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PearsonChiSquareTest

PearsonChiSquareTest[data]
tests whether data is normally distributed using the Pearson test.
PearsonChiSquareTest
tests whether data is distributed according to dist using the Pearson test.
PearsonChiSquareTest
returns the value of .
  • 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.
  • For univariate data the test statistic is given by , where and are the observed and expected counts for the ^(th) histogram bin, respectively.
  • 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:
"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:
MethodAutomaticthe method to use for computing -values
SignificanceLevel0.05cutoff 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 the Pearson test for normality:
Test the fit of some data to a particular distribution:
Compare the distributions of two datasets:
Extract the test statistic from the Pearson test:
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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Perform a Pearson 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 Pearson test:
The full test table:
A -value table:
The test statistic:
Retrieve the entries from a Pearson test table for custom reporting:
Report test conclusions using and :
The conclusion may differ at a different significance level:
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:
A power curve for the Pearson test:
Visualize the approximate power curve:
Estimate the power of the Pearson test when the underlying distribution is UniformDistribution, the test size is 0.05, and the sample size is 12:
The number of auto accidents was recorded for a city over the course of 30 days. The city council is planning on lowering speed limits in the city and wants a model of the accident rate as a baseline for later comparison:
Count data is often modeled well by PoissonDistribution:
Suppose the city collected data over another 30-day period after reducing the speed limit. Compare the distributions before and after the reduction:
The distributions are significantly different:
By default, univariate data is compared to NormalDistribution:
The parameters have been estimated from the data:
Multivariate data is compared to 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:
PearsonChiSquareTest effectively compares the observed and expected histograms:
The data is binned into approximately bins that are equiprobable under :
Under , each bin will contain an equal number of points:
Observed histograms for when is true and false, respectively:
The degrees of freedom are equal to the number of non-empty bins minus one:
One degree of freedom is removed for each parameter that is estimated from the data:
If the parameters are unknown, PearsonChiSquareTest corrects the degrees of freedom:
No correction is applied when the parameters are specified:
The fitted distribution is equivalent but the degrees of freedom and -value are corrected:
The Pearson statistic asymptotically follows ChiSquareDistribution under :
Independent marginal densities are assumed in tests for multivariate goodness of fit:
The test statistic is identical when independence is assumed:
The distribution of the Pearson test statistic:
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