A Weibull Plot is used to verify whether a particular data set follows the Weibull distribution and provides additional information about the estimated shape and scale parameters. It plots the Weibull cumulative probability on the vertical axis and data points on the horizontal axis where the axes are $\ln \left(-\ln \left(1-F\left(x\right)\right)\right)$ and $\ln \left(x\right)$ respectively. Here $F\left(x\right)$ denotes the estimated Weibull cumulative distribution function. The change of variables linearizes the cumulative distribution function and therefore if the data follows a Weibull distribution then a straight line is expected on the Weibull Plot. By default, a reference line is additionally drawn, which indicates where points following the distribution exactly would lie. The plot also contains a horizontal line at probability 0.632 and a vertical line where the horizontal line intersects the reference line. In general the scale parameter represents the 0.632-quantile of the Weibull distribution regardless of the value of shape parameter since $F\left(\mathrm{scale}\right)=1-{\ⅇ}^{\mathrm{-1}}\approx 0.632$ . Therefore, the vertical line indicates the value of the scale parameter.

The first parameter, X, corresponds to the data set to be analysed.

The options argument can contain one or more of the options shown below. All unrecognized options will be passed to the plots[display] command. See plot/options for details.

scale=deduce or realcons

The scale parameter for the Weibull distribution. If scale is set to deduce, the scale parameter will be estimated from the data sample X. The default value is deduce. See Weibull distribution for more details.

shape=deduce or realcons

The shape parameter for the Weibull distribution. If shape is set to deduce, the shape parameter will be estimated from the data sample X. See Weibull distribution for more details. The default value is deduce.

sorted=truefalse

If sorted is set to true, a sorted (ascending) copy of data is used. Otherwise, if false the original data is used. The default value is true.

reference=truefalse

If reference is set to true, a reference line is drawn, which indicates where points following the distribution exactly would lie. The default value is true.

showprobabilities=truefalse

If showprobabilities is true, then the vertical axis is labelled with the appropriate probability values. Otherwise the specially scaled probability values are used for labeling. The default value is true.

color=name or list

This option specifies colors for the points and for the reference line. When a list of colors is given, the set of points and the reference line will be colored with the corresponding colors in the list.

legend=s or list

This option specifies legends for the points and for the reference line. When a list of values is given, the set of points and the reference line will be assigned legend entries with the corresponding values in the list. For more information on how to specify s, see plot/typesetting.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Weibull}\left(1\,0.6\right)\right)\:$

$A≔\mathrm{Sample}\left(X\,100\right)\:$

$\mathrm{WeibullPlot}\left(A\,\mathrm{reference}=\mathrm{false}\,\mathrm{style}=\mathrm{line}\right)$

$\mathrm{WeibullPlot}\left(A\,\mathrm{scale}=1\,\mathrm{shape}=0.6\,\mathrm{color}=\left[\mathrm{blue}\,\mathrm{magenta}\right]\right)$

$\mathrm{WeibullPlot}\left(A\,\mathrm{scale}=0.8\,\mathrm{shape}=0.5\,\mathrm{style}=\mathrm{point}\,\mathrm{color}=\left[\mathrm{blue}\,\mathrm{green}\right]\,\mathrm{legend}=\left[''data''\,''reference''\right]\,\mathrm{color}=\left[\mathrm{green}\,\mathrm{blue}\right]\right)$

$Y≔\mathrm{RandomVariable}\left(\mathrm{Uniform}\left(1\,3\right)\right)\:$

$B≔\mathrm{Sample}\left(Y\,300\right)\:$

$\mathrm{WeibullPlot}\left(B\,\mathrm{legend}=\left[''data''\,''reference''\right]\right)$

The Statistics[WeibullPlot] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.