The test statistic is:
- (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;
- the constants are given by
- and , ..., are the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution, and is the covariance matrix of those order statistics.
The user may reject the null hypothesis if is too small.
It can be interpreted via a Q-Q plot.
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.
See also 
- Anderson–Darling test
- Cramér–von Mises criterion
- Kolmogorov–Smirnov test
- Normal probability plot
- Ryan-Joiner test
- Watson test
|This article relies on references to primary sources. (May 2012)|
||Constructs such as ibid., loc. cit. and idem are discouraged by Wikipedia's style guide for footnotes, as they are easily broken. Please improve this article by replacing them with named references (quick guide), or an abbreviated title. (May 2012)|
- Shapiro, S. S.; Wilk, M. B. (1965). "An analysis of variance test for normality (complete samples)". Biometrika 52 (3-4): 591–611. doi:10.1093/biomet/52.3-4.591. JSTOR 2333709. MR 205384.
- op cit p. 593
- op cit p. 605
- "How do I interpret the Shapiro-Wilk test for normality?". JMP. 2004. Retrieved March 24, 2012.
- Samuel Sanford Shapiro
- Algorithm AS R94 (Shapiro Wilk) FORTRAN code
- Shapiro–Wilk Normality Test in R