62 
On the Poisson Laio of Small Numbers 
In other words the binomials give a reasonable total fit, the exponentials a 
practically impossible one. 
But there is another question to be asked in such series as those of Mortax'a : 
What justification is there in cutting off at 10 cases, say of murder? A province 
may have a million inhabitants and, perhaps, 40 murders occur in a year*. Hence 
the binomial is for ten year returns 
/24,999 1 
10 X ^--V^, + 
.2.5,000 2o,000y 
but this is as close as anything can be desired to the exponential series. It may 
be reasonable to apply a separate series to districts giving 4"2 and 36"6 murders 
per annum I'espectively, but it is difficult to see why the latter district should be 
altogether excluded from treatment. If the theory of the binomial be applicable 
at all, then it applies practically as well to districts with 40 murders as to districts 
with 4; for, we need no indefinitely small q to get a closely exponential series. 
If we take the case of deaths by murder, Mortara has retained only 6 out of 16 
provinces, yet his criterion Q' (see his Table, p. 51) is not more divergent from 
unity for the rejected provinces than for those retained ; the binomials are indeed 
Reggio Treviso 10 ( '7000 + -SOOO/'^'^^^ 
Venezia 10 ( -5619 + '4381 )»''«'» 
Vicenza 10 ( '9571 + •0429y»'-'" 
Padova 10 ( -4774 + •o226)"'«« 
Pavia 10 (1-8162- -81 62)-'' '"'«^ 
Bergamo 10 ( •8857 + ■1143)^^'''«''« 
only one of which gives q small and positive and n large. 
The mean Q' for the retained provinces is "967 with a range from "7 to 1'4 and 
for the rejected 1'03 with a range from '8 to 1*4. Even if — which is not the case 
— the probability of an individual being murdered were too great for the ex- 
ponential, it ought to follow the binomial, but this, as a rule, it does not do, unless 
we give some wholly new interpretations to q and n ; the actual values render the 
theory of the binomial as stated inapplicable. 
(10) Mortara s Criterion. 
As a matter of fact the only test of whether an exponential will legitimately 
fit a given series or not is to determine the binomial {p + g)" and ascertain 
whether p is slightly less than unity. But: 
p = vpq/nq 
_ (Standard Deviation)^ 
~ Mean 
* We assume that each individual is equally likely to be murdered. But if there be a graduated 
probability for murder throughout the community, what right have we to apply Poisson's series at all ? 
The essential basis of the application — equal chance of each individual — is wanting. 
