Shapiro-Wilk test

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In statistics, the Shapiro-Wilk test tests the null hypothesis that a sample x₁, ..., x_n came from a normally distributed population. The test statistic is

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

where

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

$(a_1,\dots,a_n) = {m^\top V^{-1} \over m^\top V^{-1}V^{-1}m}$

where

$m = (m_1,\dots,m_n)^\top\,$

and m₁, ..., m_n are the expected values of the order statistics of an iid sample from the standard normal distribution, and V is the covariance matrix of those order statistics.

The test rejects the null hypothesis if W is too small.

References

• 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.