### Shapiro-Wilk

In statistics, the **Shapiro–Wilk test** tests the null hypothesis that a sample *x*_{1}, ..., *x*_{n} came from a normally distributed population. It was published in 1965 by Samuel Shapiro and Martin Wilk.^{[1]}

The test statistic is:

- $W\; =\; \{\backslash left(\backslash sum\_\{i=1\}^n\; a\_i\; x\_\{(i)\}\backslash right)^2\; \backslash over\; \backslash sum\_\{i=1\}^n\; (x\_i-\backslash overline\{x\})^2\}$

where

- $x\_\{(i)\}$ (with parentheses enclosing the subscript index
*i*) is the*i*th order statistic, i.e., the*i*th-smallest number in the sample; - $\backslash overline\{x\}\; =\; \backslash left(\; x\_1\; +\; \backslash dots\; +\; x\_n\; \backslash right)\; /\; n$ is the sample mean;
- the constants $a\_i$ are given by
^{[2]}

- $(a\_1,\backslash dots,a\_n)\; =\; \{m^\backslash top\; V^\{-1\}\; \backslash over\; (m^\backslash top\; V^\{-1\}V^\{-1\}m)^\{1/2\}\}$

- where

- $m\; =\; (m\_1,\backslash dots,m\_n)^\backslash top\backslash ,$

- 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.^{[3]}

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.^{[4]}

## See also

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

secondary or tertiary sources. (May 2012) |

## References

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## External links

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