Author . imMl Pi ^m^ imi mk Title Imprint. IBII illi ^^^1 Mm liii'iii ill m BULLETIN OF THE UNIVERSITY OF WISCONSIN No. 368: High School Series, No. 9 SCHOOL AND UNIVERSITY GRADES BY WALTER FENNO DEARBORN Sometime Assistant Professor of Education The University of Wisconsin Professor of Education The University of Chicago MADI SON Published by the University June, 1910 KotiogwwS HIGH SCHOOIi SERIES 1. Thk High School Coukse in English, by V/illard G. Bleyer, Pli. D., Assistant Professor of Journalism. 1906. 1907. 1009. 2. The High School Coukse in Gsu^rAN, by M. Blakemore Evans, Ph. D., Assistant Professor of German, 1907. 1909. 3. Composition in the High School: The First and Sec- ond Yeabs, by Margaret Aslunun, Insiructcr in Englisli. 190S 1910. 4. The High School Course in Latin, by M. S. Slaugliter, Ph. D., Professor of Latin. 1908. 5. The High School Couese in Voice Training, by Rollo L. Lyman, Assistant Professor of Rhetoric and Oratory. 1909. 6. The Relative Standing of Pupils in the High School AND IN THE UNIVERSITY, by W. F. Dearbom, Ph. D., Assistant Professor of Education. 1909. 7. A CouKSE IN Moral Instruction for the High School. by Frank Chai3man Sharp, Ph. D., Professor of Philosopliy. 1909. 8. The High School Course in Mathematics, by E. B. Skinner, Ph. D., Assistant Professor of Mathematics. 1909. 9. School and University Grades, by W. F. Dearborn, Ph. D., Assistant Professor of EdiK'atinn. 1910. Copies of these bulletins may be obtained by writing the Secretary of the Committee on Accredited SchoolS; Room liO, University Hall. Entered as second-class matter June 10, 1S98, at the post office at Madison, Wisconsin, under the Act of July 16, 1H94. SCHOOL AND UNIVERSITY GRADES BY WALTER FENNO DEARBORN Sometime Assistant Professor of Education The University of Wisconsin Professor of Education The University of Chicago MADI SON Published by the University June, 1910 \ V 3Qfl|fl||^S ;iva, RECORD AflV OF U.. AUG 1 1932 J^' CONTENTS. I. Introduction 5 II. The Distribution of Mental Ability 7 III. Inequalities in Grading 22 IV. Grades in Different School Subjects 36 V. The University Grades 43 VI. The Correlation of Schools and School Subjects 49 VII. Appendix of University Grades 54 List of Illustrations. Figure I. Figure II. Figure III. Figure IV. Figure V. Figure VI. Figure VII. Figure VIII. Figure Figure IX. X. Figure Figure XI. XI. -A -B Figure XII. Figure XIII. -A Figure XIII. -B Figure XIV. -A Stature of 1.025 English women (meas- urement by Karl Pearson 9 EfBciency of 12-year-old pupils in accur- acy and rapidity of perception 11 Memory for related words of Third Grade girls 11 General averages of High School rec- ords of 472 pupils 12 Grades of same 472 pupils in Freshman year, University of Wisconsin 13 Grades of 180 students in the Freshman and Senior years. University of Wis- consin 13 Distribution of grades assigned to 2,334 students 14 Distribution of grades according to five equal divisions — ^the base line of the theoretical curve 18 Distribution of the numerical standings of pupils in Grades III-VIII, inclu- sive ( City A) 25 Distribution of the numerical standings of pupils in Grades IV-VIII, inclu- sive (City B) 27 High School grades of pupils in English 28 High School grades of same pupils* in Mathematics 28 Grading of class in Freshman Mathe- matics 29 Ranks of Engineering students in Fresh- man English 30 Ranks of students in College of Letters and Science in Freshman English. . . 30 Ranks of Engineering students in Fresh- man Mathematics 30 Figure XIV.-B Ranks of students in College of Letters and Science in Freshman Mathemat- ics 30 Figure XV. Senior class in English (175 pupils)... 33 Figure XVI. Grades of 244 High School pupils in English 38 Figure XVII. Grades of 146 students in Freshman Mathematics 39 Figure XVIII. Grades of 79 High School students in Sophomore English and Mathematics 40 Figure XVIX. Grades of University Freshmen in His- tory and German 40 Figure XX. Grades of University Freshmen in His- tory and Mathematics 41 Figure XXI. Distribution of the average grades of students in first two years at the University 42 Figure XXII. Distribution of the average grades of students in last two years at the University 43 Figure XXIII. Distribution of the average grades of students for the four years at the University 45 Figure XXIV. Redistribution of grades shown in Chart 23 46 Figure XXV. Comparison of the distribution of the average grades of a group of stu- dents in High School and University 52 Figure XXVI. Comparison of grades of students in High School and Freshman English. 52 Figure XXVII. Comparison of grades of students in High School and Freshman German. 53 Figure XXVIII. Comparison of grades of students in High ochool and Freshman Mathe- matics 53 Plate I-A. High School Grades in Latin Facing page 49 Plate I-B. High School Grades in English... " " 49 Plate I-C. High School Grades in Mathematics " " 49 Plate I-D. Freshman Year University " " 49 Plate 1-E. Freshman Year High School " " 49 Table I-A. Percentages by Departments — Freshman and Sophomore Years 54 Table I-B. Percentages by Departments — Junior and Sen- ior Years 55 Table I-C. Percentages by Departments — All classes 55 Table II-A. Percentages of grades assigned to individual instructors to Freshmen and Sophomores.. 56 Table II-B. Percentages of grades assigned by individual instructors to Juniors and Seniors 57 I. INTRODUCTION* School grades ai'e usually regarded somewhat differently by teachers and pupils. Their assignments frequently entail upon the teacher a painstaking routine, the results of which, although generally performed conscientiously, are often un- satisfactory. The grades may appear to the teacher to stand for real differences in ability between pupils or the merely temporary successes or failures in a series of tests and quizzes; often he is not certain after all his pains that they represent the facts accurately in either case. In the hands of some teachers, marks appear to be used as incentives or as rewards and punishments. Some are generous in their assignment, others niggardly in the use of good grades. Because of these and many other differences easily suggested in the use and sig- nificance of marks, teachers generally minimize their impor- tance. It is rather exceptional, on the other hand, to find stu- dents who are not more or less concerned about the teacher's estimate of their abilities or attainments. Although the stu- dent is properly urged and warned not to work for marl^, they are, after all, about the only concrete evidence he has in many cases of his success or failure. Some students notorious- ly elect courses because they secure high grades in them and shun others because they cannot do so. Some find their inter- ests more or less determined by courses in which they appar- ently succeed, — for nothing fixes interest quite as much as suc- cess. And a high mark is often considered evidence of success and a low mark of incapacity or failure, which, of course, in any given case it may or may not be. Some students, I have no doubt, determine their election of studies, if not their per- manent scholastic interest indirectly at least by the fact that they have attained this sort of success in introductory or ele- mentary courses of study. And the grades may have been meaningless. One instructor was generous, another severe, or what not in his estimate of the pupil's work. These are ques- tions which certainly deserve some consideration, but when we *The writer is indebted to Dean E. A. Birge, Professor E. B. Skinner of the University of Wisconsin, and to Professor E. L. Thorndlke of Columbia University for suggestions and criticisms by which he has benefited during the progress of this study. 6 THE UNIVERSITY OF WISCONSIN assume to exclude pupils altogether from school or college or from certain lines of work on the basis of grades and examina- tions, the latter take on an even greater importance. Professor J. McKeen Cattell, several years ago commented as follows on this aspect of school marks: "In examinations and grades which attempt to determine individual differences and to select individuals for special pur- poses, it seems strange that no scientific study of any conse- quence has been made to determine the validity of our meth- ods, to standarize and improve them. It is quite possible that the assigning of grades to school children and college students as a kind of reward or punishment is useless or worse; its value could and should be determined. But when students are excluded from college because they do not secure a certain grade in a written examination, or when candidates for posi- tions in government service are selected as a result of a written examination, we assume a serious responsibility. The least that we can do is to make a scientific study of our meth- ods and results." "' Many cases of inequality, and occasionally of injustice, may undoubtedly be discovered in the grading of pupils from the elementary schools to the university. When, however, teach- ers are allowed to follow their own methods, and not asked to discriminate too minutely between different grades of abil- ity, it is unusual to find important differences of opinion among even very large numbers of instructors. The following study will bear out the statement that such inequalities as exist are not so much due to unfairness on the part of teachers, or to real differences in the estimation which different teachers place on the same pupil's work or abilities, as to the fact that the standards of measurement employed are not uniform, or are used differently by different men. The facts presented in regard to the prevailing conditions in the grading of pupils are based on a study of fifteen thou^ sand or more grades, assigned by about two hundred and fifty teachers in several elementary and high schools, and in the College of Letters and Science of the University of Wisconsin. Since, as just stated, the chief causes of inequality are, it is be- lieved, due to lack of uniformity in standards, certain pro- <^> Popular Science Monihli/, 66. SCHOOL AND UNIVERSITY GRADES 7 posals making toward such uniformity of practice will first be reviewed. In tiie last chapter various ways of using school marks in the investigation of several important