114 The Danger of Certain Correlation Formulae 
in terms of the standard deviation of its own distribution ; now had Mr Yule used 
the more familiar notation, there is little doubt that he would have recognised it, 
and this furnishes an excellent example of the danger of inventing new notations. 
It is interesting to note that Professor Boas*, using still another notation and 
relying on a method of proof which is of the same degree of validity as Mr Yule's, 
reached this same formula in the form 
and, like Mr Yule, failed to recognise it as an old friend. Professor Pearson 
subsequently pointed outf that of the formulae given by Professor Boas, those 
which were valid, had been in constant use for many years and remarked that 
" there is some danger, unless we see how new values for the correlation coefficient 
are related to the old values, in a multitude of formulae leading to divergent and 
possibly inconsistent results." He added that this coefficient (Mr Yule's " theoreti- 
cal value of r ") differs in the simplest cases from the true coefficient of correlation 
and often differs considerably. In the bulk of cases it does not approach r nearly 
as closely as the Q 5 J coefficient of association, and its use is liable to be misleading, 
especially if compared with the value of the true coefficient of correlation calcu- 
lated by other processes. 
Now if we make h = k and therefore b = c, the formula for Q' can be simplified 
considerably, reducing to 
„, _ ad — c' 2 
y = (a + c)(d + c) ' 
and if we express it in the form 
*y _ ad — c 2 
ad + c' 2 + c(a + d)' 
we see that Q' is always less than Q, and generally very markedly so. 
I have calculated the values of Q' for the same cases for which Q has been 
calculated and the results are given together in Table III, while in Figs. 2, 3, 
4 and 5, the comparative values are shown graphically. It is clear that while Q 
asymptotes to unity as h = k is increased, Q' asymptotes to zero. 
Now Mr Yule has used his coefficient of association in cases where h and k are 
as high as 35, and we see that, with this value of h = k, when the actual value of 
the correlation, r, is '1, his coefficient of association, Q, is '57 ; when r is actually 
"3, Q is "93 ; when r is actually - 5, Q is "99 ; when r is actually - 9, Q is greater 
than - 99. On the other hand when r is actually "1, Mr Yule's "theoretical value 
of r" Q' is '00; when r is actually - 3, Q' is - 01, when r is actually "5, Q is "03; 
when r is actually - 9, Q' is '40. 
f Science, July 2, 1909. 
