Shapiro-Wilk test

In statistics, the Shapiro-Wilk test tests the null hypothesis that a sample "x"₁, ..., "x"_"n" came from a normally distributed population. It was published in 1965 by Samuel Shapiro and Martin Wilk.

The test statistic is

:$W\; =\; \{left(sum\_\{i=1\}^n\; a\_i\; x\_\{(i)\}\; ight)^2\; over\; sum\_\{i=1\}^n\; (x\_i-overline\{x\})^2\}$

where

* "x"_("i") (with parentheses enclosing the subscript index "i") is the "i"th order statistic, i.e., the "i"th-smallest number in the sample;
* $overline\{x\}=(x\_1+cdots+x\_n)/n,$ is the sample mean;
* the constants "a"_"i" are given by

::$(a\_1,dots,a\_n)\; =\; \{m^\; op\; V^\{-1\}\; over\; (m^\; op\; V^\{-1\}V^\{-1\}m)^\{1/2$

:where

::$m\; =\; (m\_1,dots,m\_n)^\; op,$

:and "m"₁, ..., "m"_"n" 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.

ee also

* Anderson-Darling test
* Kolmogorov-Smirnov test
* Smirnov-Cramér-von-Mises test

References

* Shapiro, S. S. and Wilk, M. B. (1965). "An analysis of variance test for normality (complete samples)", "Biometrika", 52, 3 and 4, pages 591-611. [http://www.jstor.org/view/00063444/di992333/99p0027o/0]

* [http://lib.stat.cmu.edu/apstat/R94 Algorithm AS R94 (Shapiro Wilk) FORTRAN code]

* [http://cran.us.r-project.org/doc/manuals/R-intro.html#Examining-the-distribution-of-a-set-of-data Shapiro-Wilk Normality Test in CRAN]

* [http://cran.us.r-project.org/sources.html C code in CRAN (look for swilk.c)]

External links

* [http://www.analyse-it.com/blog/2008/8/testing-the-assumption-of-normality.aspx Testing the assumption of normality] .