school prob- lems are suggested. The object is to indicate ways by which the vast amount of labor expended by teachers in the grading of their students, may be made to serve some further purposes in the study of problems which concern the efficiency of school work. The discussion which follows in regard to the intelligent use of school marks leaves, the author is well aware, many points open to criticism. Some of this might be met if the scope of the article admitted of a fuller discussion. If the pro- posals appear to teachers theoretical and mechanical in their operation, considerable justification may be found for them in the fact that an examination of the systems of marking in vogue in almost any of our schools and colleges show lack of uniformity and striking inequalities. The proposals supply certain standards which will make for uniformity and equality where these are too frequently wanting. II. THE DISTRIBUTION OF MENTAL ABIIilTY Galton, Pearson, and others have held that individuals dif- fer from each other in ability in such ways that these differ- ences conform to the general biological law of variation; in 9ther words, that ability and attainment are distributed in ac- cordance with the curve of error. In discussing the early work of Galton in this field. Professor Brooks has made the follow- ing clear statement of the case in regard to physical char- acteristics: "If we select any one characteristic of a group of animals, — such a characteristic as the weight of the individuals, or the ratio between the length of their arms and legs, or anything else which admits of exact numerical statement, — it will be found that, while no two members of the group are exactly alike, they nevertheless conform to a type, and show the exist- ence of a standard, the mean or average, to which the majority adhere pretty closely, while other members of the group may be more abnormal, showing marked deviation from the mean. The deviation of these abnormal individuals from the mean is not accidental or due to chance, for it is part of the orderly 8 THE UNIVERSITY OF WISCONSIN system of nature. If the cases tabulated are numerous enough, the individuals will conform, so far as this quality is concern- ed, to what is known in statistical science as the law of fre- quency of error. This agreement will be so close, when great numbers of individuals are compared, that the number which depart from the mean to any specified degree may be computed mathematically. For example, the chest measurement of 5,738 soldiers gave the following results: — Inches Measured Computed 33 5 7 34 31 29 35 141 110 36 322 323 37 732 732 38 1305 1333 39 1867 1838 40 1882 1987 41 1628 1675 42 1148 1096 43 645 560 44 160 221 45 87 69 46 38 16 47 7 3 48 2 1 If the number of events had been five hundred thousand or five million, instead of five thousand, the agreement between the computed and observed frequency of each degree of de- parture from the mean would have been very much closer. When the number of cases is unlimited, the agreement is per- fect." "' The form of the theoretical curve, — the probability in- tegral, — corresponding to the column of results in the above table, and according to which these physical characteristics appear to be distributed is given in Figure 1. The dotted line represents the normal curve, the heavy line the approximate (1) Brooks, The Foundation of Zoolog,!/, pp. 156-157. SCHOOL AND UNIVERSITY GRADES i>7 i^ 6J ID stature of 1,025 English Women. (Measured by Karl Pearson.) distribution in stature of 1,025 English women measured by Karl Pearson. '-' There is considerable evidence that what is true of phys ical characteristics holds also for mental. In the matter of general intelligence, we speak of idiots, feeble minded, defi- cient, backward, dull, those of ordinary or average ability, the bright, the brilliant, the man of talent or genius, and of many finer distinctions. The majority of people stand between the extremes in the medium classes, — are "about average;" those who are either deficient or gifted are the exceptional, th-; greater the extent of their deficiency or their endowment, th*^ fewer there are in either case. "And in general the form of distribution is such that be- tween very many individuals the differences are little, that between many they are moderate, and that between only a few are they great. In any group of the same general class with respect to age or training, such a clustering of the cases, com- <2) Quoted by Q.zXi.&\\, Poi>ular Science Monthly , 66: 371. 10 THE UNIVERSITY OF WISCONSIN monly around a medium degree of the ability, will be the case. Individuals, that is, vary about a central type, so that we can think of any single individual's ability as a plus or a minus deviation from the central tendency of his age, sex, or grade." <"' In the more specific mental traits, as, for example, in per- ception and memory, we find similar differences. The two curves plotted below will serve as examples. The heavy lines denote the frequency of cases, the dotted the theoretical curve. The first gives the result of a test of efficiency of twelve year old pupils in the rapidity and accuracy of perception of a spe- cial sort. (Figures 2 and 3), (Figures 7 and 15, from Thorn- dike, Educational Psychology, p. 15). The second shows the memory of related words in the case of third grade girls. (Figure 3.) "' The bearing of all these facts on the question of school marks is not far to seek. Marks, representing as they do the teacher's estimate of mental abilities of various sorts, may themselves naturally be distributed according to the same frequencies as are the abilities which they are designed to represent. In so far as the teacher's judgment is correct and is made of a sufficiently large number of pupils, the frequency of the different marks given should be the same as in a "nor- mal" distribution curve. If the teacher has to do only with small classes, the results of several years' marking or of sev- eral classes in the same subject in the same year should, when put together, be similar to the marks of a larger group given at one time. This general thesis will be subjected to some modification, and will need further justification and explanation in the fol- lowing discussion. One objection to it may possibly be men- tioned at the start. Marks, it may be said, are usually given as indications of attainment and not necessarily of ability. An extremely bright boy may be lazy and accomplish little by his abilities and a dullard may by application accomplish relatively more than his abilities warrant. While this is undoubtedly a pertinent <•" Thorndike, Principals of Teaching, pp. 70-71. <^' For other examples and a concise and illuminating- discussion of this subject see the chapters in Thorndike's Principles of Teaching, and Educational Psychology, referred to. SCHOOL AND UXIVERSITY GRADES Efficiency of l:2-year-old pupils in accuracy and rapidity of perception. (Tliorndike, Ed. Psych.) !HimUm;il!!!Jl!milH:l-HHiHtfh~4 gt?: g|rftti^ip;!!Uiji:!i!iyijmUif^ Jfi^ 6* Memory for related words of Third grade girls. 12 THE UNIVERSITY OF WISCONSIN objection, such instances are the exception rather than the rule; and in the long run, those who excel do so because they are naturally superior, and those who fail, fail mainly on ac- count of inferior ability. The strongest argument, however, for such a distribution of marks as that proposed, is that it is the one usually found, especially when a fairly large number of students are graded. In Figure 4, the heavy line represents the general averages of General Averages of High School Records of 472 pupils. the high school records of 472 pupils who entered the College of Letters and Science of the University of Wisconsin from the larger high schools of the state. The dotted line shows the corresponding frequencies of the probability integral. Figure 5 gives the distribution of the grades of the same pupils in the freshman year in the university. The distribution in Figure 5 fits very closely indeed into the theoretical curve. In the freshman year, the chief difference is in the assignment of relatively too many ranl^s of "good" and too few of "fairs" and "poors" to agree with the theoretical curve. Figure 6 shows a similarly close approximation in the case of both curves. The heavy line represents the standings of SCHOOL AND UNIVERSITY GRADES 13 s 80!^ Grades of same 472 pupils (Cp. Fig. 4) in Freshman year, University of Wisconsin. 7^17^73>.75J76J57>^^^ Sd$i8Z81 Sites 8h 8) 88 %9^o^nn JM^ft, ?)n Grade of 180 students in University of Wisconsin. 14 THE UNIVERSITY OF WISCONSIN 180 students in the freshman year at the university; the dotted line the standings of the same students in the senior year. It is evident from the dotted line representing the general averages of ISO students in the senior year that the distribu- tion is about normal, although the average or median of the class has been raised to 87 (median), whereas for these same students in the freshman year it was 85. This comparison in- cludes only those pupils who remain through the four years at the university. The very poorest students are therefore not taken into account, but considering the same 180 pupils in their freshman and senior years, it appears that this grading is not remarkably different, and is in each case according to a normal distribution. A fairly close approximation is also found in a curve pub- lished by Professor W. S. Hall, giving the distribution of the marks assigned by him to 2,334 medical students in the course of ten years. The dotted line in Figure 7 represents the bi- nomial curve. SOtbO 70 7S SO i<> ^0 % JDO SCHOOL AND UNIVERSITY GRADES 15 Professor Hall makes the following statements in explana- tion of his curve: "That the curve derived from the rating of 2,334 students is really a binomial curve no fair-minded judge would for a moment question or doubt. We have, therefore, demonstrated beyond cavil that examination data is biologic data and obeys the laws of distribution of biologic data. "Certain important divergences from strict coincidence re- main yet to be explained. Why does the apex of the curve stand to the right of the symmetrical binomial curve; i. e., why is the curve of my ratings unsymmetrical? The