122 Influence of 11 Broad Categories" on Correlation 
exactness will be reached by equal ranges. The differences, however, between 
equal ranges and equal frequencies are after six classes so small — within the value 
of the usual probable error of random sampling — that either method will give 
practically quite good results. 
(7) It is interesting to note the relation of the present method to the usual 
correction for grouping in the value of the correlation. In that method we take 
the groups at their mid-points, and we do not correct the product, but only 
the standard deviations of the two variates by the usual Sheppard's correction, 
T 1 ¥ th of the sub-range squared being subtracted from the raw second moment 
coefficient. Now we have 
r X y = S{n st x s y t )la x a y , 
and we have to replace x s , y t by x s and y t . 
But we have seen that approximately with equal ranges 
_ 1 -I 1ls— i — n s+i 
24 n s 
-k 
1 , ttt-j - 7l t+1 
Now 
8 (n st x s y t ) = S (» st x s y t ) - g| hS I"-- 1 - >l * +1 n st y\ 
- /. o [ n t.—i ~ n t+i „ „| , 1 f.„ "s-i — n s+i >h—i ~ 'h+i 
Consider first the last term*, it contains not only the product of hk, but also the 
product of differences, and is of the form, when we divide by Na x a y , 
1 h k ^ hist x h (q» - a«-i) + \ (a t+1 - g,) i (<*'« - a' t -i) + i (a'«+i - «'«) 
36 a-3 cr ;/ (iV n s « t 
which will be of the fourth order of small quantities if hja x and k/a y be small and 
may be neglected as compared to the square of small quantities. 
* As a matter of fact for Gaussian frequency 
8 ( n st '^f^ 1 7 l^^)=±hkS { n st x s y t ) 
to our degree of approximation. Thus the fourth term may be written ^ h 2 k" S (n st x s y t ), which gives 
S ( n a x s y t ) = S fa ,r s y t ) ( 1 - i fc« - i ft* + A AW 
^ 1 - ^ * s ) I I 
