Karl Pearson and J. F. Tocher 
167 
(5) The previous methods are all more or less inadequate ; they test whether 
a certain single character of the distribution does or does not present significant 
difference in the two populations compared. If we want to test the distributions 
as a whole, we must adopt a modification of the method given in Biometrika, 
Vol. VIII. p. 250 for determining the probability that two systems of frequency are 
random samples of the same population. It has been shown in a memoir dealing 
with partial contingency* that the proper test is to determine 
entering the Tables of Goodness of Fit with = s + 1. Here and j)' are the 
numbers in the sampled populations. Now the factor 
is for most practical purposes unity, as we shall illustrate in the sequel. Hence for 
such purposes 
of our earlier investigations. Thus to obtain the probability of the two districts 
being random samples of the same population all we have to do is to look up Q'^ 
for n' = u + 1, u being the number of age groups, in the Tables of Goodness of 
Fit. It will be seen that Q has a wide meaning, and generally speaking we may 
treat it as a constant, which can be used in a variety of criteria for testing the 
existence of differential deathrates. We propose to illustrate this in the following 
sections, applying in each case the four tests discussed above, namely : 
(a) The probable error of the "corrected" deathrates' difference reduced to 
the standard population of maximum difference. 
(b) The significance of the means of the 2u deathrate classes. 
(c) The significance of the squared standard deviations of the 2w deathrate 
classes. 
(d) The general test of partial contingency with its approximate value Q^. 
The following illustrations have been used : 
(i) General Deathrates of Liverpool and Birmingham. 
(ii) Cancer Deathrates of Edinburgh and Dundee. 
(iii) — (viii) Cancer Deathrates for all England and Wales divided into four 
groups : (a) London, (b) County Boroughs other than London, (c) Urban Districts 
other than (a) and (&), {d) Rural Districts. 
(ix) — (xiv) Diabetes Deathrates precisely as in the preceding Cancer case. 
(xi) 
Biomelrika, Vol. xi. p. 157. 
