Shapiro-Wilk test: Difference between revisions
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Overview
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
- <math>W = {\left(\sum_{i=1}^n a_i x_{(i)}\right)^2 \over \sum_{i=1}^n (x_i-\overline{x})^2}</math>
where
- x(i) (with parentheses enclosing the subscript index i) is the ith order statistic, i.e., the ith-smallest number in the sample;
- <math>\overline{x}=(x_1+\cdots+x_n)/n\,</math> is the sample mean;
- the constants ai are given by
- <math>(a_1,\dots,a_n) = {m^\top V^{-1} \over (m^\top V^{-1}V^{-1}m)^{1/2}}</math>
- where
- <math>m = (m_1,\dots,m_n)^\top\,</math>
- and m1, ..., mn 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.
See also
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. [2]