lxxvi Tables for Statisticians and Biometricians [L — LI 



The multiplication can therefore be done very rapidly and it suffices to re-examine 

 not the whole of the arithmetic but only those rows which do not agree with the 

 table. 



Illustration. Calculate the first four raw moments of the distribution of head 

 lengths in 1306 non-habitual criminals on the previous page and test whether 

 they are correct. 



This was an actually worked out case, and it will be seen that in this instance 

 only one slip was made — that of a wrong multiplication by 5 in the contribution 

 to the fourth moment of the frequency of head lengths 196. Often far more 

 serious blunders are found. Correction would be made and the columns then 

 added up on the adding machine. Two points should be noticed. First it is not 

 in practice necessary to copy out the results from Table L, — they are merely 

 compared on the table itself with the items in column (vii) and any divergence 

 noted. Secondly in actual practice, it would be quite sufficient to take 20 instead 

 of 40 sub-groups in this case. Sheppard's corrections would fully adjust for the 

 difference. 



Table LI (pp. 113—121) 



Tables of the General Term of Poisson's Exponential Expansion (" Law of 

 Small Numbers"). (H. E. Soper, Biometrika, Vol. x. p. 25.) 



The limit to the binomial series 



p n + np n-, q + "ft*" 1 ) p»-Y + "^"^t^V -Y + Oxxxi), 



when q is very small, but nq = m is finite, was first shewn by Poisson to be 



m' m? m 



X 



.. lit no ill/ \ ,i ..\ 



1+w + i.2 + T7273 + - + .n + "J ( lxxxll >- 



The present table provides the value of the terms of this series, i.e. e~ m m x jx ! 

 to six decimals for m = O'l to m = 15 by tenths. 



A previous table for m = O'l to m = 10 to four decimals has been published by 

 Bortkewitsch*, but his values are not always correct to the fourth decimal. 

 Poisson's exponential limit to the binomial has been termed the " Law of Small 

 Numbers " by Bortkewitsch, but there are objections to the term. The approxi- 

 mation depends on the smallness of q (or, of course, p) and the largeness of n, so 

 that the mean m is finite. Thus 100 murders per annum might be quite a "small 

 number," if they occurred in a population of 40,000,000, for n would be large and 

 q would be small. It is therefore space and time which has limited the present 

 table to m = 15, not the idea of m being small of necessity. 



Illustration (i). The number of monthly births in the Canton Vaud being 

 taken as 662, and one birth in 114 being that of an imbecile, find the chance of 12 

 or more imbeciles being born in a month. 



* Das Gesetz der kleinen Zahlen, Leipzig, 1898. 