answer is to be sought in two directions: "1. Either the examiner was too generous and habitually rated his students above their equitable deserts; or "2. The students were (in a sufficient number of indi- vidual cases to influence the totals) guilty of raising their rating above what it should be by nature through dishonest mea,ns or extraneous aids in quizzes, examinations and the preparation of note books. "I am convinced that both of these factors were at work, and in the same direction, i. e., both tended to raise the rating of students and thus to throw the curve out of symmetry. To- gether these two factors have made a difference of approxi- mately five per cent — the actual median value of the rating of the 2334 men being 85.15 per cent when theoretically it should have been 80 per cent." The reasons why the distribution of Professor Hall does not follow more closely the normal distribution need not in- clude suggested unfairness or cribbing, on the part of students. On this basis it would be hard to account for those who do poorer than the normal distribution would call for. A con- siderable number of the latter cases exist, although not as many as of those who secure higher ranks in excess of the nor- mal requirement. The more acceptable explanation is, simply that the instructor in question gave more high grades and somewhat more low ones than were to have been expected in a normal distribution. The factor of selection may of course also enter as an explanation of the larger number of high marks; that is, college students are a somewhat selected group and thus should possibly be graded somewhat higher than a 16 THE LIXIVERSITY OF WISCONSIN group of people selected at random would be. It is hardly possible, however, as noted below, to take account of this factor under usual conditions. Professor M. Meyer has also recently made a similar study of the methods of grading at the University of Missouri. There appears, however, to be some contradiction or incon- sistency in his preliminary discussion, if the writer has under stood the matter at all rightly. In criticising and rejecting the proposal of Professor Hall, mentioned above, that the dis- tribution of marks should conform to the binomial curve. Pro- fessor Meyer presents the results of a test made by himself of the native musical ability of seventy-one students. The dis- tribution actually found does not conform at all to the bi- nomial curve nor to the probability curve, or, in the terms of the statistician, to the distribution of frequencies of the law of error, which Professor Meyer makes the basis of his pro- posal. The following statement of the case is made: "It seems plausible to start from the assumption that the combined mental and moral ability which we want to measure is distributed among different people in accordance with the probability curve, which describes, e. g., the distributions of accidental errors in scientific observation." It would seem that the reason ^Dhy it is plausible to start from this assump- tion, is that those mental tests and characteristics which lend themselves to more or less precise measurement, do. as the results of Galton, Pearson, Cattell. Thorndike, and others, show, conform fairly closely to this so-called normal distribu- tion. The native individual differences in various mental abili- ties tested, when the usual caution, as to a sufficient number of cases, sex, age, training, and selection, etc., is taken, appear to conform to the general biological law of variation nearly as closely as do the similar physical or anthropological measure- ments of height, weight, cephalic index, etc. Professor Meyer's tests of the native musical abilities on the contrary, appear as far as the number of cases measured is concerned, to be an exception to the general rule and by itself invalidate the assumption on which his argument is based. Marks, representing as they do, the teacher's estimate of mental abilities of various sorts, may themselves naturally, as noted above, be distributed according to the same fre- SCHOOL AXD UMVERSITY GRADES 17 quencies as are the abilities wliich they are designed to repre- sent. We may find instances where this has been done, and Professor Meyer possibly does Professor Hall scant justice in failing to note in connection with his criticism of the latter's contentions, that, in spite of the unequal percentile units em ployed in the scale of marks, to which attention is called, the distribution of grades assigned by Professor Hall to some 2,300 students, is a fairly close approximation to the probabil- ity curve. It is not, to the writer's mind, an important, al- though a proper, criticism, that the percentile units are not exactly equal amounts, since it is apparent that the main differentiation is made on the basis of the nine general grades employed; and it would probably make little difference in the actual grading, whether or not the percentile amounts of each grade are exactly the same, since the distinction into nine diffei-ent ranks is about as fine a scale as most can employ; and a finer differentiation usually does not mean much. These criticisms do not affect the validity of Professor Meyer's contention, but are concerned with what seems to be an inconsistency between the facts presented in the criticism of Professor Hall and his own proposal. It is fair to assume that marks may properly be distributed according to the fre- quency of the probability integral, because the individual differences in native capacity are according to most studies approximately so distributed. Such a distribution of marks as has been proposed finds, therefore, justification both from theoretical considerations, and from the fact that it is used in actual practice. Several proposals have been made in regard to the division of the dis- tributions into groups corresponding to the usual grades of ex- cellent, good, fair, poor, and failure. What proportion of stu- dents should be found in each grade, assuming that their marks are to be distributed according to a normal distribu- tion? If we divide the base line of a theoretical curve into five equal parts in order to secure the same range of abilities, we should secure the following percentages in each grade in a normal distribution: A B C D E Excellent Good Fair Poor Failure 2% 23% 50% 23% 2% 18 THE UNIVERSITY OF WISCONSIN This distribution is shown in the upper distribution of Figure 8. Figure 2 of Cattell. '•" Professor Cattell believes a somewhat different distribution more convenient for practical use, as indicated in the following quotation from his discussion: "" "If the performances of students in examinations are as- sumed to vary in the same way as their height, then we can, if we like, place them in classes which represent equal differ- ences. Thus by the Harvard-Columbia method of grouping into five classes, if we put half of the men into the middle class, C, and let B and D represent an equal range, we should have about 23 per cent of both B's and D's and about 2 per <^> Popular Science Monthly. 66: 372. <«> CsLtteU, Examinations, Grades and Credits in Popular Science Monthly 66: 371 seq. SCHOOL AND UNIVERSITY GRADES 19 cent of H's and F's. This, however, gives too few men in the H and F classes for our purposes. If we make the range of the unit 20 per cent smaller, we obtain the distribution shown in Figure 3, (reproduced in Figure 8, lower chart), according to which of ten men four would receive C, two B, two D, one H. and one F. It departs slightly from the theoi-etical dis- tribution but certainly not so much as the theoretical distribu- tion departs from the actual distribution. It appears to be the most convenient classification when five grades are used; one in ten being given honors, and one in ten being required to repeat the course, corresponding fairly well with the average practice and being a convenient standard." The following table and figure showing the grades given to 200 students in each of five courses in Columbia College furnishes another example of the fact that the general average of grades usually approximates the theoretical distribution: Percentages of Students Receiving A B C D E English A 4.5 41.5 44.5 4.5 5. English B 4. 40. 39. 6.5 10.5 Mathematics A 11. 24. 24. 22. 19. History A 10.5 28. 28.5 20. 13. Economics A 9. 36. 33. 17.5 4.5 Average 8. 33.9 33.8 14.1 10.4 The variations in the separate subjects will be discussed in Chapter IV. The following statement is made by Professor Cattell : "The average grade is a little above C, the median grade is nearly midway between C and B, and more than two-thirds of all the grades are either C or B. Eight per cent of the grades are A and 10 per cent are F, which approximates closely to the standard recommended above. The average of the grades assigned in these courses does not vary considerably, but the distribution is different. In the courses in English the dis- tribution tends to follow the normal curve of error, with the failures as a separate group or species. In the courses in mathematics and history the groups are more nearly equal in size, except in the case of excellent. Here the range of ability 20 THE UNIVERSITY OF WISCONSIN is presumably greater in D and F than in B and C. The dis- tribution in economics is intermediate. The fact that the courses in English, though given by different instructors, corre- spond closely, shows that within a department certain stand- ards may be followed; and that this would be possible for the whole college or for the educational system of the country. It is only necessary to adopt the standards and then to teach peo- ple how to apply them." '"* Professor Meyer in the discussion '^' of the grading of pupils in the University of Missouri, just mentioned, argues for the use of the theoretical distribution, and the division into three grades, the median grade containing 50 per cent of the whole group, the remaining 50 per cent being divided equally into the grades of superior and inferior students. As regards further subdivision of these groups, the following quo- tation may be made from his discussion: » "We have divided all students taking a particular kind of work into three-groups, medium students, inferior students and superior students. Should we subdivide these