70 
Oti the Poisson Laiv of Small Numbers 
death by accident is graduated with occupation. At any rate until those who 
support the use of the "law of small numbers" demonstrate its application on 
material where the probable errors are sufficiently small for us to measure the true 
value of q and n, no advance can be made. Nor until we have clear ideas of the 
population n in which the chance is q, is it possible to assert that it may be used 
for the suicides of school children, and the marriage of uncle and niece, and must 
not be used for the deaths of aged people, which certainly occur in "smaller" 
numbers. 
In the illustrations of deaths we have taken, certainly the Poisson-Exponential 
is not the rule, although the distributions appear to approach it, as towards a limit, 
when the number of deaths approach zero. But our data which show the rule of 
the negative binomial appear to show it in no more marked manner than much of 
the data selected by Mortara himself indicate the negative binomial, although owing 
to the sparsity of his material his results are far more erratic and unreliable. Nor 
is Bortkewitsch much behind Mortara in the evidence he produces for a negative 
binomial being as reasonable a description — possibly owing to inherent lumpiness — 
as a positive binomial of these "small number" frequencies. 
(15) Conclusions. 
{a) The Poisson-Exponential gives a fairly reasonable method of dealing with 
the probable deviations of small sub-frequencies in the case of random sampling. 
When the average value of a sub-frequency is not more than 3 of a population, 
then Poisson's formula suffices in most practical cases to determine the range of 
error likely to be made. Tables are given to assist its use. 
{h) The application of the Poisson-Exponential to various data by Bortkewitsch 
and Mortara has hardly been justified by those writers, for they have not tested 
whether the probability q is small and positive and the power n large and positive 
in the cases considered by them. When this is actually done, it is found that 
their hypotheses, having regard to the probable errors of q and n, are largely 
unjustified in the case of their illustrations. Even in such cases where it is 
justified, a binomial gives a better result as measured by the test for goodness 
of fit. 
(c) Negative binomials repeatedly occur and give just as good fits, where 
they occur, as positive binomials. In the illustrations taken by Mortai'a, the 
frequency 10 used is so small that it is not possible to assert that either positive 
or negative binomials are demanded by the data. Still the average p of his results 
is very significantly in excess of unity. 
{d) Mortara like Bortkewitsch cuts out of his data straight off all districts 
with, on the average, more than 10 cases in the year. But the q obtained from 
20, 40, or even 100 cases in a population of 100,000 is a small q in the sense that 
the resulting binomial is adequately expressed by a Poisson-Exponential. There 
