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PearsonChiSquareTest
✖
PearsonChiSquareTest

✖

tests whether data is normally distributed using the Pearson test.

✖

tests whether data is distributed according to dist using the Pearson test.

✖
PearsonChiSquareTest[data,dist,"property"]

returns the value of "property".

Details 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 {x1,x2,…} or multivariate {{x1,y1,…},{x2,y2,…},…}.
  • 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 ^(th) 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 "property".
  • Properties related to the reporting of test results include:
  • "DegreesOfFreedom"the degrees of freedom used in a test
    "PValue"-value
    "PValueTable"formatted version of "PValue"
    "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 "TestData"
    "TestStatistic"test statistic
    "TestStatisticTable"formatted "TestStatistic"
  • 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 Automaticthe 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 "TestConclusion" and "ShortTestConclusion" properties is controlled by the SignificanceLevel option. By default, is set to 0.05.
  • 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.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Perform the Pearson test for normality:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-e2z9oi
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-b6snhw
Out[2]=2

Test the fit of some data to a particular distribution:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-erxs41
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-dofsl4
Out[2]=2

Compare the distributions of two datasets:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-b8s07g
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ckwifh
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ht6wkq
Out[3]=3

Extract the test statistic from the Pearson test:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-m3chh3
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-cm7ucc
Out[2]=2

Scope  (9)Survey of the scope of standard use cases

Testing  (6)

Perform a Pearson test for normality:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-lqxbgq

The -value for the normal data is large compared to the -value for the non-normal data:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-mks25
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-kjd2s
Out[4]=4

Test the goodness of fit to a particular distribution:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-budcm8
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-dtqxca
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ci5vf
Out[3]=3

Compare the distributions of two datasets:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-f1whrm
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-2qof
Out[2]=2

The two datasets do not have the same distribution:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-inhd0f
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bbarau
Out[4]=4

Test for multivariate normality:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-67n6h
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-eooal3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-cn5yan
Out[3]=3

Test for goodness of fit to any multivariate distribution:

In[1]:=1
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https://wolfram.com/xid/0dbjf29egf0adulmli-gtwavq
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-n3thk6
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-hlhupv
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ut5c
Out[4]=4

Create a HypothesisTestData object for repeated property extraction:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-cdnr2n
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bolz27
Out[2]=2

The properties available for extraction:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-e40fsc
Out[3]=3

Reporting  (3)

Tabulate the results of the Pearson test:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-cqxopz
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-be6kk1

The full test table:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ef4et7
Out[3]=3

A -value table:

In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bfqugt
Out[4]=4

The test statistic:

In[5]:=5
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-oi9r56
Out[5]=5

Retrieve the entries from a Pearson test table for custom reporting:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-s3qsd
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ga3bij
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-gg2n22
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-65k71
Out[4]=4

Report test conclusions using "ShortTestConclusion" and "TestConclusion":

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bkir6y
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-cd96sm
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ggy9zn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-el9mb

The conclusion may differ at a different significance level:

In[5]:=5
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https://wolfram.com/xid/0dbjf29egf0adulmli-c53cri
In[6]:=6
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-byyexa
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bc67bi

Options  (3)Common values & functionality for each option

Method  (3)

Use Monte Carlo-based methods or a computation formula:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-b56tvj
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-eyrfe
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-evuhgg
Out[3]=3

Set the number of samples to use for Monte Carlo-based methods:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-xg6xc
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-499mh

The Monte Carlo estimate converges to the true -value with increasing samples:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-eli8sg
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ba5c5u
Out[4]=4

Set the random seed used in Monte Carlo-based methods:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ccet45
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ip8pt1

The seed affects the state of the generator and has some effect on the resulting -value:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-go8plt
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-pfg0ok
Out[4]=4

Applications  (2)Sample problems that can be solved with this function

A power curve for the Pearson test:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-f9ry9j
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bhp5v
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-fyqopk

Visualize the approximate power curve:

In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-eel2vq
Out[4]=4

Estimate the power of the Pearson test when the underlying distribution is UniformDistribution[{-4,4}], the test size is 0.05, and the sample size is 12:

In[5]:=5
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-z6f7c
Out[5]=5

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:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-pvmuv
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bphwu6
Out[2]=2

Count data is often modeled well by PoissonDistribution:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-id8isa
Out[3]=3

Suppose the city collected data over another 30-day period after reducing the speed limit. Compare the distributions before and after the reduction:

