Karl Pearson 
119 
Thus l/(r x c x r,jQ ) appears to be the corrective factor when we group the 
variates x and y both into broad categories, the ranges of which are of any 
nature. 
WehaVe _ 8{n st x st y st )lN 
and ,,„ = ■ ' \ N 
where n st is the frequency of the cell in the sth category of x and tt\i category 
of y. 
It is obvious that r xy takes its usual value when the classes of x and y become 
very numerous, for both factors in the denominator are then unity. 
In the particular case of normal distributions, on the assumption that the 
product x st y st is sufficiently close to x s y t to be replaced by it: 
1 
Vx 'J~ (Tr ^ Tat 7 ( X11 )> 
which admits of fairly ready determination from Sheppard's Tables. 
(5) An approximate value of r x Q can be reached in the following manner, if 
we suppose the range of the broad classes are all equal and given by h. 
Assume a parabola y = a + bx + cx 2 
to give by its area the three class frequencies « s _ 1; n g) n s+1 , the origin being at the 
start of the n g ^ class. Its equation will be found to be 
_ 2n s+1 -7n s + lift,-! _ n s+1 - 3n s + 2w s _ , /x\ n^. ^2?v+ fwV ... 
V ~ ' 6h h W 2h [hj •••^ XU1 > 
r ih h 
But n s x s ' = J yxdx = ^ (n s+1 + 36w s - n s _{), 
-, 3, h w g+ i - n 4 _i 
or x s = -h + ?tt • 
2 24 n s 
Thus, if Xg be the distance from the origin to the middle of the sth class, 
, , _ J_ , >h+i - rh-i 
s 24 n s 
aud if x s and x s be measured from the mean of the whole population 
- 1 ? ^S— I ~ n s+i 
