This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# ShapiroWilkTest

 ShapiroWilkTest[data] tests whether data is normally distributed using the Shapiro-Wilk test. ShapiroWilkTest returns the value of .
• ShapiroWilkTest performs the Shapiro-Wilk goodness-of-fit test with null hypothesis that data was drawn from a NormalDistribution and alternative hypothesis that it was not.
• By default a probability value or -value is returned.
• A small -value suggests that it is unlikely that the data came from dist.
• The dist can be any symbolic distribution with numeric and symbolic parameters or a dataset.
• The data can be univariate or multivariate .
• The Shapiro-Wilk test effectively compares the order statistics of data to the theoretical order statistics of a NormalDistribution.
• Properties related to the reporting of test results include:
 "PValue" -value "PValueTable" formatted version of "ShortTestConclusion" a short description of the conclusion of a test "TestConclusion" a description of the conclusion of a test "TestData" test statistic and -value "TestDataTable" formatted version of "TestStatistic" test statistic "TestStatisticTable" formatted
• The following properties are independent of which test is being performed.
• Properties related to the data distribution include:
 "FittedDistribution" fitted distribution of data "FittedDistributionParameters" distribution parameters of data
• The following options can be given:
 Method Automatic the method to use for computing -values SignificanceLevel 0.05 cutoff for diagnostics and reporting
• For a test for goodness of fit, a cutoff is chosen such that is rejected only if . The value of used for the and properties is controlled by the SignificanceLevel option. By default is set to .
Perform a Shapiro-Wilk test for normality:
Perform a test for multivariate normality:
The full test table:
The test statistic and -value:
Perform a Shapiro-Wilk test for normality:
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Perform a test for multivariate normality:
The full test table:
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The test statistic and -value:
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 Scope   (6)
Perform a Shapiro-Wilk test for normality:
The -value for the normal data is large compared to the -value for the non-normal data:
Test for multivariate normality:
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Tabulate the results of the Shapiro-Wilk test:
The full test table:
A -value table:
The test statistic:
Retrieve the entries from a Shapiro-Wilk test table for custom reporting:
Report test conclusions using and :
The conclusion may differ at a different significance level:
 Options   (3)
Use Monte Carlo-based methods or a computation formula:
Set the number of samples to use for Monte Carlo-based methods:
The Monte Carlo estimate converges to the true -value with increasing samples:
Set the random seed used in Monte Carlo-based methods:
The seed affects the state of the generator and has some effect on the resulting -value:
 Applications   (2)
A power curve for the Shapiro-Wilk test:
Visualize the approximate power curve:
Estimate the power of the Shapiro-Wilk test when the underlying distribution is a CauchyDistribution, the test size is , and the sample size is 12:
The boiling point of water was measured at varying altitudes in the Alps. The barometric pressure was recorded for each boiling point. Determine if a linear model is appropriate for use in predicting boiling points given pressure:
A plot of the model and the data:
For the model to be appropriate, the residuals should be normally distributed:
A QuantilePlot confirms that the linear model is not appropriate for this data:
ShapiroWilkTest compares the order statistics of the data to their expectations under :
Expected values of the order statistics and an estimate of their covariance matrix:
These are used to compute weights:
The statistic using the estimated covariance matrix is slightly different from the reported value:
For tests of multivariate normality, a transformation to univariate data is made:
The data has been transformed to approximate univariate normal data:
Perform the test on the transformed data:
The result agrees with a test of the original data:
The Shapiro-Wilk test requires sample sizes be less than 5000 for -values to be valid:
The distribution of the Shapiro-Wilk test statistic:
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