304 ◾ Simple Statistical Methods for Software Engineering
Standard Weibull Curve
If the scale parameter is 1 and the location parameter is 0, the Weibull curve
assumes a standard form shown below, which is completely controlled by the shape
parameter as follows:
W x x e
x
( ) =
− −
α
α
α
1
(19.5)
We have plotted three standard Weibull curves for three different values of
shape factor in Figure 19.2.
NIST defines the several useful properties for the standard Weibull distribu-
tion. e one we need to look at is the formula for median. is relationship is of
technique of using explosive charges to determine the type of ocean bottom
sediments and their thickness. e same technique is still used today in off-
shore oil exploration. In 1939, he published his paper on Weibull distribution
in probability theory and statistics. In 1941, BOFORS, a Swedish arms fac-
tory, gave him a personal research professorship in Technical Physics at the
Royal Institute of Technology, Stockholm.
Weibull published many papers on the strength of materials, fatigue, rup-
ture in solids, bearings, and of course the Weibull distribution, as well as
one book on fatigue analysis in 1961. In 1951, he presented his most famous
paper to the American Society of Mechanical Engineers (ASME) on Weibull
distribution, using seven case studies.
Weibull worked with very small samples at Pratt & Whitney Aircraft and
showed early success. Dorian Shainin, a consultant for Pratt & Whitney,
strongly encouraged the use of Weibull analysis. Many started believing in
the Weibull distribution.
e ASME awarded Weibull their gold medal in 1972, citing him as “a
pioneer in the study of fracture, fatigue, and reliability who has contributed
to the literature for over thirty years. His statistical treatment of strength and
life has found widespread application in engineering design.”
Weibull’s proudest moment came in 1978 when he received the great gold
medal from the Royal Swedish Academy of Engineering Sciences, personally
presented to him by King Carl XVI Gustaf of Sweden.
e Weibull distribution has proven to be invaluable for life data analysis
in aerospace, automotive, electric power, nuclear power, medical, dental, elec-
tronics, and every industry.
Weibull Distribution ◾ 305
immense help for curve fitting. From the median value of data, we can directly
calculate shape factor as follows:
Median = ln( )2
1
α
(19.6)
Box 19.3 an impReSSive Range
We can find several applications of the Weibull application in the literature;
here are a few instances.
e strength of yarn is not a single-valued property but a statistical vari-
able. e statistical distribution of yarn strength is usually described by the
normal distribution. As an improvement, Weibull statistics was used by Shi
and Hu [2] as a tool to analyze the strength of cotton yarns at different gauge
lengths to find the relationship between them.
Propagation delay in CMOS circuits is characterized by Weibull distribu-
tion. e experiments of Liu et al. [3] on large industrial design demonstrate
that the Weibull-based delay model is accurate, realistic, and economic.
Fire interval data are known to belong to the Weibull family. A study by
Grissino-Mayer [4] shows that two- and three-parameter Weibull distribu-
tions were fit to fire interval data sets. e three-parameter models failed to
provide improved fits versus the more parsimonious two-parameter models,
indicating that the Weibull shift parameter may be superfluous.
Reliability can be predicted only with the help of suitable models. Sakin
and Ay [5] studied reliability and plotted fatigue life distribution diagrams
of glass fiber-reinforced polyester composite plates using a two-parameter
Location
=
0 alpha
=
5 beta
=
1
Location
=
0 alpha
=
2 beta
=
1
Location
=
0 alpha
=
1 beta
=
1
–1 0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
1 2 3
x
Location = 0, scale = 1
Weibull probability
4 5 6
Figure 19.2 Weibull curves with different shape factors.
306 ◾ Simple Statistical Methods for Software Engineering
Three-Parameter Weibull
We can think of a general Weibull curve by including a location parameter as
follows:
W x
x
e x
x
( ) ,=
−
> >
−
−
−
α
β
µ
β
µ α β
α
µ
β
α
1
0, (19.7)
where the fitted values of the parameters are as follows: μ is the location parameter,
α is the shape parameter, and β is the scale parameter.
is equation has been generated by substituting x by (x − μ) in Equation 19.2.
