Andrew W. Young and Karl Pearson 
223 
(7) First Application. Contingency. 
As mentioned in (1), if we regard the sth division as the (u, v) cell of a contin- 
gency-table and if we take 
then 
and l(n^^ - "-—-"j 1 
cf)^ = S j j = Mean square contingency. 
Accordingly, with the notation 
and 
I \ M f 
= '5 Ua J = 
the equation (xiv) gives the standard deviation of the mean square contingency, 
when M is very large as compared with N. 
The terms enclosed in the first bracket of (xiv) are exactly those of Pearson's 
1914 paper in Biometrilia, so that the second and third brackets contain the terms 
arising from the squares and higher products of Sn,. 
Of the total correction due to the higher approximation it is of interest to find 
how much is due to the change of mean and consequently the change of origin of 
8^2, when the square of Sn, is not neglected. The true mean is given by 
1 ^ /«c^\ 1 ^ f«o /, "c 
and using the observed values of n^ as the best approximation available for 
so that the difference between the true mean and the approximate mean obtained 
by neglecting squares of hn^ is 
In accordance, then, with the formula for change of second moment with change 
of origin we get the effect of the change of mean on a'^^i by subtracting 
from the approximate value. 