groups? "Little can be said in favor of subdividing the medium group. That this group is the largest, is, in itself, no reason for subdividing it. A strong argument against subdivision is the fact that this would bring about unjust grading of a large number of students. The curve is highest for medium ability. If we divide the area by a vertical line, we must have a large number of students on one side differing by an almost in- finitesimal amount of ability from a larger number on the other side. If the teacher, nevertheless, has to give them different grades, the probability is that a considerable number will re- ceive grades either too high or too low. This probability of injustice must be avoided as much as possible. It can be largely avoided if we make subdivisions only where the curve is comparatively low; and it is best, therefore, to give all the students within the central area of 50 per cent the same grade." "" "More advisable than a division of the medium group of students seems a subdivision in the group of superior students. Id. pp. 373-374. Science N. S., 28: 246-2.50. Id. p. 248. SCHOOL AND UNIVERSITY GRADES 21 To belong to the group of the 25 per cent best is not a great distinction. It would be well, therefore, to separate from the group those wlio possess unusual ability. The manner of sub- dividing the group is a matter of convenience. We may pro- ceed in the following way. In the probability curve (Figure 2) the point of extreme ability, where the height of the curve is practically zero, is chosen as 3. Tlie point of the vertical line which separates the superior from the medium students, is then .68, as can be read off from any table containing the values of the probability integral. It suggests itself to divide the ability-difference between this point and the extreme point. 3, into two equal parts. The result of this division is the point 1.84. To the left of this point are then found 3 per cent of all the students, as can again be read off from any table of the probability integral. We have thus divided the group into two parts in such a way that the best possible student is as much better than the best student of the second class, as this one is better than the best of the medium class. Let us, then, call the three per cent just separated by the name of 'excellent' and retain the name of 'superior,' for the 22 per cent following. "In the same manner we may subdivide the group of in- ferior students, calling the three per cent worst, 'failures.' and retaining the name of 'inferior' for the other 22 per cent. "I expect to meet with opposition when I restrict failures to such a small percentage. But I believe that three per cent is a sufficient number in order to weed out those who have suc- ceeded in entering college, but are entirely unable to do the work which they have chosen. I can not regard it as just to grade the other 22 per cent as failures. But I do not mean by this that they ought to be permitted to take advanced work in the same line of study, or to enter courses of other depart- ments for which this particular study is required, or that they should receive credit for the whole number of hours. The teacher who gives these advanced courses, and the teacher who gives the courses of the other department, must have the power to admit or to exclude these 22 per cent as he deems best. And the faculty should decide what fraction of the regu- lar number of hours of credit they should receive. Similarly, the faculty should, as Professor Cattell has proposed, give more than the usual number of hours of credit to those stu- 22 THE UNIVERSITY OF WISCONSIN dents who have excelled the medium 50 per cent. To make all this possible the teacher must place each student in the group to which he belongs according to his rank. But those whose rank puts them in the fourth group should not be called failures in every possible sense — should not be regarded as having accomplished nothing. If a teacher instructs his class in such a manner that according to his own judgment 25 per cent of them accomplish nothing, then the conclusion is justifi- able that the teacher as a teacher has not accomplished any- thing, either." <^''* III. INEQUALITIES IN GRADING Having these facts before us, an examination may now profitably be made of distributions which do not conform to the theoretical curve and to the many resulting inequalities in the marking system. These examples are taken from a study of the grading in the University of Wisconsin and in several of the high and elementary schools of the state. We may begin with the latter, and with an example that is not unusual in its occurrence. There are two classes of the same grade, e. g., the seventh, in the same school taught by different teachers. The assign- ment of pupils to these rooms is made on purely incidental grounds, e. g., alphabetically, with no differentiation as to scholarship. The superintendent finds, however, that the marks assigned in one room average ten to fifteen points high- er than those given in the other room. In one class, half of the pupils receive grades above 85, in the other room half of the pupils are graded below 70. In such a case as this it would seem to be one of the functions of the superintendent or prin- cipal to find out what is the cause of this difference. Not in- frequently he may find that the fault is to be laid at his own door. He may discover that he has not made clear to the teachers either what the prevailing standards of marking are in his school or what they should be. Several causes might be assigned for the above difference. The pupils in one class are actually that much superior to those in the other class. This is possible but very unlikely under the conditions named; (lo^ Id. p. 249. SCHOOL AND UNIVERSITY GRADES 23 it certainly would not be likely to happen twice in succeeding years. Secondly, one teacher is very much superior to the oth- er — is able to get more and better work out of her pupils, etc. The superintendent can ascertain whether this is actually the case; if it proves to be so, it is questionable whether such un- equal work on the part of teachers can be tolerated in the same school. Another explanation is that these are purely arbitrary differences in the standards of the marking used by the two teachers; one teacher is accustomed to mark all pupils low and the other to rate them all high; the teachers would or do not differ in their judgment as to what pupils do the better work, but simply as to the absolute mark to be assigned to them. This is a purely artificial difference, but it is also either in whole or in part the most frequent cause of the apparent inequality in marks. The same difficulty is found elsewhere. If we are considering one hundred pupils in each of two high school subjects, e. g., English and history, we may anticipate some slight difference from year to year, but in the long run there should not be any difference at all in the percentage of pupils who receive the same grades. For every pupil who ex- cels in English there will be one who excels in history, or in mathematics or in any other school subject. Are there not, therefore, certain rules which a superinten- dent may lay down, or certain suggestions which he may offer to teachers, in addition to the usual ones in regard to the rela- tive weight of the examination and daily recitations, in the making up of the total mark, etc? It is not enough to tell a new teacher that "in this school we mark on the scale of one hundred" (which, as a matter of fact, they seldom or never do, the scale usually being from about 50 or 60 to 100, frequently from 75 to 100, or in other words, on the basis of 25 to 50 points of difference rather than 100). Nor is it enough to say "there are five grades 'A,' 'B,' 'C,' 'D,' 'E,' or 'Ex.,' 'G,' 'F,' 'P,' and 'Failure,' and if you think a pupil's work is good, give him the grade 'G.' Is it a perfectly proper question to ask what is good in this school? And one of the best answers which can be given to that question is to indicate the percent- age of pupils who from year to year attain that grade. In view of the considerations in the preceding sections, it would be well if a principal could say to a new teacher: "In 24 THE UNIVERSITY OF WISCONSIN most of our classes, taking them year after year, about one- half, or 50 per cent of the pupils, secure the grade of 'F' and about 25 per cent of them get above that grade and 25 per cent of them below^ it. Of course in any individual case there will be variations from year to year especially in small classes, but if the teacher puts together the grades assigned to him for several years back there should be a very close approximation to this result, and at any rate the marks of all the teachers in this school when taken together for any given month or year, give, in general, this result." This gives some fairly definite notion of what the grade "F" means in this school. If the superintendent, instead of adopting standards some- what in accord with the theoretical requirements, wishes simply to maintain more uniformly the standard actually prevailing in his school and not attempt any great change in the system as he finds it, it is more likely that his statement will need to be: "In this school we find that taking into con- sideration the grading of all the teachers, about 60 per cent of the pupils secure grades of 'good' or above. About 20 per cent secure the rank of 'excellent.' about 40 per cent the rank of 'fair' or below, and about 10 per cent the grade of 'poor;' and one or two per cent fail. We wish to make this the general practice." Such a definite understanding would make for uniformity in the marking system. There will be exceptions, some of which are discussed below, but this proposal is better than leaving it to the individual teacher to decide for herself not simply as to which pupils are the better pupils — which it is her business to determine — but to determine that no pupil in her classes, however good, shall receive a grade higher than 95 when the teacher in the next room employs the grade of 99 and 100, or to give the poorest pupil a rank of 40 when an- other teacher would use 60, or to grade half of her class "good" when another teacher having the same pupils to deal with assigns to half of them the grade of "fair." Another example may be instanced which might well re- ceive the attention of the principal or superintendent. When marks are assigned according to the principles which have been discussed above, we should not only expect to find rela- tively little variation from year to year in the same class or grade, but the distribution of marks in the various grades in Tvgure 3, 6^ /Pupils of the E/ahth Grade X xxxxx X XXX x;^x ^ }^^6 X XXXXXXXXX XXXXXXX X K S3 Pujails cfthe i Sewenth QradQ t1edm=83 M ^ x ^ >. »,x>x ^X X A?CX X. X X X XA JfX^^X 66 Sixtjf) Grade Rjpils x x 5 nedm-Sg » » x^^ x J Jx J J x x *^X X xxxx> )(xJrxxxx xxx x x { 9^ PuD/Zs cftk Fihh Grad^Q | X XX X X XX X X X xxxxx X AX X V, / „^ ^ > X XX X >< X XX SSfoudh Brade Pvjoils j I X XX X XXX X XX X i XXX X )< X XX XX XX xW JUJ/upils of the Thrd Grade xJ Med/aio=86 X >«lf^x A XA ^ X x/xxxxxxx X XX.X ^ X A X xxxxx xxx: ^ 'f'f*,^ w.