360 On Hierarchical Order among Correlation Coefficievits 
coefficients concei'ned, in the two columns. In using their formula, its authors do 
not apply it to all the pairs of columns in the square table. They say : " In any 
case the correction must be kept within limits : as usual, the larger the correction 
the less it is to be trusted. If the sampling errors are large enough, they 
eventually will quite swamp the true differences of magnitude upon which the 
observed correlation should be based. In this case, the true correlation is beyond 
ascertainment ; any attempt at correction is merely illusory. To avoid this, and at 
the same time to ensure impartial treatment of all data, it is necessary to fix before- 
hand some definite limit to the feasibility of correction. We have here adopted 
the following standard : in order to attempt to estimate the correct correlation 
between columns, it is required that in each of these columns the mean square 
deviation should he at least double the correction to he applied to that deviation." 
That is to say, the equation (5) is not to be used unless, in each factor of the 
denominator, S{p') is at least double its correction {n — 1) S--. This condition (the 
" correctional standard " ), will be found to be important. 
It is clear that the accuracy of this formula (5) could be conveniently tested 
were we in possession of material in which all the true correlations were known 
a priori, in addition to the observed correlations found in samples. Such material 
is supplied in perfection by correlated dice throws. 
First Example. The first experiment with dice of the above nature which 
I carried out was described in the Brit. Journ. Psychol. 1916, vill. There ten 
variates were artificially made up of group factors and specific factors, without any 
general factor, so as to make a very good hierarchy, which gave the following 
results when tested by the Hart and Spearman criterion. 
TABLE II. 
Columns 
passing 
standard 
Observed columnar 
correlation R 
True columnar 
correlation 
The Hart and Spearman 
corrected columnar 
correlation R' 
ah 
0-95 
1-00 
1-04 
ac 
0-89 
0'99 
1-00 
bo 
0-91 
TOO 
roi 
cd 
0-90 
1-00 
i-u 
Means 
0-91 
1-00 
1-04 
Here the exaggeration of the Hart and Spearman R' is not very noticeable, for 
the hierarchy is in any case almost perfect. Indeed in this case I took some pains 
to make the arrangement of group fact(jrs imitate a perfect hierarchy very closely, 
for the sake of emphasising the point I then wished to make, viz., that such group 
factors can, unaided by any general factor, approach exceedingly close to perfection 
of hierarchical order. I did not then realise that the pains I took over this point 
were hardly necessary, for random sampling of the group factors gives good 
