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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.[1]
The test statistic is:
where
- x(i) (with parentheses enclosing the subscript index i) is the ith order statistic, i.e., the ith-smallest number in the sample;
- x = (x1 + ... + xn) / n is the sample mean;
- the constants ai are given by[2]
- where
- 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.[3]
It can be interpreted via a Q-Q plot.
Interpretation
editRecalling 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.[1]
See also
editReferences
edit- ^ 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 0205384.
- ^ op cit p. 593
- ^ op cit p. 605
- Razali, Nornadiah Mohd (2011). "Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests". Journal of Statistical Modeling and Analytics. 2 (1): 21–33.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - Rowston, Patrick (1982). "An extension of Shapiro and Wilk's W test for normality to large samples". Applied Statistics (31): 115–124. doi:10.2307/2347973. JSTOR 2347973.
- Rowston, Patrick (1982). "Algorithm AS 181: The W test for Normality". Applied Statistics (31): 176–180. doi:10.2307/2347986. JSTOR 2347986.
- Rowston, Patrick (1995). "AS R94: A remark on Algorithm AS 181: The W test for normality". Applied Statistics (44): 547–551. doi:10.2307/2986146. JSTOR 2986146.
- Shapiro, S. S. (1968). "A Comparative Study of Various Tests for Normality". Journal of the American Statistical Association. 63 (324): 1343–1372. doi:10.1080/01621459.1968.10480932.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - Chen, Edwin H. (1971). "The Power of the Shapiro-Wilk W Test for Normality in Samples from Contaminated Normal Distributions". Journal of the American Statistical Association. 66 (336): 760–762.
- Leslie, J. R. (1986). "Asymptotic Distribution of the Shapiro-Wilk W for Testing for Normality". The Annals of Statistics. 14 (4): 1497–1506. doi:10.1214/aos/1176350172.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
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