Karl Pearson 
125 
Let us now pass to the consideration of the correlation r xy as obtained by 
correction from the class index correlation, i.e. 
„ _ r Cx C v 
The difficulty is to know how to determine the product 8{n st x st y st ) for such a 
slender division as a fourfold table. If we assume it still to be equal to 
SiristXtyt) 
we can then find r xy . 
We have 8 (n st x s y t ) = n^y^ + n 2 x 2 y 2 + n 3 x 3 y 3 + n^y^ 
where we take for our fourfold table : 
% + «2 
n 3 
«2 + «3 
N 
But clearly on the above assumption 
X x — <T 4 , 
Thus 
then since 
we easily deduce 
Similarly we find 
8 (n st x s y t ) = n.x.y, + n 2 x 2 y, + n^y^ + n 4 a^ 4 , 
(«i + "4) x\ + (»2 + n s ) x 2 = 0, 
(fh + n 2 ) % + (n 3 + n 4 ) y t = 0, 
8(n st x s y t ) 
"a; 2 y 4 (w 1 ?? 3 -TC 8 n 1 ) 
(«i + n 4 ) (nj + « 2 ) 
^2 + T h - ,\ 
»i + rc 4 ' 
?l 3 + « 4 . 
fti + n 2 
y* 
And accordingly 
8(n st x s y t ) 
.(xxii). 
.(xxiii). 
.(xxiv). 
v(»i + » 2 ) (»i + ««) (^a + n 3 ) (h 3 + » 4 ) 
This is not really the true value of the class index correlation unless we assume 
x s y t = x st y st , which is not generally true. It is really the correlation of the means 
used as class-marks, and as shown long ago by me it is the correlation of errors in 
the means of the two variates, and also equals the basis of the mean square 
contingency. If we correct to obtain r xy by the factors 
