Shapiro-Wilk test

In statistics, the Shapiro-Wilk test tests the null hypothesis that a sample x₁, …, x_n came from a normally distributed population. It was published in 1965 by Samuel Shapiro and Martin Wilk.

The test statistic is

<math>W = {\left(\sum_{i=1}^n a_i x_{(i)}\right)^2 \over \sum_{i=1}^n (x_i-\overline{x})^2}</math>

where

• x_(i) (with parentheses enclosing the subscript index i) is the ith order statistic, i.e., the ith-smallest number in the sample;
• <math>\overline{x}=(x_1+\cdots+x_n)/n\,</math> is the sample mean;
• the constants a_i are given by

<math>(a_1,\dots,a_n) = {m^\top V^{-1} \over (m^\top V^{-1}V^{-1}m)^{1/2}}</math>

where

<math>m = (m_1,\dots,m_n)^\top\,</math>

and m₁, …, m_n are the expected values of the order statistics of independent and identically-distributed random variables sampled from the standard normal distribution, and V is the covariance matrix of those order statistics.

The user may reject the null hypothesis if W is too small.

• Anderson-Darling test
• Kolmogorov-Smirnov test
• Smirnov-Cramér-von-Mises test

• Shapiro, S. S. and Wilk, M. B. (1965). “An analysis of variance test for normality (complete samples)”, Biometrika, 52, 3 and 4, pages 591-611. [1]

• Algorithm AS R94 (Shapiro Wilk) FORTRAN code

• Shapiro-Wilk Normality Test in CRAN

• C code in CRAN (look for swilk.c)

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