46 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Solution:
e ultimate tensile strength is a random variable. Per Equations (2.27), (2.28), and (2.29) or
the functions/commands in Excel or MATLAB, the mean, standard deviation, and coefficient
of variance of ultimate tensile strengths of two materials are listed in Table 2.5.
Table 2.5: e mean, standard deviation, and coefficient of variance of ultimate tensile strength
Ultimate Tensile
Strength
Mean Standard deviation Coeffi cient of variance
Material A 30.9 (ksi) 3.080 0.100
Material B 70.2 (ksi) 5.056 0.072
From Table 2.5, the mean of the ultimate tensile strength of material B is much bigger
than that of materials A. So the material B is much stronger than material A. e standard
deviation of ultimate tensile strength of materials B is 5.056 ksi, bigger than that of the material
A. It means that the absolute variation about its mean for material B is bigger. However, the
coefficient of variance of material A is 0.100 and is bigger than that of the material B. is says
that the relative variation of the ultimate tensile strength of material A is much bigger than that
of material B.
2.8 HISTOGRAM
2.8.1 DEFINITION OF A HISTOGRAM
When the sample size is big, such as more than 30, we can still use the mean, standard deviation,
and coefficient of variance to describe them, but we can dig out more useful information from
such big set of data. One typical tool is to build the histogram based on the data.
A histogram is a display of statistical information that uses rectangles to show the frequency of
data items in successive numerical intervals of equal size.
An example of a histogram is shown in Figure 2.6, which is a histogram of a material’s
ultimate tensile strength. In a histogram, the width of a rectangular bar represents the range
value of a bin, and the height of the bar is the frequency of sample data inside the bin’s range.
e bin in a histogram can be considered as a “bucket” with a lower boundary and an upper
boundary. After a set of sample data of a random variable x are provided, the minimum and the
maximum of the samples can be determined as x
min
and x
max
. en, the bin size and values of
2.8. HISTOGRAM 47
the bin ranges can be determined by the following equations:
x D
.
x
max
x
min
/
J
; J 6; (2.30)
B
j L
D x
min
C
.
j 1
/
x; j D 1; 2; : : : ; J; (2.31)
B
j U
D x
min
C j x; j D 1; 2; : : : ; J; (2.32)
where J is the number of bins, which is typically larger than or equal to 6. x is the range of
the bin values. e symbol j represents the j th bin. B
j L
and B
j U
are the lower boundary
and the upper boundary of the j th bin.
rough all the sample data x
i
, we can use the following recurrence formula to determine
the values of frequency in each bin:
n
j
D
8
<
:
n
j
C 1 for j D 1 if B
j L
x
i
B
j U
n
j
C 1 for j ¤ 1 if B
j L
< x
i
B
j U
;
(2.33)
where n
j
is the frequency in the j th bin, that is, the number of sample data in the j th bin. x
i
is the ith collected sampling data.
e histogram is a very useful tool to help us to smoothly understand an important concept
of the PDF of a random variable and help us to tentatively select the type of distribution for a
random variable, which will be discussed in Sections 2.9 and 2.13.
Now, we will use one example to show how to create and to explain a histogram.
Example 2.27
A set of 238 ultimate tensile strength data of material is listed in Table 2.6. Create and explain
the histogram of the data with 15 bins.
Solution:
According to the data in Table 2.6, the minimum and maximum of the sampling data are 48.4
(ksi) and 76.8 (ksi). e range of a bin value per Equation (2.30) with 15 of bins is:
x D
.
x
max
x
min
/
K
D
.
76:8 48:4
/
15
D 1:893 .ksi/:
Per Equations (2.31) and (2.32), we can get the lower boundary and upper boundary of each
bin as shown in the second column of Table 2.7. Per Equation (2.33), we can determine the
frequency in each bin as shown in the third column of Table 2.7.
e histogram based on the data in the second and third column of Table 2.7 is shown
in Figure 2.6. e histogram clearly shows that the frequencies in each bin are different even
though the width or the range of each bin is the same. For example, there is 48 of sampling data
in the 7th bin, but only 13 of sampling data in the 4th bin.
