334 071 Linearity of Regression 
Genercd Theorem. Let 11.2,... be the number of individuals in any number 
of mutually exclusive groups and let quantities X, Fbe defined by the equations 
X = A{ii^ + A.2ih + A-^n., + . . . , 
Y = B,n, + + B.n., + . . . , 
where the ^'s and B's are quantities which remain constant as the frequency 
distribution is changed. Then 
8X = A^Sn, + ^oS?io + A-ihi-i + 
S F = BM + B,8)u + BMs + ■■■■ 
Multiply these two ex^jressions together, sum for all random samples and divide 
by the number of such samples; then using (i) and (ii) we get immediately 
XY 
Ix^yRxv = - + A^BiU^ + A^B^n^ + A^B-^n^ + (iii). 
Problem I. To determine the correlation between the deviations due to 
random sampling in the values of nx^y^ and y^^, . 
We have ij^^, = S {n^^, y^y,]. 
Hence n^^, 8y,^, ^ S [Sn^^, y^y,] - hn^^.y^^, , 
Multiply these two expressions togethei", sum for all random samples and 
divide by the number of such samples ; then 
(a) p and p different 
(/3) p equal to p 
We shall need, besides, the results proved by Karl Pearson as Propositions III, 
IV, VIII in the memoir previously cited, which we will state here for reference. 
Proposition III. 

Proposition IV. 
Ry.^n^ , = 0 (p' not equal to p) (vii). 
Proposition VIII. 
where %i is a quantity defined by the equation 
s [n^^ (Tn^l {yx^ - y)"] = Na;- ( i - r) x ^-i , 
and is obviously such that, in the case of normal distributions when cr,j is constant 
and r = we get %i = 1. 
