Miscellanea 
191 
IX. On the Probable Error of Mean-Square Contingency. 
By JOHN BLAKEMAN and KARL PEARSON. 
Let there be any contingency table and be the total fi'equency in the ^;th row, in the 
qi\\ column, n,,g the frequency of the constituent common to the pih row and ^th column, 
iV the total frequency dealt with in the table. Then it is known* that the total mean square 
contingency of the table is given by : 
'N 
"pi ]\[ 
the sum being for all values of p and q. 
Let ^'pg be the contribution to c^^ of the p, q constituent frequency, i.e. 
1 V / 
.(ii) 
Let be the contribution to from a single row, and <^f/ from a single column, i.e. 
pq 
Hp Uq 
■n 
n 
(«,„) ^ lip 
N 
+ ji\Sq{nq) 
smce 
Thus : 
and similarly : 
It follows that : 
Let us write 
Then : 
N 
Sq{npq) = '>h and Sg{tig) = ]V. 
<l)p^ = S,, 
"11 '"q. 
n„n, 
n~ 
Tip 
npUg 
-1. 
Upn — ■ 
■ pq 
.(iii) 
.(iv) 
..(V) 
l+4>-=Sp,j(2ipg) (vi) 
We shall now proceed to find the probable error of ?^,„^. We state the f(jllowing preliminary 
propositions t, where cr denotes the standard deviation of random sampling : 
1 _ i?'' 
= n„ 1 
"4 " « l^'^ n) ' 
..(vii) 
.(viii) 
...(ix) 
S) "^"4 ^^'■pnq — ''h> 
'■p "-q 
N '■ 
.(X) 
* Drapers' Research Memoirs, Biometric Series, i. Ou the Theory of Contingency, etc., p. 6. 
Dulau & Co. 
t Drapers' Research Memoirs, Biometric Series, ii. Ou the Theory of Skew Correlation, etc., 
pp. 11—17. Dulau & Co. 
