218 On the Probable Error of a Coefficient of Contingency 
of for fis constant is clearly given by the sub-hypergeometrical series with 
N' = N — — Sn, as total population and 
so that the Mean Sn^.- for Sn, constant is 
[N - n, - 8n,) - n,. = - f,""^-'-!- *. 
^ « " N ' -/V - lis 
Hence 
Mean Sn^Sw,- = — Mean Sn, 
Mean Sm,2,^ 
iV - ?i. 
= -xiif (ft). 
(c) Mean (Sw,,)'''. Directly from (vi) and (vii) 
Mean S^/ = ^1X2 (l " 7^) (l " (^)- 
(f?) Mean (Sh^)^ Sks-. Using the process of double summation as in {b) we 
have 
Mean {8n,)nn,' = - Mean {8n,)\ 
'N - ¥u 
2n, 
^.Xa.«.(l-|)(l-f)from (0) 
= -xa.f^(i-f) (^). 
(e) Mean (Sn^)*. As in (0) we have immediately 
Mean (8n,y = Xi^l. (l - ^) (Sxa'ls (l - 7^) + Xa) (e)- 
(/) Mean (Sw.,)' (S^'s )"- again use the double summation as in (6), but in 
this case the algebra is much more troublesome. From the constants of the 
sub-hypergeometrical as given in (h) we have 
x.'*>'»' = - * ) - - - ^».) N^,i} - F^.) • 
* Since <S (5m,) must be zero, we can regard the Mean 811^' for a given Sw, as being the result of a distri- 
bution of a deviate - dn^ distributed over all the divisions except the sth. The portion due to the 
s'th is then x ( — 5?iJ, as obtained above. 
N-n, ° 
