FOOTNOTES:
[1] An interesting review of the origin and development of phrenology and other systems of character analysis is given by Joseph Jastrow, in an article in Popular Science Monthly, June, 1915.
[2] To make clear the way in which these figures are secured, and to show concretely what they mean, suppose that the twenty photographs are lettered A, B, C, D, etc. They are to be arranged in an order by each judge according to his judgment of the intelligence of the individuals, the individuals being unknown to the judges. Suppose that the ten judges place photograph A respectively in the following positions: 9, 11, 5, 8, 9, 12, 7, 8, 7, 14. The average of these ten positions is 9, which we then take as the standard or most probable position of photograph A. Only two of the judges actually place A in the ninth position. The other eight judges all vary more or less from this position. We then find how much each judge varies from the average of the group, and the ten variations are respectively 0, 2, 4, 1, 0, 3, 2, 1, 2, 5 positions. The average of these individual variations is 2.0 positions. This figure indicates how closely the ten judges agree in their estimates of photograph A, a small average deviation indicating close agreement. In this way we find for each of the twenty photographs its average deviation; and if the twenty figures thus secured are in their turn averaged we secure an approximate measure of the disagreement of the judges when estimating the intelligence suggested by the photographs. Similarly we may compute average deviations for any other trait which is judged. These final figures are the ones which are given in the table, each of them being the average of twenty photographs as judged by ten persons.
[3] In such experiments the actual magnitude of the measure of variation becomes larger as the number of judges is reduced, the number of photographs increased, or the photographs so selected as to resemble one another more closely.
[4] Since such coefficients of correlation will be frequently used throughout the book as measures of the amount of correspondence or relationship between two things, it may be well at this point to indicate briefly how they are computed. Suppose that, as arranged in order on the basis of their final averages, the photographs stand in the following positions for the two traits—courage and kindliness.
| Photo | Courage | Kindliness | d | d2 | |
| A | 2 | 5 | 3 | 9 | When the several values under d2 are added their sum is 376. This, multiplied by 6, according to the formula, gives 2256. The denominator of the fraction is, since there are 20 cases, 7980. Dividing 2256 by 7980 gives us .28; for 7980 is 20 times 399, which in turn is 202—1. When this is subtracted from 1.00 it gives us .72, which is the measure of correlation between the two orders. Since it is very high it suggests that the two traits are judged in much the same way. |
| B | 5 | 1 | 4 | 16 | |
| C | 10 | 13 | 3 | 9 | |
| D | 1 | 4 | 3 | 9 | |
| E | 7 | 6 | 1 | 1 | |
| F | 11 | 8 | 3 | 9 | |
| G | 14 | 10 | 4 | 16 | |
| H | 20 | 15 | 5 | 25 | |
| I | 16 | 12 | 4 | 16 | |
| J | 4 | 2 | 2 | 4 | |
| K | 8 | 14 | 6 | 36 | |
| L | 3 | 3 | 0 | 0 | |
| M | 12 | 20 | 8 | 64 | |
| N | 15 | 11 | 4 | 16 | |
| O | 17 | 18 | 1 | 1 | |
| P | 9 | 7 | 2 | 4 | |
| Q | 6 | 17 | 9 | 81 | |
| R | 13 | 9 | 4 | 16 | |
| S | 18 | 16 | 2 | 4 | |
| T | 19 | 19 | 0 | 0 |
A formula is provided by mathematicians which enables us to compute the degree of resemblance between these two orders. There are, in fact, several formulae for such purposes, all of which yield substantially the same results. The one used in this case was r = 1.00-(6Σd2)/(n(n2-1)). In this formula r stands for the coefficient of correlation for which we are working; d is the difference between the positions which each of the photographs receives in the two traits; Σ means the sum of these differences when each has been squared or multiplied by itself; n means the number of cases, which is in this case 20, since there are that number of photographs. When these substitutions are made and the equation solved, the result will be the measure of resemblance, which will lie somewhere between +1.00 and -1.00, as explained in the text. This calculation is carried out here for the two sample traits, for the convenience of readers who may not be familiar with statistical methods.
[5] These experiments were conducted by Lucy G. Cogan, M. A., to whom I am indebted for permission to use the results in advance of their more detailed publication in her forthcoming paper on "Judgments of Character on the Basis of Photographs."