Karl Pearson and David Heron 
173 
Let us examine the effect of " adjusting " the tables in Mr Yule's cases of 
vaccination and death at Sheffield, Leicester and Homertou-Fulham. He asserts 
that it is "natural" to take 50% of vaccinated; we fail to understand why 50°/ o 
is more or less "natural" than 70°/ o or 95°/ 0 or than the percentage which actually 
occurs in the smallpox cases of those towns. Even if he takes 50% of vaccinated, 
why it should be " natural " to take 50% of deaths also is to us equally mysterious, 
and we believe must be to that juryman " any man of ordinary intelligence " to 
whom Mr Yule appeals. The following table gives the values that would arise 
from different methods of "adjusting" the tables: 
Smallpox — Vaccination and Death. 
Yule's 
Association 
Q 
Boas-Yulean <f> 
a 2 
P 
Percentage of Deaths 
Percentage of Vaccinations 
(a) 
50 7 
50 % 
(&) 
50% 
50% 
(«) 
Actual 
70 % 
(d) 
Actual 
Actual 
W 
Actual 
95 % 
Actual 
Actual 
Actual 
Sheffield 
Leicester ... 
Homerton -Fulham 
■902 
•862 
•804 
•630 
•572 
•504 
■531 
•249 
■423 
•479 
■233 
•409 
•383 
•190 
•084 
•769 
•611 
•662 
•432 
•187 
•379 
5-15/10 23 * 
2-46/10 4 
1-22/10 414 
It is clear that the Boas-Yulean method will give any results whatever between 
zero and those in the (b) column, according to what percentage we choose to take 
in the adjusted table of deaths and vaccinations. We can also change the perfectly 
arbitrary order that Mr Yule has given for the three towns. It appears to us 
that his statement that " it should have been an obvious matter that the 
ordinary theory of correlation, once that theory had been freed from any 
necessary relation to the theory of normal correlation, was applicable in its 
modify it by a Yulean selection using the factors I and m into 
la 
6 
la + b 
line 
md 
m (le + d) 
I (a+ mc) 
b +md 
Q remains unchanged. But 0 now takes the value 
Im (ad-be) , . , . / If , " a~b ,\ 7 " , bd , \ 
— ; = (ad - be) / . / ad + bc-i vmcd ad + bc + -,- + lac . 
Jlm{a + mc) (b + md)(la + b) (Ic + d) /V V m J\ 1 J 
This is a minimum when I and m are indefinitely great ; it is a maximum when 
m= »Jab/ Jed, l=JbdJJac, 
or <p— v ^ — ^^ = w, the coefficient of colligation. 
J ad + J be 
