500 
Correlation between Vaccination and Smallpox 
Since p and q are essentially positive and r" < 1 we see from (VIII.) that it is 
impossible for r to be numerically greater than r. If, however, r = 0 or 1, then 
r' also =0 or 1. 
The same can be shewn by differentiating p — h. 
The equation (VIIT.) can be written thus : 
.(IX.). 
In the case of smallpox correlations r' is the calculated apparent correlation. 
r is unknown, and cannot be calculated since h is also unknown. But it will be 
seen that when r is constant r is greater when 
— lipq _ fp\ 
is less. 
Calling this 
values of h : 
"(f) 
quantity 
z, the following gives 
values 
of 
k 
z 
h 
2 
-2-0 
+ ■119 
+ 0-1 
+ 
658 
1-5 
-m 
0'2 
•662 
1-0 
•371 
0^3 
•700 
0-5 
•514 
0^4 
•715 
0-4 
•540 
0^5 
•729 
0-3 
•565 
1-0 
•796 
0-2 
■592 
1-5 
•848 
0-1 
■613 
2^0 
•854 
-0-0 
+ •637 
+ 3^0 
1 
■30 
z for a range of 
It will be seen that, as h increases, z and consequently the discrepancy between 
r and r' also increase*. 
* A simple geometrical interpretation of the quantity z is obtained tlius. Let C, Fig. 2, represent the 
centroid of the whole figure, CA', CY the axes, P2V the dividing line, and 0 the centroid of the curtailed 
figure. Then CW^=/)(r^. Draw Oil/ perpendicular to CA. 
1 
i 
( 
D 
c 
N 
M 
Fig. 2. 
CM-- 
: C3r- - CM . CN= CM . MN. 
