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# Shapiro-Wilk test

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 Title: Shapiro-Wilk test Author: World Heritage Encyclopedia Language: English Subject: Normal distribution Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Shapiro-Wilk test

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

The test statistic is:

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

where

• $x_\left\{\left(i\right)\right\}$ (with parentheses enclosing the subscript index i) is the ith order statistic, i.e., the ith-smallest number in the sample;
• $\overline\left\{x\right\} = \left\left( x_1 + \dots + x_n \right\right) / n$ is the sample mean;
• the constants $a_i$ are given by
$\left(a_1,\dots,a_n\right) = \left\{m^\top V^\left\{-1\right\} \over \left(m^\top V^\left\{-1\right\}V^\left\{-1\right\}m\right)^\left\{1/2\right\}\right\}$
where
$m = \left(m_1,\dots,m_n\right)^\top\,$
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

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