184 On Criteria for the Existence of Differential Deathrates 
is the difference between the corrected deathrates, corrected to a standard popula- 
tion which gives the maximum difference between those rates. But this standard 
population is, if the individual age group deathrates are not all greater in one 
district than the other, an algebraical fiction and not a real standard population. 
But with the single possible exception of London and County Boroughs there is 
no approach in the case of diabetes mortality to greater deathrates in all the 
age groups of one district class. This was far more nearly the case in the cancer 
mortality. Hence the second test was more reasonable in that case. 
We have retained this maximum difference of corrected deathrates to the end 
of our illustrations, because we think it serves to indicate the danger of any 
argument as to significant differences in mortality based on "corrected deathrates." 
In the case of diabetes, our district classes give no such significant differences. But 
our ;yo^ test shows that these differences actually exist, as indeed might be suspected, 
although their numerical valency could not be adequately tested by a mere 
examination of the age group deathrate table on p. 182. The reason for this 
failure of the corrected deathrate dift'erence lies largely in the fact already insisted 
on that the significance of the difference in the corrected deathrates depends largely 
upon the standard population selected — a point often overlooked. What then is 
to be the standard population selected? Clearly it should be such as (i) to make 
the corrected deathrate difference a maximum and (ii) at the same time be a real 
population, i.e. not one with negative age classes. At present we do not see how 
to reach the maximum of a certain function of variates, subject to the condition 
that the variates are to take positive values only. The corrected deathrates for 
diabetes in certain districts show no significant differentiation when reduced to the 
general population of England and Wales as standard. They show a marked 
differentiation when reduced to the standard populations of maximum difference. 
It is true that these populations are merely algebraic fictions, but how far should 
we approach this marked differentiation, if we could discover the real maximum 
difference population ? We cannot say ; and in view of this uncertainty, it seems 
to us needful to drop for the present any criterion of mortality differentiation 
depending on the so-called corrected deathrates. 
We can only conclude that the proper test for differentiated mortality is the 
test used for the first time in the present paper. For this test does not depend 
on the measure of the divergence between two means — "corrected" it may be, — 
but on the general diff'erence between two frequency distributions as wholes 
and this appears to us the essential feature of any true measure of differential 
mortality. 
We have to thank our colleagues Mr A. W. Young, Mr I. Horwitz and 
Mr Greorge Rae for much assistance in a piece of arithmetical work more arduous 
than may appear on the face of this paper. 