Figure 19.3 shows three Weibull curves with three location parameters, the
shape and the scale are kept constant. It is evident how changing the location
parameter shifts the curve along the x-axis. When location = 0, we obtain the basic
Weibull curve, which stays on the positive side of the x-axis. is by itself offers an
advantage of modeling nonzero values. When location value is increased, the curve
has a potential to model real life data with higher positive numbers that stay at
Weibull distribution function. e reliability percentage can be found easily
corresponding to any stress amplitude from these diagrams.
Robert et al. [6] used the Weibull probability density function as a diam-
eter distribution model. ey stated, “Many models for diameter distribu-
tions have been proposed, but none exhibit as many desirable features as the
Weibull. Simplicity of algebraic manipulations and ability to assume a variety
of curve shapes should make the Weibull useful for other biological models.”
Location
=
10 alpha
=
1.4 beta
=
4
Location
=
5 alpha
=
1.4 beta
=
4
Location
=
0 alpha
=
1.4 beta
=
4
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0 5 10 15 20
x
Weibull probability
25 30 35
Figure 19.3 Weibull curves with different location parameters.
Weibull Distribution ◾ 307
some distance from the origin, such as software productivity. First, we should park
the curve at an optimal location and the adjusted scale and shape until we obtain
minimum error in fitting.
e use of this three-parameter model is illustrated in the following paragraphs.
In his 1951 path-breaking paper “A Statistical Distribution Function of Wide
Applicability,” Weibull presented seven case studies of the Weibull distribution
application [7]. e case for BOFORS steel strength is interesting; hence, we
studied this case to illustrate the modern version of the Weibull distribution.
Weibull presented steel strength data that we convert into a histogram, as shown
in Figure 19.4.
We have fitted the three-parameter Equation 19.7 to Weibull’s data using the
following steps:
1. e location parameter was fixed at the minimum value of reported steel
strength, 32 units.
2. e choice of shape parameter α is based on the shape of the histogram shown
in Figure 19.4. e choice of 2 is obvious.
3. e scale parameter was adjusted to obtain the minimum least square error.
We begin the iteration by keeping the initial value of the scale factor at the
median departure from the minimum value. e initial value of the scale
factor thus obtained is 3.5. We calculate the mean square error at this stage
between the predicted and the actual probability value. Tentative perturba-
tions of scale factor indicate that moving up reduces error. We increase the
scale factor in steps of 0.1 until the minimum error is achieved. e best value
of scale parameter happens to be 4.7.
0
32 33 34 35 36 37
Yield strength × 1.275 kg/mm
2
Frequency
38 39 40 42
10
20
30
40
50
60
70
80
90
Figure 19.4 Historical steel strength data used by Weibull, the scientist.
308 ◾ Simple Statistical Methods for Software Engineering
e fitted curve is shown in Figure 19.5.
e highlight of the model lies in introducing a location parameter that is nec-
essary, in this case, where data have a large minimum value.
Software Reliability Studies
Defect discovery during the life cycle follows a Weibull curve. e curve math-
ematically extends to beyond the release date. e tail area depends on discovery
rates before release and is governed by the Weibull equation. Figure 19.6 shows a
two-parameter Weibull PDF with a shape parameter of 2 and a scale parameter of
20 days, a defect discovery curve, with release date marked; the tail beyond release
denotes residual defects.
Figure 19.7 is the cumulative distribution function (CDF) of the same Weibull,
but the interpretation is interestingly different.
e y-axis represents the percentage of defects found and is likened to product
reliability at any release point. Using this model, we can predict reliability at delivery.
Both Figures 19.4 and 19.5 provide approximate but useful judgments about
defect discovery.
Above all they provide valuable clues to answer the question, to release the
product or test it further?
An experienced manager can make objective decisions using these graphical
clues. Some organizations have attempted to declare the CFD value at release date
as software reliability and use this numerical value to certify the product.
Vouk [8] reported the use of Weibull models in the early testing (e.g., unit test-
ing and integration testing phases) of a very large telecommunication system. It is
0.25
0.20
0.15
0.10
0.05
0.00
30 32 34 36 38 40 42 44
Yield strength × 1.275 kg/mm
2
Weibull, location 32 shape 2 scale 4.7
Weibull fit
BOFORS steel data
Figure 19.5 Historical Weibull model of steel strength.
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