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Classic Case Studies in Reliability Analysis: The Weibull Distribution

It has been exactly 50 years since Professor Waloddi Weibull published his groundbreaking paper, "A Statistical Distribution Function of Wide Applicability," in the Journal of Applied Mechanics. Dr. Bob Abernethy has called this paper "the hallmark paper of Weibull analysis." Although Weibull had published his initial work in 1939 in the Proceedings of the Royal Swedish Institute for Engineering Research, his 1951 paper was the one that would expose his work to a wide audience.

The 1951 paper contained seven case studies. Three of the case studies were modeled using a three parameter Weibull distribution and four were modeled using a two population mixed Weibull. It is interesting that only one of the seven examples involved life data; the others involved the distribution of physical characteristics, such as strength and size. We would like to revisit the first of these examples using modern computer tools.

Weibull's Strength of Bofors Steel Example
Professor Weibull was Scientific Advisor to AB Bofors, the major Swedish armaments company. The first example in his paper concerned the yield strength of Bofors steel, measured in kg/mm2. Professor Weibull had the advantage of working with very large sample sizes; in this case he had 388 observations. Professor Weibull made an unusual choice of scale for the strength values. He modeled these strengths using the distribution we now call the three-parameter Weibull distribution.

As modern day analysts, we enter this data set into Weibull++ 6 and our first point of interest is the goodness of fit of the three-parameter Weibull distribution. We can use the Distribution Wizard to determine that the three-parameter Weibull distribution is an excellent model for this data, dominating the other possible choices handily. Professor Weibull used a cumbersome chi-squared test, computed by hand, to convince his readers of the goodness of fit of the distribution.

After confirming the applicability of the three-parameter Weibull distribution for the data, we fit the recommended model and obtain results very similar to those found by Professor Weibull. The small differences can be attributed to our modern use of median ranks in the calculations.

It is interesting to examine the graphs used by Professor Weibull to present his results. In Figure 1, we see an original graph from Weibull's 1951 paper. The vertical axis uses the ln(ln(1/(1-p)) transformation and the horizontal axis uses the ln(x) transformation. This agrees with modern practice, except that we now re-label those scales in the original units. This makes it much easier to read the graph.

 Figure 1: Prof. Weibull's probability plot for Bofors steel

Note that Professor Weibull has plotted the translated values (strength minus the threshold parameter) and that the data is remarkably linear, indicating an excellent fit. Also note how difficult it would be to extract useful information, such as the median yield strength, from this graph.

The Weibull++ 6 plot of the Bofors steel data is presented in Figure 2. You can see that it is much easier to read the Weibull++ 6 graph than the original graph from Professor Weibull's paper. The essentials (choice of scale, estimation of parameters) are the same, but the user interface is greatly simplified. Both the translated data (on the left, represented by red circles) and the original data (on the right, blue triangles) are presented in the Weibull++ 6 plot. The median yield strength can be obtained from the plot and is 45.45 kg/mm2.

Professor Weibull did not present an analysis of the fitted model in his paper. Using Weibull++ 6, we are able to quickly learn that the median yield strength is 45.45, the mean yield strength is 45.52 and that only 0.68% of the units in the population are expected to have a yield strength of 40 or lower. The percent failing below a yield strength of 40 can be easily calculated using the Weibull++ QCP, also shown in Figure 2.

 Figure 2: Weibull++ probability plot and QCP results

It is unpleasant to imagine how long it took Professor Weibull, working by hand, to generate Figure 1. It was but the work of a few minutes using the Weibull++ software, and most of that was data entry.

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