118 Influence of "Broad Categories" on Correlation 
Hence r y ^ - r xy r x ^ = 0, 
r yc x r \ 
or ^, 7/ = ? r- ( 1V )- 
xC x 
In other words, to find the true correlation of x and any other variate y, divide 
the correlation of y and the class-mark of x, by the correlation of x with its class- 
mark. 
From (iv) we can deduce the correction for number of arrays when we find r) xy 
the correlation ratio of x and y on the supposition that r x;/ equals sufficiently 
closely t) xi/ , i.e. when we "find r by rj methods." Let H yC equal the value of rj 
found when y is finely classified and the mean of an array can be determined for y, 
but the arrays of x are broad classes *. Then 
= KyC 3= H yC x (v) 
Vxy r x0x </S(n g x s *)l(N<rJ) h 
E hC x , 
= ' j (vi), 
IN 
V *,^~**? 
if we suppose the distribution normal. 
(4) Now let us consider two variables x and y given by their class-marks C x 
and G y . If we correlate y with any variable u we have at once by (iv) 
r^--^- (vii). 
r yc y 
Now as u is quite arbitrary put it equal to the class-mark of x or G x , then 
r l 
Substitute this in (iv) and we havef 
r yCx = -W* (viu). 
r xy = rC " Cx (ix). 
r xC x r yC y 
* The primary establishment of equation (v) is due to "Student." His paper published in this 
number reached me, as I was writing this paper. I had obtained (iv) and (ix) and used them to correct 
contingency, but not to correct -q. 
t A somewhat different proof of this formula may be obtained as follows : the partial correlation 
q q is clearly zero, for when x and y are constant C x and C v do not vary. Hence 
xy u x u,j 
r C x C v 0-- ''V " >'x C x r x C v ~ r y Cy r y C x + r xy C x r y C u + r x C y r y C.) = °" 
But r xy = r ycJ r xC= r xC y l r yC u > 
and if we substitute for r ,i and r a we have 
r Cx C v (! ~ r \y) - (>-,y ~ **xy) r y O v r xC x ^ 
