A High Value of Beta is Not Necessarily Cause for Concern
Although often viewed with fear and loathing by reliability engineers, a high value of the Weibull distribution’s shape parameter beta is not necessarily a cause for concern.
Weibull Distribution and Beta
Since many organizations perform reliability analyses of mechanical items, the phenomenon of data sets with beta values greater than 1 is not uncommon. However, when the calculated value of beta turns out to be substantially larger than 1 (for example, 6 or larger), a sense of unease may set in for the reliability engineer. Such high beta values are sometimes regarded as an indication that the product or the test is flawed in some way. As the following example serves to illustrate, however, this is not necessarily the case.
of a "Harmless" High Beta
Using ReliaSoft’s Weibull++ and the maximum likelihood estimation (MLE) analysis method, the engineer estimates the Weibull parameters for the data as eta = 1665.5 hours and beta = 11.6. This may alarm the engineer as the beta value of 11.6 is relatively high. However, when he doggedly perseveres with his analysis, the engineer discovers that the 90% confidence reliability value at 1000 hours is 96.3%. This result is more than enough to meet the specification and, despite the high value of beta, there was nothing wrong with the test or the product.
of High Betas that are not a Problem
In fact, for repairable systems, components with high beta values may actually be preferred because the lack of variability can increase the efficiency of a preventive maintenance program. Less variability means that failures occur in a more "controlled" manner and therefore a better optimum replacement interval for preventive maintenance can be quantified. For example, it would be ideal for a preventive maintenance program to have a component that always fails at exactly 1,000 hours of operation. The optimum replacement time would therefore be just before the expected failure, at 999.9 hours.
Another concern that has been associated with reliability data sets with high values of beta is that the relatively steep slope makes it difficult to discern patterns in the data, such as outliers or breaks in the data, on the probability plot. While this is true, it is not a sufficient reason to reject a set of data. Most of the problems that could potentially be discerned by viewing the pattern in the probability plot would hopefully be detected elsewhere in the engineering and testing process. For example, breaks in the pattern of data may indicate that multiple failure modes are active. While a steep slope on a probability plot may tend to obscure such a pattern, the presence of multiple failure modes would likely have been observed during the test or by a concerted failure analysis program. Similarly, outliers can often be identified merely by looking at the raw data. In general, it is not a good idea to depend on the pattern of a probability plot to supplant more dedicated engineering and analysis efforts.
Although there may be genuine concerns about data sets with high values of beta, the fact that a data set has a high value of beta is not necessarily cause for alarm as long as the associated value of h is high enough to offset the lack of variability inherent in data sets with high beta values. In the end, it is more important to analyze the overall behavior of the data and whether or not the product’s test results meet the requirements than to focus solely on the value of a single parameter.