 253, 256, 282, 294, 
 313 
 
 Coefficient of correlation, cal- 
 culation of, 184, 185; ex- 
 amples of, 183; meaning of, 
 181, 182 
 
 Coefficient of variability, 170 
 
 Coffman, L. D., 314 
 
 Collecting data with high school 
 students, 86-88 
 
 College degrees, error in com- 
 parisons with, 8 
 
 College salaries, error in getting 
 average, 19 
 
 Columbus Dispatch, 290 
 
 Committee on standards for 
 graphic presentation, 267 
 
 Committee on uniform records 
 and reports, 58 
 
 Comparisons, simple, 240 (see 
 also Relationships, 164-186) ; 
 using component parts, 14, 266, 
 267; using percentages, 13, 
 14 ; using indefinite units, 
 3-16; using relative position, 
 269 ; using unsound treat- 
 ment, 3-16; with bar graphs, 
 239, 240, 243, 244, 253, 266, 
 
 267, 269 ; with circle graphs, 
 250, 256, 258; with cartoon 
 effects, 256, 258; with curves, 
 104, 262, 263, 266, 267; with 
 triangle graphs, 257 
 
 Component part graphs, 236, 
 238, 239, 242, 245, 254, 255 
 
 Composition scales, 46, 53 
 
 Composition tests, frequency 
 table for, 131 ; graph for, 168 
 
 Concentric circle graph, 252 
 
 Constant errors, 188 
 
 Contests, judging, 11-12, 52, 
 192 ff. 
 
 Continuous series, 48 ; graph for, 
 165 
 
 Conway, Ark., Survey, 314 
 
 Correlation, 173 (see also Co- 
 efficient of Correlation) ; 
 graphic devices for showing, 
 175-181 ; Hke-signs, table for, 
 178-179; translation for, 316 
 
 Cost of instruction, 16, 17, 18, 
 48, 55, 70, 96, 101, 104, 105, 
 130, 150, 222, 223, 244, 307, 
 308 
 
 Cost of maintenance, 218, 264 
 
 Cost per pupil, 263 
 
 Cost records, cartoon effect to 
 show value of, 287 
 
 Costs, cartoon for, 290 ; errors 
 in computing, 16; graphs for, 
 288, 292; sampling for, 68; 
 translation for, 307, 308 ; units 
 and scales for, 55. 
 
 Courtis arithmetic tests, 46, 47, 
 142, 143, 149, 199; frequency 
 table for, 134 ; graphs for, 172, 
 266 
 
326 
 
 Index 
 
 Crayons for graphs, 296 
 
 Crelle tables, use of, 191 
 
 Cross-section paper, for computa- 
 tion, 190 ; for large graphs, 296 
 
 Cubberley, E. P., 263, 265, 314 
 
 Current school reports, 200 
 
 Curve effects with bar graphs, 
 263, 266 
 
 Curves, arbitrary signs for, 166; 
 for comparisons, 262 ; for dis- 
 crete scales, 104; standards 
 for, 266-268 
 
 Data, carelessness in securing, 
 15 ; collection of, 33-89 ; econ- 
 omies in collecting, 82-88 ; 
 sources of, 58-61 
 
 Dearborn, W. F., 106, 107, 108, 
 180 
 
 De Voss, J. C, 271 
 
 Defectives, unanalyzed total for, 
 3 ; number of, translation for, 
 309 
 
 Degrees, college and university, 
 error in comparisons with 8 
 
 Des Moines, 207, 210, 219, 257 
 
 Detroit, 217 
 
 Deviation, coefficient of, 170 ; 
 measures of, 149-163 ; graphic 
 representation of, 101, 137 ; 
 measure for given distribution, 
 162 ; translation for, 315 
 
 Dimmitt, R. L., 298 
 
 Discrete scale or series, definition, 
 48 ; graphic representation of, 
 103-105, 164 
 
 Dispersion (see Deviation) 
 
 Distance, map to show by time 
 elements, 280 
 
 Distribution, of cases on a map, 
 277 ; of pupils, age-grade blank 
 for, 73; of school moneys, 
 error in, 5; of time in ele- 
 mentary grades, graph for, 236 ; 
 tables, 106-111 ; graphing. 111 
 
