282 ◾ Simple Statistical Methods for Software Engineering
TBF seconds
Scale 15.464 Shape 0.789
Scale 16.404 Shape 0.105
0
0
5,000,000
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25,000,000
30,000,000
5E-08
1.5E-07
2.5E-07
3.5E-07
0.0000001
0.0000002
0.0000003
Figure 17.10 Log-normal PDF of TBF clusters.
TBF seconds
Frequency
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
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More
2
4
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16
Figure 17.9 Histogram of TBF.
Software Size Growth ◾ 283
Review Questions
1. Compare normal distribution with log-normal distribution.
2. Provide an example of the logarithmic scale used in practice.
3. What is the formula for the mean of log-normal distribution with shape σ
and scale β?
4. What is the formula for the variance of log-normal distribution with shape σ
and scale β?
5. Who invented the log-normal distribution?
Box 17.3 AnALogy—AcceLerATed LiFe TesT
If we record time served by computer hardware before the first failure occurs,
what we obtain is life data, and the distribution is called life distribution. Life
data manifest log-normal distribution, both for machines and for humans.
Log-normal distribution for machine failure is used to measure and improve
reliability. Log-normal distribution for human life is used to calculate life
insurance premiums. In both the cases, we estimate or “measure” life proper-
ties using log-normal distribution.
In accelerated life tests of systems, extreme conditions are created, mim-
icking real-world scenarios, to stimulate failure events much earlier than nor-
mal. Moreover, the tests are stopped either after a certain time or after a
certain number of failures occur. Tests are neither conducted indefinitely nor
till all failures occur. Life data thus obtained are truncated or “censored.” e
picture obtained is partial, but the full picture can be constructed by statisti-
cal analysis. One such attempt is to fit log-normal distribution to censored
life data and see even the unseen part of the full story of failure probabilities.
Accelerated life tests are faster and cheaper.
Dube et al. [8] discussed the problem of applying log-normal distribu-
tion to censored life data, particularly parameter extraction. ey analyze car
failure data from Lawless [9] for this purpose. e Lawless data “shows the
number of thousand miles at which different locomotive controls failed in a
life test involving 96 controls. e test was terminated after 135,000 miles,
by which time 37 failures had occurred.”
Dube et al. took up and answered the question whether the data fit the log-
normal distribution or not. ey showed that “the data fits reasonably well.”
Analogically, software is stressed by usage testing (e.g., user acceptance
testing), triggering failure events. A record of failure times is called life data.
Tests are not indefinitely conducted but terminated at some point of time,
either after a certain number of defects have been found or after the lapse of
certain time, depending on estimation and strategy of testing. Life data thus
obtained can be fitted to log-normal.
284 ◾ Simple Statistical Methods for Software Engineering
Exercises
1. Download reliability data from CSIAC website [7]. Select postrelease failure
events for any one project. Draw a histogram of the failure interval data. Fit
the failure interval data to a log-normal distribution.
2. Use the above model to predict the current reliability related to the software
project data you have selected.
References
1. Available at http://nasa-softwaredefectdatasets.wikispaces.com.
2. E. Limpert, W. A. Stahel and M. Abbt, Log-normal distributions across the sciences:
Keys and clues, BioScience, 51(5), 341–352, 2001.
3. NIST/SEMATECH Engineering Statistics Handbook. e National Institute of
Standards and Technology (NIST) is an agency of the U.S. Department of Commerce.
Available at http://www.nist.gov/itl/sed/gsg/handbook_project.cfm.
4. J. Aitchison and J. A. C. Brown, On criteria for descriptions of income distribution,
Metroeconomica, 6, 88–107, 1954.
5. Journal of the Reliability Analysis Center, Volume 13, Second Quarter, RAC, New York,
2005.
6. P. V. Varde, Role of statistical vis-à-vis physics of failure methods in reliability engineer-
ing, Journal of Reliability and Statistical Studies, 2(1), 41–51, 2009.
7. Available at https://sw.thecsiac.com/databases/sled/swrelg.php.
8. S. Dube, B. Pradhan and D. Kundu, Parameter estimation of the hybrid censored
log-normal distribution, Journal of Statistical Computation and Simulation, 81(3), 275–
287, 2011.
9. J. F. Lawless, Statistical Models and Methods for Lifetime Data, Wiley, New York, 2003.
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