182 
Miscellanea 
affected by a certain disease or associated with a certain characteristic. A sub-sample marking 
a class or locality is found to have q' of its members thus differentiated. Does the group marked 
by the sub-sample differ significantly from the general sample out of which it is drawn ? Or, 
again, do children of a particular parentage differ in physique from those of the general 
population, the test being made on a sample and a sub-sample of the school population ? 
I would suggest the following method of approaching the problem. Consider the general 
sample {N, M, 2) to consist of two component samples, the sub-sample {n, m, a) and all the 
remainder {N', M', 2'). Then if the whole sample be homogeneous and random, and the 
two components also homogeneous and random, their difference of types m — M' will have for its 
probable error : 
^(«,- 3/') = -67449 
2 ^'-1 
The test therefore of the difference being due to random sampling is the relative magnitude of 
m-i/' and ^(,„_^/,). 
But if we consider the general sample we have at once : 
N=N' + n, or: N' = N-n, 
M= {N'M' + nm)IN, or : M' = M+ i,M- m), 
iV22 = m {o-2 + {m - Mf] -H N' {2'^ -|- (J/- J/')2}, 
N^^-no-^ nN_ 
N 
Accordingly : m - 3/' = — - {m -M), 
(_N 
-n) n \ n) N{N-n))' 
Or we must compare the relative magnitude of : 
{in - M) and -67449 k/ -at+~ 1-^) - -TrTTr- — n • 
nn,,^ N- 0-2 /. 2?A n(M-mf 
•67449 A,/ir. + — - „ , — r 
In other words, the probable error of the difference in type of the general sample and . the 
sub-sample, or of m — M, is : 
^ — in)^ 
This expression satisfies the requisite condition of becoming zero as the sub-sample increases 
in magnitude up to the value of the general sample. 
Now if N be large as compared with n, clearly the important term in this expression is o-^jj 
and M—m will be of the order vJa'^jn, where v is a small integer, 1, 2 or 3, say. Hence the 
order of the last term in the root is : 
or, since a will not differ very widely from 2, we may say v^^'^jN'^. Now the probable error of 2 
is -67449 — and accordingly if we put 2 ( 1 H 7=^ ) for 2 we should not alter significantly 
the first term under the radical ; thus 2^/iV' may be read : 
