Glossary term
Weibull Distribution
A probability distribution widely used to model life, strength, and failure data.
Definition
modelThe Weibull distribution is a flexible probability distribution used to model time-to-failure, strength, fatigue life, and weakest-link behavior.
Weibull models are common in reliability engineering because the shape parameter can represent decreasing, constant, or increasing failure rate behavior. They are used for life data, warranty analysis, brittle strength, fatigue scatter, bearing life, electronic reliability, and accelerated test interpretation.
For the two-parameter Weibull distribution, the reliability function is commonly written as:
where \beta is the shape parameter and \eta is the scale parameter. A three-parameter form adds a location parameter \gamma that shifts the distribution.
Engineering interpretation
The shape parameter carries important engineering meaning. If \beta < 1, the failure rate decreases with time, often associated with early-life or infant-mortality failures. If \beta = 1, the distribution reduces to the exponential case with constant failure rate. If \beta > 1, failure rate increases with time, which can represent wear-out, fatigue, corrosion progression, or aging mechanisms.
Weibull analysis is useful only when the data correspond to comparable populations, failure definitions, and operating conditions. Censored data, suspended tests, mixed failure modes, small samples, and accelerated testing require careful treatment.
Data handling
Engineering Weibull work starts with a consistent definition of exposure and failure. Time, cycles, distance, stress level, or operating hours may be the right life variable depending on the mechanism. Suspensions and right-censored observations should not be discarded, because they contain information about items that survived the test interval. When several mechanisms are present, separate failure-mode analysis is usually more useful than forcing all data into one fitted curve.
Common mistakes
A common mistake is fitting a Weibull line to mixed failure modes and then interpreting one shape parameter as a physical mechanism. Another is reporting a characteristic life without confidence bounds or censoring assumptions. A strong Weibull review states population, failure definition, time or stress metric, censoring method, sample size, fitted parameters, confidence intervals, goodness of fit, and whether the fitted model is being used for prediction, comparison, or warranty risk.