H. Waite 
437 
We have, therefore, the value of the correlation ratio of restricted Tables 
given by 
1 - 7] J' + ?// ' 
or 7] = — ^ . 
VI - 7)^ + V 
The correlation ratio has been found by the method described above for all the 
restricted Tables ; it has also been determined by the ordinary method for a few 
of the other Tables, but no correction for number of arrays has been applied. 
The results, together with the coefficients of correlation and of contingency, 
are given in Tables 8 and 9. 
Regression curves for all the restricted Tables are given on Plates {a — e). The 
continuous line is the independent probability curve and the broken line the 
curve of the observed means. It follows that the area between the curves, 
weighted, of course, with the marginal totals, gives a measure of the correlation 
ratio between the two characters. 
Each set of three figures for two particular characters, namely, those for the 
right hand, left hand, and both hands respectively, will generally be found to 
resemble each other closely. Irregularities occur chiefly with composites but this 
is not surprising if we consider the nature of this class. 
8 c. Coefficients of Correlation of Restricted Tables. A glance at the diagrams 
of means of the restricted Tables, Plates (a — e), shows that the regression is 
generally non-linear; it is also evident that a sensible value of r is introduced by 
the restriction*. Hence the value of r as found by the ordinary product-moment 
method is (i) too small because of the skevvness of regression and (ii) too large on 
account of the restriction. These two contrary causes render the coefficient of 
correlation of restricted Tables unreliable and therefore quite valueless ; for even 
if it sometimes agrees fairly closely with the correlation ratio and the contingency 
coefficient, this agreement is probably due to the fact that the two sources of 
error counterbalance each other. 
In the remaining Tables, for which the results are given in Table 9, the 
regression is frequently skew ; for this reason and for those given above, I have 
rejected the values of the coefficient of correlation in the sequence and have based 
my conclusions on the contingency coefficients, confirmed in general by the corre- 
lation ratio. 
* For example, in small loops and large loops, left, the ease in which the difference between r and c 
is the greatest, the independent probability numbers have the correlation coefficient - "512 (instead of 
the theoretical value zero), as compared with - "507 of the observed numbers. In the case of arches 
and small loops, both hands, ?• for the independent probability numbers is - -MB, as against + -147 of 
Table 8. 
5G— 2 
