194 journal of Comparative Neurology and Psychology. 



independent probability of occurrences for each compartment of 

 the table had been calculated that there were very considerable 

 and regular deviations of the observations from what would be 

 expected if intellectual capacity and head circumference were in 

 no way associated in this sample. If these two characters were 

 entirely independent we should expect the frequency in each com- 

 partment of Table II to be, within the limits of error due to ran- 

 dom sampling, the same as that given by the ordinary figures. 

 But it is clear that the italic figures and the ordinary ones differ 

 from each other in a definite way and by considerable amounts. 

 Thus, neglecting the arrays of low frequency as too small to be 

 significant, we see that in the left hand column (intelligence above 

 average) individuals with small heads occur in defect of the 

 expected proportions, and those with large heads occur in excess 

 of expectation. The reversed relation holds for the right hand 

 column (intelligence below average). 



From Table II as it stands I find the value of the mean square 

 contingency (cf. Pearson, loc. ctt.) to be: 



<lp2 = .0385 

 which leads to a value of the contingency coefficient of 



Ci=.i925 

 Using the method of mean contingency 



r=.0397 

 whence C2 = .I3. 



Now, if the variation follows the normal law and the grouping is 

 not too fine, the value of C calculated by the two methods should 

 be equal. This is clearly not the case. Hence, it is necessary to 

 determine whether, by taking a somewhat coarser grouping, these 

 values may not be brought nearer together.^ Accordingly, a table 

 was formed in which the size of the head circumference classes 

 was doubled, thus reducing the number of rows from twelve to 

 six. This new table then had eighteen compartments with some 

 observed frequency in each. Working from this table I found the 

 following values : 



<?>2 = .0203 



Ci = .i4io (mean square contingency coefficient.) 



¥ =.0426 



C2 = .i4 (mean contingenc)^ coefficient.) 



^A full discussion of the effect of too fine grouping on the value of the contingency coefficient is 

 given in the original memoir on the contingency method, to which the reader is referred. 



