2.7. MEAN, STANDARD DEVIATION, AND COEFFICIENT OF VARIANCE 43
A random variable is a variable that associates a unique numerical value with every outcome of
an experiment. e value of a random variable will vary from trial to trial as the experiment is
repeated. For example, the diameter of a machined shaft is a random variable. In this example,
the diameter of a machined shaft is an experiment. Before the shaft has been completely ma-
chined, the actual value of the diameter of the shaft is unpredicted. For another example, if we
use a variable with a value 0 and 1 to represent the occurrence of “tail” and “head” in tossing a
coin experiment, this variable is also a random variable.
In mechanical design, almost all of the design parameters can be described or expressed
by random variables. For example, material strengths such as yield strength, ultimate strength,
and fatigue strength are random variables. e loadings such as axial loading, bending moment,
and torsion are random variables too.
2.7 MEAN, STANDARD DEVIATION, AND COEFFICIENT
OF VARIANCE
When a set of sample data of a random variable has been observed or collected, three typical
statistical characteristics—mean, standard deviation, and coefficient of variance—can be used
to describe or represent the sample data.
Mean is a measure of the central value of a random variable. It is also termed as the expected
value, mathematical expectation, or average. e mean can be calculated by the following equa-
tion when a set of sampling data of a random variable is collected:
x
D
iDn
X
iD1
x
i
=n; (2.27)
where x
i
is the ith sampling data of the random variable x and n is the number of sampling
data.
x
is the mean of random variable x.
Since the random variable will inherently have different sample values, the variation of
the random variable should be considered.
Standard deviation is a measure of variation or dispersion of a set of sampling data around its
central value:
x
D
8
ˆ
<
ˆ
:
q
P
n
iD1
.
x
i
x
/
2
=.n 1/ n < 30
q
P
n
iD1
.
x
i
x
/
2
=n n 30;
(2.28)
where
x
represents the standard deviation of a random variable x. e rest of the symbols in
Equation (2.28) are the same as those in Equation (2.27).
e coefficient of variance is a standardized nondimensional measure of variation or dispersion
of a set of sampling data around its central value. It is also known as relative standard deviation