Godfrey H. Thomson 
359 
small sample, even if p and e were really uncorrelated, it would be most unlikely 
for S{p'e) to be negligible. As the sample increases the signs tend to settle down 
to the above arrangement, and S(p'e) does not tend to disappear compared with 
S(e€), but only to take on one or other of alternative values. It will only be zero 
when all the errors are zero, i.e. when no corrections are needed to R. The 
distribution of S{p'6) about zero in a number of samples of the same size will not, 
that is, show a maximum at zero, but a minimum, as is shown qualitatively in 
Fig. 1. 
Fig. 1. 
To show the order of magnitude of these neglected quantities, consider the 
following example, in which the true correlations are known a priori, and with 
their observed values were as follows : 
^ ca 
= 0-730, 
n-a 
= 0-703, 
e = 
-0-U27, 
r'da 
= 0-598, 
= 0-708, 
e = 
+ 0-115, 
^' ea 
= 0-356, 
' ea 
= 0-367, 
e — 
+ 0-011, 
= 0-174, 
^> 
= 0-337, 
e = 
+ 0-163, 
J 
3" 
= 0167, 
' !/" 
= 0-281, 
e = 
+ 0-114, 
fia 
= 0-120, 
I'/ia 
= 0-371, 
e 
+ 0-251, 
'''k(i. 
= 0-116, 
= 0-112, 
■ = 
- 0-004, 
r'la 
= 0-112, 
ri„ 
= 0-183, 
e = 
+ 0-021. 
The variates here were made up of dice throws, and the sample was one of 36 
cases. Here, knowing as we do the actual true correlations* which would be given 
by the whole population or by a sufficiently large sample, we can form the 
quantities *S (e\-„) and ^Sipxa^xa)- They prove to be -064 and --116. It is 
clearly unwise to neglect the latter of these in comparison with the former. 
(4) Experimental Demonstrations in Cases ivhere the True Values of the 
Columnar Correlations are known a priori. 
The formula at which Dr Hart and Professor Spearman eventually arrive, after 
neglecting these quantities and making various other assumptions, is 
j^' _ _ ^_ (Pxap xb) - O'l — ^)'>~a hO'xaO'xh 
Vf'S ip-^a) - {n - 1) o-\.„} Vl'S {p%!,) - {n - 1 ) a\^i,\ 
where the cr's are standard deviations of the correlation coefficients, the bar 
indicates mean values for the column, and n is the number of pairs of correlation 
* G. H. Thomson, "A Hierarchy without a General Factor," Brit. Juurn. Psychol. 1916, viii. p. 271. 
