User:Protonk/SWtest

In statistics, the Shapiro–Wilk test tests the null hypothesis that a sample x₁, ..., x_n came from a normally distributed population. It was published in 1965 by Samuel Shapiro and Martin Wilk.

The test statistic is:

$W={\left(\sum _{i=1}^{n}a_{i}x_{(i)}\right)^{2} \over \sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}$

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 + ... + x_n) / n is the sample mean;
• the constants a_i are given by

$(a_{1},\dots ,a_{n})={m^{\top }V^{-1} \over (m^{\top }V^{-1}V^{-1}m)^{1/2}}$

where

$m=(m_{1},\dots ,m_{n})^{\top }\,$

and m₁, ..., 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.

It can be interpreted via a Q-Q plot.

Interpretation

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.[1]

See also

• Anderson–Darling test
• Kolmogorov–Smirnov test
• Cramér–von Mises criterion
• Normal probability plot
• Q-Q plot

References

• 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= suggested) (help)