 Dollar circle graphs for component 
 parts, 254 
 
 Dollar graph for proportionate 
 parts with cartoon effects, 258, 
 259 
 
 Dollar proportionate parts table, 
 230, 231 
 
 Dollar-sign graph, 260, 289 
 
 Dots for bar graphs, 292 
 
 Double distribution table, 72 
 
 Double entry table for receipts 
 and payments, 211 
 
 Economies, in collecting data, 82- 
 88 ; in graphs, 296 
 
 Education, value of, graph to 
 show, 247 ; translation for, 311 
 
 Educational investigations, knowl- 
 edge needed for, 94 
 
 Efficiency record of teachers, 
 summarizing graph for, 720 
 
 Elimination, bar graph for, 243; 
 of high school students, table 
 for, 217 ; problems of, 34 
 
 Elliff, J. D., 314 
 
 Emphasis in tables, 227 
 
 Enrollment, errors in, 3-5; 
 graphs, 248, 249, 252, 253, 261, 
 294; map devices to show 
 distribution, 277, 278 ; N. E. A. 
 blank for, 74; translation for, 
 310 ; units and scales for, 57 
 
 Errors, constant, 188 ; variable, 187 
 
Index 
 
 327 
 
 Estimating reliability, 189 
 Exaggerations in graphs, 283 
 Expenditure graphs, 245, 259, 
 
 265, 269 
 
 Expenditures, average, 62 ; errors 
 in comparisons, 4 ; problems 
 of, 34 ; proportionate units 
 and scales for, 54 ; sampling 
 for, 68, 70; tables, 214, 215; 
 translation for, 308 (see also 
 Costs) 
 Extreme range variation, 149 
 Eyestrain, avoiding, 218, 227 
 
 Fisher, Irving, 242 
 
 Florida, 20 
 
 Ford automobile translation, 306 
 
 Fractions, errors in getting too 
 
 small, 17 
 Frederic, Wis., 88 
 Frequency, surface of. 111, 116, 
 
 123 ; multi-modal surface of, 
 
 125; normal surface of, 117; 
 
 skew surface of, 118-122 ; 
 
 tables, 106-111 
 
 Georgia, University of, 279 
 
 Giles, J. T., 200 
 
 Grading pupils, 10, 11, 21, 22, 
 
 63, 66, 107, 108, 170, 196; 
 
 blank for studying teachers' 
 
 standards, 76 
 Graphs, 234-301 ; check list for, 
 
 274 ; economies in making, 
 
 296; examples of good, 288; 
 
 size of, 282 ; standards for, 
 
 266, 272, 281; summarizing, 
 268 
 
 Gray, W. S., 48 
 
 Grounds, size of, cartoon graph 
 
 for, 260; translation for, 310 
 Grouping, 41, 107-110, 210 
 Gummed letters for graphs and 
 charts, 297 
 
 Haggerty, M. E., 171, 172, 176, 
 199 
 
 Hammond Survey, 220 
 
 Handwriting scales, 45, 50, 53 
 
 Handwriting tests, distribution 
 table for, 128 ; surface of fre- 
 quency at Cleveland, 118 
 
 Hanus, Paul, 25, 27, 200, 201 
 
 Harvard-Newton composition 
 scale, 53 
 
 Headings, for graphs, 281 ; for 
 tables, 209, 219; form for 
 printing, 216 
 
 Health education, lack of, trans- 
 lation for, 311 
 
 Heating, 22 
 
 Herrick, W., 26 
 
 High school enrollment, graph 
 for, 293 
 
 High school training, value of, 
 graph to show, 247 
 
 Hillegas composition scale, 53, 
 131 
 
 Histogram or column diagram, 
 111-116, 122 ; check form, 114, 
 115 
 
 Illiteracy, cartoon graph to show, 
 288 ; error in treatment of, 13, 
 16 
 
 India ink for graphs and charts, 
 297 
 
 Indiana, 199, 200 
 
328 
 
 Index 
 
 International Harvester Company 
 graph, 247 
 
 Jingle fallacy, 47 
 Judging contests, 11, 12, 52, 
 192 ff. 
 
