[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 |d^{2} ------+---------+----------+---+----+--------------------------------- A | 2 | 5 | 3 | 9 | When the several values B | 5 | 1 | 4 | 16 | under d^{2} are added their C | 10 | 13 | 3 | 9 | sum is 376. This, multiplied D | 1 | 4 | 3 | 9 | by 6, according to the E | 7 | 6 | 1 | 1 | formula, gives 2256. The F | 11 | 8 | 3 | 9 | denominator of the fraction G | 14 | 10 | 4 | 16 | is, since there are 20 cases, H | 20 | 15 | 5 | 25 | 7980. Dividing 2256 by I | 16 | 12 | 4 | 16 | 7980 gives us .28; for 7980 is J | 4 | 2 | 2 | 4 | 20 times 399, which in turn is K | 8 | 14 | 6 | 36 | 20^{2}--1. When this is subtracted L | 3 | 3 | 0 | 0 | from 1.00 it gives us M | 12 | 20 | 8 | 64 | .72, which is the measure of N | 15 | 11 | 4 | 16 | correlation between the two O | 17 | 18 | 1 | 1 | orders. Since it is very high P | 9 | 7 | 2 | 4 | it suggests that the two Q | 6 | 17 | 9 | 81 | traits are judged in much R | 13 | 9 | 4 | 16 | the same way. 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-(6 Sigma d^{2})/(n(n^{2}-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; Sigma 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.