Karl Pearson and Egon S. Pearson 
129 
respectively. The numerical values of hg and A'^' can be easily ascertained from the 
table published recently of ordinates of normal curve to permilles of area*. Care 
must be taken in every case to give the correct sign to and k^'. 
Now if there were no correlation, hg and kg' combined would give the mean of 
the group n,,,,', and they give a fair approximation to the result if there are numerous 
categories, that is if the range of the categories be small. 
The correlation found from these marginal centroids would then be 
rc = S {nss'hsks')/^' (ii), 
but as Ritchie-Scott has shown ■]■ this diverges much more than r,j, the mean 
square contingency value from the true correlation, and considerably more than 
the tetrachoric or polychoric coefficients do. The reason for this is clear and was 
pointed out by one of us in 1913|. Namely hg and do not give the coordinates 
of the mean of ngg-. In fixct ngg'hgkg' is not the contribution of ngg' to the product- 
moment. 
We propose in the present paper to give first the actual contributions of n^s' to 
the means and product-moments of the two variates and then to apply these results 
in order to obtain (a) a polychoric coefficient, and (6) a graph of the relation of the 
two variates. 
The essential assumptions that will be made are the following : 
(i) The marginal totals having been reduced to a normal scale, and the corre- 
lation being supposed to be we shall calculate what the contents of the sth-s'th 
cell would be on the assumption that the frequency surface is the normal surface 
represented by the given correlation and the marginal totals reduced to normal 
scales. We shall further calculate the ^'-moment, the ^/-moment and the product- 
moment of the sth-s'th cell on the same hypothesis. 
(ii) From these data we shall determine the most suitable value to give to ?■, 
so that the actually observed frequencies differ least from those that would be given 
by such a correlation surface. We shall also obtain a formula for calculating the 
mean value of y for the array of i?-variates, rig. in number, which corresponds to 
the sth category of A. We shall thus be in a position to plot, the regression line of 
B on ^'and test at the same time the closeness with which it fits the thus calcu- 
lated array means, both variates being represented on a normal scale. 
We shall write the real coefficient of correlation of the population r, the 
coefficient as found from a single sth-s'th cell, as r.,,,-, and those found from the 
and n.g' arrays as Vg. and r.g' respectively. 
Ks', kss' will be the A- and 5-variate means of the sth-s'th cell and TVgg' the 
product-moment, per unit of the population, of the frequency in the sth-s'th cell 
about the mean axes as determined from the marginal totals on the normal scale. 
* See Biometrika, Vol. xiii. pp. 426-8. 
t Biometrika, Vol. xii. p. 122. 
J Biometrika, Vol. ix. p. 138. 
Biometrika xiv 9 