^ X X xxxxxxxxx^^ 26 THE UNIVERSITY OF WISCONSIN school from the lowest to the highest school should not vary- materially. If there should be any difference, we might expect to find, as we advanced through the grades, a somewhat higher percentage or frequency of high grades because the poorer students fall behind or drop out of school more frequently than do the better students. In order to see in how far this is true in practice, I have plotted on the following pages the standings of a group of students from the third to the eighth grades, inclusive. This has been done in two different cities, A and B. The following simple method of plotting has been followed, in the horizontal scale of marl^ ^ XXXXXXXAX/K ?^ A XX y X gy yScxxxxxxx^xxxxxxxxxxAxx x ho (>S 70 7S So 8S ^ IS ioH Sixth Grade Pupik. ;xxxpx , X ?S, xxxxxxxx'x^cJcxx^xxx 6i) 6S Id t ^ 65 1o is tiGclm==y7 S7/^oyrU Grade Ri/o/ls X ;< X XX ^ X X^ XX X xxxx X XXXX XX xx;7^ % yjf 7i'!^8bBJ 8S 8i> 8^ 8S 8(, 87 88 8f ^d i/ n 717^ -B. tlediayi^SO t^^ Matheivatics^78Jiipils r^r-L,r^ X 707/1 Of^ 757j^7S7i>7J^y87f 8o8t 82 8i8i^8S 86 8!^88 899o 9J 9!^ 91 S^ 2s 36 3^98 High School grades of same pupils in English and Mathematics tributions of grades in high schools and university, which are open to many points of criticism. In Figure 11, A and B are the distributions of practically the same pupils in high school classes in English and mathematics. It does not appear likely that as a groxip there should exist such a difference in the abilities involved in these two school subjects as seem to be in- dicated by the marks given. It is a condition due to a merely arbitrary difference in the standards of the teachers con- cerned. Chart 12 is a distribution of a class in freshman mathe- matics. In this case the relatively large number who receive the grade of 70, tends to make the distribution of marks in the other grades very unequal. A range of over 30 marks is em- ployed, seeming to indicate that degree of differentiation, whereas, as a matter of fact, no differentiation is made in the case of nearly one-third of the students graded. Charts 13 (a and b) and 14 (a and b) show the distribu- 1 1 ; Ml 1 _i- 4- 4-1- ^^ Till 11 -ir ■^' 1 ; 1 ■ 1 — S-J J 1' ========--1=====^^^ 1^ — 1 — — 1 — — 1 — — 1 — — \ "T- = : S; ; Mill 1 Ti — r "1 — ^ . ....... -,„ J 1 M ^ ' 1 1 1 1 1 1 M 1 1 i 1 ' " ^ 1 — \"J~\ t M ' 1 1 1 1 1 1 T ■ ■ f f----""-^"l i i 1 ' ~^ Ml 1 + j:i ± : ^1 ff 'lllftTliMffli I, iiiiinina4 ._± .^1 :=Ft:ii-2=i::-:ii:: t. >r/g.y3-/7. nOnriJ 1 \A^ fyti Corii 707Jj^l5M%'J^78:&SO&f;^8lBfiSSkei8»990^l^^^l^^S%^;8f9 jFVd Con'd 70)l)niWS%7V82fScSi8^8imiSb8JSS8^70»^^7i3ih?s;n>%lJi8:if FTd ComV 7o7in7i7n27an8j^^&ihnuid^s^sm^oj(/jm^fj^^!i.fffH II fl,J')--B. J oji^ri [UL^ V jFTd CohV 7^7^7^M}^M/S2^^9/S^S58/8SSl,&889/'^/^/i/J';ai^/;8 SCHOOL AND UNIVERSITY GRADES 31 tion of grades in the case of about equal numbers of students from the College of Engineering and the College of Letters and Science of the University of Wisconsin in two freshman sub- jects, — English and mathematics. The chart makes possible, in the first place, a ready comparison of grades given to two different classes of students of engineering and of letters and science; and, in the second place, of the grading in two typical freshman studies. In the first connection it is interesting to note that there is always danger of considerable error, when persons attempt to make comparison of large groups of in- dividuals purely on the basis of general impressions. It seems to be the prevailing impression, for example, that the engineering students at the university do much poorer work in the study of English than do the students of the Col- lege of Letters and Science. This, as the chart indicates, is to some extent true, but not to the extent that is commonly believed. In such cases as this, the general impression is apt to be unduly influenced by the extreme cases, e. g., by a some- what larger proportion of inferior or poorly qualified students. In making comparison of large groups, some statistical method is almost always requisite in order to secure an accurate state- ment. The large percentage of failures and conditions given in the freshman engineering mathematics, is apparently a conscious attempt at the elimination of the poorer students on the basis of the standings in mathematics. This is a higher percentage of failure than is usually justifiable, provided that only properly qualified students are admitted. The conclusion would seem to be that either students, who were not properly prepared or qualified, were admitted to the work, or else the grading is somewhat unfair. The practice of admitting a large number of students and selecting the more efficient is, of course, a method which has many advantages, as well as dis- advantages. The more frequent fault of the system is that the standing in one subject, — not infrequently mathematics, — is apt to be made the chief basis of elimination. "While it is true that the general average of viost of the students who are drop- ped from the university is low, — although this is by no means always the case, — this result is in itself sometimes due to the greater demands made on the student's time by the subjects in which he is deficient. 32 THE UNIVERSITY OF WISCONSIN The second comparison suggested by Charts 13 A and B and 14 A and B is, perhaps, the more important. The differenci? in the grading of students in English as compared with mathe- matics is very apparent. The distributions in English tend to be "normal" with the larger proportion of students in the median grades. In mathematics there is a clear tendency to group students as either "good" or "poor" with relatively few ■of "average" ability. This, as pointed out elsewhere, is prob- ably not a true representation of the facts. In these charts, as well as in several which precede and follow, a marked tendency to use certain grades in preference to others is evident, as, for instance, the division of five, viz., 70, 75, 80, 85, and 90, and, in the university grades, the turn- ing points of the grades, as 78, and 93, as well as some other marks, as 88. This coupled with the fact that certain grades are correspondingly little used, or in some cases practically not at all, seems to indicate that the actual range of marks here employed is greater than can be used with discrimination. Teachers either cannot or will not differentiate to the extent of 25 to 30 points in the work of the students of a class. The fact that one teacher marks 80 and another 83 is largely a matter of individual predilection. Ten to twelve degrees of differentiation is the maximum that most can employ with any meaning, and it would, on the whole, cause greater uni- formity and less inequality if that was all which was at- tempted. The use of five grades, with possibly a further dif- ferentiation of a plus and minus in each grade, would really be a more accurate method than a somewhat random use of 30 apparently more precise grades. The latter is apt to lead to careless marking, and apparently greater differences of in- dividual judgment between teachers than may actually exist. Probably no particular harm is done, but teachers are deceived into thinking that they are securing much greater precision than as a matter of fact results when the system as a whole, rather than any one individual's grading is considered. Another fact that sometimes leads to inequalities in mark- ing is that students in advanced courses are apt to be graded higher than in elementary courses. Chart 15 gives the dis- tribution of the standings of 175 students in an advanced •course in English Literature. The average is. as may readily Zr j^ ? a 4 ^ § ri § 1 vi c ES 34 THE UNIVERSITY OF WISCONSIN be seen, much higher than in the case of the freshman class just cited. There is no reason, as it seems to the writer, why this should not be the case, but it not infrequently leads to inequality chiefly because individual instructors do not realize that this is the general practice. Professor Meyer has raised certain points in the article already referred to that bear upon this fact. One justification for this high grading of students in the advanced course, may be made on the basis that the students are a more selected group. Professor Meyer does not attempt, in the short article re- ferred to, any discussion of this somewhat involved question. There is, as yet, little evidence as to what extent the selection of the university makes for the higher grades of ability. There has undoubtedly been considerable selection operative as between the lower grades of the grammar school and the university, but it is doubtful if there is very much difference in this respect between the university and the high school. <^> It would be attempting much more precision than the grading of pupils is ever likely to attain, to take much account of this factor in its general aspects. The question does arise, how- ever, in another form, namely, whether, in some instances, superior students do not elect certain lines of work or in- dividual courses, and that at the same time these courses are avoided by average or inferior students. If there are such cases, the grading should be higher in these latter courses. The trenchant criticism of Professor Meyer in this connection is quite to the point, however. The high grading is more