Page:American Anthropologist NS vol. 1.djvu/511

 45 2 AMERICAN ANTHROPOLOGIST [n. s., i, 1899

We can show, in the same manner, that

2 *, x n = nq XXi // n f. Therefore

��I ?m Ml* = 4 in /C

��We will call

�� ��I Then q ltl = r— 1, and

��2? a:, jr n = «r/i, // n

(3) /- = 4^ n

We call r the coefficient of correlation, q the coefficient of regression, because it measures the regression of the correlated value toward the average. 1

It remains to determine the variability of the array of values x n which are correlated to a given value x x. According to (1) this variability does not depend upon x l9 since the e are not functions of x l% but are entirely independent. The variability of each array is equal to the mean square of the differences between the members of the array and their average. The latter is 4 + ^m «*i» while the value for each member of the array is In + **• Their difference is

X n y 111 «*|

and the mean square variability of the array

2(x a — q ni *,) * = 2x n * + 2g n * x t 2 — 2x n q nt x x.

= 2 x n * — 2q u * x } *.

Since the value on the left-hand side of this equation remains the same for all values of x ly we may substitute for it /o, the variability of the array. If we form the sum of these equations for all the values of x lt we have

2 X n 2 = tf^n 2 ,

2 x x 2 = ;/ 1*,*.

��Therefore

��n p* = nu n 2 — nqj /*. 8

��P =^nV 1 — r 2

��1 See Galton, Natural Inheritance, pp. 95 H.

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