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
44 2. FUNDAMENTAL RELIABILITY MATHEMATICS
and is a much better tool to measure or describe the degree of variation of a random variable.
e following equation can calculate the coefficient of variance:
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.
When the number of sample size is less than 30, the mean, standard deviation, and co-
efficient of variance are the only three statistical characteristics for describing/representing this
random variable. e mean is the expected value of the random variable, that is, the true value
of the random variable if the uncertainty related with this random variable is eliminated. Both
the standard deviation and the coefficient of variance can describe the variation of the data. e
bigger the standard deviation and coefficient of variance are, the bigger the data scatter of the
random variable is. However, the standard deviation is a measure of the absolute variance of the
data around its mean. e coefficient of variance is a relative variance of data around its mean
per unit of the mean. For example, the random variable X has a mean 10 (psi) and standard de-
viation 5 (psi). e random variable Y has a mean 100 (psi) and standard deviation 5 (psi). For
these two random variables, they have the same standard deviation 5 (psi), but the coefficients
of variance of X and Y are 0.5 and 0.05. is result means that the random variable X has a
much bigger relative variation compared with the random variable Y .
When the number of a sample size of a random variable is more than 30, it is worthy
of finding a statistical-related function for describing/presenting it, which will be discussed in
Section 2.9.
Microsoft Excel and MATLAB are two available tools for engineers and engineering
students. We can use them to run the calculations of the mean and the standard deviation.
In Microsoft Excel, the function for calculating the mean of sampling data x
1
; x
2
; : : : ; x
n
is
average
.
x
1
; x
2
; : : : ; x
n
/
for calculating the mean:
In Microsoft Excel, there are several functions available to calculate the standard deviation
of sampling data x
1
; x
2
; : : : ; x
n
:
STDEV:S
.
x
1
; x
2
; : : : ; x
n
/
n < 30 for calculating the standard deviation
STDEV:P
.
x
1
; x
2
; : : : ; x
n
/
n 30 for calculating the standard deviation.
In MATLAB, if the sample data is stored in a matrix A, the command for the mean and the
standard deviation are:
mean.A/ for calculating mean
std.A/ for calculating standard deviation:
2.7. MEAN, STANDARD DEVIATION, AND COEFFICIENT OF VARIANCE 45
Example 2.25
e diameters of machined shafts by two machine operators are listed in Table 2.2. Analyze data
and draw some comments about these two operators.
Table 2.2: e diameter of machined shafts
Operator e Diameter of the Machined Shaft (inch)
Operator A 1.013, 0.944, 1.045, 1.022, 1.008, 1.069, 0.989, 1.009, 1.084, 0.967
Operator B 1.011, 1.024, 1.023, 1.007, 1.011, 1.007, 1.027, 1.009, 1.015, 1.016
Solution:
e diameter of a machined shaft by each operator is a random variable. Per Equations (2.26),
(2.27), and (2.28) or the functions/commands in Excel or MATLAB, the mean, standard devi-
ation, and coefficient of variance for operator A and B are listed in Table 2.3.
Table 2.3: Mean, standard deviation, and coefficient of variance of diameters of machined shafts
Operator Mean Standard Deviation Coeffi cient of Variance
Operator A 1.015˝ 0.0431 0.0425
Operator B 1.015˝ 0.0073 0.0072
From Table 2.3, the means of machined shafts by both operators are the same. e stan-
dard deviation of the diameter of machined shaft by operator A is much bigger than that by
operator B. e coefficient of variance of diameter of machine shaft by operator A is much big-
ger than that by the operator B. erefore, we could say that operator B has a much better skill
in machining because the diameter of machined shaft by him has a much smaller variation.
Example 2.26
Ultimate tensile strength of two different material’s tensile test specimens lists in Table 2.4.
Calculate the mean, standard deviation, coefficient of variance, and make some comments about
them.
Table 2.4: Ultimate tensile strength of two materials
Material Ultimate Tensile Strength (ksi)
Material A 34.3, 31.9, 30.6, 34.8, 25.6, 33.1, 32.5, 29.3, 30.6, 26.5
Material B 67.4, 71.4, 73.9, 81.3, 68.3, 71.7, 70.7, 63.3, 69.2, 64.8
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.