240 " Goodness of Fit " in Statistics and Physics 
But. ■,.(!_ I), ^5i._S«.J . (i _ ^.j (ii), 
and accordingly 
= mean of the array in the sampled population. 
Thus to a high order of approximation at least the mean of the array means is 
the mean of the corresponding array in the sampled population. This result 
cannot be taken as obvious, as the size of the array in the sample varies f. 
We can now write to a second approximation: 
"i, V «p / Tip \ Vp J 
_ S {Srigp {Xg - nip)} ( 
\ "3,/ 
1 2 
'p 
•(111) 
since Sup = S (8/?„ 
We shall now rind the mean value of {SmpY as far as third order terms. Let 0., 
and O3 give the second and third order terms ; let S denote a summation for all 
samples and A their number ; write for x^ — nip. Then 
0., 
A A 
But = (1 - 1^)' 
S {Sn^'pSng^'p) _ n^-pHg-p 
.(iv). 
Thus O2 
A N 
iip^N Tip^N 
0% _ \ \ S {VgpXg) 
rip N{ rip 
2 
= since S (jv^"',) = 0 (v). 
/( p 
* Biometrika, Vol. ix. p. 2. 
t The assumption that the mean value of a character in a number of samples is the value in the 
sampled population is often made, hut is nearly as often erroneous. Thus the mean value of the 
correlation of a character in samples is not the correlation of these characters in the sampled population. 