 Kelley, F. J., 184, 271 
 
 Key numbers, for blanks, 75; 
 
 for tables, 219 
 King, W. I., 36, 38, 39, 41, 91, 
 
 92, 95, 97, 145, 313 
 Kirk, Jno. R., 65 
 
 Length of school year, bar graph 
 for, 291 ; translation for, 307 
 
 Lettering on graphs, 296 
 
 Library Bureau, 59 
 
 Library statistics, error in in- 
 definite units, 10 
 
 Lighting space, graphs for, 286, 
 295 
 
 Lines, clear in graphs, 297; 
 dividing, in tables, 218, 219 
 
 Louisville, 6, 231, 244, 258, 283 
 
 Lowry, F. C, 289 
 
 MacAndrew, Wm., 300, 304 
 
 Maclin, E. S., 300 
 
 Maintenance cost, per cent of, 
 
 table for, 218 
 Maps, 275 ff. ; use of black on, 
 
 293 
 Marking pupils, 10, 11, 21, 22, 63, 
 
 66, 107, 108, 170, 196; blank 
 
 for studying teachers' standards 
 
 on, 76 
 Masonic pin translation, 306 
 Maxwell, Wm., 25, 26, 37 
 
 Median, advantages of, 138; 
 computation of, 129-137; def- 
 inition of, 128; disadvantages 
 of, 139 ; graphic representa- 
 tion of, 137 ; translation of, 313 
 
 Median deviation, 155 
 
 Medical inspection, unanalyzed 
 totals in, 3 ; problems in, 35 
 
 Membership, table for showing 
 growth in, 232 
 
 Memphis, 166, 265 
 
 Meyer, Max, 11, 63 
 
 Minneapolis, 63, 70, 145 
 
 Minnesota, 9 
 
 Missouri, University of, grading 
 system, 11 ; State Department 
 of Education, 279 
 
 Mobile, 27 
 
 Mode, advantages of, 126; def- 
 inition, 124 ; computation of, 
 125; disadvantages of, 127; 
 graphic representation of, 125 ; 
 translation of, 315 
 
 Monroe, W. S., 136, 271 
 
 Monument graph, 249 
 
 Multimodal surface of frequency, 
 -125 
 
 Nashville, Tenn., 77, 119-121, 
 
 280 
 National Education Association, 
 
 58, 74, 78 
 Neatness in tables, 220 
 Negroes, schooling for, 15, 307 
 Newburgh Survey, 259, 260, 261 
 Newton, Mass., 40, 102, 103, 
 
 261, 263, 288, 300 
 New York Survey, 27, 303 
 Normal schools, 2,7, 150 
 
Index 
 
 329 
 
 Normal surface of frequency, 117- 
 119 
 
 Oakland Survey, 218 
 
 Ohio Survey, 259 
 
 Old Man Ohio cartoon, 290 
 
 Omitting important factors, 15 
 
 Order of items for bar graph, 241 ; 
 
 for tables, 221, 223 
 Over-age, error in method of 
 
 determining, 5 
 
 Paper letters for charts and 
 
 graphs, 297 
 Peabody (George) College for 
 
 Teachers, 277 
 Pearson coefficient of correlation, 
 
 158, 185 
 P. E., or probable error, 155 
 Penmanship (see Handwriting) 
 Percentage tables, graphing, 167 
 Percentages, errors in, 13-15 
 Percentile deviations, 153 
 Percentiles, 153 
 
 Perpetual attendance graph de- 
 vice, 298 
 Phelps, S. J., 86 
 Phi Beta Kappa, 11 
 Planning statistical treatment, 38- 
 
 43 
 Playgrounds, cartoon graph for, 
 
 260; size of, translation for, 
 
 310 ; units and scales for, 53 
 Population of school district, 
 
 units and scales for, 56 
 Portland Survey, 51, 52, 224, 227, 
 
 230, 282 
 Presentation of school statistics 
 
 to public, errors in, 24 
 
 Probability surface of frequency, 
 
 118 
 Problem, statistical, how to state, 
 
 39 
 Property, value of, translation 
 
 for, 309 ; value of, graph for, 289 
 Proportionate parts table, 231 
 Public indifference, 26-27 
 Puckett, W. F., 5 
 
