Reliability HotWire: eMagazine for the Reliability Professional
Reliability HotWire

Issue 45, November 2004

Tool Tips

* For the Weibull distribution, what is the difference between the characteristic life and the mean life?

In general, these two values are not the same for the Weibull distribution. Consider the 2-parameter Weibull cdf equation given below.


  • β = shape parameter

  • η = scale parameter

If you assume that t = η then the above equation reduces to:

Therefore, η, the characteristic life, is the time at which 63.2% of the units will fail. Now, the mean is the expected life and this is calculated by:

where Γ() is the Gamma function. You can see that the mean life is actually a function of the characteristic life, η. If β = 1 then the mean life equation reduces to:

since Γ(2) = 1. This is the only case for the Weibull distribution where the characteristic life and the mean life are equivalent. Additional information on the Weibull distribution statistical properties can be found at

* When considering complex repairable systems, what is the difference between the Weibull distribution and the Power Law model?

In order to address the reliability characteristics of complex repairable systems, a process is often used instead of a distribution. The most popular process model is the Power Law model. This model is popular for several reasons. One is that it has a very practical foundation in terms of minimal repair. This is the situation when the repair of a failed system is just enough to get the system operational again. Second, if the time to first failure follows the Weibull distribution, then each succeeding failure is governed by the Power Law model in the case of minimal repair. From this point of view, the Power Law model is an extension of the Weibull distribution. In other words, the Weibull distribution addresses the very first failure and the Power Law model addresses each succeeding failure for a repairable system.

Additional information on analyzing repairable systems can be found at


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