Editorial 
271 
(iii) For a sample of constant size the following are the moment-coefficients 
of the variation in the mean * : 
pj^-i _ (' "lip 
n.,r 
Now let >S ( — ^, j equal the sum of — for all values of j) which may occur in the 
V'/Jjj / ')lp 
samples of N. If therefore f,,^ is the frequency with which h^, occurs, the whole 
problem reduces to finding the values of 
for various values of s. 
Now the frequencies of the are simply the terms of the binomial. The term 
in which )ip = 0 must not we think be taken into consideration, for in this case 
there is no variability in /i/ as there is no frequency in the array, i.e. jj/Uo must be 
put zero. Thus in the notation of a binomial {p + 7)" we require to find : 
ppn-iq H, (n — 1) -5- n{ii — l){n — 2)p 
V ^ r72 ¥ ^ 1.2. 3 
.(F) 
and to divide the result by { 'p + q)" - p" = 1 — j>". This finite series we have not 
succeeded in summing. Before indicating how we may approximate to it by the 
mean-powers of Srip, we can look at the problem from two other standpoints. 
(i) If np/N=q be not small the binomial approximates to a Gaussian of 
standard-deviation squared <t^ = npq = Tip (1 — TipjN). Hence 
1 - 
1.t"- 
dx 
= ^ 1 + 
n. 
s{s + l) (T-^ s (,9 + 1) (s + 2) (s + 3) 3cr^ 
1.2 
Thus 
1 + 
1.2.3.4 
+ 
+ 
0 / 1 1 
1 
'N. 
+ 15 
J,) = ^.|l + 3(| 
+ 
+ 
.(G) 
Here is the sth moment-coefScient of the array about its mean in tlie sampled population. 
