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Reliability, Life Testing and the Prediction of Service Lives - For Engineers and Scientists (Springer Series in Statistics)
"CHAPTER 5 Applicable Life Distributions (p. 75-76)
Often a number of parametric distributions can be used to summarize a given sample of life-length data. Sometimes several of them can do it quite well. For example, if we take the Data-Set VII in Chapter 9 (101 observations of the fatigue-life of aluminum coupons) we find there are several unimodal, skewed to the left, two-parameter life distributions that will fit it adequately in the region ofcentral tendency. These include the Galton, Weibull, Gamma, and fatiguelife distributions; certainly there are others.
How does one decide which of these distributions is most appropriate? In certain instances it makes little difference which of these families of distributions is adopted for use. But if the life of airframe components, made of the same material as that tested, must be predicted under many different loading conditions, all at some fraction of the maximum stress applied during the test, great differences arise among the families in their realistic predictive capability when the service-life is extrapolated from test data. Obtaining fatigue-life data at unrealistically high stress levels is necessitated by having to complete the testing within a small fraction of the design life.
After all, time is money. This is called an accelerated test since the stress level is beyond that encountered in service. What is desired is a method to calculate a safe-life for critical components when the maximum stress in service is, say, one-hundredth of that imposed in the test. That is, we must have a statistical model in which the parameters of the life distribution are constructs of the physical factors, such as the stress regime and the type of material (both of which are known to be of primary importance) so that if these physical factors are changed the appropriate modifications to the distribution of service life are possible, with valid predictions over the range of applicable service-life conditions. This is especially true whenever public health and safety are at risk.
5.1. The Gaussian or Normal Distribution
Under what conditions should the normal distribution be used? It is applied so universally and so uncritically that, simultaneously, it is the most used, and misused, distribution in statistics. The Central Limit Theorem (the limit theorem which is central to so much of statistical theory) is given by the classical LindebergFeller normal convergence criterion."