Most physical properties exhibit a lower bound in the probability distribution, which the normal distribution fails to accurately describe. The Weibull distribution was originally proposed for describing fatigue life, but it has been used to model many different engineering properties, such as brittle fracture of ceramics and the life of electronic components. The probability density, p( x), for the Weibull distribution is given by

The shape of the distribution curve is controlled by the value of m and is referred to as the Weibull modulus. Example distributions, with varying values of m, are shown in Figure 6.4. The population distribution narrows rapidly as the value of m increases, and measurements with a high Weibull modulus are thought of as more reliable because there is less scatter in the data. The scaling parameter ? is called the characteristic value; at x = ? the population is divided into 63.2 percent below and 36.8 percent above ? for all values of m.

Figure 6.4: A schematic plot showing the Weibull distribution function with different values of m. In this plot ? = 1 and x ₀ = 0.

Calculation of a mean, Equation (12), and of a variance, Equation (13), for a Weibull distribution is not straightforward; these calculations involve the standard gamma function, ?. However, the main reason for using Weibull statistics is not to report means or variances, but rather to report the probability of...