Karl Ppurson 
241 
Hence to this order we have o^^^ = of^jViij,, precisely the value of the standard 
deviation of the mean on the hypothesis that the array is of constant size equal 
to the mean size «„. We next turn to the third order terms Oo. Here 
= - 2 
'S S (Sn2,j,Swj,.Tg2) Si: (Sn, 
An/ 
But the summations marked by S can be expressed by aid of 
p. 244 below. We have 
and (/) on 
9? ^ 
n X 
p{S {n„\)\^ 
N 
0. 
Accordingly 
= o, + o,= 
2 f 1 - 
K \ n) 
VI 
The corrective term O3 is, I think, of marked importance for it indicates that the 
standard deviation of the mean of an array in a sample is not the same thing as 
the standard deviation of an array of constant size. It is not therefore legitimate 
to assume, as some authors have done, that the standard deviation of the mean of 
an array is given by the same formula, i.e. ojjjVfij,, as for an array of constant 
size equal to the mean number in that array for a large number of samples. 
Had we included the terms of the next order in (vi) we should have obtained 
in the curled brackets terms of the order 1//!^^. For a small array this might 
be equally important with the term already given in 1/N. Hence some caution 
must be exercised in retaining that term and dropping terms in l/n^^; at the 
same time for the larger arrays v^/N may be commensurable with unity. 
We may throw (vi) into the form 
t2- 
\bis 
, for many purt)oses. 
+ 2 
This result agrees only with that given by me in 1905*, i.e. 
rr II 
^m,, — /— , 
to a first approximation, i.e. when we may neglect 2/iV as compared with unity 
and 2 as compared with the mean number in the array. To assume without proof 
* "On the General Theory of Skew Correlation," p. 14. Drapers' Company Research Memoirs, 
Cambridge University Press. 
Biometrika xi 1(3 
