1.3.3.30. Weibull Plot

1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.3. Graphical Techniques: Alphabetic

1.3.3.30. Weibull Plot

Purpose:
Graphical Check To See If Data Come From a Population That Would Be Fit by a Weibull Distribution

The Weibull plot (Nelson 1982) is a graphical technique for determining if a data set comes from a population tha would logically be fit by a 2-parameter Weibull distribution (the location is assumed to be zero).

The Weibull plot has special scales that are designed so that if the data do in fact follow a Weibull distribution, the points will be linear (or nearly linear). The least squares fit of this line yields estimates for the shape and scale parameters of the Weibull distribution. Weibull distribution (the location is assumed to be zero).

Sample Plot

This Weibull plot shows that:

the assumption of a Weibull distribution is reasonable;
the shape parameter estimate is computed to be 33.32;
the scale parameter estimate is computed to be 5.28; and
there are no outliers.

Definition:
Weibull Cumulative Probability Versus LN(Ordered Response)

The Weibull plot is formed by:

Vertical axis: Weibull cumulative probability expressed as a percentage
Horizontal axis: LN of ordered response

The vertical scale is ln-ln(1-p) where p=(i-0.3)/(n+0.4) and i is the rank of the observation. This scale is chosen in order to linearize the resulting plot for Weibull data.

Questions

The Weibull plot can be used to answer the following questions:

Do the data follow a 2-parameter Weibull distribution?
What is the best estimate of the shape parameter for the 2-parameter Weibull distribution?
What is the best estimate of the scale (= variation) parameter for the 2-parameter Weibull distribution?

Importance:
Check Distributional Assumptions

Many statistical analyses, particularly in the field of reliability, are based on the assumption that the data follow a Weibull distribution. If the analysis assumes the data follow a Weibull distribution, it is important to verify this assumption and, if verified, find good estimates of the Weibull parameters.

Related Techniques

Weibull Probability Plot
Weibull PPCC Plot
Weibull Hazard Plot

The Weibull probability plot (in conjunction with the Weibull PPCC plot), the Weibull hazard plot, and the Weibull plot are all similar techniques that can be used for assessing the adequacy of the Weibull distribution as a model for the data, and additionally providing estimation for the shape, scale, or location parameters.

The Weibull hazard plot and Weibull plot are designed to handle censored data (which the Weibull probability plot does not).

Case Study

The Weibull plot is demonstrated in the airplane glass failure data case study.

Software

Weibull plots are generally available in statistical software programs that are designed to analyze reliability data. Dataplot supports the Weibull plot.