ON POLYCHOTIIC COEFFICIENTS OF CORRELATION. 
By KARL PEARSON, F.R.S. and EGON S. PEARSON. 
(1) One of the difficulties which are constantly recurring in statistical practice 
is that of the correlation or contingency table in which the two variates are 
classified in broad categories. We may indeed proceed by the method of mean 
square contingency and correct for the grouping of both variates by the class 
index corrections on the assumption that the marginal totals for both variates 
may be assumed to follow appro.ximately normal distributions. Such a procedure 
gives reasonable satisfixctory results*, provided the marginal totals are not in very 
unequal groupings and the correlation is not intense (say, '85 and above). The 
polychoric table has been discussed by Ritchie-Scott and he has described a method 
of reaching a polychoric coefficient of correlation from the weighted mean of the 
possible tetrachoric valuesf. Such a process is, however, so laborious that it can 
hardly establish itself in practice. From the theoretical standpoint, however, 
Ritchie-Scott's paper was of great interest (i) as guiding us by the size of the 
probable errors to discriminate between the valuable and worthless dichotomies in 
tetrachoric determinations of the correlation (ii) as providing standard values by 
which those obtained by other procedures could be directly tested. 
We shall endeavour to reach in this paper another form of polychoric co- 
efficient, — that is a correlation coefficient which does use all the information given 
in a polychoric table, — but which requires less analysis than Ritchie-Scott's weighted 
mean coefficient. Thus what may be lost in exactness will possibly be repaid by 
practical efficiency. There is another point also of very considerable illustrative im- 
portance ; we desire wherever the data are suitable actually to exhibit in the form 
of a graph the relation between the two variates. This should be possible in the 
case of a polychoric table, and in the past has frequently been done by approximate 
methods of more or less validity. 
We can indeed take such methods as our present starting point as they will 
directly indicate to the reader our line of approach. 
We start with the hypothesis that the marginal totals of our polychoric table 
can be represented on a normal scale. This is no great assumption in itself. If a 
true quantitative scale ever becomes available it can be attached at once and with 
little trouble to the normal scale. To exhibit a variate on a normal scale makes 
* By " reasonably satisfactory results," we mean that in cases which can be directly checked by the 
product moment method the difference is within the range of practical insignificance as judged by 
probable error. 
t Biometrika, Vol. xii. pp. 93—133. 
X Thus in a 3 x 3 table it is possible for two of the corner dichotomies, i.e. those unassociated with 
the diagonal in the sense of the correlation, to have even negative weights, so that they should be omitted 
in finding the mean. 
