Editing Chi-squared test (section)

{{Short description|Statistical hypothesis test}}
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[[File:Chi-square distributionCDF-English.png|thumb|right|300px|[[Chi-squared distribution]], showing {{math|χ<sup>2</sup>}} on the ''x''-axis and ''p''-value (right tail probability) on the ''y''-axis.]]

A '''chi-squared test''' (also '''chi-square''' or '''{{math|χ<sup>2</sup>}} test''') is a [[Statistical hypothesis testing|statistical hypothesis test]] used in the analysis of [[contingency table]]s when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables (''two dimensions of the contingency table'') are independent in influencing the test statistic (''values within the table'').<ref>{{Cite web |title=Chi-Square - Sociology 3112 - Department of Sociology - The University of utah |url=https://soc.utah.edu/sociology3112/chi-square.php |access-date=2022-11-12 |website=soc.utah.edu}}</ref> The test is [[Validity (statistics)|valid]] when the test statistic is [[chi-squared distribution|chi-squared distributed]] under the [[null hypothesis]], specifically [[Pearson's chi-squared test]] and variants thereof. Pearson's chi-squared test is used to determine whether there is a [[Statistical significance|statistically significant]] difference between the expected [[frequency (statistics)|frequencies]] and the observed frequencies in one or more categories of a [[contingency table]]. For contingency tables with smaller sample sizes, a [[Fisher's exact test]] is used instead.

In the standard applications of this test, the observations are classified into mutually exclusive classes. If the [[null hypothesis]] that there are no differences between the classes in the population is true, the test statistic computed from the observations follows a {{math|χ<sup>2</sup>}} [[frequency distribution]]. The purpose of the test is to evaluate how likely the observed frequencies would be assuming the null hypothesis is true.

Test statistics that follow a {{math|χ<sup>2</sup>}} distribution occur when the observations are independent. There are also {{math|χ<sup>2</sup>}} tests for testing the null hypothesis of independence of a pair of [[random variable]]s based on observations of the pairs.

''Chi-squared tests'' often refers to tests for which the distribution of the test statistic approaches the {{math|χ<sup>2</sup>}} distribution [[asymptote|asymptotically]], meaning that the [[sampling distribution]] (if the null hypothesis is true) of the test statistic approximates a chi-squared distribution more and more closely as [[Sample (statistics)|sample]] sizes increase.