2.11. STANDARD DEVIATION AND COEFFICIENT OF VARIANCE 61
where Var
.
X
/
refers to the variance of a random variable X and
x
is the mean of a random
variable X.
For a continuous random variable, the variance of a random variable X is
Var
.
X
/
D E
X
2
2
x
D
Z
1
1
x
2
f
.
x
/
dx
2
x
: (2.49)
For a discrete random variable, the variance of a random variable X is
Var
.
X
/
D E
X
2
2
x
D
X
All i
x
2
i
p
.
x
i
/
2
x
: (2.50)
In the previous section, Equation (2.28) is used to calculate the standard deviation of sampling
data. After the definition of the PDF and the PMF are defined, the variance of a random variable
can define the standard deviation.
Standard deviation is a measure of variation or dispersion of a set of data values around its
central value and is defined as the square root of the variance of a random variable.
For a continuous random variable, the standard deviation of a random variable X is
x
D
p
Var
.
X
/
D
q
E
Œ
X
2
2
x
D
s
Z
1
1
x
2
f
.
x
/
dx
2
x
: (2.51)
For a discrete random variable, the standard deviation of a random variable X is
x
D
p
Var
.
X
/
D
q
E
Œ
X
2
2
x
D
s
X
All i
x
2
i
p
.
x
i
/
2
x
; (2.52)
where
x
and Var
.
X
/
are the standard deviation and the variance of the random variable X,
respectively. f
.
x
/
is the PDF of a continuous random variable X. p
.
x
i
/
is the PMF of a discrete
random variable X .
x
is the mean of random variable X.
After the mean and standard deviation of a random variable X have been determined, we
can define the coefficient of variance.
e coefficient of variance is the ratio of the standard deviation to the mean of a random variable
and can be directly calculated by Equation (2.29), from page 44, which is repeated here:
x
D
x
x
; (2.29)
where
x
is the coefficient of variance of a random variable x.
x
and
x
are the standard devi-
ation and the mean of a random variable X.