68 
On the Poisson Law of Small JVumbers 
Table XVI shows that the announcements of deaths over 70 years of age only 
amount to o'7-i per day for males and 3"52 for females. These are certainly " small 
numbers," but " small " with regard to what ? Are we to consider n as the number 
of the population which embraces, (i) all the individuals of the limited classes of 
the same range of ages as the defunct, (ii) all the individuals announced as dead 
oa the same day, (iii) all the individuals of whatever ages of the class which 
announces deaths in the Times 1 Or, should we refer to all the individuals in the 
community of that range of ages, or the whole community at large, i.e. the chance 
that in a population of so many millions an individual over 70 or 80 as the case 
may be will die and have their death announced in the Times newspaper? Well, 
it really does not matter, because if for any one or all of these populations the 
binomial {p + 5)" applied, we should get if q were small and n large, the Poisson 
series 
m- m^ \ 
e-(l+«.+ 2-, + 3-, + ...). 
and this quite regardless of the size of n. If therefore we did find a series in 
which q was very small and n large, we might not be able to say to which, if any 
of the above populations 71 applied. On the other hand the mere fact that m is 
small is no justification for the use of the " law of small numbers " as is sometimes 
implied. If it be argued that the small number of people who die over 80 and 
have their names recorded in the Times are drawn from a small population, we 
reply so it may be argued are the school children who commit suicide, the uncles 
who feel any inclination to marry their nieces, or the men liable to die of chronic 
alcoholism ; and we can in the case of the announcement of deaths test the values 
of q and 11 on fairly adequate numbers. As a matter of fact we do not know, in 
attempting to apply the Poisson formula, what is the population from which we 
are drawing our individuals, and the justification of the Poisson formula lies only 
in showing that there actually does exist a binomial for which q is small a.nd 
n large. We might imagine that as we got to the higher ages practically every 
person of that age would die, or that in our notation q would be 1 nearly and p be 
a very small quantity ; thus an approach might be made to the Poisson-Exponential. 
But the approach to the Poisson-Exponential arises not through q approaching 
unity but from q becoming very small. Nor again in the lower age groups do we 
find ourselves left with a positive binomial. 
In all cases except women over 90 years of age, we find that a negative 
binomial best fits the observations. Even in the case of the announcements of 
deaths of women over 90 years, we find that the approach of the binomial to the 
Poisson exponential depends on 
/ ]^ \. 53 -3333 
being measured with sufficient approximation by e = 2'71828. But 
(l-01875)'^-^^^ = 2'69323, 
