2(i0 On Theories of Association 
table for eye-colour in brothers, for which he discusses the relative fluctuations 
of his coefficients and tetrachoric r t by a method which from the standpoint of 
scientific statistics is wholly inadequate. He states that "as soon as we leave the 
narrow field within which normal or ' strained normal ' correlation holds good, the 
normal coefficient fluctuates as we change the axes of division quite as largely 
as any other coefficient" (loc. cit. p. 633). How does this "narrow field" tally 
with the " many cases " in which these other coefficients fluctuate more largely ? 
Mr Yule has purposely selected three or four markedly skew distributions to 
show how the tetrachoric r, fluctuates; why did he not test adequately and 
completely its fluctuation as against those of the coefficient of association on 
this material ? Would he ever have written his above dogmatic assertion had 
he done so ? 
Now we do not for a moment agree with Mr Yule that there is no importance 
in the fluctuations of a coefficient of association for different divisions of the same 
surface. On the contrary we assert that a true coefficient of association should 
be as stable as possible, that is to say that for a given surface it should have 
fluctuations which are within a reasonable range indicated say by twice its probable 
error. Indeed, for most practical purposes fluctuations of "05 are of small im- 
portance. It is purely idle to do as Mr Yule has done, namely proceed to test 
such fluctuations by their ranges or their standard deviations obtained from the 
raw values. Each value must be considered in conjunction with its probable 
error. Q = l and r = "70 are not comparable if the probable error of the first is 
zero and of the second "05, the weight of the first observation is infinite and of 
the second finite. Mr Yule has compared such results without any regard to their 
relative weights. 
Starting with tetrachoric r t we have a definite surface, the Gaussian, either in 
its original or strained condition, for which there are no fluctuations in r t , except 
such as might arise from random sampling. No such surface has been discussed 
by Mr Yule for Q or <f>. But is an approximation to the Gaussian surface a 
rarity ? Does it only occur in <: a narrow field " ? On the contrary it covers 
within the approximations required in practice a very wide field, namely nearly 
all distributions in anthropometry and many characters in plant, insect and 
animal forms. Even the irregularities of the eye-colour data may well turn out 
to be due to neglect of the age-corrections and not to failure of the Gaussian 
system. Can Mr Yule produce, even with careful seeking, actual distributions, 
in number one-tenth of the Gaussian cases just referred to, for which Q and 
therefore « are practically constant for all divisions? There is no reasonable 
and logically consistent theory of deviations which leads to a surface of constant Q, 
although there is such a theory leading to a surface of constant r t . We hold that 
stability of an association coefficient is not only a desirable, but an essential part 
of any true theory of association, and the fact that one theory does give a relatively 
stable coefficient for a large section of material is immensely in its favour. 
