ON THE DEGREE OF PERFECTION OF HIERARCHICAL 
ORDER AMONG CORRELATION COEFFICIENTS. 
By GODFREY H. THOMSON, D.Sc, Armstrong College, 
in the University of Durham. 
PAGE 
(1) Introduction ....... ... 355 
(•2) A Criterion for Hierarchical Order 356 
(3) Tire Kelationship between Correlation Coefficients and their 
Sampling Errors ......... 357 
(4) Two Experimental Demonstrations of the Effect of the Criterion 
in Cases where the True Values of the Colunmar Correlations 
are known a priori ........ 359 
(5) Tlie Effect of the " Correctional Standard " .... 364 
(6) Conclusion ........... 366 
(1) Introduction. 
When several mental tests are applied to a group of subjects, and the correla- 
tions between the tests (taken in pairs) are worked out, the coetHeients are as a 
rule found not to be arranged entirely in haphazard order, but to show a certain 
degree of what has become known as hierarchical order. This means that if the 
total correlation of each test with all the others is found by adding together its 
coefficients, and if the tests are then arranged in sequence according to the order 
of magnitude of this total correlation, they are found to be also in sequence, or 
nearly so, according to the order of magnitude of their correlations with any one of 
their number. 
If the correlation coefficients are set out, as is convenient, in a square table 
such as the following, the letters x^, x^, etc. being the names of certain mental 
tests, and the quantities rjo, Vy,, etc. the correlations between the marks scored in 
these tests, then hierarchical order shows itself in the fact that each coefficient is 
smaller than that on its right or than that below it, provided the tests have been 
arranged in sequence according to the magnitude of the total correlation of each 
with all the others. 
• 
• 
• 
»'13 
ru 
• 
''12 
• 
'23 
•'•24 
• 
''13 
• 
'•31 
• 
x^ 
''l t 
• 
• 
% 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
The observed numbers in an actual experiment naturally do not in any case 
come out in perfect hierarchical order, and it becomes important to have a measure 
