Shapiro-Wilk test

In statistics, the Shapiro-Wilk test tests the null hypothesis that a sample x1, ..., xn came from a normally distributed population. 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;
 * $$\overline{x}=(x_1+\cdots+x_n)/n\,$$ is the sample mean;
 * the constants ai are given by


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


 * where


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


 * and m1, ..., mn are the expected values of the order statistics of an iid sample from the standard normal distribution, and V is the covariance matrix of those order statistics.

The test rejects the null hypothesis if W is too small.