Calculate Weibull Mean and Variance

For our use of the Weibull distribution, we typically use the shape and scale parameters, β and η, respectively. For a three parameter Weibull, we add the location parameter, δ.

The scale or characteristic life value is close to the mean value of the distribution. When β = 1 and δ = 0, then η is equal to the mean. The characteristic life is offset by δ when it is not equal to zero, such that when β = 1 and δ = x, then the characteristic life or mean is η + δ.

We often use θ to represent the characteristic life. The mean and characteristic life are not the same when β ≠ 1.

Given a set of Weibull distribution parameters here is a way to calculate the mean and standard deviation, even when β ≠ 1.

 

The Gamma Function

First we will need the Gamma function. It is often tabulated in reliability statistics references. The function is

$$ \large\displaystyle \Gamma \left( n \right)=\left( n-1 \right)!$$

For positive non-integers, we use the smooth function

$$ \large\displaystyle \Gamma \left( n \right)=\int _{0}^{\infty }{{{t}^{n-1}}}{{e}^{-t}}dt$$

A table is often an easier route for a quick calculation.

 

Calculate the Weibull Mean

The mean of the three parameter Weibull distribution is

$$ \large\displaystyle\mu =\eta \Gamma \left( 1+\frac{1}{\beta } \right)+\delta $$

 

Calculate the Weibull Variance

The variance is a function of the shape and scale parameters only. The calculation is

$$ \large\displaystyle \sigma ={{\eta }^{2}}\left[ \Gamma \left( 1+\frac{2}{\beta } \right)-{{\Gamma }^{2}}\left( 1+\frac{1}{\beta } \right) \right]$$

Datasheets and vendor websites often provide only the expected lifetime as a mean value. This is deceptive as the variance matters. For the same mean, a higher variance would generally provide a lower reliability performance at the same point in time.

---

Related:

Weibull Distribution (article)

Lognormal Distribution (article)

The Normal Distribution (article)