498 
Correlation between Vaccination and Smallpox 
Using fourfold tables the value of r' varies very much according to the point 
chosen for division. Thus I have calculated the following three tables : 
TABLE XXX. 
Head Length. 
-o 
03 
CD 
CD 
m 
a 
QJ 
03 
CU 
19-0 or less 
Over 19-0 
Totals 
Less than 15'0 ... 
15-0 
701 
109 
599 
119 
1300 
228 
Totals 
810 
718 
1528 
?-' = -0837. 
TABLE XXXI. 
Head Length. 
19-0 or less 
Over 19-0 
Totals 
Less than 14-9 ... 
14-9 or 15-0 
619 
191 
458 
260 
1077 
451 
Totals 
810 
718 
1528 
r' = -2266. 
TABLE XXXIL 
Head Length. 
10-0 or less 
Over 19-0 
Totals 
Less than 14'8 ... 
14-8 to 15-0 
526 
284 
348 
370 
874 
654 
Totals 
810 
718 
1528 
r' = -2595. 
It is seen that in every case r' is less than r. 
If we use product moments a mathematical formula can be obtained. 
"\-r-)\<Tj' <T^<jy 0-// (I. 
Let 
27r Vl - r= cTr 
. e 2(1- 
represent the equation to the correlation surface, and x = ]i = hcTx be the equation 
to the bounding line. 
Let X, y be the coordinates of the centroid, r the correlation, both the centroid 
and the correlation applying to the curtailed surface only. 
Then 
r — 
J 
/• + 00 
{x - X) (y - 
- y) zdxdy 
{/J 
{x 
— CO 
— x}- zdxdy X 
(y -yy- zdxdy 
■(II-) 
