40 
On the Foisson Law of Small Numbers 
Secondly (ii) can only be justified as an assumption by actually ascertaining 
the form of the binomial from the data and testing whether n is large and q small 
and positive. It appears absurd to base our formula on an approximation to 
a binomial of a particular kind when, on testing in the actual problem, such a 
binomial does not describe the results. As a merely empirical formula, the 
Poisson-Exponential of course can be tested by the usual processes for measuring 
goodness of fit, but no such test nor any discussion of the probable errors of their 
results have been provided by Bortkewitsch himself nor by Mortara, who has 
followed recently his lines in a work to be considered later. As a matter of fact in 
the cases dealt with by Bortkewitsch, by Mortara and by " Student," n will be found 
almost as frequently small and negative as large and positive, and q takes a great 
variety of values large and negative and large and positive, as well as small 
and positive. Thus the initial assumptions made from which the "law of small 
numbers" is deduced are by no means justified on the matei'ial to which it has so 
far been applied. 
(5) ApjMcation of the Lavj of Small Numbers to determine the Probable Errors 
of Small Frequencies. Given a distribution of frequency for a population N let Ti^t 
be the frequency in the cell of the sthrow and tth column of a contingency table 
(or if we drop t, Vt, would stand for the frequency of any class). Then if we take 
a random sample of N individuals from this population, the chance that an indivi- 
dual is taken out of the n^t cell is fl^;/-^, and that it is not is 1— Therefore if 
the original population be so large that the withdrawal of an individual does not 
affect the next draw, the frequency of individuals in M random samples of N will 
be given by the terras of the binomial : 
Now, if nst/N be very small, and N large this will approximate to the 
Poisson series : 
where ^ ^- ^^'^ /'sj/iV' will approximately be the mean proportion of the 
whole in the st ceil of the sample itself = 'Hje/iV, or ?h = n^f. Thus if in any cell of 
a contingency table, or in any sub-class of a frequency whatsoever, we have a 
frequency n^t small as compared to the population N, then in sampling, this small 
frequency will have a distribution approximating to the Poisson Law, and tending 
as Ust becomes larger to approach the Gaussian distribution*. It would appear, 
* Such approach is usually assumed when we speak of 
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as the probable error of the frequency ii^f. But such a "pmbable error" liaa really no meaning if 
be very small and the exponential law be applied. 
