Karl Pearson and J. F. Tocher 
165 
deathrate at all. Two such deathrates might show no significant difference 
although the aged were dying in one population and the young in the other in 
excess*. We might work only with the groups for which cV — dgja^ is of 
the same sign, and this would indicate, by the same test, differentiation or its 
absence in the manner of dying ; but of course we should then be dropping the 
idea of a "corrected deathrate"; that idea is, however, essentially imperfect and 
does not really distinguish effectually between differences in the manner of dying. 
(3) We now ask whether it is not feasible to interpret the value of Q in 
Eqn (ix) as a measure of the probability of a differential mortality without regard 
to the theory of "corrected deathrates." 
As before let be the number of deaths in the age group of size cig in one 
district or class. We suppose this to be a sample of a population of which 
is the chance of dying in this age group, then if there be u age groups in each 
district or class, we have 2u deviations, all of which are independent, and given by 
~ Ps^^s and d' g — fga' g. These deviations have respectively standard deviations 
'Va^fsls s-iid ^('''sfsls- Accordingly if every deviation be measured in terms of 
its standard deviation we shall have a second moment coefficient given by 
u 
d', - p,a' 
and 2^ ought to be unity. 
Now in the above expression p^, are the unknown values of the death and 
survival chances in the population from which the two districts or classes are 
supposed to be sampled, and the best values we probably can take for them are 
Ps = I — qs = {dg + d's)j{as + a'J. But inserting these values in the above 
expression, we find with the value of Q given in Eqn (ix) : 
Z'^IqK or S^Q.^l. 
But the standard deviation of a second moment coefficient is V/x^ — fx.^^jVn 
and this in the case of a normal distribution (^li^ = Sjj,^^) equals /\^^ H-z> in our 
case, since = 1 ^nd n = 2u, the standard deviation of 2- equals IjV'U. Thus 
we have to measure — 1 in terms of 1/Vu, or to ascertain the probability of 
a ratio of deviation to standard deviation of magnitude greater than {hQ^ — u)jV u. 
We see again therefore Q arising as a constant which naturally determines the 
mortaUty resemblance or difference of the two districts. But while in the 
previous approach to the solution of the problem from the corrected deathrates 
we found Q alone sufficed to obtain our criterion, we require in the case of 
* Another factor, sometimes overlooked when deathrate after correction is taken as a measure of 
local health, is emigration. If the ath age class tend to migrate from A to B, it is usually the healthy 
who migrate; thus the deathrate of the «th class in A will be inflated and that in B reduced. If the 
migration be chiefly that of males, as to mining districts and colonies, a spurious correlation between 
deathrate and sex ratio may be created. 
