166 Oil Criteria for the Existence of Differential Deathrates 
this second criterion to make use explicitly as well as implicitly of the number 
of age classes. We can thus give a physical meaning to Q, is the ratio of 
the standard deviation of the 2w age classes' deathrates — each supposed to be 
drawn from the same population and measured in terms of its own standard 
deviation — to the standard deviation of this standard deviation, i.e. to 
- ^ It by no means follows that this second criterion will give the same 
V 2 X 2m 
result as may be drawn from the first. But this new aspect of Q frees us from 
many of the difficulties essentially associated with "corrected deathrates" and 
the indefinite category of a standard population. We must observe, however, 
that to evaluate the probability of the occurrence of Q we have again to justify 
the assumption that will follow a normal distribution. We could not justify 
this for the distribution of deaths in any single age group, nor even for the 
distribution of factors like [d^ — 2Js('-s)^/((>'sPs(ls) ^-iid {d's — p^a' s)^l{a' sp^qg) summed 
for our two populations on one occasion, but we can do this for the distribution 
of on the basis of a number of random samples*. 
(4) We can again approach the problem by considering quantities like 
Va,p,q, Va'sp.qs 
The mean of these deviations measured each in terms of its own standard 
deviation should be zero ; and the standard deviation of this mean should be 
~= , since there are 2u variates and, each being measured in terms of its own 
standard deviation, the standard deviation of the series is as before unity. Thus if 
hi 
V a,p,q, J V V a',p,q, ' J 
m I ^,^/^) = V'2w X 777 may be looked up in the tables of the probability integral, 
and the probability of the system, as a result of random sampling from a population 
p^, qs, thus again determined. If we use the values of p^ and q^ so often adopted 
above, the quantity with which to enter the probability table, i.e. the ratio of 
deviation to standard deviation, is expressed by 
V2u X m = -j=^- ,Si« - • ' , V (x). 
V2u /{d, + d\) {I, + r,) 
} 
where 1^ = ff^ — d^ and l\ = a', — d\ are the survivors in the sth age groups. 
* We can show in a manner similar to that on p. 161 that the distribution of approaches the 
normal when each of the constituent x'^'s is drawn from different populations, none of these populations 
being in themselves accurately normal. 
