242 
''Goodness of in Statistics and Physics 
the above value to be true is not legitimate as there is no a priori reason why 
the standard deviation of the mean of an arrav of variable size should take, even to 
a first approximation, the value of the standard deviation of the mean of arrays 
of constant size, that constant size being the reduced frequency (i.e. VpN/M) 
of the sampled population. Actually a better approximation to appears to 
be obtained if we add two units to the mean frequency — a correction which may 
be of considerable importance for small arrays. 
We next turn to the correlation of means and we have to evaluate the product 
of two expressions like (iii) for 8/»j, and SdIj,' where is not equal to p'. We shall 
obtain, summing for A samples and dividing by A, the value of o-^^^o-^^,i?^p^^,, where 
-^TOpWi,,' is the correlation of the means. We shall again treat the square and cubic 
order terms separately : 
Square order terms in o-„^^a„,^,R„^^,^^^. 
I Xrjpfip' j * I A«j,?!j,' 
N ' N ' N N 
= 0. 
Thus we see that as far as square order terms are concerned: 
^Wpm„. = (vii), 
notwithstanding that there exists a correlation between the numbers in the two 
arrays on which the means are based. So far this result is only true to a first 
approximation*. But we now tuin to the cubic terms given by the product of the 
square terms in Snij, with the hnear terms in S;^^.. There results the four separate 
terms : 
_ S { S (Sn . ^'X,) S (8n,^Sn^ x,)} m^S {E {hn^^hn^hn^.x,)} 
+ 
1 have evaluated each of these terms in succession by the use of formulae similar 
to those on p. 244, and all these terms give the same result, i.e. their sum with 
proper signs to its constituents 
Nn^ V N ) Nn^ V N J 
Nn^ \ N Nn.„ { N 
* I have given this result in memoir just cited, p. 13. 
