TheShapiro–Wilk test is atest of normality. It was published in 1965 bySamuel Sanford Shapiro andMartin Wilk.^[1]

The Shapiro–Wilk test tests thenull hypothesis that asamplex₁, ...,x_n came from anormally distributed population. Thetest statistic is

$${\displaystyle W=\frac{{\left(\sum _{i=1}^n{a}_i{x}_{(i)}\right)}^2}{\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2},}$$

where

• ${\displaystyle x_{(i)}}$ with parentheses enclosing the subscript indexi is theithorder statistic, i.e., theith-smallest number in the sample (not to be confused with${\displaystyle x_i}$).
• ${\displaystyle \overline{x}=\left(x_1+\cdots +x_n\right)/n}$ is the sample mean.

The coefficients${\displaystyle a_i}$ are given by:^[1]$${\displaystyle (a_1,\dots ,a_n)=\frac{m^{\mathsf{T}}{V}^{-1}}{C},}$$whereC is avector norm:^[2]$${\displaystyle C=\Vert V^{-1}m\Vert ={\left(m^{\mathsf{T}}{V}^{-1}{V}^{-1}m\right)}^{1/2}}$$and the vectorm,$${\displaystyle m=(m_1,\dots ,m_n{)}^{\mathsf{T}}}$$is made of theexpected values of theorder statistics ofindependent and identically distributed random variables sampled from the standard normal distribution; finally,${\displaystyle V}$ is thecovariance matrix of those normal order statistics.^[3]

There is no name for the distribution of${\displaystyle W}$. The cutoff values for the statistics are calculated throughMonte Carlo simulations.^[2]

Thenull-hypothesis of this test is that the population is normally distributed. If thep value is less than the chosenalpha level, then the null hypothesis is rejected and there is evidence that the data tested are not normally distributed.^[4]

Like moststatistical significance tests, if the sample size is sufficiently large this test may detect even trivial departures from the null hypothesis (i.e., although there may be somestatistically significant effect, it may be too small to be of any practical significance); thus, additional investigation of theeffect size is typically advisable, e.g., aQ–Q plot in this case.^[5]

Monte Carlo simulations have demonstrated in practically relevant settings that Shapiro–Wilk has the bestpower for a givensignificance, followed closely byAnderson–Darling when comparing the Shapiro–Wilk,Kolmogorov–Smirnov, andLilliefors.^{[6][unreliable source?]}

Royston proposed an alternative method of calculating the coefficients vector by providing an algorithm for calculating values that extended the sample size from 50 to 2,000.^[7] This technique is used in several software packages includingGraphPad Prism, Stata,^[8][9] SPSS and SAS.^[10] Rahman and Govidarajulu extended the sample size further up to 5,000.^[11]

• Anderson–Darling test
• Cramér–von Mises criterion
• D'Agostino's K-squared test
• Kolmogorov–Smirnov test
• Lilliefors test
• Normal probability plot
• Shapiro–Francia test

• Worked example using Excel
• Algorithm AS R94 (Shapiro Wilk) FORTRAN code
• Exploratory analysis using the Shapiro–Wilk normality test in R
• Real Statistics Using Excel: the Shapiro-Wilk Expanded Test

• Normality tests

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• Short description matches Wikidata
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• Articles lacking reliable references from October 2024