David Heron 
113 
these distributions with the actual value of the Coefficient of Correlation, without 
however restricting myself to median divisions. 
It is clear that, if we take a Gaussian frequency surface of given correlation 
and divide it by two planes parallel to the axes of x and y at distances h and k 
from the origin, h and k being measured in terms of the standard deviations of 
their respective variables, then the value of r is independent of h and k. The 
point to be determined is this : To what extent is the Coefficient of Association 
dependent on the values of h and k 1 
If a, b, c, d are the frequencies in the four quadrants of the frequency 
surface, it has been shown* that 
d b + d c+ d ™/r n ,,„_ _ 
N N " N i \nl 
b + d c + d " „„/ Mr 2 (h 2 - 1)(& 2 - 1) , 
= ~W ' ~W +s 1 ' HK [ r+ ^r + 3> r +etc ' 
Now the coefficients of the r's on the right have been determined by 
Mr Everitt for values of h and k from 0 to 3"09, in his most valuable " Tables of the 
Tetrachoric Functions for Fourfold Correlation Tables f." It is clear then that in 
a normal frequency surface of definite correlation, ~ , and hence a, b, c, d can be 
determined by summing the series on the right, and we can at once compare 
Q and r for any values of h and k. I have throughout made h = k, and it should 
be noted that this will reduce rather than accentuate the differences between 
Q and r. I have calculated Q for a series of values of h = k from 0 to 3"09 in 
some cases extending this to h = k = 5 in order to make certain of the form of the 
curve. The results are given in Table III, but I shall delay their consideration 
until I have discussed Mr Yule's " theoretical value of the correlation coefficient." 
This coefficient 
V(^)(«)(jB)oj) 
is reached by Mr Yule in his Text-Book\ by a process which is nothing short of 
extraordinary. It would be interesting to know what justification he can give for 
the assumptions he has made in the proof. 
If we express Q' in the more usual notation we obtain 
„, _ ad — be 
V(a + b) (c + d) (a + c) (b + d) 
Now this formula was given ten years ago by Professor Karl Pearson§, not as 
the correlation between the two characters under consideration, but as r hk , the 
correlation between errors in the position of their means when each is measured 
* Pearson, Phil. Trans. A. 262, p. 6. t Biometrika, Vol. vn, No. 4, p. 437. 
t p. 213. § Phil. Trans. A. 262, p. 12. 
Biometrika vm 15 
