142 
0)1 Polychoric Coefficients of Correlation 
Before we consider the graph due to this solution, let us investigate the value 
of r to be found from (xvi). The values of nss'jngs' are already provided in 
Table IV, but we need a table corresponding to Table III giving the product 
^sTjj^s'Tp instead of the product "^g r^, t/. This is provided in Table VI. 
Further if 
K,,. = T, t: + r ^, T, t; + ^, T., t:+ 
Table VII (p. 143) provides «:.,,,' for the same three values of r, i.e. 0"45, 0-50 and 
0'55. Finally Table VIII (p. 143) gives «:s,,y(7ij,,y/i.s,s'), whence by summing we obtain 
v = r - 8,s' {Kss'/in.s'lns,')], 
for the three cases. 
Using the same interpolation formula as before in order to discover the value 
of r for which v = 0 we find : 
r - -5204. 
There is thus a difference of '0170 between the two methods. The probable 
error found for the product-moment r is "0160 and the result by the usual product- 
moment process may be given : 
r= -5189 + -0160. 
Thus either of the values reached by the methods of this paper differ by less 
than the probable error from the true product-moment value. 
(4) If we work out the results by mean square contingency we find : 
a = -480,690, 
and the class index correlations are*: 
For fathers : ?v,^= -962,329. 
For sons : r,., = -964,523. 
Hence correlation from mean square contingency 
r=aj{r,^r,,)= -5179, 
which is in excellent agreement with the product-moment value. 
It would therefore be quite reasonable for such a table as the present to use 
mean square contingency and class index corrections, and save the heavy labour 
of Equation (xvi bis) or (xvii). At the same time we cannot assert that this 
process would always be equally satisfactory for tables with but few broad 
categories and with much higher correlation. 
Our two processes seem to give values slightly in defect and in excess of the 
true value of?', and we might use their mean, i.e. -5118, to obtain our graph. We 
shall, however, first proceed to compare the actual results of solving (xiv) and 
substituting in (xv) with the result of such approximative processes. 
Table IX (p. 145) gives the products of ^4. '^s' V and will therefore enable us 
by aid of Table IV (p. 140) which gives the values of nss'/rtss' to obtain hs. for any 
value of r. Let 
X,, = % n X' T„' + r T, t/ + X 1\ + (xxii). 
* Using the values of and ;/s in Tables I and II respectively. 