often due to too great leniency, etc., on the part of the instructor than to the presence of superior stu- dents in his courses. Teachers sometimes, too, appear, as Professor Meyer states, "guided by the conviction that the very fact of a student electing his work under his instruction proves that he is a superior student and that he ought to obtain high- er than the average grade." Still I question whether the proposal that "If a student ex- cels, this means, of course, that he excels among the students who are taking the same instruction which he is taking," can *Por some data on this question, see an article by the writer on "Qualitative Elimination from School," Elementary School Teacher. Sept. 1909. SCHOOL AND UNIVERSITY GRADES oO be universally applied. While they may be exceptional, it vi^ill, I think, be generally recognized that there are usually some courses in a university which, from year to year, secure only an inferior grade of pupils, and other lines of work which, for various reasons, secure a disproportionate number of superior students. Classical students in the high school and university, and students in the advanced courses in mathematics are often examples of such selected groups of students. The above principle would not be equitable in these cases. A more important exception occurs in the case as just noted, of all advanced courses as compared with elementary courses. In the University of Wisconsin students in advanced courses are graded about 20 per cent higher than in the el«:^ mentary courses. Juniors and seniors are similarly graded much higher than freshmen and sophomores. This practice holds consistently for all departments as a whole and for most individual instructors, and since the prime requisite for equi- table grading is uniformity, I see no reason, as just stated, why an attempt should be made to change the arrangement and to secure the same distribution of marks in the advanced courses as in the elementary. There are, on the other hand, many reasons why the ar- rangement is a natural one. In the first place many students have been dropped from the university during freshman and sophomore years. Those who remain are presumably on the whole better qualified to meet the university requirements, and might naturally expect to receive at least as high grades as before. If the student were graded as Professor Meyer suggests, solely on the basis of the relative rank he attains "among the students who are taking the same instruction that he is taking," some students must because of the elimination of the poorer students, receive lower grades than they did in the elementary courses. They presumably have in many cases elected the advanced courses because of interest or success in the elementary work. Students in the advanced courses have thus been more or less sorted out and differentiated as regards their interests and abilities, and are undoubtedly as a whole better qualified for their work. There is no reason why they should not on this account be graded higher. In estimating, in a following section, the extent of in- 36 THE UNIVERSITY OF WISCONSIN equality existing among the various instructors in the uni- versity, I have, therefore, taken account of this factor. If one instructor has a larger percentage of advanced students than another, it may be expected that he will give a larger percent- age of high marks. Since this is the prevailing practice, no inequality results. The results of Professor Meyer in regard to the University of Missouri may very possibly on account of this factor, represent a larger extent of inequality than in reality exists. Furthermore, in the case of any given class it may be approximately determined whether or not as a group the stu- dents do differ much from the average, by finding out whether they rank similarly in other subjects of study which they are following. Mental abilities are certainly not so specialized that any considerable group of men are found doing superior work in one subject, who will not take somewhat similar rank in other subjects of university instruction which they may also be following. IV. GKAllES IN DIFFERENT SCHOOL SUBJECTS It was stated above that we had reasons to believe that the distribution of grades in different subjects should not be un- like. Many instances from the ranking of pupils in the high school and university might be presented where this is the case. The distribution of marks in such subjects as history, German, English, and mathematics, should, when a fairly large number of pupils is concerned, ordinarily be similar. It may be assumed that in general a hundred or more stu- dents in one course do not differ as a group much in general intelligence or in their ability to succeed in given subjects from a hundred students in another subject. Some courses or lines of study may select superior students, and others inferior students, but these are the exception rather than the rule, and may be dealt with as suggested in the last chapter. We may expect then that the grades or marks assigned to a large group of students in different subjects of study will be similar, and that they will be distributed in the various grades in about the same frequencies. Some instances, both of where this is true and of where it is not true, will be cited below. In order to recall to mind the close relation which fre- SCHOOL AND UNIVERSITY GRADES 37 quently exists between the distribution of ttie marlis of a large group of students and tlie normal distribution, tliis comparison is made in Figure 16. Tliis shows a distribution of 244 pupils in high school English. Its approximation to the normal dis- ti-ibution is indicated by the dotted line. Figure 17, on the other hand, shows a distribution of the standings of 146 stu- dents in freshman mathematics. It varies widely from normal distribution as indicated by the dotted line. Both for the theoretical reasons outlined in a previous chapter, and because of the fact that the sort of distribution shown in Figure 16 is closely approximated in the majority of cases of grading, it should, as it seems to the writer, be taken as the standard. If, then, a large group of students are graded as a group very differently in two school subjects,- — as in the case of English and mathematics cited in the last paragraph, — we shall consider that distribution more nearly right which approxi- mates the so-called "normal" distribution as seen in Figure 16. Figure 18 presents another example of this sort of varia- tion which exists in the grading of the different school sub- jects. The continuous line of the chart shows the distribution of the grades of 79 pupils in the sophomore year of a high school class in English. The dotted line shows the distribu- tion of the grades of the same pupils in a class in mathematics taken during the same year, (74 pupils). It is hard to believe that as a group the class would differ in the way indicated. Figure 19 gives a similar comparison of the standings of university freshmen in German and histoi-y. The groups are made up of practically the same individuals, (226 freshmen). Although there is a tendency to grade a considerable number of the pupils in German higher than in history, as a whole the two distributions are not unlike. This, as just stated, should be the case. There may be a definite reason for the higher grading of some pupils in German due to the fact that the freshman class is composed both of those who have had con- siderable preliminary training and those who have had none. In Figure 20 a similar comparison is made of the stand- ings of 218 pupils in history and mathematics. These are all practically the same students as were compared in Figure 19. The difference here between the form of distribution be- tween history and mathematics is more striking. Since the 38 THE UNIVERSITY OF WISCONSIN I t r ,1 \ ^ '4: o5^ § I ^ ^^ Ox \ 1 1 1 1 >f> 1 ■ ; 1 ^ ^ 03 ^ — 1- - ^ ^ ^ \ 1 r. ■o L-V] h L \ \ — ^ -HS 1 »o 1 42 THE UNIVERSITY OF WISCONSIN distribution in the case of history is much closer to a normal distribution, it must, for the reasons given above, be consider- ed the better distribution. As noted above, it would appear that there was a tendency in the case of the classes in mathe- matics to consider pupils as either good or bad; there is less halfway ground, or in other words, fewer medium or mediocre students than in other subjects. It is hard, however, to be- lieve, as said above, that, as a matter of fact, the mathematical abilities are distributed any differently than others. These apparent differences must, as it seems to the writer, be largely attributed to artificial methods of grading. Some further facts in this connection are discussed in the succeeding chapters. 300- ^J5o 200- ,£V^ /50- 700- ^O [nl tfi ■i. F C '^o 7S ao s5 90 55 5T" SCHOOL AND UNIVERSITY GRADES 43 V. THJE UNIVERSITY GRADES The following charts show the actual distribution of the averages of the grades attained by students in the first two years of the university, (Figure 21), and in the last two years, (Figure 22). There are about 5,500 averages plotted in Figure 21 and a little less than 6,500 in Figure 22. These marks are combined in Figure 23, giving' a total of about 12,000 general averages. In these cases all the marks of a student in the chief subjects of instruction, were, in each case, averaged. The marks taken were given in three different semesters and they may, therefore, be considered as representing what is the gen- eral standard of grading in the university. In order to remove the irregularities in the disti'ibution of Figui'e 23 which are J30- 300- 25a- 200- /JO- /OO- .J^'^JL. SO o C 1 r C70 J.J W 93 95 100 44 THE UNIVERSITY OF WISCONSIN due simply to the predominance of certain grades as 85, 90, and 93, the grades have been regrouped in Figure 24 under ten divisions as there indicated. Ten, as noted above, are about as many grades of difference as usually have much sig- nificance. This procedure results, as may be seen, in a dis- tribution which is "heavier" in the higher grades and not "balanced" about the median point as in the case of a normal distribution. This fact is mainly due to the higher grading of the junior and senior years, as compared with the first two years. This difference in the first two and, last two years of the college course may be seen by comparing Figures 21 and 22. If, as Professors Cattell and Meyers have suggested, the high schools and colleges would publish occasionally, or dis- tribute to the members of their faculties, some such record of the actual practice of marking in the institution concerned, as is here given, this in itself would tend greatly to promote uniformity in grading. For further comparison, the j^ercentages of marks assigned to the five divisions employed, — excellent, good, fair, poor, and condition or failure, — is also given in the following table. Those conditioned or failed have been put into one group. Column 1 gives the percentages for the freshman and sopho- more years, 2 for the junior and senior years, and 3 for all four years. Percentages of Grades Assigned in the University No. of 1 Freshman and Ex. Good Fair Poor Failure Cases Sophomore 13.8 33.2 28.7 16.8 6.9 5494 2 Junior and Senior 18.3 44.7 24.2 9.G 3.2 6397 3 All four years.. 