W. R. Macdonell 
181 
in Mr G. Udny Yule's paper " On the Theory of Correlation " in the Journal of the 
Royal Statistical Society, Vol. LX., Part iv., December 1897. We have modified 
Mr Yule's formula for the square of the standard deviation by diminishing it by 
the quantity J^, in accordance with Mr W. F. Sheppard's paper in the Proceedings 
of the London Mathematical Society, Vol. xxix., Nos. 634/5. 
The results are as follows : 
TABLE 1. 
3000 Criminals. 
Standard Deviation 
Mean 
Head Length (cm.) ... 
Head Breadth (cm.)... 
L. M. Finger (cm.) ... 
Height (ins.) ... 
•6046 + -0053 
•5014 + ^0044 
•5479 +'0048 
2-5410 + -0221 
19-1663 + -0075 
15-0442 + -0062 
11 -5474 + -0068 
65-5355 + -0313 
TABLE 2. 
3000 Criminals. 
Coefficient of Correlation 
Head Length and Head Breadth 
Head Breadth and Height 
L. M. Finger and Height 
-4016 + -0103 
•1831 + -0119 
-6608 + -0069 
(6) Tests of Normality. At the foot of Tables IL and IIL (see pp. 215, 216) 
are shown the mean Finger Length and mean Head Breadth of each column array ; 
these means are plotted on Figs. 1 and 2, and the lines are drawn which show the 
theoretical regression of Finger on Height, and Head Breadth on Height. The 
slope of these lines is calculated from the formula tan 6 = — -, where r is the 
coefficient of correlation, and cr,, o-o, the standard deviations of the correlated organs. 
(See Yule, loc. cit.) The means of the column arrays at the extremities of the 
tables are not included, as they are based on such small frequencies. 
It will be observed that the regression lines fit the observations very well, i.e., 
the regression is very closely linear. 
