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

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