Answer: for this problem, it is reasonable to expect that errors introduced in
the making of a three-foot tape measure again are accountable for inaccuracies
in the three-yard tape measures. It is then reasonable to assume that the
coefficient of variation is constant for tape measures of all lengths
manufactured by this company. Thus
and the standard deviation for a three-yard tape measures is
feet.
This example illustrates the fact that the coefficient of variation is often
used as a measure of quality for products of different sizes or different weights.
In the concrete industry, for example, the quality in terms of concrete strength
is specified by a coefficient of variation, which is a constant for all mean
strengths.
Central moments of higher order reveal additional features of a distribution.
The coefficient of skewness, defined by
gives a measure of the symmetry of a distribution. It is positive when a uni-
modal distribution has a dominant tail on the right. The opposite arrangement
produces a negative 1. It is zero when a distribution is symmetrical about the
mean. In fact, a symmetrical distribution about the mean implies that all odd-
order central moments vanish.
The degree of flattening of a distribution near its peaks can be measured by
the coefficient of excess, defined by
A positive 2 implies a sharp peak in the neighborhood of a mode in a unimodal
distribution, whereas a negative 2 implies, as a rule, a flattened peak. The
significance of the number 3 in Equation (4.12) will be discussed in Section 7.2,
when the normal distribution is introduced.
4.1.3 Conditional Expectation
We conclude this section by introducing a useful relation involving conditional
expectation. Let us denote by that function of random variable Y for
Expectations and Moments 83
v/mv0 : 03
3
0 : 01 ;
0 : 01 9 feet)
0 : 09
1
3
^3
4 : 11
2
4
^4
3 : 4 : 12
EfXjYg