ShapiroWilkTest

ShapiroWilkTest is a statistical test used to assess whether a sample comes from a normally distributed population. Named after Samuel Shapiro and Martin Wilk, it was proposed in 1965. The test evaluates the null hypothesis that the data are drawn from a normal distribution with unknown mean and variance against the alternative that the distribution is not normal.

The Shapiro-Wilk statistic W is defined as W = ( (sum_{i=1}^n a_i x_(i))^2 ) / ( sum_{i=1}^n (x_i - x̄)^2 ), where x_(i) are the ordered sample values, x̄ is the sample mean, and a_i are constants determined by the sample size n and the expected order statistics of a normal distribution. Larger values of W indicate closer adherence to normality. P-values are obtained from the distribution of W under H0, using approximations or precomputed tables; a small p-value leads to rejection of normality.

ShapiroWilkTest is particularly powerful for detecting departures from normality in small to moderate samples and is widely used as a preliminary check before applying parametric tests that assume normality. It is most sensitive to deviations in the center of the distribution and may be less informative about tails or skewness alone.

Limitations include sensitivity to outliers and non-independence among observations. With very large samples, trivial deviations can produce small p-values even when deviations are of little practical concern. The test does not directly test symmetry and should be interpreted alongside other diagnostics of distribution shape.