200 
Miscellanea 
Now if \,Q naean square contingency of Tahle I. and if (^^^ the contribution to it of 
the b row we have : 
= 0«' + 06' + </)/+... +^.2 
where 
Or, we have : 
But: 
Thus we have the following rule : 
N 
2 ^ ^2 
.(ii). 
Start with Table I. and determine the contributions t^a''^, </)(,^, (^^^ ... of each sub-group or 
locality to the total mean square contingency of this table. Then C„, C^, ... determined as 
above are the "coefficients of divergence" of the respective sub-groui)s or classes or localities 
from the general population, and their relative magnitudes measure the relative divergency of 
such groups or localities. 
(2) If the Class b were, for example, merely a random sample of the general population, we 
should have (^6^=0 and C(,=0. It becomes accordingly of importance to determine the probable 
error of on the assumption that Class 6 is a random sample. If Ci, differs from zero by 
several times its probable error, the divergence of the Class is almost certainly significant. The 
general expression for the probable error of a coefficient of mean square contingency has been 
dealt with in another paper*. In the notation of that paper 
.(iv). 
02 will now be xb" and we have to perform the summations for the two rowed table. Table II, 
above. The q summation will be from a to co and the p summation for the two rows of 
our table. I take the terms in order. 
(i) Spq (i^^pq ) ■ This in our present notation stands for 
Tlh 
since (^^p^ for any constituent of the second row is by the line above Equation ({) = — <t>\a.'> 
1\ 71}) 
where is the contribution to the mean square contingency from the first row constituent 
immediately above. Let us write vi = niij{N—ni,). Then we have, if we write 
rb^ = S {<t>\a,nj,J{iii,n^)} (v), 
a 
("t^'p" irt) (1 - + "b'^t^b'/nb (vi). 
(ii) 2'^pq(4>p^<pq^-"^l. For the first line (p^ = <^i? and for the second line =V(,(f)b'^. Hence 
* " On the Probable Error of Mean Square Contingency," see Equation (xxxvi), Biometrika, Vol. v. 
p. 195. 
