Shapiro–Wilk test
The Shapiro–Wilk test is a test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a sample x1,..., xn came from a normally distributed population. The test statistic iswhere
- with parentheses enclosing the subscript index i is the ith order statistic, i.e., the ith-smallest number in the sample.
- is the sample mean.
where C is a vector norm:
and the vector m,
is made of the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution; finally, is the covariance matrix of those normal order statistics.
There is no name for the distribution of. The cutoff values for the statistics are calculated through Monte Carlo simulations.
Interpretation
The null-hypothesis of this test is that the population is normally distributed. If the p value is less than the chosen alpha level, then the null hypothesis is rejected and there is evidence that the data tested are not normally distributed.Like most statistical significance tests, if the sample size is sufficiently large this test may detect even trivial departures from the null hypothesis ; thus, additional investigation of the effect size is typically advisable, e.g., a Q–Q plot in this case.