In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-gkxnd6
In[5]:=5
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-c8ii08
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-q0toii
Out[6]=6

The distributions are significantly different:

In[7]:=7
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-h7dit2
Out[7]=7

Properties & Relations  (10)Properties of the function, and connections to other functions

By default, univariate data is compared to NormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-in3dv2
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-loot1k
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ekgpjw
Out[3]=3

The parameters have been estimated from the data:

In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-f3jfr4
Out[4]=4

Multivariate data is compared to MultinormalDistribution by default:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-dhffw
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-cddu5a
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-hpjfy5
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bstr67

The parameters of the test distribution are estimated from the data if not specified:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-d47mlh
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-io3pky
Out[2]=2

Specified parameters are not estimated:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-hxkimc
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-nx9q2m
Out[4]=4

Maximum likelihood estimates are used for unspecified parameters of the test distribution:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ob999
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ccvtx8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-lczwnz
Out[3]=3

PearsonChiSquareTest effectively compares the observed and expected histograms:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-tvky4
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-dlds3

The data is binned into approximately bins that are equiprobable under :

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-gnqlxb
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-foynro

Under , each bin will contain an equal number of points:

In[5]:=5
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-c5nty4
Out[5]=5

Observed histograms for when is true and false, respectively:

In[6]:=6
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-e1yw92
Out[6]=6

The degrees of freedom are equal to the number of non-empty bins minus one:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-lnhoig
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ijz3mx
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-o9xubq
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-d9ictj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-dl3eb9
Out[5]=5

One degree of freedom is removed for each parameter that is estimated from the data:

In[6]:=6
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-f48lz3
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-dmv6le
Out[7]=7

If the parameters are unknown, PearsonChiSquareTest corrects the degrees of freedom:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-cyr6pp
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-cdmjix
Out[2]=2

No correction is applied when the parameters are specified:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-j8jo0o
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-e87ev

The fitted distribution is equivalent, but the degrees of freedom and -value are corrected:

In[5]:=5
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-voh29
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-bx464
Out[6]=6

The Pearson statistic asymptotically follows ChiSquareDistribution under :

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-goz612
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-pbcxh
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-rlsxq
In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-eh38ry
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-duxa4p
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-r7sr0j
Out[6]=6

Independent marginal densities are assumed in tests for multivariate goodness of fit:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-et8fgp
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-plqmfp
Out[2]=2

The test statistic is identical when independence is assumed:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-fpwagd
Out[3]=3

The Pearson test works with the values only when the input is a TimeSeries:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-7py5sj
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-57bf57
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-vvd88
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

Compute the statistic when the null hypothesis is true:

In[1]:=1
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-2qqg3c
In[2]:=2
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-ywy3ty

The test statistic given a particular alternative:

In[3]:=3
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-c5cy2n

Compare the distributions of the test statistics:

In[4]:=4
✖
https://wolfram.com/xid/0dbjf29egf0adulmli-87eb6q
Out[4]=4
Wolfram Research (2010), PearsonChiSquareTest, Wolfram Language function, https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html.
✖
Wolfram Research (2010), PearsonChiSquareTest, Wolfram Language function, https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html.

Text

Wolfram Research (2010), PearsonChiSquareTest, Wolfram Language function, https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html.

✖
Wolfram Research (2010), PearsonChiSquareTest, Wolfram Language function, https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html.

CMS

Wolfram Language. 2010. "PearsonChiSquareTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html.

✖
Wolfram Language. 2010. "PearsonChiSquareTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html.

APA

Wolfram Language. (2010). PearsonChiSquareTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html

✖
Wolfram Language. (2010). PearsonChiSquareTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html

BibTeX

@misc{reference.wolfram_2025_pearsonchisquaretest, author="Wolfram Research", title="{PearsonChiSquareTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html}", note=[Accessed: 22-April-2025 ]}

✖
@misc{reference.wolfram_2025_pearsonchisquaretest, author="Wolfram Research", title="{PearsonChiSquareTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html}", note=[Accessed: 22-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_pearsonchisquaretest, organization={Wolfram Research}, title={PearsonChiSquareTest}, year={2010}, url={https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html}, note=[Accessed: 22-April-2025 ]}

✖
@online{reference.wolfram_2025_pearsonchisquaretest, organization={Wolfram Research}, title={PearsonChiSquareTest}, year={2010}, url={https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html}, note=[Accessed: 22-April-2025 ]}