366 On Hierarchical Order among Correlation Coefficients 
however, give no reasons for their choice of this particular standard, upon which 
depends so much the values they obtain. The standard which they thus arbitrarily 
adopt begins admitting the criteria at just such a distance above unity as to 
balance the cases which give a criterion below unity, and entirely explains the 
remarkable unanimity with which this average value unity is obtained by them in 
their calculations. 
(6) Goncliisiov. 
A criterion suggested by Dr Hart and Professor Spearman has been widely 
used by psychologists for the pixrpose of ascertaining the degree of " hierarchical " 
order among theoretical correlation coefficients of which only experimental values 
are known, and a Theory of General Ability has been based on the results In the 
present paper it is however shown theoretically that an assumption made in 
deducing this criterion, namely that p and e are uncorrelated and the sums S{p'e) 
negligible, is incorrect. The quantity e taken regardless of sign is strongly corre- 
lated with p', and its signs tend to be either all the same as, or all different from, 
those of p. The distribution of the sums S{p'e) shows a minimum, not a maximum, 
at zero. 
Otherwise the paper is empirical, and applies the criterion in question to 
correlated dice throws. In the cases tried, this criterion exaggerates the perfection 
of the hiei-archy considerably, claiming a quite poor hierarchy formed by random 
group factors as being perfect ( true mean columnar correlation 0'.59, the Hart and 
Spearman i?' = l'00). The reason for this exaggeration, and for the unanimity 
with which in so many experiments the average value unity has been found for the 
Hart and Spearman criterion, appears to be mainly the peculiar distribution of 
this quantity, combined with the action of the " correctional standard " adopted, 
which commences admitting the criteria at such a distance above unity as to 
balance those which are less than unity. 
