Andrew W. Young and Karl Pearson 
217 
Sum all such equations for a large number of samples and divide by the number 
of samples. Then, since Mean = 0 and Mean 80^ = 0, we have 
The expression Mean (8n,)" is typical of several which we have to use in what 
follows and it will be useful to state or prove all the needful formulae before 
proceeding further. 
(3) Formulae regarding Products of Deviations. 
The deviations Bn^ arrange themselves according to a hypergeometrical series 
and the moment coefficients of this series are known to be* 
where 
H-2 -■ 
F-3 ■ 
1U4 = 
X2 
X3 
Xi 
XiNpq 
XxX-z^PI iP - 9) 
XiNpq i^X-sNpq + Xi) 
N - 1 
M - 1 
2{N-1) 
1 
.(vi), 
M - 2 
= 1 
N -1 fN -10 
= 1-6 
M 
M-2\N -2 
^ iV-2- 
+ 
M 
M 
and in the present case 
P = 
M 
N 
M 
= 1 
N 
.(vii). 
When M is very large as compared with N, as in the majority of cases in 
practice, we may write ^1 = X2 = Xa^ Xi== 1- 
We can now make use of these formulae to derive the following: 
{(i) Mean (S»,)2 This is jx^ in the notation of (vi) and 
Xi>h 1 
N 
.(«). 
(5) Mean Sw,,8h,/ where s and s' differ. Suppose first that 8n, remains constant 
and investigate the mean of 8n,' for this constant value of 8n, . Now the distribution 
* Pearson, Phil. Mag. 1899, p. 239; Biometnl.-a, Vol. v. p. 174. 