15. 9 39.5 2G.4 13.3 4.9 12278 In order to show the variations which exist between de- partments and individual instructors within the university, which have been arranged in the above distributions, tables are given in an appendix of the grading of various depart- ments and of the instructors in them. The synopsis given in the following tables of this chapter is sufficient to indicate the general extent of this variation. SCHOOL AND UNIVERSITY GRADES 45 ^ L or ? 06 ^ ^ ^ ^ 46 THE UNIVERSITY OF WISCONSIN >%. ^f. ti ttaaWTtyt^ 'TT i ' MniTHl iI n Zodp- r+f- f* G-cff E- E+ SCHOOL AND UNIVERSITY GRADES 47 The following table gives the range, i. e., the maximum and minimum percentages of the various grades assigned by in- structors in six departments during freshman and sophomore years. The range is larger in the last two years: Range of Marks Given in Single Departments (Freshman and Sophomore Years) Ex. G F P X max. min. max. rnin. max. iiiin. max. min. max. min. History 16.7 3.4 52.9 25.2 3S.9 27.3 26.S 2.3S ILS English 19.3 1.98 53.9 22.6 45.5 19.3 38.7 7.69 10.2 Mathematics 24.1 12.1 27.8 18. 28.5 15.7 27.4 14.25 23.3 L29 French 23.9 14.5 43.8 24.2 27.5 21.4 27.4 7.5 9.67 2.31 Latin 26.1 11.7 47.6 46. 26. 15.9 16. 7.48 8.04 German 34.3 14.47 49.1 27.4 33.5 19.4 22.95 2.98 10.95 2.27 A simple method of presenting this variation in grading is to compare for each instructor the percentages of the first two grades — "excellent" and "good" — with the percentages of the remaining grades given. This comparison, for the reasons given in the last chapter, should be made separately in the case of the first two and last two years, since, as there stated, the grading of the last two years is uniformly higher than is the first two years. Considering the general averages of prac- tically all the marks given in the three semesters under study, (see following table), 47 per cent of the marks given in the freshman and sophomore years, and 63 per cent in the junior and senior years are either "excellent" or "good." Consider- ing the four years together, about 54 per cent of all the marks given are either "excellent" or "good," and but 46 per cent are of the remaining grades, "fair," "poor," and "condition" or "failure." This may therefore be considered the general prac- tice, and those instructors grading most closely to this aver- age, are under normal conditions grading most fairly. Percentages of Grades Assigned in the University (1) The sum of "excellent" and "good." (2) The sum of the remaining marks, i. e., "fair," "poor," and "condition" or "failure." (1) (2) Freshman and Sophomore 47% 53% Junior and Senior 63 37 All years 54 46 48 THE UXIVERSITY OF WISCONSIN In the following table the percentages of "excellent" and "good" of the whole number of marks given are presented for several departments. The two highest and the two lowest percentages for each department are given, the results of the first two years and the last two being stated separately. More detailed comparisons may be made by referring to the tables in the appendix. As has been shown in the above charts and tables, the per- centages of high grades is higher than should be found if they were distributed according to a "normal" distribution. The largest proportion of grades according to this system should l3e of the grade "fair." In the university more grades of "good" are given than of ■"fair." There can be no important objection to this practice provided it is somewhat uniformly followed; but, if one in- structor believes that the majority of his students should re- ceive simply the grade of "fair" for the same work for which the majority of instructors will assign the grade of "good," inequality, of course, results. For this reason it is consider- ed that some knowledge on the part of all instructors of the general practice in the matter of grading will promote greater uniformity. In the case of the university grades, it has al- ready been noted that the students in the last two years are, as a general rule, graded considerably higher than those in the first two years of the university course. This is largely a, difference between "advanced" and "elementary" classes. The Two Highest (a and b) and the Two Lowest (x and y) Percentages of the First Two Grades (Excellent and Good) Given by Instructors in the Same Departments A B Freshman and Sophomores Juniors and Seniors abxyabxy English 67.5 62.9 35.2 31.6 80.4 74.3 52.4 39.0 History 69.1 63.7 31.2 25.4 77.7 75.8 47.3 33.3 German 74.6 63.6 39.7 30.5 77.5 76.7 60 52.8 French 65 61.3 47.5 38.7 85.6 74.5 60 58 Mathematics 49.1 44.4 ... 30 76.6 73.4 66.1 52.1 Latin 73.7 62.1 . . . 57.9 78.9 78.0 73.5 65.9 Philosophy 74.3 52.4 46.7 40.2 ^ SS 74.7 62 53.1 89.2 71.3 52.4 39.0 83 66.2 62.5 85.5 71.4 61.1 84 65.1 51 36 SCHOOL AND UNIVERSITY GRADES 49 Education Political Economy Political Science. Physics Chemistry VI. THE CORRELATION OF SCHOOLS AND SCHOOL • SUBJECTS It was noted in Chapter IV, in discussing the grades in dif- ferent school subjects, that there should normally be approxi- mately about the same sort of distribution where there are large groups of students concerned. This, of course, does not mean that any given student will necessarily hold the same relative position in the different subjects, but simply that the distributions as a whole will be similar. The question of how widely students' work differs in various subjects is itself an interesting problem, although it is only indirectly involved here. Its bearing was suggested in the last chapter in dis- cussing the question of the election of courses by students. The following charts on Plate I indicate something of this relationship in the case of high school subjects, and are pre- sented here largely to indicate a simple method of study by which such a problem may be approached. The method used is as follows: All the pupils in the class are numbered consecutively. These numbers are then distributed on the scale of grades as seen in Plate I, Chart A, which shows the standings of 113 high school pupils in Latin. Nos. 1, 7, and 2. above the grade 94 signify that the three pupils indicated by these numbers secured the grade of 94 in their class work in Latin. The numbers of the first quarter of this class i. e., of those whose rank places them in the highest quarter, are then colored red. Those in the second quarter are colored purple; those in the third are colored green; and those in the fourth or lowest quarter of the class are colored black. These colors are re- tained in the distributions with which this is to be compared; that is, the colored numbers of these pupils are now dis- tributed according to their standings in English and mathe- matics. If they all received exactly the same grade in these two subjects which they received in Latin, there would be no change in the color relations; all the "reds" would be in the FLHTE L R li>5 Hi IS !t /»7 151 ISi I5t> lt-5 111 ISI /^7 171 lol lf-3 m I 151 IIS I'M, 1^2 Hi^h School Grades in Latm 1)2 Pupils 8S 7^ n 7o S7 hi h yf 'ii __v /B & ipTTio 3fe 88 5b III loo 110 1^ 51 67 7/ III ?8 loS M. ill S2 //•/ \^8 3? V9 t3 75 bi 3/ 55 ■"^f 38 ^/ ?f . 9/ 37 /7 Zb f^ 3o 5 n 27 23 ^ 82 ^8 /» 5!* 3 /3 /O 7^ 11 7>-;5'7^ ;/ 7^^/ 5c s; ^^ ^^ ^y- ^^ S6 «; ^e g| ^o sj n n^!h fi iSI ni lU lol If7 111 in lol- Ibi IbS lor ihi l¥5 /SJ /¥(, loS 15b 89 /5^/27 I'^f Rl 95 III 8b /oc /<'3 fe7 /V? 65 //3 lf>8 87 71 V. 3? ?5 ^8 bl lit 7o II 80 ¥1M. 'TV 5fc H/^/> ScW Grades in English JJZ Rjioils III lob SS 81 III ^0 /y/ 32 m io ill lio io 38 15 15 Hi^ h '7n87^ 808j 8Ui 8^ 8S 86 8y 88 8^ ^c yy ^^ ^5 ?j^^S Hi^h School Graces in tloihemtic^ J/3 Pupils ibf Jib IbS f ISl in 117 /63 ISI 159 /^^ //y 11/ 151, loZ 131 Is?, log o;. w Ibl 91 /,?/ So 9F 81 58 18 II Si 38 6 3/ /^ '0 28 /¥ 93 /7 2/ 9 22 ^^ 1 10 ^ 5 3 ^^ ^<^ 76 7n87?80 8j Sm8^8i86 8)88Bi^o^^^^Z^^fSf(,^ c V 5i I ^ K •!»