128 Influence of "Broad Categories" on Correlation 
for the second and third sums obviously vanish if we suppose zero frequencies 
at the boundaries of each row or column. Hence by (xvi) bis we have 
8 (jf fyy^) = S (n st x s y t ) x(l+ + J^.) (xxxi). 
Thus we cannot replace $ (n st x 3t y 3 () by S (n 3t x 3 y t ) unless h and k are very small 
relative to * x and a y respectively. 
-r. , -i 8 (n x ,,xu) , 
But now consider r xv = A7 - , the accurate value. 
JS* x * y 
We may replace 8{n xy xy) by S(n st x s y s ), there being no Sheppard's correction 
for the product moment. Hence 
r xy = = — by (xxx) 
<T X *y a x *y 
= 'S(jjf— (l + t^—„ + fi— ,) by (xxxi) 
\N a x * y ) \ I2a x a 12* y *) J v 
\N <T X (Ty) 
S 
12* J) V 12<7, 
to the same degree of approximation 
s 
'■st a* y±\ 
— — by (xvn) bis (xxxii). 
N **) \N * y - 
This formula is free from all Sheppard's corrections, and in form it does not 
involve the equality of the subranges. Hence it seems likely to give moderately 
good results even for unequal subranges, provided they do not differ very widely, 
and there are not too few of them. It leads as before in the case of normal 
frequency to 
[n s n t Sl 
- %) K - 
- v} 
but it is clear that this cannot be pressed so far as to make only a very few, and 
those very unequal, divisions for each variate. 
(10) The present method of correction seems likely to give good results in 
the case of the method of mean square contingency*, when we may reasonably 
assume the distribution not widely divergent from the Gaussian. 
* A much fuller discussion of all the corrections for contingency has been some years in progress 
and will shortly appear. 
