The Normal Curve of Distribution.

If Not, Why Not?1

FRANCIS LEWIS CARDOZO,* WASHINGTON, D. C.

T

HE Normal Curve of Distribution is the line which touches the ends of levels among individuals in unselected groups. Since disparity in age will necessarily make for variations of all kinds, the scale of measurement referred to is more significant, in some instances, when the subjects approximate the same age. Some elements can be measured more accurately than others. Men vary in multitudinous ways as to moral and physical organization, but markedly with respect to intelligence, the outward expression of mental organization, which for present consideration may be defined as the sum total of related consciousnesses by the use of which one is able to master a given experience or adapt himself to a situation with which he is confronted, readily and surely. In short, intelligence is best shown in the conquest of environment.

The realization, belated though it be, that there are others inferior as well superior to himself in any sort of measurements, and the reflection that causes an observant person to note that, in the last analysis, Nature nevertheless delights in orderliness and symmetry, enable one to appreciate how and

1 Other new procedures than those already successfully inaugurated during the last few years by the Superintendent are said to be under serious consideration at the national capital. If the methods involved in the Normal Curve of Distribution are deemed practicable it will be to the decided advantage of Washington that Dr. Ballou was formerly Director of the Bureau of Standards and Measurements in the Boston public school system with the rank of Assistant Superintendent. Thomas H. Briggs, the distinguished professor of Education at Teachers College, Columbia University, New York City, recently observed during a course of lectures to a large group of teachers at Washington, "Dr. Ballou may be regarded as one of the half dozen able and progressve superintendents of public schools in this country."

*Author of The Junior High School, its Origin and Trend. A Study. Education, June, 1923; The Consolidation of Rural Schools, Involving the Transportation of Children at Public Expense. Education, May, 1924.

why scientists long since ventured formulae as to certain characteristic trends in groups of mankind, apparently conforming to law in respect to persistent continuity. The scales, pictorial and otherwise, illustrative of this theory, will show that, out of whatever examinations of traits or characteristics have been made, the largest number of human beings invariably cluster about the center, with as many of those remaining found in approximately equal degree next on either side and very few at the ends.

Boyce makes specific application that happily suits the purport of this article. "If we had a large number of teachers classified as to any one quality, we should expect to find the largest number medium, not so many good or poor, and still fewer excellent or very poor. The exact number filling in each division would depend on our arrangement of the limits. Various investigations have proposed different percentages for these divisions. We give here the number out of 100 recommended by Professors Meyer,2 Dearborn,3 and Cattell,1 as proper for each division.

The publication of data from extensive scientific investigation over a long period, especially during the World War, convinces one that intelligence is by no means a common possession, in fact, it would seem that reckoning 70 per cent of the population in this respect medium or worse should not be considered an extravagant estimate."

The knowledge that comes from the means by which the

2 Meyer, M. The Grading of Students, Science, No. XXVIII, 243.

3 Dearborn, W. F. School and University Grades, Bulletin of Univ. of Wis., No. 36, 1910.

4 Cattell, J. McKeen. Examinations, Grades and Credits, Popular Science Monthly, LXVI, 367.

5 Boyce, A. C. Methods for Measuring Teachers' Efficiency, The Fourteenth Year Book of the National Society for the Study of Education, Part II, page 48.

6 Bronk, M. The Problem of the Darned Fool, North American Review, April, 1924. Goddard, H. H. Psychology of the Normal and Sub-Normal, Dodd, Mead and Company, New York, 1919. "Twelve per cent. of the drafted army of the United States was found to have too low intelligence to be sent over seas.

Normal Curve is formed-the determination of intelligence and as accurate a measurement as possible of certain achievements previously decided upon and of which the subjects have been definitely advised-and a full understanding of the mathematical theory upon which the Curve is based, can be made of practical value. If one cannot so well examine himself or his output by such means as he has at hand and is able and willing to use, it will be helpful for an expert, unbiased and disinterested, to accomplish it in order that results thereof for his own sake as well as that of others, may be compared to a standard. The fact should be emphasized, however, that there should be a full understanding on the part of all concerned with the process before it begins to mechanize, otherwise it were better that the matter had never been thought of. As some one very wisely put it: "By a study of differences the causes which make men good and bad, wise and foolish, skillful and clumsy, efficient and futile-education may hope to learn about the means of making all men wiser, more skillful and efficient. The causes of the differences between one man and another, as things now are, will lead to knowledge of the causes whereby all men may be made to differ from their former selves."

