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			File:Chi-square_distributionCDF-English.png\|Cumulative distribution function of the chi-square distribution
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'''Pearson's chi-squared test''' ([[Chi-squared test]]), also known as the '''chi-square goodness-of-fit test''' or '''chi-square test for independence''', is a statistical hypothesis test used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table. In the field of [[statistics]], it is one of the most common tests for analyzing categorical data.

==Overview==
The test is applicable in situations where the data can be categorized into a contingency table, and the sample size is sufficiently large. The chi-squared test provides a method to gauge the discrepancy between observed and expected frequencies under the null hypothesis that no difference exists. It was developed by [[Karl Pearson]] in the early 20th century, hence the name.

==Assumptions==
Before applying Pearson's chi-squared test, certain assumptions must be met:
* Observations are independently drawn from the population.
* The sample size is large enough. As a rule of thumb, all expected counts should be at least 5.
* The data are categorical rather than numerical.

==Calculation==
The test statistic is calculated as:
\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \]
where \(O_i\) is the observed frequency for category \(i\), \(E_i\) is the expected frequency for category \(i\), and the summation is over all categories.

==Applications==
Pearson's chi-squared test is widely used in two major scenarios:
* '''Goodness-of-fit test''': To determine how well an observed distribution fits with an expected distribution.
* '''Test of independence''': To determine if there is a significant association between two categorical variables.

==Limitations==
While widely used, the test has limitations:
* It is not suitable for small sample sizes.
* It can only be used on categorical data.
* The test is sensitive to the sample size, meaning that with very large samples, even trivial differences can appear statistically significant.

==Examples==
An example of a goodness-of-fit test would be comparing the observed color distribution of M&Ms to the expected distribution claimed by the manufacturer. For a test of independence, one might analyze data from a survey to see if there is an association between gender and preference for a particular type of music.

==See Also==
* [[Fisher's exact test]]
* [[Likelihood-ratio test]]
* [[Yates's correction for continuity]]

==References==
<references/>

[[Category:Statistical tests]]
[[Category:Nonparametric statistics]]

{{Statistics-stub}}

Pearson's chi-squared test - Revision history

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