BUILT-IN MATHEMATICA SYMBOL

# PearsonChiSquareTest

PearsonChiSquareTest[data]
tests whether data is normally distributed using the Pearson test.

PearsonChiSquareTest[data, dist]
tests whether data is distributed according to dist using the Pearson test.

PearsonChiSquareTest[data, dist, "property"]
returns the value of .

## Details and OptionsDetails and Options

• PearsonChiSquareTest performs the Pearson 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 Pearson test effectively compares a histogram of data to a theoretical histogram based on dist. The bins are chosen to have equal probability in dist. »
• For univariate data the test statistic is given by , where and are the observed and expected counts for the histogram bin, respectively.
• For multivariate tests, the sum of the univariate marginal -values is used and is assumed to follow a UniformSumDistribution under .
• PearsonChiSquareTest[data, dist, "HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
• PearsonChiSquareTest[data, dist, "property"] can be used to directly give the value of .
• Properties related to the reporting of test results include:
•  "DegreesOfFreedom" the degrees of freedom used in a test "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 .
• With the setting Method->"MonteCarlo", datasets of the same length as the input are generated under using the fitted distribution. The EmpiricalDistribution from PearsonChiSquareTest[si, dist, "TestStatistic"] is then used to estimate the -value.

## ExamplesExamplesopen allclose all

### Basic Examples (4)Basic Examples (4)

Perform the Pearson 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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Extract the test statistic from the Pearson test:

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