Karl Pearson and David Heron 
167 
have the probability 1, or a deviation from the normal probability I - p. Those 
cases in which the event 1 does not occur have the probability 0, or the deviation 
from the average probability of — pi" 
Pearson at the time read this many times through ; both of the present writers 
have read and re-read it since, and they fail utterly to grasp how an event can 
have at the same time a probability of 1 and of 0 and a normal probability 
of 1 — p ! Mr Yule says that Professor Pearson failed " to understand what Dr Boas 
was doing*." We still fail completely to understand what Dr Boas means, or 
how Mr Yule justifies the assertion tliat Dr Boas was demonstrating a formula 
which applied to two values of a character differing by a unit. What did come 
out of Dr Boas' investigation of 1909 when it was translated into the fourfold 
table terminology 
a 
b 
a + b 
c 
d 
c + d 
a + c 
b+d 
N 
ad — be 
WaS r " <S(b + d)(a + c)(c + d)(a + b) 
a value already known (i) as the correlation r/, k between the means, each measured 
in terms of their standard deviations, of two variates of a fourfold Gaussian tablef 
or (ii) as the square root of the mean square contingency of a fourfold table 
without any assumption of a Gaussian distribution J. 
Now it is known that the correlation of errors in two means, r%y, is equal to the 
correlation of deviations, r xy , in the two variates of which x and y are the means. 
But is not r xy -, for h = x/a x and k = yja- y and <r x and a y are correlated as well as 
x and y. It is therefore clear that if x and y are continuous variables of any kind, 
yjij is not a " theoretical value of the correlation " of variates. It is a correlation of 
ranks where the ranking consists of only first and second, and is wholly uncorrected 
for class-index. It becomes a true value of the correlation when the two classes 
differ by a unit quantity as in the units of theoretical Mendelism. There is not a 
word of this in Dr Boas' paper ; he speaks of his formula as applicable when things 
can be counted but not measured. Mr Yule speaks of it as applicable in its 
entirety to the 2 x 2-fold table ! — When pressed he says it would have been 
thought that no one acquainted with the work of the Biometric School — on 
Mendelism — could fail to understand what his passage signified. We wholly fail 
to understand it now. Is r^, i.e. <£, applicable to every fourfold division ? If so, 
why does not Mr Yule use it and drop his coefficient of association ? — In truth he 
* Journal of R. 8. S. Vol. xlv. p. 608. 
t Phil. Trans. Vol. 195 A (1900), p. 12. 
X Drapers' Company Research Memoirs, " On the Theory of Contingency," 1904, p. 21. 
