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Statistics

 WeibullPlot
 generate Weibull plots

 Calling Sequence WeibullPlot(X, options, plotoptions)

Parameters

 X - data options - (optional) equation(s) of the form option=value where option is one of scale, shape, sorted, reference, showprobabilities, color or legend; specify options for generating the Weibull plot plotoptions - options to be passed to the plots[display] command

Description

 • 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 $\mathrm{ln}\left(-\mathrm{ln}\left(1-F\left(x\right)\right)\right)$ and $\mathrm{ln}\left(x\right)$ 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 $F\left(\mathrm{scale}\right)=1-{ⅇ}^{-1}\approx 0.632$ . Therefore, the vertical line indicates the value of the scale parameter.
 • The first parameter, X, corresponds to the data set to be analysed.

Options

 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.

Examples

Get a random sample of a Weibull distribution

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Weibull}\left(1,0.6\right)\right):$
 > $A≔\mathrm{Sample}\left(X,100\right):$

Estimate the parameters.

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

Using the original parameters.

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

Plot it against a Weibull distribution with different shape and scale parameters.

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

Compare sample generated from a different distribution to the Weibull distribution

 > $Y≔\mathrm{RandomVariable}\left(\mathrm{Uniform}\left(1,3\right)\right):$
 > $B≔\mathrm{Sample}\left(Y,300\right):$
 > $\mathrm{WeibullPlot}\left(B,\mathrm{legend}=\left["data","reference"\right]\right)$
 > 

Compatibility

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