ON CRITERIA FOR THE EXISTENCE OF 
DIFFERENTIAL DEATHRATES. 
By KARL PEARSON, F.R.S. and J. F. TOCHER, D.Sc. 
(1) To determine whether the general deathrates or the deathrates for special 
diseases in two towns or in two classes of the general population are significantly 
difEerent is a problem of very great importance. It is now generally recognised 
that crude deathrates are of- little service for this purpose, the age distribution 
in the two districts or in the two classes of the population may be widely different, 
and the general deathrate or the special deathrate, i.e. the deathrate for a special 
disease, usually is a marked function of age*. The deathrate is therefore corrected 
by reduction to a "standard population." We ask what would be the deathrate 
in the given district or class if its age distribution were constituted in a given 
or "standard" manner. This deathrate is spoken of as the "corrected deathrate." 
Now we usually suppose the deathrate to be subject to "probable error." In 
other words in a population of size n, if be the chance of a person dying in the 
year, q the chance of a person surviving, then the standard deviation of the number 
of deaths is Vnpq, or the probable error of the corresponding deathrate rn is 
•67449 / 
^ V pq. If there be a second town of deathrate m' and population n', and 
V n 
chances p' and q', we are tempted to compare without very full consideration 
m' — m with -67449 a / — + — to obtain a measure of significant differentiation 
\ n n ° 
probably using a table of the probability integral. This method is for several 
reasons unreliable and fallacious. In the first place if it be applied to the crude 
deathrates, p and p can hardly be applied to the individual, they are so markedly 
a function of age. In the next place for many diseases, or for many ages in 
a general deathrate, p and p' will be very small; accordingly there will be no 
approach to Gaussian distribution, but the binomials will approach Poisson's 
Exponential Limit, in which case the meaning of the standard deviation of a 
difference requires much further consideration and probabilities will not be given 
by a table of the probability integral. 
* It is also well known to be a marked function of class. This is customarily disregarded in a 
comparison of local deathrates. But to test real sanitary efficiency a standardised class population 
may be as important as a standardised age population. 
