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)

Hammad Awan says

Indeed an informative post. So one can use the variance formula to calculate the standard error of the mean and then calculate the confidence interval for the mean, right!….

K.Sukkiramarhi says

The mean of the three parameter Weibull distribution is

μ=ηΓ(1+1β)+δ…….can you please derive ??

Fred Schenkelberg says

I did a quick check on Reliawiki (by the folks at Reliasoft) where they do list many of the reliability related formulas.

http://reliawiki.org/index.php/The_Weibull_Distribution

They have quite a bit about Weibull, of course, yet not the derivation.

Check with Google Scholar for a paper or book that may show the derivation. I’ve not done it myself that I recall. And, haven’t had need for the mean of any Weibull distribution in practice.

Cheers,

Fred

Nicholas Wamaniala says

Is there an on-line calculator or equivalent for the Confidence bounds or interval for Weibull distribution in cases where lifetime data is a mixture of complete and right-censored data?

Fred Schenkelberg says

sure, try this beta tool, we’re exploring how to improve – let us know what you think

https://accendoreliability.com/weibull-analysis-tool/

Cheers,

Fred