174 THEORY OF STATISTICS. 
are associated with large values of y, and conversely (as in 
Tables I.-IV.), negative if small values of « are associated with 
large values of ¥ and conversely (as in Table V.). The numerical 
value cannot exceed +1, for the sum of the series of squares 
in equation (7) is then zero and the sum of a series of squares 
cannot be negative. If r= +1, it follows that all the observed 
pairs of deviations are subject to the relation x/y=o,/o,: this 
Fathers stature 
ol 61 -~ 66 68 i i2 
we Ey 
66 
Cc 
67 E 
Sh 
3 
69 
« 
a) 
R 
S 
“wv 77 
73 
75 - 
Fig. 37.—Correlation between Stature of Father and Stature of Son (Table 
III.) : means of rows shown by circles and means of columns by crosses : 
r= +0°51. 
would be the case if the circles and crosses in such a diagram as 
fig. 33 all lay on one and the same straight line. From these 
properties 7 is termed the coefficient of correlation, and the 
expression (4), 7 =p/o,0, =3(zy)/N.0.0 should be remembered. 
It should be noted that, while r is zero if the variables are 
independent, the converse is not necessarily true: the fact that 
r is zero only implies that the means of rows and columns 
lie scattered round two straight lines which do not exhibit 
Te 2