At the risk of being repetitious, it is as obvious as it is equitable, that the examined, whoever they may be, should be previously and fully advised as to the requirements that are to be made of them, that is, the bases around which the Curve is to form. In the case of teachers, aside from the all-important elements of technique of teaching and results thereof, as observed and secured in the classroom through competent supervision, such desiderata as the following will articulate closely with the nature of their employment and place them at their best, thus subserving the real purpose of an examination: Theses on aspects of education (so many during a given period); professional as well as general reading (evidences shown as required); increase in professional equipment through evening, summer or correspondence schools of approved standing; definite evidences of benefits derived

from membership in educational or quasi-educational organizations, etc. The effect upon the teaching body of such method of procedure will be instant and wholly beneficial, in that it will be more tangible and therefore more fully understood than that which has ordinarily obtained.

Aside from the foregoing considerations, of what value to a school system is the Normal Curve of Distribution, granting that it has been executed efficiently and equitably? When in full operation and effect it will bring about the proper placement of teachers, officers, and all other employees who have been overrated or underrated-taking round pegs out of square holes, and the reverse, and putting them where they belong-thus affording through reorganizations larger service to students, to parents, and eventually to the community in general.

Ample reasons having been given for the adoption of this plan, we now come to an aspect of somewhat grave importance in the discussion of the matter, which however should be fairly but frankly treated. Is it in general use? If not, why not?

The following comments seem to bear upon the answers to this query:

"A much greater source of difficulty has already been mentioned in our discussion of differing abilities of the judges to discriminate between teachers. This lack of discrimination is due in largest measure to actual ignorance on the part of the rating officer, first as to what constitutes excellence in many of the aspects of teaching; and second, as to what teachers are actually doing. In response to our request to note any difficulties in filling out the record, Technique of Teaching is set down most often as the set of qualities most difficult to judge. There is little hesitation or uncertainty in expressing opinion about the teacher's personal or social qualities."7

"The obvious remedy for these difficulties is to have a body of competent judges trained to a common standard and have them judge the same teachers many times."8

7 Boyce again, page 64. 8 Boyce again, page. 75.

Is there any probability of becoming lost in the mathematical labyrinth in which the rating official will find himself before turning in the last chapter of his book of accounts? Let

us see.

From Rugg:9 "The school man . . . has said frankly that his mathematical training has been limited to high-school algebra. He has told us that 'graphs' are mysterious things to him; that equations of lines and formulae have no significance; that the use of 'frequency distributions,' 'probability curves,' 'medians,' 'measures of variability,' and 'coefficients of correlation,' can hardly be said to lend clearness to his thinking. . . . Three courses are open to the writer who wishes to acquaint such persons with statistical methods of treating facts. First, he can say that the school man's lack of familiarity with college algebra, analytic geometry, the calculus and least square is his own lookout, and that it is impossible to write 'statistical methods' and to give the student of education a manual of formulae and rule-of-thumb methods of computing the various coefficients without any explanation of the derivation of these constants. . . . Third, the writer in this field can assume that it is necessary to equip school men generally with a thorough-going knowledge of statistical methods; that in order for them to be discriminating in the use of the various methods in improving their school practice, this large background of knowledge must be developed. . . . This book has been written with a deeprooted conviction that the third of these three courses is the proper one; with a complete recognition of the limitations in mathematical equipment of the 'average' school administrator.

In view of impending changed procedures in the educational world, the immediate concern of teachers is whether there are forces without as well as within "that will exalt the spirit and stretch the mind."

Further than this deponent sayeth not.

9 Rugg, H. O. Statistical Methods Applied to Education, Preface, pp. vii, viii, Houghton Mifflin Company, New York City, 1917.