122 The Correlation Coefficient of ci Polychoric Table 
Mean weighted square deviation of calculated from observed" or product 
moment vahies of r. 
"V2 
22 (omitting H) 
Mean contingency 
•00138 
•00102 
Mean square contingency 
>■■/> 
•00089 
•00060 
Enncachoric r 
re 
•00364 
•00036 
Polychoric r 
r,, 
•00004 
•00002 
Tetrachoric r 
rt 
•00018 
•00016 
Mean tetrachoric r 
^1)1 
•00005 
•00003 
Mean weighted tetrachoric r 
r„ 
•00002 
•00002 
Three row tj from mean dispej'sion* 
Three rowr; from "individual" dispersion 
•00020 
•00019 
Vn 
•00151 
•00144 
Marginal centroids 
rc 
•00215 
•00255 
I have given the value of including and omitting Table H, which gives very 
anomalous results, as yet unexplained. Broadly the best results are given by r^, 
r,m and r„, and. Table H aside, the best result is by rj,. In the case of r^ the results 
are not quite satisfactory. The figure given was arrived at by taking the mean of 
the raw figure from the curve and the same corrected for broad categories as 
suggested in Tables for Statisticians and Biometricians. An attempt was made to 
find an empirical formula which would give better results with the tables here de- 
scribed, but the result was not worthy of record. With three row r), although strictly 
the method is quite inapplicable to 3 x 3 tables, it may be useful to notice that 
when so applied the best results on the whole were got from assuming the 
distribution to be homoscedastic and using the mean dispersion of the arrays. This 
was largely due to several of the tables being divided so that some of the arrays 
contained very small frequencies which had therefore large probable errors, giving 
an undue efiect on the result when squared. When such small frequencies are 
avoided the results appear to be about equally good. Of course our theory fails, 
as we have already pointed out, when any cell frequency is of the same order as 
its variation. 
Comparing the probable errors of r„^ , r,„ and r^, (tabulated for convenience in the 
Appendix on page 133) it will be seen that on the whole they are in descending order 
of magnitude. They differ very little from each other and, considering the labour 
involved in finding r^, would in most cases give a result with a sufficiently low 
probable error. 
The method of marginal centroids as already known is unsuited for tables with 
so few categories. 
An interesting and important relation which is not shown in the tables of 
numerical results (§ 13) is the degree of correlation between r^^, r^^, etc., viz. 
•^11 • 1-2 5 -^n • 2n 6tc. These are collected in the table on p. 123. 
All the enneachoric tables are arranged so that reading from to the right, and 
downwards, r is positive so that the values R may be compared among each other. 
It will be seen on examination that i?^ . Ru ■ 2i) -^12 • 22 > -^21 • 22 are on the whole 
greater than R^^ . 22 and R^^ ■ 21 and of the two latter R^^^ • 21 is usually the greater. 
* See § 13, 8. 
