ON THE PKOBABLE ERROK OF A COEFFICIENT 
OF CONTINGENCY WITHOUT APPROXIMATION. 
By ANDREW W. YOUNG, M.A. and KARL PEARSON, F.R.S. 
(1) Introductory. 
There have been two memoirs dealing with the probable error of a coefficient 
of contingency, namely that by Blakeman and Pearson in 1906* and that by 
Pearson in 1914 1. In the former paper the authors started from the expression 
for the mean square contingency 
and varied Ws,,', w^. ^"^^ '^.s' but neglected the squares and products of these varia- 
tions. The result was lengthy, and the arithmetical work laborious. In 1914 
Pearson gave reasons for considering rig. and n.^- as constant during the sampling 
and got a much simpler value for a^... The result in actual numerical cases did 
not differ widely from the much more elaborate formula of the earlier memoir. 
Recent work in other directions has, however, shown that caution must be used 
in neglecting the square and product terms of the variations due to random sampling, 
and the object of the present paper is to consider the variation of cf)'^ on the hypothesis 
of Pearson's 1914 note but without approximation. 
Let a population of size M be grouped into c divisions — for example, the cells 
of a contingency table — and let the contents of the sth division be . Let a 
sample of size N be taken at random from the population and let Wj,. be the contents 
of the sth division according to the same grouping. 
We shall here consider the variation of the quantity (f)~ defined by 
'^^'-'(M) 
where is a number connected with the sth division and is for the present restricted 
only by the condition 
S{X,) = N (ii), 
— a condition which enables us to write 
as equivalent to (i). 
* Biometrika, Vol. v. p. 191. f Biometrika, Vol. x. p. 570. 
