Student 
215 
environments which will vary the chances of an individual falling into them, and 
so we may expect that as a rule negative binomials will occur in place of the 
exponential. 
Finally, suppose that the presence of an individual in a division influences 
the chance of other individuals falling in that division. 
Clearly it may do so either by way of increasing the chance or diminishing it. 
If the chance be increased it is clear that we shall get for the same mean 
number of individuals per division a larger number of divisions containing high 
numbers of individuals and a larger number of zero divisions. In other words, for 
the same mean we shall get a larger Standard Deviation, so that fi^jvi will be 
greater than 1 and a negative binomial will fit better than the exponential. On 
the other hand, if the chance of other individuals is decreased by the presence 
of one already in a division yu-s/i'i will become less than unity and the best fitting 
binomial will be positive. The first of these two cases includes linking or clumping 
of events or bacteria, the second such a thing as the counting of large cells on a 
haemacy to meter whose divisions are comparable in size with them. 
We have now shown that a population which might be expected at first sight 
to follow Poisson's law 
(1) Will do so if the only deviation from the ideal conditions is that the 
chances of different individuals falling into the same division are not equal, as 
long as these chances are all small. 
(2) If in addition to this the chances of some individuals are large a positive 
binomial will fit the results better than the exponential. 
(3) If the different divisions have different chances of containing individuals, 
as is usual, a negative binomial will fit the results better than the exponential, 
except in so far as (2) may interfere. 
(4) If the presence of one individual in a division increases the chance of 
other individuals falling into that division, a negative binomial will fit best, but if 
it decreases the chance a positive binomial. 
Generally speaking (3) is the operating deviation from Poisson's conditions and 
accordingly most statistics give negative binomials. 
Finally I should like to jwint out that the object of my original paper {Biometrihi , 
Vol. v) was to give the user of the haemacy tometer a guide to the error which 
he may expect from its use, and that the net result was that the probable error of 
his count was '6745 ViV" where N was the total number counted* and that if N be 
a reasonably large number tables of the probability integral may be used, otherwise 
the exponential (or better still go on counting). This result is not affected by 
slight deviations from the Poisson law, any more than slight deviations from the 
normal law affect our use of the probability integral tables. 
* Biometrika, Vol. v, p. 355. The probable error of mean is -6745 slmjl\[ where m is the mean and 
M the number of unit areas counted. If in this we put il/=l, then m = N and the total count is 
JV±-6745Vw' as above. 
