174 On Criteria for the Existence of Differential Deathrates 
The following table gives the values of p^, for the various age groups in 
the case of (a) Liverpool and Birmingham for General Deathrates, (6) Edinburgh 
and Dundee, Cancer Deathrates, (c) Rural and Urban Districts, England and 
Wales, Cancer Deathrates. 
Illuslrations of Approximate Deathrates for unknown sampled Populations. 
Age 
Group 
(a) Liverpool and 
Birmingham, General 
Deathrates 
(b) Edinburgh and 
Dundee, Cancer 
Deathrates 
(c) Rural and Urban 
Districts, England and 
Wales, Cancer Deathrates 
Ps 
Ps 
Ps 
Ps 
Ps 
0—5 
5—15 
15—25 
25—35 
35—45 
45—55 
55—65 
65—75 
75—85 
85 and over 
•065,272 
•003,550 
•004,236 
•006,905 
•011,845 
•021,155 
•042,690 
•079,718 
•158,531 
•329,670 
•065,332 
•003,611 
•004,299 
■007,039 
•011,937 
•021,335 
■042,905 
•080,269 
•158,498 
•328,756 
•000,046 
•000,022 
•000,060 
•000,182 
•000,700 
•002,347 
•004,725 
•007,511 
•008,235 
•006,962 
•000,045 
•000,021 
•000,058 
•000,187 
•000,698 
•002,360 
•004,726 
•007,532 
•008,252 
•006,405 
1 -000,025 
•000,033 
-000,116 
■000,404 
•001,500 
•004,256 
•007,787 
•009,653 
•008,856 
•000,025 
•000,033 
•000,116 
•000,404 
•001,500 
•004,256 
•007,788 
•009,646 
•008,847 
It will be seen that the corrective factor contains the inverses of the total 
number {ds + d'^) of deaths and the total numbers of individuals {a^ + a'^) in 
the combined age groups s and s'. Hence for big districts as in (c) the 
corrective factor is of small importance even for special diseases. For the 
general deathrates in two large towns as in («) the difference between p^ and 
ps is as a rule less than 1 %. Even in special diseases in towns of moderate 
size, it is only where the total number of deaths in the combined age groups 
s and s' is very small that any substantial divergence between p^ and 
arises, e.g. in (b) for the child or extreme old age groups. Thus the value of 
is hardly likely to be modified practically, if we replace p^ by pg. Accordingly 
— ^tiL* ^ besides being easy to calculate, is a reasonable approximation to the 
better value p^. Of course p^ itself is only an approximation* to the " best value" 
and this "best value" also depends on the accuracy of replacing the binomial 
by a normal curve. Thus it is by no means certain that, if we obtained a true 
best value for the deathrate in the unknown sampled population, it would be 
markedly nearer to p^ than to p^. We content ourselves by remarking that 
neither for practical nor theoretical reasons does there seem likelihood of a great 
gain resulting from taking any other value than {d^ + f^'s)/(«s + a'g) for jh- 
* We should have to solve a cubic for each age group to find the accurate "best value," on the 
above hypothesis of normality. Doubts may also be raised as to the legitimacy of the theory which 
makes 11 in (xii) a maximum. They are discussed in another paper. 
