Karl Pearson and David Heron 
263 
It will be seen that the values we have obtained are not wholly in agreement 
even to the second place of figures with the values obtained by Mr Yule, who has 
selected only 8 out of the 15 diagonal cases. Diagrams XVI and XVII show that 
Diagram XVI. Correlation of Barometer at Laudale and Southampton. Actual values 
of tetrachoiic r t . 
Height at which division was taken. 
Diagram XVII. Mr Yule's representation of tetrachoric r t . 
29-15 29 25 29 45 29-65 29-95 30-25 30-45 30-55 
Height at which division ivas taken. 
deduce from one of Mr Yule's type of coefficients a second which has greater or less or no significance 
as the case may be. Further : 
1 1-P 2 
P n 1 _ r i i " 
1 e V« (l + e)2 
where u = ~ — —. , but as n increases this tends to take the limit : 
»(l + e V»)2 e 
1 (1 + k)* 
n 4/c ' 
or can be made less than unity. In other words, given )• or Q, we can always choose a new quantity p 
or O, which has a smaller absolute probable error. If colligation for a given series has a smaller 
probable error than Q or a smaller one than r t , it is always possible to choose new functions p or fi, 
which are absolutely determined by r t or by Q and w, which have still smaller probable errors, and 
yet are true Yulean coefficients of association, ranging between 0 and 1 ! The fact is that the term 
"probable error" has no meaning for these coefficients until Mr Yule has discussed the nature of their 
frequency distribution, which is certainly not Gaussian. In the text we have compared r t and Q with 
their probable errors because it probably gives some rough estimate of their relative stability, and it is 
the test Mr Yule has himself chosen. 