■ 5S,-~ C d > ,0 A o, ' ^5 -1 ^- - rv> 00 cv^ "-o c^^ JN 1§^^ S^c^ d^' - - 3s? 00 to ^ ^ ^ ^ n LU iHSiUliiiPt Ci 50 THE UNIVERSITY OF WISCONSIN first quarters of the classes in mathematics and English, as they were in the first quarter of the class in Latin and so on. A red colored number in another quartile than the first in- dicates therefore at once the fact that although this student was in the first quarter of his class in Latin, he does not stand as high in these other subjects. The amount or extent of such change will give, therefore, at a glance, the general correla- tion of standings in the case of these three subjects. There is, as was to be expected, some interchange of position, but in gen eral it is readily apparent from the chart that those who stand well in Latin tend to do very nearly equally as well in the study of English and mathematics, and on the other hand, those who are the poor scholars in Latin are apt also to be poor in their other subjects. Some exceptions, of course, ap- pear. Nos. 80, and 91, for example, rank in the first quartile in Latin, but are in the lowest quarter of the class in mathe- matics; but these are the exceptions, the rule in this particu- lar school at least, being a rather close correlation of stand- ings in these subjects of study. A similar problem of interest for the study of which the grades of pupils may be made use of is the relation between standing or rank of pupils in high school and their relative rank in the university. '^' A test in this way may be made of the success of the method of accrediting schools as a basis for admission to college. Such a study might also be of value in determining the closeness of relation between the work of pupils in the various grammar schools of a city and that done by them later in the high school. Such a study would furnish the best evidence of the efficiency of the various schools con- cerned, at least, in so far as preparation for the high school is a measure of efficiency. As a further example' of this method of study, there are presented in Charts 25-28 inclusive, the general distribution in the high school and the university of a group of 174 stu- dents who entered the College of Letters and Science from the Madison high school during the years 1902-1905 inclusive. In Figure 25 the dotted line represents the distribution of the general averages of these students in all their high school This problem has been studied In detail in Bulletin No. 312, Hig-h School Series No. 6 of the Bulletin of the University of Wisconsin. SCHOOL AND UNIVERSITY GRADES 51 work; the continuous line indicates liow they stood in the uni- versity. Their ranlvs, were, as may be seen somewhat higher, although this is an arbitrary difference of not much impor- tance. In Figure 26, the comparison is made between their ranks in the English studies of the high school and their rank in freshman English. The average, or median of the two groups, is practically the same (Medians^83.4 and S3. 5). Figure 27 shows a similar comparison in the case of German for 125 pupils. The freshman grades assigned these pupils average very much higher than the grades which they re- ceived in the high school. Finally, Figure 28, shows the com- parison in the case of a group of 112 pupils from the same school in high school and freshman mathematics. The fresh- man grading averages in this case much lower than in the high school, as may be seen by a glance at the chart. These four charts would appear then to indicate that there is closer correlation between the high school in question and the Col- lege of Letters and Science in some subjects of study than in others. This may be the case, although it is quite as likely due to arbitrary differences in the scale of marks used as has been indicated in the preceding chapters. In order to study this question satisfactorily it is necessary to follow the record of the individual student by some such method as has been suggested above in this chapter. For pur- poses of illustration this has been done in the case of the gen- eral averages of this group of pupils from the Madison high school. The method is the same as outlined above, and is shown in Charts D and E of Plate I. In this case, those who ranked in the first quarter of the class in high school are colored red, those in the second, purple, etc. (See p. 49.) The standings of these pupils may be easily followed by the colors which remain the same in the freshman distribution. If there were no changes between high school and university standings this would be indicated by an absence of change in the color arrangement, i. e., all the "reds" would remain in the first quarter of the group in the university, all the "purples" in the second quarter, etc. And, as a matter of fact, it may be seen that there is a very close correlation between the stand- ings of pupils in the high school and their standings in the university. ^ ^' /^ ^ ■>. ^ 54 THE VXIVERSITY OF WISCOXSIN A final problem in this connection may be suggested, name- ly, as regards the character or quality of the elimination from school and university. There has, during the last few years been considerable attention given to the amount of elimina- tion from school. The sort of elimination is also of interest. In order to indicate a simple method for the study of this ques- tion a square has been placed about the high school grades of those students who did not finish the sophomore year at the university. While, as may readily be seen, the larger number who drop out for one reason or another are from the lower half of the class, there is a fair proportion from the upper half. This is a problem which has greater significance in the high school where the sort of elimination from school is prob- ably not as much related to scholastic attainment. VII. APPENDIX OF UNIVERSITY GRADES (See Chapter V.) TABLE I— A.<^' Percentages by Departments (Freshman and Sophomore Years) Ex. History 5.91 English 8.4 Mathematics 16.3 French 19.75 Physics 27.95 Chemistry 21.1 Latin 17.9 German 19.9 Per Cent of Total. . 14 No. of G. F. P. X. Cases 29.6 34.9 21.5 8.1 1353 36.3 33.8 16.3 5.26 1180 23.9 23.1 19.7 17.1 633 36.0 24.5 14.3 5.46 567 37.8 21.1 lO.S 2.45 204 35 24.6 15.6 3.81 280 46.7 21.1 11.2 3.2 313 36.3 25.0 14.13 4.61 955 29 17 5494 (1) The following abbreviations are used,— Ex. for "Excellent," G. for "Good," F. for "Fair," P. for "Poor," and X. for "Condi- tioned" and "Failure" combined. SCHOOL AND UXIVERSITY GRADES 55 TABLE I— B. Percentages by Departments (Junior and Senior Years) Ex. 1. English 18.8 2. Political Economy. 17.2 3. Grerman 17.4 4. Philosophy 12.8 5. History 13.1 6. Education 14.6 7. French 26. 8. Political Science... 20.3 9. Latin 19.3 10. Mathematics 34.2 11. Chemistry 21.7 12. Physics 34.6 13. Botany 17.5 14. Geology 12. G. F. P. X. Cases 42.3 26.9 9.3 2.4 1154 35.9 25. 16.4 5.2 889 49.4 24.4 6.5 1.7 796 38.3 27. 15.5 6.2 709 48.1 27.9 7.3 3.4 660 53.2 25.9 4.7 1.3 443 45.8 20. 5.3 2.6 410 53.6 16.8 7.9 1.1 339 57.3 16.5 4.7 1.9 314 35.7 15.5 9.8 4.6 193 40.7 23.4 10.6 3.3 179 39.2 13.8 9.2 3. 130 51.7 19.2 8.7 2.6 114 43.1 34.4 5.1 5.1 58 Per Cent of Total. 18.3 44. 24.2 9.6 6397 TABLE I— C. Percentages by Departments (All Classes) Ex. History 8.3 English 13.5 Mathematics 20.4 French 23.4 Physics 30.5 Chemistry 21.3 Latin 18.6 German 18.7 Biology 12.7 Geology 5.5 <^> Philosophy 12.8 "> Education 14.7 •1' Political Science... 20.4 <^^ Political Economy. . 17.2 Per Cent of Total. . 15.9 No. of G. F. P. X. Cases 35.7 32.5 16.8 6.5 2022 39.2 30.4 12.8 3.8 2334 26.6 21.3 17.4 14.1 826 40.1 22.5 10.5 4.2 977 38.3 18.2 10.1 2.6 334 37.1 24.1 13.6 3.6 468 51.9 18.8 7.9 2.5 627 42.4 24.7 10.6 3.3 1751 40.1 27.6 14.5 4.9 344 39.1 36.2 14.4 4.6 215 38.4 27.1 15.5 6.21 709 53.3 25.9 4.74 1.35 443 53.6 16.8 7.96 1.18 339 36 25.1 16.4 5.3 889 39.5 26.4 13.3 4.9 12278 'i> Juniors and seniors only. 5G THE UNIVERSITY OF WISCONSIX TABLE II— A Percentages of Grades Assigned by Individual Instructors TO Freshmen and Sophomores Historv Ex. 1 4.91 2 9.84 3 3.4 4 7. 48 5 1G.7 6 9.1 7 6.33 G. 26.2 52.9 22 25.2 52.4 39.4 27.4 F. 32.8 31.6 38.9 37.4 28.6 27.3 30.8 P. 26.2 3.11 26.7 23.4 2.38 18-2 23.6 X. 9.84 2.59 8.96 6.54 6.06 11.8 No. of Cases 183 193 558 107 42 33 237 1353 English 8 19.3 22.6 19.3 38.7 31 9 12.5 30 37.5 12.5 7.50 40 10 6.42 45 31.2 14.7 2.75 109 11 1.98 33.2 45.5 12.9 6.43 202 12 7.44 30.6 38.0 19.0 4.96 121 13 6.31 33.7 40 14.7 5.26 95 14 4.08 38.8 32.6 14.3 10.2 49 15 16.7 22.8 36.0 17.55 7.01 114 16 7.3 39.6 37.5 13.5 2.08 98 17 16.9 50.6 19.5 11.7 1.3 77 IS 11.4 42.9 27.2 14.3 4.29 70 19 8.99 53.9 21.8 7.69 7.69 78 20 6.12 25.5 23.5 37.8 7.14 98 IISO Mathematics 21 16.6 27.8 28.5 14.25 12.9 302 22 24.1 25 15.74 19.45 15.74 108 23 12.1 17.9 19.3 27.4 23.3 223 633 Prench 24 21.2 43.8 22.5 7.5 5 80 25 17.5 43.8 27.5 7.5 3.75 80 26 22.3 35.7 21.4 12.5 8.03 112 27 15.53 32 27.2 19.4 5.82 103 28 14.5 24.2 24.2 27.4 9.67 62 29 23.9 35.4 24.6 13.85 2.31 130 507 SCHOOL AND UNIVERSITY GRADES 57 Phvsics and Chemistry 30 27.9 37.8 31 21.1 35 Latin 32 16.1 46 33 11.77 46.2 34 26.1 47.6 German 35 26.3 34.2 36 12.03 49.1 37 34.3 40.3 38 11.47 34.4 39 17.33 37.3 40 17.9 35 41 14.74 29.5 42 12.32 27.4 43 29.03 30.1 44 21.6 42.03 45 22.4 39.64 21.1 10.8 2.45 204 24.6 15.6 3.81 289 20.7 9.20 8.04 87 26 16 119 15.9 7.48 2.804 107 21.9 12.3 5.26 114 24.1 11.1 3.702 108 19.4 2.984 2.984 67 27.9 22.95 3.28 61 29.3 6.66 9.34 75 26 17.1 4.06 123 33.5 18.95 3.16 95 27.4 21.95 10.95 73 22.6 14 4.3 93 21.6 12.5 2.27 88 20.7 15.53 1.726 58 955 TABLE II— B. Percentages of Grades Assigned to Juniors and Seniors No. of English Ex. G. F. P. X. Cases 1 .. 14.6 37.8 33.5 11.9 1.9 301 2 . . 29.5 50.9 18.5 0.9 210 3 .. 22.1 43 25.5 7.6 1.7 172 4 .. 13.2 25.8 35.7 14.5 10.6 151 5 . . 14.7 50 21.3 13.9 122 6 . . 20.9 43 24.4 10.4 1.1 86 7 .. 13.5 60.8 22.9 2.7 74 Political Economy 9 .. 11.8 30 26 22.8 9.45 127 10 . . 25.2 46.1 20 7 1.74 115 11 .. 20.7 21.6 27.05 18.9 11.7 111 12 .. 8.7 45.6 27.2 16.5 1.94 103 13 .. 43.4 45.8 3.6 7.2 83 14 .. 15 33.8 47.7 5 2.5 80 . 15 .. 33.4 26.9 25.6 14.1 78 16 .. 31.6 25 29.7 14.05 11.8 64 17 .. 6.98 32.6 4.7 46.5 9.3 43 18 .. 14.6 43.9 26.8 12.2 2.44 41 19 .. 43.2 41 15.9 44 58 THE UNIVERSITY OF WISCONSIN German 20 29.9 47.6 17.7 48.8 164 21 8.7 68 22.45 0.73 138 22 9.3 56.6 24 10.7 129 23 24.55 41.5 27.35 5.66 .94 106 24 14.6 38.2 30.4 11.2 5.63 89 25 10 50 31.3 7.5 1.25 80 26 17 35.8 32.1 7.55 7.55 53 27 27 37.85 16.2 13.5 5.4 37 Philosophy 28 12.1 34.6 22.7 23.3 7.38 339 29 8.8 41.4 40.4 3.1 6.22 193 30 15.9 36.5 23.35 18.7 5.6 107 31 22.9 51.4 17.15 7.15 1.43 70 History 32 14 51.2 31.9 1.9 .97 207 33 5.15 47.4 27.75 11.34 8.25 97 34 6.46 40.8 34.4 10.75 7.52 93 35 17.15 58.6 22.85 1.43 70 36 15.25 49.15 32.2 3.4 59 37 19 41.3 20.7 17.25 1.7 58 Latin 38 17.65 61.25 13.72 4.9 2.45 204 39 39 39 17.05 4.88 41 40 2.86 63 28.6 5.71 35 41 23.5 50 20.6 2.94 2.94 34 Mathematics 42 34.6 42 12.35 9.88 1.235 81 43 33.9 32.2 20.35 5.08 8.48 59 44 36.7 36.7 13.32 10 3.33 30 45 30.4 21.7 17.38 21.7 8.7 23 Chemistry 46 31.8 33.3 16.7 16.7 1.51 66 47 18.35 32.65 32.65 6.12 10.2 49 48 36 48 16 25 49 16 68 12 4 25 Political Science 50 29.4 53.6 13 3.9 177 51 4.4 58.2 22 12.1 3.3 91 52 18.3 47.9 19.7. 12.7 1.4 71 French 53 44.1 41.5 6.0 5.0 1.7 118 54 26.4 48.1 19.8 1.9 3.8 106 55 14.8 43.2 33.3 4.9 3.7 81 56 10 50 31.4 7.1 1.4 70 57 22.8 51.4 11.4 11.4 2.8 35 SCHOOL AND UNIVERSITY GRADES 59 Education 58 15.1 59.6 23.1 1.7 .35 285 59 10.6 42.5 31.9 11.3 3.5 141 60 11.6 50.4 36.4 1.6 85 Physics 61 50 35.5 6.45 8.06 62 02 20.4 40.7 18.5 12.95 7.4 54 Botany 03 20 42.9 21.4 11.4 4.29 70 64 13.64 65.9 15.9 4.55 44 Geology 65 12.05 43.1 34.5 5.17 5.17 58 m LIBRARY OF CONGRESS 021 338 827 7