What You Need to Know About Weibull Distributions

The second unexpected characteristic of the Weibulls is seen in the top curve of Figure 3, which shows the areas within three standard deviations of the mean. While these areas do drop slightly at first, they stabilize for the J-shaped Weibulls at about 98%. This means that a fixed-width, three-standard-deviation central interval for a Weibull distribution will always contain approximately 98% or more of that distribution.

So, what is changing as you select different Weibull probability models? To answer this question, Figure 2 considers 23 different Weibull models. For each model we have the skewness and kurtosis; the areas within one, two, and three standard deviations on either side of the mean; and the z-score for the most extreme part per thousand of the model.

Fortunately, by understanding the properties of probability models, a simpler approach is possible. Once we realize that virtually all mound-shaped and J-shaped probability models will have 98% to 99.9% within three standard deviations of the mean, we no longer need to laboriously fit a particular probability model to our data. We can use three-sigma limits centered on the average to filter out virtually all of the routine variation, and treat anything left over as a potential signal.

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and the variance of a Weibull distribution is

Figure 8 summarizes how this approach works with lognormals, gammas, and Weibulls. As long as our processes are subject to signals that have an economic impact, we need not be concerned with the exact risk of a false alarm. We collect data to take action when appropriate. The one-size-fits-all approach of three-sigma limits is sufficiently conservative to minimize the risk of false alarms while allowing us to detect signals of economic importance, and they do this regardless of the shape of the histogram.

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Figure 3: How the coverages vary with skewness for Weibull distributions

This approach, where you specify the amount of noise to be filtered out, is equivalent to using the table in Figure 6. So, here you fit a model, choose a specific area to filter out, and then find the exact width of interval to use in packaging the noise. (Unfortunately, the apparent precision of this approach is offset by the uncertainty inherent in fitting a specific model to the data.)


What You Need to Know About Weibull Distributions

The more you know, the easier it becomes to use your data

Each column in Figure 6 is used to create a curve in Figure 7 by plotting the radii in a column vs. their skewness values. The bottom curve shows that the middle 92% of a Weibull will shrink with increasing skewness. The 95% fixed-coverage intervals are remarkably stable until the increasing mass near the mean eventually begins to pull this curve down. The 98% fixed-coverage intervals initially grow, and then they plateau near three standard deviations.

The spread of the top three curves shows that for the Weibull models it’s primarily the outermost 2% that gets stretched into the extreme upper tail. Although 920 parts per thousand are moving toward the mean, and another 60 parts per thousand get slightly shifted outward and then stabilize, it’s primarily the outer 20 parts per thousand that bear the brunt of the stretching and elongation that goes with increasing skewness.

The purpose of analysis