356 On Hierarchical Order aiiioiHi Correlation Coeffi,cumts 
of the degree of perfection present, and some means of estimating from what 
" true " correlations the observed numbers are most probably derived, and the 
degree of hierarchical order among these " true " correlations. The importance of 
this matter arises in the Theory of General Ability which has been proposed by 
Professor Spearman, for that theory can only be considered proved if the correla- 
tions are derived from an absolutely perfect hierarchy. A merely high degree of ' 
hierarchical order can be attained without any General Factor whatever, by the 
random selection of Group Factors. The very difficult question therefore arises of 
deciding (if possible) whether the hierarchies actually observed in experimental 
psychology are more ^^I'obably derived from perfect hierarchies such as are postu- 
lated in the Theory of General Ability, or from the good but not perfect hierarchies 
which arise in the Theory of (iron p Abilities*. 
A criterion which, it was hoped, would give such a measure of the perfection 
of the true hierarchy from which the observed numbers were derived by experiment, 
and which has been widely adopted for this purpose, was worked out by Dr Bernard 
Hart and Professor C. Spearman in the British Juurnal of Psychology for March, 
1912. The object of the present paper is to inquire into the accuracy of that 
criterion. 
(2) A Criterion for Hierarchical Order. 
The underlying idea was that if the above square table of correlation coefficients 
shows hierarchical order in any degree, there will be correlation between the 
columns of that table taken in pairs, and that when the hierarchical order is 
perfect the columnar correlation R will rise to unity, except in so far as it is blurred 
by the sampling errors, which obviously cannot increase an already perfect correla- 
tion, but can only decrease it. Let us write dashed letters throughout for the 
true values of the various quantities, which in ordinary experiment are unknown, 
reserving undashed letters for their measured values. We then have : 
/■' = true correlation coefficient, 
e = its sampling error on one occasion, so that 
r = r + e, 
r = mean of the column of true values r, 
r = mean of the column of observed values r. 
In finding these means, that coefficient is omitted which has no partner in the 
column with which correlation is being found. Write also 
p' = measured from the mean of the true column, i.e. 
= /•' — and similarly 
p = r measured from the mean of the observed column, i.e. 
= /■ - r, 
6 = p — p', = e ~e, 
where e is the mean of the column of e's. 
* See G. H. Thomson, " The Hierarchy of Abilities," Brit. Journ. Psychol. 1919, ix. p. 337 and 
" The Cause of Hierarchical Order among the Correlation Coefficients of a Number of Variates taken in 
Pairs," Roy. Soc. Proc A, xcv. p. 400 (April 1st, 1919). 
