# Chi-squared test

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Chi-squared test
"Chi-square test" is often shorthand for Pearson's chi-square test.

A chi-square test, also referred to as chi-squared test or χ2 test, is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-square distribution when the null hypothesis is true, or any in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chi-square distribution as closely as desired by making the sample size large enough.

Some examples of chi-square tests where the chi-square distribution is only approximately valid:

• Pearson's chi-square test, also known as the chi-square goodness-of-fit test or chi-square test for independence. When mentioned without any modifiers or without other precluding context, this test is usually understood (for an exact test used in place of χ2, see Fisher's exact test).
• Yates's correction for continuity, also known as Yates' chi-square test.
• Cochran–Mantel–Haenszel chi-square test.
• Linear-by-linear association chi-square test.
• The portmanteau test in time-series analysis, testing for the presence of autocorrelation
• Likelihood-ratio tests in general statistical modelling, for testing whether there is evidence of the need to move from a simple model to a more complicated one (where the simple model is nested within the complicated one).

One case where the distribution of the test statistic is an exact chi-square distribution is the test that the variance of a normally-distributed population has a given value based on a sample variance. Such a test is uncommon in practice because values of variances to test against are seldom known exactly.

## Chi-square test for variance in a normal population

If a sample of size n is taken from a population having a normal distribution, then there is a well-known result (see distribution of the sample variance) which allows a test to be made of whether the variance of the population has a pre-determined value. For example, a manufacturing process might have been in stable condition for a long period, allowing a value for the variance to be determined essentially without error. Suppose that a variant of the process is being tested, giving rise to a small sample of product items whose variation is to be tested. The test statistic T in this instance could be set to be the sum of squares about the sample mean, divided by the nominal value for the variance (i.e. the value to be tested as holding). Then T has a chi-square distribution with n – 1 degrees of freedom. For example if the sample size is 21, the acceptance region for T for a significance level of 5% is the interval 9.59 to 34.17.

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