124 The Correlation Coefftdent of a Polychoric Table 
rji = -8046 ± •0126, 
= -7190 ± -0162, 
= -7190 ± -0162, 
= -8028 ± -0132. 
• C^-, = 1, 6'i2 = - -26399, 6*21 = - -26399, C22 = -82793. 
= -8382 ± -0122. 
These negative weights require further investigation, particularly the conditions 
for the existence of zero weights, but it is clear that certain divisions are to be 
avoided in determining r from a 3 x 3 table. 
On the whole C^, C22, C^a, C^i are in descending order of magnitude. 
§ 13. Precis of the methods of finding the coefficient of correlation. 
1. r^. Mean contingency, corrected for class index correlation. 
2. r^. Meaii square contingency, corrected for class index correlation and 
where necessary for the number of cells. 
3. fg. By selecting the central cell, the method first described in this paper. 
As its use treats any table as virtually 3 x 3, it may be called enneachoric r. 
4. Tp. By weighting the r's so that the p.e. shall be a minimum, the second 
method described in this paper. As it is applicable to tables of any size it may be 
called polychoric r. 
5- ^11 J ■''12 ! *'2i > ^"22 • Tetrachoric r of the various quadrants. The probable errors 
were calculated by the complete formula (p.e.) and also by the approximate method 
(a.p.e.). {Tables for Statisticians and Biomelricians , p. xl.) 
6. r^. The unweighted mean of r^j, r^a? *'2i> ^22- 
7. r^. The mean of the r^, etc., weighted by the reciprocals of the squares of 
their standard deviation. 
8- V^i^ Vjc^} "Qh-^i ftg- Three row -q calculated from each of the dividing planes as 
planes of reference with a class index correction on the foot of the columns. Since 
the standard deviation may be found in this case from the individual arrays or, 
assuming the distribution sufficiently homoscedastic, may be given the mean 
value crVl — yf) I have used both methods for the purpose of comparison. These 
are distinguished by the headings "individual dispersion" and "mean dispersion" 
respectively*. 
9. fc- By marginal centroids. 
The probable error of r.^ and was obtained as follows: 
Let the correlation coefficients r^^, r-^^.^ ■■■> 
have the s.D.'s o-^i, ai2, 
and the weights /i2) in, ■■■■ 
* The probable error of Biserial (or three row r?) has now been given (Biometrika, Vol. ix, part iv), 
but too late for use in the present paper. 
