WeibullPlot

WeibullPlot is a graphical diagnostic tool used in statistics and reliability engineering to assess whether a data set follows a Weibull distribution. It is a form of probability plot in which observed data are transformed so that, if the data are Weibull, the plotted points align roughly along a straight line.

Construction of the plot typically proceeds as follows. Arrange the data in ascending order x_(1) ≤ x_(2) ≤ … ≤ x_(n). Choose plotting positions F_i, for example F_i = (i - 0.3)/(n + 0.4). Compute the transformed coordinates y_i = ln(-ln(1 - F_i)) and x_i = ln(x_(i)) (one common form). Plot y_i against x_i. A linear relationship supports the hypothesis that the data come from a Weibull distribution. If a straight line is found, a linear fit y = a + b x yields a slope b equal to the Weibull shape parameter k, and an intercept a related to the scale parameter λ by a = -k ln λ, so that λ = exp(-a/b).

Alternative forms exist, such as plotting on axes of ln x versus ln(-ln(1 - F_i)) or using different plotting-position conventions. The key idea is that for X ~ Weibull(k, λ), the transformed coordinates should exhibit linear behavior.

Interpretation and caveats: Departure from linearity suggests the data may not be Weibull, or there may be outliers, censoring, or multiple subpopulations. Censored data require specialized plotting positions. Weighing the plot with model fitting or complementary goodness-of-fit tests strengthens conclusions. WeibullPlot is widely used in life data analysis, reliability studies, and quality control.