126 Influence of " Broad Categories" on Correlation 
we find 
»I»3-'»o»4 , x 
r xy = =; =■ (xxv). 
(n- 2 + n. 3 ) (n 3 + n 4 ) x ^ x ^ 
a x (Ty 
Mr Yule has termed (xxiv) the "theoretical value of the correlation*." 
It will be quite obvious from the present discussion that it is only an approximate 
value even to the class index correlation, and cannot represent r xy at all, unless 
the frequency distributions each consist of two isolated points. Generally 
n x n s - n 3 n x / Oy Oy 
1 XV ~ + h 8 ) (n, + n 4 ) (m s + m.) (n, + n 4 ) V a, «»■ ft % 
= r x /Zl^°JL;<h ( XX vi). 
V x x x 2 y 1 y x 
This corrective value on tq q is always true, although the above value 
of fc x c u ^ s on ly approximate. 
This is brought out at once by considering what (xxv) reduces to in the case 
of Gaussian frequency. Here 
_ 1 *L 
1 2 2 
(re 2 + ?? 3 ) £ 2 = Nz 2 <r x = Na x ~j== e * , 
V Z7T 
_ 1 fc2 
{n. i + n i )y i = Nz i a y = N(7y-^^e 2 *» , 
where and A" are the distances of the dividing lines from the mean. Hence if 
H and K have their usual meanings (xxv) gives us 
*>y = JfqjjZ =6 (xxvn). 
e is the expression used in my memoir on the correlation of characters not 
quantitatively measurable"]". Thus the physical meaning of e is r c c ^ corrected 
for the use of class indices, but not for the assumption that x st xy st may be 
replaced by x s xy t . We may therefore anticipate that e or, failing Gaussian 
frequency, (xxv), will give a better approximation than r c c to the true value of 
the correlation of x and y. 
I shall discuss elsewhere the possibility of any absolute identity in the 
members of a class; the only thing in which I have personally come across it 
is in theoretical investigations — never in practical — on Mendelian units. When 
we are absolutely ignorant of the nature of the frequency, then I feel sure that 
in dealing with either physical or mental characteristics in living forms, the 
Gaussian distribution will in the long run be found closer than any single type 
of distribution hitherto discussed, and that accordingly even e — to say nothing 
* Introduction to the Theory of Statistics, p. 212. 
t Phil. Trans. Vol. 195, A, p. 7. 