 Q or Quartile Deviation, 151, 153 ; 
 graphic representation, 137 ; 
 translation for, 315 
 
 Quartiles, 151; graphic repre- 
 sentation, 137 
 
 Questionnaire method, value of, 
 59 ; blank for, 65 ; sampling 
 for, 66 
 
 Range, measures of, 149-163 
 Rank-order combinations of data, 
 
 192-198 
 Rating of teachers, summarizing 
 
 graph for, 270 
 Receipts and payments, problems 
 
 of, 34 ; table, 211 
 Records, 58, 287 
 Red Cross, 279 
 Relationships, 164-186 
 Relative position, bar graph for, 
 
 269 
 Rehabihty, 187-190 
 Reports, 25-27, 40 
 Reproducing graphs for the public, 
 
 297 
 Retardation, error in determining, 
 
 6; graph for, 166, 237, 257; 
 
 problems of, 35; translation 
 
 for, 306, 310 
 
330 
 
 Index 
 
 Revenues, errors in comparison, 
 15 
 
 Rice, Dr., 9, 45 
 
 Rockford Review, 231, 236, 252, 
 264, 279, 285 
 
 Rubber stamps for graphs and 
 charts, 297 
 
 Rugg, H. 0., 67, 79, 182, 316 
 
 Ruled blank book, 77 
 
 Ruler strip device for use on re- 
 ports, 85, 86 . 
 
 Rural school work, map for, 279 
 
 Russell Sage Foundation, 166, 
 243,244,257, 259, 260, 261 
 
 Salaries, janitors', 257 ; principals', 
 257; teachers', 22, 109, 110, 
 125, 150, 154, 290; teachers', 
 error in indefinite units, 6, 7 ; 
 teachers', units and scales for, 
 54 
 
 Salt Lake City Survey, 122, 125, 
 168, 219, 232, 237, 282, 293 
 
 Sampling, 22-24, 62-71 
 
 San Antonio Survey, 17, 22, 86, 
 167, 225 
 
 San Francisco Survey, 66 
 
 Scales, 24, 43-53, 100; discrete 
 and continuous, 48; examples 
 of, 53-57; graphic represen- 
 tation of, 101, 285; objective, 
 44 ; subjective, 44 
 
 Schedules, teachers', error in com- 
 puting with indefinite units, 7 
 
 Schooling, value of, translation 
 for, 311 
 
 School problems, 34-37 
 
 School statistics, errors in, 1-24 ; 
 errors in presentation of, 24 ; 
 
 need for better, 1-32 ; value 
 
 of, for superintendents, 30-31 
 Scoring data, economies in, 82-84 
 Semi-inter-quartile range, 151 
 Sequence in tables, 221 
 Seymour, F. 0., 68 
 Sharp, L. A., 316 
 Shaw-Walker Com.pany, 59 
 (T = Greek sigma = abbreviation 
 
 for Standard Deviation 
 Signs, arbitrary, for graphs, 
 
 299 
 Skew distribution, 148 ; devia- 
 tions for, 159 
 Skewness, 118-122 
 Skew surface of frequency, 118- 
 
 122 
 Smoothed surface of frequency, 
 
 119 
 Smoothing graphs, 113 
 Snedden, David, 25, 27, 33, 72, 
 
 78, 201, 202, 213, 216 
 South Bend Survey, 209, 221 
 Spaulding, F. E., 40, 43, 63, 70, 
 
 71, 103, 145 
 Special classes, 35 
 Spelling, errors in lack of units 
 
 for, 9 ; scale, 106 ; tests, 45, 
 
 122, 167, 169 
 Spread, measures of, 149-163; 
 
 translation for, 315 
 Springfield Survey, 211, 243, 257, 
 
 313 
 Standard Deviation, 158 
 Standard tests, sampling in, 66 
 Standards, errors in striving for, 
 
 17-19 
 Statistical method, reliability of 
 
 results with, 187, 189, 190; 
 
Index 
 
 331 
 
 value of, and when to use, 19- 
 24, 28, 29, 30, 33, 36, 37, 38- 
 40, 91-93, 95, 97-99, 100, 200 
 
