160 Oil Criteria for the Existence of Differential Deatlirates 
But the method outlined above has this importance: it suggests that the 
deathrate obtained is only a "sample" deathrate, and subject to the variations of 
sampling ; thus it forces the problem upon us in a very definite form : Can two 
populations dying in a known manner during a given period be considered as 
samples drawn at random from the same material ? Here again the age distribution 
difficulty arises, for by hypothesis we admit the age distributions of our two samples 
are not the same. Let us fix our attention for a time on a fairly narrow age group, 
say that d,. deaths occur in an age group of size in a certain population and that 
the chance of death in the age group is and the chance of survival in the 
population out of which the sample is supposed to be drawn. Then undoubtedly 
the standard deviation of samples would be V a^fsls) but as before the distribution 
would hardly be Gaussian. Now let us suppose the standard population, size A, 
to consist of the age groups A-^, A^, ... As; then the "corrected" deathrate 
M will be given by 
fd g. A s 
«- 
and if 8 denote a variation due to random sampling, we shall have 
^8 . d„ A, 
h.M = S 
( i>-ds M 
\ as A J' 
Now speaking generally we do not "draw" our deaths in such a manner that 
their total remains constant. A shot so to speak fired at one age group is not 
to be supposed if it misses that group to have a chance of hitting a second age 
group. There will thus not be any of the usual negative correlation between a 
variation in d^ and one in d^'. Of course epidemics which in a given period attacked 
individuals in certain age groups only might show a positive correlation between 
Sds and Sd^'. But in general it will be sufficient to suppose the variations of the 
(Z<,'s independent, and measured by the probability of death in each group. Thus 
we should have 
2 Q { ^s^\ 
Vsqs A, 
= ^[T-T') <"'• 
Similarly if there be another population with age groups a'^ and corrected deathrate 
M', then 
We do not write p'g and q'^ in this population, because we are supposing both 
to be samples from one and the same population. Now clearly 
