SUPPLEMENTS—GOODNESS OF FIT. 373 
the reduction of which, its integration, can be effected in terms 
of x by methods described in text-books of the integral calculus. 
Everything turns, therefore, upon the computation of the function y. 
As we have seen, x? is determined by evaluating the standard 
deviations of the n variables and their correlations two at a time 
(the higher partials being deducible if the correlations of zero 
order are known). 
By an application of the method of p. 257, we have 
= {1 R2\"p 
ts V y1-3)% 
for the standard error of sampling in the content of the pth class ; 
while by a similar adaptation of the reasoning on p. 342 we reach 
J npg 
Eps No yo, 
for the correlation of errors of sampling in the pt* and ¢'® classes. 
With these data, x? can be deduced (the actual process of reduc- 
tion is somewhat lengthy, but the student should have no difficulty 
in following the steps given in pp. 370-2 of ref. 74, infra). Its 
value is 
=F nl 
n=0 Nin 
the summation extending to all n+ 1 classes of the frequency 
distribution. 
Values of the probability that an equally likely or less likely 
system of deviations will occur, usually denoted by the letter 
P, have been computed for a considerable range of x? and of 
n' =n + 1 =the number of classes, and are published in the Tables 
for Statisticians and Biometricians mentioned on p. 358. 
The arithmetical process is illustrated upon the two examples 
of dice-throwing given on p. 258. 
There are three points which the student should note as regards 
the practical application of the method. In the first place, the 
proof given assumes that deviations from the expected frequencies 
follow the normal law. This is a reasonable assumption only if 
no theoretical frequency is very small, for if it is very small the 
distribution of deviations will be skew and not normal. It is 
desirable, therefore, to group together the small frequencies in 
the “tail” of the frequency distribution, as is done in the second 
illustration below, so as to make the expected frequency a few 
units at least. In the case of the first illustration it might have 
been better to group the frequency of 0 successes with that of