 Step method, economy for com- 
 putation, 191 
 
 Step on a scale, meaning of, 24, 49 
 
 Stone tests, 117 
 
 Strayer, G. D., 62, 79, 119 
 
 Street maintenance, table for, 
 225 
 
 Students, use of, for statistical 
 work, 192, 299 
 
 Subjective scales, 46 
 
 Summarizing data on blanks, 75, 
 76 
 
 Summary tables, 221 
 
 Summer session enrollment, error 
 in indefinite units, 3 
 
 Supervision, problems of, 35 
 
 Surface of frequency, 111-122 
 
 Symmetrical distribution, 117 
 
 Tables, distribution, 106; of 
 
 frequency, 106 ; series of, 220 
 Tabulation, 209, 233; for the 
 
 public, 203, 204, 206-233 
 Taxes, increase in, bar graph for, 
 
 261 ; increase in, translation 
 
 for, 311 
 Tax rate, errors in comparison 
 
 with, 7, 8 ; graphs for, 102, 
 
 265 ; table for, 224 ; units and 
 
 scales for, 56, 57 
 Teachers College, use of statistical 
 
 method at, 28 
 Teaching staff, units and scales 
 
 for, 54 
 Technical methods needed in 
 
 school statistics, 19-24, 90-99 
 
 Tests, rank-order combinations 
 
 for, 192 ; summarizing graphs 
 
 for, 168, 271 
 Textbooks, graph for cost of, 242 
 Thermometer graph, 102, 288 
 Thorndike, E. L., 30, 45, 46, 47, 
 
 48, 49, 79, 92, 109, 144, 189, 
 
 191, 206 
 Thorndike handwriting scale, 45, 
 
 53 
 Tie rankings, 195 
 Title of graph or chart (see 
 
 Headings) 
 Time charts, 261 
 Time spent on each subject, units 
 
 and scales for, 56 
 Time unit for translations, 306 
 Totals, forms for emphasizing, 
 
 216 ; unanalyzed, error in, 2-3 
 Training of teachers, graphs for, 
 
 256, 259 ; units and scales for, 
 
 54 
 Translation of statistics for the 
 
 pubHc, 303-316 
 Triangle graphs, 254 
 Truancy, problems of, 35 
 Two-way tables, 72 
 Type, measures of, 124-147 
 
 Unclassified items table, 214, 215 
 Unequal things, errors in con- 
 sidering equal, 9-10, 47 
 Uniform records and reports, 233 
 Units, errors in, 3-12, 16; ex- 
 amples of, 53-57; how to de- 
 termine, 43-57 
 Updegraff, Harlan, 70 
 U. S. Bureau of Education, 59, 
 120, 247, 248, 311 
 
332 
 
 Index 
 
 1 
 
 Uselessness of statistics in school 
 reports, 200 
 
 Valedictorian, determining, 192 
 Valid scale, 47-48 
 Vanderbilt University, 279 
 Variability, of children, graph 
 
 for, 293; coefficient of, 170; 
 
 translation for, 315 
 Variable errors, 187 
 Variation, measures of, 149-163 
 Variations, errors in neglecting, 
 
 21-22 ; in size of type for 
 
 tables, 232 
 Variety in graphs, 286 
 Virginia, 9, 286 
 
 Waste, in school statistics, 24- 
 26; in teaching, translation 
 for, 311 
 
 Wax crayons for graphs and 
 charts, 296 
 
 Wealth, units and scales for, 56 ; 
 
 real and assessed behind each 
 
 $1 spent on schools, table for, 
 
 51,52; real, table for, 223 
 Webb, H. A., 306 
 Weighing machine cartoon, 289 
 Weighting factors in rank-order 
 
 combinations, 197 
 Wisconsin, Experiment Station, 
 
 287; State Department of 
 
 Public Instruction, 88 
 Withdrawals, error in determining 
 
 age, 6 
 Womack, J. P., 314 
 
 Y. M. C. A., 280, 308, 309 
 
 Zero line in graphs, 263, 283, 
 
 284 
 Zero point on scale, 46, 47 
 Zeros, use in tables, 216, 217 
 Zone of safety, 18, 315 
 
 5477 
 
LIBRARY OF CONGRESS 
 
 021 334 61 
 
 67