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

# KolmogorovSmirnovTest

 KolmogorovSmirnovTest[data] tests whether data is normally distributed using the Kolmogorov-Smirnov test. KolmogorovSmirnovTest tests whether data is distributed according to dist using the Kolmogorov-Smirnov test. KolmogorovSmirnovTest returns the value of .
• KolmogorovSmirnovTest performs the Kolmogorov-Smirnov goodness-of-fit test with null hypothesis that data was drawn from a population with distribution dist 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 Kolmogorov-Smirnov test assumes that the data came from a continuous distribution.
• The Kolmogorov-Smirnov test effectively uses a test statistic based on where is the empirical CDF of data and is the CDF of dist.
• For multivariate tests, the mean of the univariate marginal test statistics is used. -values are computed via Monte Carlo simulation.
• 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 Kolmogorov-Smirnov test for normality:
Test the fit of some data to a particular distribution:
Compare the distributions of two datasets:
Perform a Kolmogorov-Smirnov test for normality:
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Test the fit of some data to a particular distribution:
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Compare the distributions of two datasets:
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 Scope   (9)
Perform a Kolmogorov-Smirnov test for normality:
The -value for the normal data is large compared to the -value for the non-normal data:
Test the goodness-of-fit to a particular distribution:
Compare the distributions of two datasets:
The two datasets do not have the same distribution:
Test for multivariate normality:
Test for goodness-of-fit to any multivariate distribution:
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Tabulate the results of the Kolmogorov-Smirnov test:
The full test table:
A -value table:
The test statistic:
Retrieve the entries from a Kolmogorov-Smirnov test table for custom reporting:
Report test conclusions using and :
The conclusion may differ at a different significance level:
 Options   (4)
Use Monte Carlo-based methods for 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:
Set the significance level used for and :
By default is used:
 Applications   (2)
A power curve for the Kolmogorov-Smirnov test:
Visualize the approximate power curve:
Estimate the power of the Kolmogorov-Smirnov test when the underlying distribution is a UniformDistribution, the test size is 0.05, and the sample size is 12:
A sample of 31 sheets of airplane glass were subjected to a constant stress until breakage. Investigate whether the data is drawn from a NormalDistribution or a GammaDistribution:
Compare the quantile-quantile plots for the candidate distributions:
The data appears to fit a GammaDistribution slightly better than a NormalDistribution:
By default, univariate data is compared to a NormalDistribution:
The parameters have been estimated from the data:
Multivariate data is compared to a MultinormalDistribution by default:
The parameters of the test distribution are estimated from the data if not specified:
Specified parameters are not estimated:
Maximum-likelihood estimates are used for unspecified parameters of the test distribution:
If the parameters are unknown, KolmogorovSmirnovTest applies a correction when possible:
The parameters are estimated but no correction is applied:
The fitted distribution is the same as before and the -value is corrected:
When parameters are estimated Lilliefors' correction is used:
Estimate the parameters prior to testing to perform the classical Kolmogorov-Smirnov test:
Conceptually, the Kolmogorov-Smirnov test computes the maximum absolute difference between the empirical and theoretical CDFs:
Plot the CDFs, showing the maximum absolute difference:
Independent marginal densities are assumed in tests for multivariate goodness-of-fit:
The test statistic is identical when independence is assumed:
The Kolmogorov-Smirnov test is not intended for discrete distributions:
The Kolmogorov-Smirnov test is not valid for some distributions when parameters have been estimated from the data:
Provide parameter values if they are known:
Alternatively, use Monte Carlo methods to approximate the -value:
Ties in the data are ignored:
Differences may be more apparent with larger numbers of ties:
The distribution of the Kolmogorov-Smirnov test statistic:
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