110 The Banger of Certain Correlation Formulae 
where N is the total number of observations and 8 = (AB) — (A) (B)/N. This 
latter formula is said in some cases to give a theoretical value of the correlation 
coefficient*, and it is suggested that it might be used instead of the Coefficient of 
Association, but no indication is given of any limitation to the application of the 
method, nor is any example given of its use. 
The most cursory examination of these two methods makes it quite clear that 
they lead to results which differ very widely indeed, and to illustrate this point 
a few examples may be given. 
Thus Mr Yule has discussed f the relationship between deaf-mutism and 
imbecility and finds that the Coefficient of Association, Q, is "91. If, however, the 
degree of association be calculated by Mr Yule's second method, we find that Q', 
Mr Yule's " theoretical value of the correlation coefficient," is "02 ! 
To take a further example, the most extensive use of the Coefficient of 
Association has been made by Mr Yule in his discussion of the Association of 
Defects in Children and AdultsJ, based on material collected by Dr Warner. 
I have already § given fully my reasons for considering these data to be of little 
value, and, if there be any doubt on that point, there is certainly none as to the 
unsoundness of the methods adopted by Mr Yule in his analysis of those data. The 
Coefficient of Association has been used throughout to determine the relationship 
between the various forms of defect, mental and physical. Most stress has, 
however, been laid on the relationship between Low Nutrition and Mental 
Dullness. It is therefore of some interest to calculate Mr Yule's " theoretical value 
of the correlation coefficient " and to compare the results with the Coefficients of 
Association already given by Mr Yule. 
The results are as follows : 
TABLE I. The Relationship between Low Nutrition and Mental Dullness as 
determined by (1) Mr Yules Coefficient of Association, and (2) Mr Yules 
" Theoretical Value of the Correlation Coefficient." 
Boys 
GlELS 
Coefficient 
of 
Association 
"Theoreti- 
cal Value 
of r" 
Coefficient 
of 
Association 
" Theoreti- 
cal Value 
of r" 
Irrespective of the Presence of other Defects... 
Where no other Defect is present 
Nerve Defects also present 
Nerve and Developmental Defects also present 
■82 + -01 
■95+ -01 
•08 + -05 
•20 ± -06 
•22 
■23 
•02 
•07 
•84+ -01 
•94 +-01 
•01 + -06 
- -03 + -06 
•24 
•21 
•00 
- -oi 
* The Correlation-coefficient for a two-x two-fold Tahle. ..."In some cases, however, a theoretical 
value is obtainable for the coefficient, which holds good even for the limiting cases when there are only 
two rows and two columns," p. 212. 
| Text-book, pp. 33 and 38. + Phil. Trans. A. Vol. 258, pp. 257 et seq. 
§ Mental Defect, Mai-Nutrition, and the Teacher's Appreciation of Intelligence. Questions of the 
Bay and the Fray, No. 2, Dulau and Co. 
