120 The Danger of Certain Correlation Formulae 
The various approximations are as follows : 
_ . 7r ad — be 
Q 1 = sin 
2 (a + b)(b + d)' 
ad — be 
ad + be 
(Mr Yule's Coefficient of Association, Q), 
„ . 7r VW — V6c 
2 \'ad + wbc 
1 
Qi = sin 
Q 5 = sin 
1 + 
2bcN 
, ad > be, 
(ad - be) {b + c) 
2 vrr# 
t'here 
4abcdN- 
(ad - be) 2 (a + d) (b + c) 
TT TT 
Q e = sin — (r Afc ) = sin ^ ((/), 
and 
V(a + b) (c + d) (a + c)(b + d)' 
Mr Yule's " theoretical value of r." 
Now in the work already done, I have always taken h = k and hence b — c. 
With this restriction, some of the formulae are considerably simplified. Thus 
Qi = Qt = Q s . 
I have given in Table IV the values of these coefficients for various values of 
h = k in the case of the frequency surface for which r — '5. 
TABLE IV. 
Showing the Comparative Degree of Accuracy of Various Approximations 
to the Coefficient of Correlation, '50. 
h 
Q 2 
Q 3 
Qo 
Q' 
0 
•50 
■60 
•50 
•50 
•33 
•52 
•48 
•62 
■52 
•51 
•32 
1-03 
•42 
•67 
•57 
•50 
•28 
1-63 
■32 
•77 
•67 
•46 
•21 
2-51 
•16 
•91 
•85 
•35 
•10 
3-09 
•08 
•97 
•94 
•24 
•05 
These results are also given graphically in Fig. 7. It will at once be seen 
that Mr Yule's coefficients, Q and Q' are very poor approximations, whatever be 
the values of h and k. The other approximations suggested by Professor Pearson 
