60 
On the Poisson LaiD of Small Numbers 
Examining- these we see that there are only two in which q and n are positive 
and onl}^ one in q is small and positive and ii moderately large. The probable 
error of q for 10 observations on the assumption that n is very large and q very 
small is + 'SOIG and is quite inconsistent with the last four districts being samples 
from exponentially distributed frequencies. The other four districts may or may 
not belong to such frequencies — the data are wholly inadequate to determine 
whether they do or not. Reggie Calabria and Foggia have the lowest Q's, 
i.e. 0"9 and I'O. But that six districts out of an already selected eight give 
negative q and a seventh a relative large q and small n suggests the inapplicability 
of the hypothesis ad(jpted. If we seek for " goodness of fit " of the totals, we find : 
Binomial Exponential 
;^2 = 25-12 47-92 
P = -0336 -0000 
Thus the odds against the binomial system are 28 to 1, but the odds against 
the exponential are enormou.s. It dues not seem possible to justify the treatment 
of such data by the use of the Poisson-Exponential. 
Let us turn to a second nf Mortaia's illustrations, that of deaths from small- 
pox. He rejects first six out of the 17 districts, the remaining ten are given in 
Table XIII. The districts give the following binomials : 
Venezia 
10 ( -9500 + 
•0500)'« 
Bologna 
10 ( -9889 + 
•0111)8' 
Treviso 
10 ( 2-2000 - 
l-2000)-'8^-« 
Pavia 
10 ( 1-8000- 
•8000)-' •^«°'' 
Gagliari 
10 ( 4-5190- 
3-5190)--^^''« 
Padova 
10 ( 3-6833 - 
2^6833)-8s« 
Verona 
10 ( 5-6000- 
4^6000)--^-" 
Brescia 
10 ( 9-9727 - 
8-97 27)- •3'^™ 
Bergamo 
10 ( 2-3821 - 
l-3821)-^-8-^" 
Catanzai'o 
10(15-6128 - 
14-6128)---™ 
Vicenza 
10 ( 3-4854- 
2-4854)-i''^«' 
Out of the eleven cases only two give q small and positive; not a single one 
gives for q anything like the chance of a death from small-pox in the district, nor 
for n anything like the population of the district. There is an increasing divergence 
from the positive binomial as Mortara's Q' increases in value. We see that in nine 
cases, however, a negative binomial not the exponential is required to describe the 
frequencies. The probable error of q, for insignificant q is as before + "3016, and 
therefore it is improbable that q is zero in at least 9 out of these 11 districts. 
Examining the totals we find 
Binomial Exponential 
X^^Q-Q^ ' 570-79 
P= -67 -000,000 
