Editing Weibull distribution (section)

===Ordinary least square using Weibull plot===
[[File:Weibull qq.svg|thumb|right|Weibull plot]]
The fit of a Weibull distribution to data can be visually assessed using a Weibull plot.<ref>{{cite web|url=http://www.itl.nist.gov/div898/handbook/eda/section3/weibplot.htm|title=1.3.3.30. Weibull Plot|website=www.itl.nist.gov}}</ref> The Weibull plot is a plot of the [[empirical cumulative distribution function]] <math>\widehat F(x)</math> of data on special axes in a type of [[Q–Q plot]]. The axes are <math>\ln(-\ln(1-\widehat F(x)))</math> versus <math>\ln(x)</math>. The reason for this change of variables is the cumulative distribution function can be linearized:
:<math>\begin{align}
F(x) &= 1-e^{-(x/\lambda)^k}\\[4pt]
-\ln(1-F(x)) &= (x/\lambda)^k\\[4pt]
\underbrace{\ln(-\ln(1-F(x)))}_{\textrm{'y'}} &= \underbrace{k\ln x}_{\textrm{'mx'}} - \underbrace{k\ln \lambda}_{\textrm{'c'}}
\end{align}
</math>
which can be seen to be in the standard form of a straight line. Therefore, if the data came from a Weibull distribution then a straight line is expected on a Weibull plot.

There are various approaches to obtaining the empirical distribution function from data. One method is to obtain the vertical coordinate for each point using
:<math>\widehat F = \frac{i-0.3}{n+0.4}</math>,
where <math>i</math> is the rank of the data point and <math>n</math> is the number of data points.<ref>Wayne Nelson (2004) ''Applied Life Data Analysis''. Wiley-Blackwell {{ISBN|0-471-64462-5}}</ref><ref>{{Cite journal |last=Barnett |first=V. |date=1975 |title=Probability Plotting Methods and Order Statistics |url=https://www.jstor.org/stable/2346708 |journal=Journal of the Royal Statistical Society. Series C (Applied Statistics) |volume=24 |issue=1 |pages=95–108 |doi=10.2307/2346708 |jstor=2346708 |issn=0035-9254}}</ref> Another common estimator<ref>{{Cite ISO standard | csnumber = 69875 | title = ISO 20501:2019 – Fine ceramics (advanced ceramics, advanced technical ceramics) – Weibull statistics for strength data}}</ref> is 
:<math>\widehat F = \frac{i-0.5}{n}</math>.

Linear regression can also be used to numerically assess goodness of fit and estimate the parameters of the Weibull distribution. The gradient informs one directly about the shape parameter <math>k</math> and the scale parameter <math>\lambda</math> can also be inferred.