48 On the Poisson Law of Small Nvmbers 
PART II. CRITICISMS OF PREVIOUS APPLICATIONS OF 
POISSON'S LAW OF SMALL NUMBERS. 
(7) We now turn to the illustrations which various authors have given of 
the Law of Small Numbers. 
"Student's " Cases. We take first the series given by " Student " in his memoir 
on counting with a Haemacytometer*. They are of special importance beca\,ise 
the series at fir.st appear of fairly adequate size, namely consisting of 400 
individuals, and further we should anticipate that the Law of Small Numbers 
would hold in his cases. He obtains better fits with the binomial than with the 
exponential but, as he remarks, he has one more constant at his disposal. On the 
other hand, if the exponential be a true approximation, the binomial ought to come 
out with a large n and a small but positive q. " Student " finds for his four 
series : 
L 400 X (!• 1893 - ■1898)-^ ''"^ 
II. 400 X (-97051 + -02949 
III. 400 X (1-0889 - •0889)--"'=^". 
IV. 400 X (-9525 + miof''''"'^ 
II. and IV. may, perhaps, be held fairly to satisfy the conditions, although it 
is not certain if 46 is to be considered a large n or '05 a very small q. 
I. and III. fail to satisfy the conditions at all, unless the probable errors of q 
and n are such that q might really be a small positive quantity and n really large 
and positive. The following are the values for the four series of n and q and their 
probable errors : 
I. (/ = - -1898 ± -0647, 11 = - 8-6054 + 1-2209. 
II. q = + -0295 + -0457, n = 46-2084 ± 71-7373. 
III. g = - -U889 ± -0534, n = - 20-2473 + 12-1165. 
IV. (y = + -0475 ± 0452, 98-5263 + 93-7494. 
Now while these results are very satisfactory for II. and IV., they are not 
wholly conclusive for I. and III. We can approach the matter from another 
standpoint; the probable error of q for p = l is 
•67449 V2 = -67449 x -0707 
V A 
in " Student's " cases. Thus the deviation of q from q a very small quantity is for 
I. 2-68 times the S. D., and for III. r26 times the S. D. Since q may be either 
positive or negative, we may reasonably apply the probability tables and the odds 
against deviations occurring as great as these ai'e in one trial about 250 to 1 and 
9 to 1 respectively. Hence in four trials we should still have large odds against 
their combined appearance. 
* Bioinetrika, Vol. v. p. 356. 
