A. ElTCHIE-SCOTT 
121 
In place of calculating a and R it will be found easier to employ the /S's directly 
by writing the equations for C in the form 
ri '^ii'ii I n '^ii'i2 I n '^ii"i3 L — 
Uji Y + ^^12 r '^IS i- ... — A 
X11X12 XnXi3 
C„ + Ci, ^-'^ + 6*13 + ... = A 
XnXi2 " X12 Xi2?(:i3 
+ etc (144). 
Eliminate the A by subtracting one equation from each of the others; put 
Cn = 1 and solve by successive elimination for the remaining C"s. This is preferable 
to using the determinant as it is at least no more laborious and lends itself to various 
checks for accuracy. The A should be determined from each of the equations as a 
further check. Then we have \ 
Or^=^ (145). 
By putting = Xii, ^^12 = - Xi2> C'21 = - X2i> ^-'22 = X22. we may derive the 
enneachoric standard deviation from the polychoric r in a form convenient for 
computation in terms of the ;S's, 
(Xll ~ X12 ~~ X21 + Xvif'^Te = ^WW + *Si2-i2 + '^21.21 + 'S'22.22 + 2 ('Sii-22 + 'S^12-2l) 
- 2 {Sxx'xi + + 'Si2-22 + '^'21.22) (146). 
§ 12. Comparative results of various methods op finding r from a 
3x3 TABLE. 
In testing the methods developed in this paper upon actual material it was 
thought desirable to try them side by side with all the other methods of finding 
the correlation coefficient so that some indication could be got of their comparative 
accuracy. Each of the tables was therefore dealt with by nine methods which are 
indicated in § 13. These tables were selected at the beginning of the investigation, 
and had the course which the research has taken been foreseen probably a different 
selection might have been made. Two of them, I and III, are normal tables with an 
arbitrary population of 1000. In Table I the frequencies have been taken to the 
nearest integer and in III to the nearest two places of decimals, so that any irregu- 
larity in them is due to the roughness of the approximation to the true figures. 
In the r^, we have an additional lack of approximation in taking from the curve* 
for determining and also in r^, and from finding the class index correlation 
from a small number of marginal groups. In II and IV we have actual samples. 
A rough test of the value of the various methods may be made by finding 
the mean square deviation of the calculated from the "observed" value of r, each 
constituent being merely weighted with its total frequency, regarding the product 
moment values of r as the "observed" value. 
Thus let n^, = total frequency in Tables I, II; R^, -R2 = product moment 
value of the correlation coefficient in Tables I, II; 7\, r^, = correlation coefficient 
calculated by one of the methods, then writing 
(wi + W2 + ...) = (i^i - ?-i)2 + ^^2 (i?2 - r. 
\2 
we shall have in a measure of the goodness of the various methods. This gives 
the following values of 2^. 
* Tables for Statisticians and Biometricians, p. Ivii and p. 65. 
