364 On Hierarchical Order amoug Correlatioti Coefficients 
without as yet reaching a point where any of the cohimns pass the correctional 
standard. I feel that working a large number of such examples is beyond the 
power of an individual, with other claims on his time, and rather a task for a 
statistical laboratory with experienced computers and mechanical aids. 
(5) Tlie Effect of the Correctional Standard. 
Clearly the fact that the criterion is apparently too large in a majority of cases 
requires further explanation beyond the error already pointed out of neglecting 
the terms in p. 
The other approximations made in obtaining the criterion do not appear to be 
so erroneous as this one, though their cumulative effect may explain some 
anomalies. Leaving them on one side let us consider the '' correctional standard " 
required by Dr Hart and Professor Spearman before they admit any pair of columns. 
It is this correctional standard, combined with the peculiar distribution of R' , 
which chiefly is responsible for the exaggeration of perfection produced by this 
criterion, and for the regularity with which an average value of unity is arrived at. 
Let us examine first the actual distribution of the Hart and Spearman R' in a 
psychological hierarchy, viz., that of Wyatt already referred to, and calculate R' 
not only for those columns which pass the correctional standard, but also for other 
pairs of columns. What we find is that its value rises as we descend the hierarchy, 
I'ushing asymptotically to infinity, remaining for a time imaginary, and then 
returning. The value reaches infinity when one of the corrections in the denomi- 
nator becomes as large as the term to be corrected, and remains imaginary until 
the other term is likewise passed by its correction, when both quantities under the 
square root are negative and an arithmetically possible but meaningless value is 
again calculable. Specimen values from Mr Wyatt's hierarchy are given in this 
Table. 
TABLE VIII. 
Pairs of Columns Values of the Hart and Spearman R' 
Analogies and Wordbuilding 0-93 \ 
Completion and Wordbuilding 0-97 Passed by the 
Completion and Part-wholes 1'05 r correctional 
Wordbuilding and I'art-wholes 0'99 standard 
Part-wholes and Memory (delayed) 0'92 ' 
Rearranged letters and Missing digits ri7 
Wordbuilding and Test 1-26 
Sentence construction and Fables 1-33 
Rearranged letters and JS B Test Practically infinity 
Nonsense syllables and Dissected pictures Imaginaiy 
Crossline test and Letter squares 0'35, both factors in the denominator 
being now negative. 
Expressed in diagraimnatic form this and similar calculations lead to the 
conclusion that in actual practice the criterion is distributed as in Fig. 2, where 
the curve is to be understood as a " best fitting " curve among the values of B' 
scattered, with a very considerable dispersion, on both sides of it. The line, in 
fact, ought to be a broad smudge. 