When the frequency of each bin is divided by the total number of sampling data, in this
case, 238, we can get the relative frequency of each bin as listed in the 4th column of Table 2.7.
48 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Table 2.6: A set of ultimate tensile strength data
Ultimate Tensile Strength (ksi)
63.8, 69.3, 51.7, 65.2, 62.8, 55.8, 59.6, 62.9, 76.8, 73.4, 55.6, 74.5, 64.6, 61.2, 64.5, 59.6, 60.9, 69.9, 71.5,
67.5, 64.3, 58.3, 64.5, 68.5, 63.6, 65.9, 64.6, 59.1, 62.7, 58.1, 65.3, 55.5, 56.9, 58.0, 48.8, 67.6, 62.8, 58.2,
67.3, 54.1, 61.0, 60.4, 62.8, 62.8, 58.7, 61.3, 60.7, 64.1, 66.1, 66.2, 57.7, 61.8, 56.2, 56.7, 61.4, 68.0, 58.1,
63.0, 60.5, 66.3, 56.8, 61.6, 63.8, 66.2, 68.1, 61.8, 55.0, 58.3, 56.9, 71.6, 58.8, 64.7, 60.6, 65.3, 58.2, 55.4,
55.3, 63.5, 60.7, 60.6, 67.6, 62.7, 62.3, 68.3, 58.0, 64.4, 65.1, 60.4, 62.4, 56.4, 56.5, 61.9, 64.6, 72.6, 58.6,
62.3, 61.1, 53.1, 59.6, 53.7, 65.1, 57.6, 61.9, 59.1, 62.8, 58.9, 63.6, 64.6, 68.8, 60.6, 52.3, 57.8, 67.3, 56.8,
65.6, 62.0, 67.6, 53.0, 60.6, 56.3, 73.9, 65.0, 67.4, 56.9, 59.4, 60.3, 66.2, 60.3, 64.5, 52.6, 59.9, 57.9, 54.7,
63.6, 62.7, 61.6, 55.7, 66.3, 63.0, 60.2, 61.5, 60.3, 53.9, 60.2, 57.9, 57.2, 56.5, 59.2, 52.8, 65.6, 63.7, 61.4,
61.3, 58.0, 65.8, 60.9, 58.4, 67.3, 60.5, 58.9, 60.2, 57.8, 56.6, 72.3, 68.6, 62.8, 56.0, 57.7, 60.7, 64.9, 55.7,
51.4, 55.2, 62.9, 63.1, 63.4, 60.9, 62.2, 59.4, 65.2, 55.6, 63.4, 57.8, 60.0, 63.8, 65.9, 56.6, 66.9, 64.3, 61.2,
60.6, 60.5, 60.1, 61.5, 61.7, 65.0, 68.0, 63.5, 60.5, 64.1, 62.2, 57.0, 65.5, 62.8, 62.0, 63.7, 62.6, 57.4, 60.8,
60.8, 59.2, 69.7, 57.7, 59.4, 58.4, 56.4, 60.6, 60.3, 68.0, 60.4, 56.9, 68.3, 66.8, 60.5, 55.0, 59.5, 60.8, 62.6,
60.3, 63.4, 63.1, 56.1, 57.4, 58.3, 59.3, 60.1, 61.5, 48.4,
Table 2.7: Statistical data of a histogram
Bin# Bin’s range Frequency Relative Frequency Relative-Density Frequency
1 48.400–50.293 2 0.0084 0.0044
2 50.293–52.187 2 0.0084 0.0044
3 52.187–54.080 7 0.0294 0.0155
4 54.080–55.973 13 0.0546 0.0288
5 55.973–57.867 28 0.1176 0.0621
6 57.867–59.760 31 0.1303 0.0688
7 59.760–61.653 48 0.2017 0.1065
8 61.653–63.547 35 0.1471 0.0777
9 63.547–65.440 30 0.1261 0.0666
10 65.440–67.333 17 0.0714 0.0377
11 67.333–69.227 14 0.0588 0.0311
12 69.227–71.120 3 0.0126 0.0067
13 71.120–73.013 4 0.0168 0.0089
14 73.013–74.907 3 0.0126 0.0067
15 74.907–76.800 1 0.0042 0.0022
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.