HN THEORY OF STATISTICS. 
say, when we fulfil the conditions of simple sampling, is uniform 
over the whole range from 0 to 1. Thus for a rough grouping 
into four classes the above series of trials gave :— 
Number of Tables giving a Value of P 
lying between the Limits on the Left. 
Expected. Observed. 
1:00-075 25 23 
075-050 25 30 
0°50-0°25 25 22 
025-0 25 : 25 
The value of x? for this comparison is 1:52, giving P=068, or 
we should expect a worse fit roughly twice in every three trials. 
COMPARISON FREQUENCIES BASED ON THE 
OBSERVATIONS. 
Contingency Tables.— Attention was specially directed above 
to the fact that the theoretical frequencies were assumed to be 
given a priori. The theory of the more general case, in which 
comparison is made with frequencies determined by the aid of the 
observations themselves, has only recently been fully worked out 
(Fisher, ref. 76). The most important practical case of the 
kind is that of association or contingency tables in which the 
observed frequenciescare compared with the independence-values 
obtained from the totals of rows and columns—that is, the values 
A4,.)(B 
(AmBp)y = AmB) 
of Chapter V. § 6, p. 64, and in which the differences 
Smn= (Am Br) i (Am Bn) 
are used as an indication of the divergence from independence. 
The rule to which the theory leads is a very simple one: the x? 
method is still applicable, but the tables must be entered with 2’ 
equal to the number of algebraically independent frequencies (or 
values of 8) increased by unity, and not with »’ equal to the 
number of compartments in the table. Now, if in any column 
of the contingency table we are given all the values of 6 but one— 
say, the marginal value at the bottom,—the remaining one can be 
determined. because the sum of the &'s for every column must be 
“78 
p