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by Fred Schenkelberg 5 Comments

Calculate Weibull Mean and Variance

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)

 

Filed Under: Articles, CRE Preparation Notes, Probability and Statistics for Reliability Tagged With: Discrete and continuous probability distributions

About Fred Schenkelberg

I am the reliability expert at FMS Reliability, a reliability engineering and management consulting firm I founded in 2004. I left Hewlett Packard (HP)’s Reliability Team, where I helped create a culture of reliability across the corporation, to assist other organizations.

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Comments

  1. Hammad Awan says

    April 3, 2015 at 1:07 PM

    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!….

    Reply
  2. K.Sukkiramarhi says

    March 16, 2017 at 7:29 AM

    The mean of the three parameter Weibull distribution is

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

    Reply
    • Fred Schenkelberg says

      March 16, 2017 at 7:53 AM

      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

      Reply
  3. Nicholas Wamaniala says

    June 20, 2019 at 3:20 AM

    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?

    Reply
    • Fred Schenkelberg says

      June 20, 2019 at 6:46 AM

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

      https://lucas-accendo-site-speed.sprod01.rmkr.net/weibull-analysis-tool/

      Cheers,

      Fred

      Reply

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CRE Preparation Notes

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