Descriptive Statistics 337
of component y on the occurrence of value v of component x. The influence,
as will be shown later, is mutual. Hence, one is interested in the distribution
of one component given a certain value for the other component. This dis-
tribution is called the conditional frequency distribution. The conditional
relative frequency of x conditional on w is defined by
(^) fv==fxw
fvw
fw
() ()
(,)
()
xwxy
y|, (A.6)
The conditional relative frequency of y on v is defined analogously.
In equation (A.6), both commonly used versions of the notations for the
conditional frequency are given on the left side. The right side, that is, the
definition of the conditional relative frequency, uses the joint frequency at v
and w divided by the marginal frequency of y at w. The use of conditional
distributions reduces the original space to a subset determined by the value
of the conditioning variable. If in equation (A.6) we sum over all possible
values v, we obtain the marginal distribution of y at the value w, fy(w),
in the numerator of the expression on the right side. This is equal to the
denominator. Thus, the sum over all conditional relative frequencies of x
conditional on w is one. Hence, the cumulative relative frequency of x at
the largest value x can obtain, conditional on some value w of y, has to be
equal to one. The equivalence for values of y conditional on some value of
x is true as well.
Analogous to univariate distributions, it is possible to compute mea-
sures of center and location for conditional distributions.
independence
The previous discussion raised the issue that a component may have influ-
ence on the occurrence of values of the other component. This can be ana-
lyzed by comparing the joint frequencies of x and y with the value in one
component fixed, say x = v. If these frequencies vary for different values of
y, then the occurrence of values x is not independent of the value of y. It
is equivalent to check whether a certain value of x occurs more frequently
given a certain value of y, that is, check the conditional frequency of x con-
ditional on y, and compare this conditional frequency with the marginal
frequency at this particular value of x.
The formal definition of independence is if for all v,w
(^) fvxy,(,wf)(= xyvf)(⋅ w) (A.7)