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Shapiro-Wilk test

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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.[1]

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[2]
(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.[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

References

External links

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