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:

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

where

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

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

where

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

and $m_1$, ..., $m_n$ 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.

It can be interpreted via a Q-Q plot.

Interpretation

Recalling that the null hypothesis is that the population is normally distributed, if the p-value is less than the chosen alpha level, then the null hypothesis is rejected (i.e. one concludes the data are not from a normally distributed population). If the p-value is greater than the chosen alpha level, then one does not reject the null hypothesis that the data came from a normally distributed population. E.g. for an alpha level of 0.05, a data set with a p-value of 0.32 does not result in rejection of the hypothesis that the data are from a normally distributed population.

See also

• Anderson–Darling test
• Cramér–von Mises criterion
• Kolmogorov–Smirnov test
• Normal probability plot
• Ryan-Joiner test
• Watson test

External links

• Samuel Sanford Shapiro
• Algorithm AS R94 (Shapiro Wilk) FORTRAN code
• Shapiro–Wilk Normality Test in R