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Not to be confused with the special case: Pearson's chi-squared test

Preface: The Chi-Square distribution is the distribution of the sum of the squares of a set of normally distributed random variables. Its value stems from the fact that the sum of random variables from any distribution can be closely approximated by a normal distribution as the sum as samples size increases. Thus the test is widely applicable for all distributions.

A chi-squared test, also referred to as chi-square test or ${\displaystyle \chi ^{2}}$ test, is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. Also considered a chi-squared test is a test in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chi-squared distribution as closely as desired by making the sample size large enough.

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

• Pearson's chi-squared test, also known as the chi-squared goodness-of-fit test or chi-squared test for independence. When the chi-squared test is mentioned without any modifiers or without other precluding context, this test is usually meant (for an exact test used in place of ${\displaystyle \chi ^{2}}$, see Fisher's exact test).
• Yates's correction for continuity, also known as Yates' chi-squared test.
• Cochran–Mantel–Haenszel chi-squared test.
• McNemar's test, used in certain 2 × 2 tables with pairing
• Tukey's test of additivity
• 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-squared 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-squared 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 n 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-squared 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.

See also

• Chi-squared test nomogram
• G-test
• Minimum chi-square estimation
• The Wald test can be evaluated against a chi-squared distribution.

References

• Weisstein, Eric W. "Chi-Squared Test". MathWorld.
• Corder, G.W., Foreman, D.I. (2009). Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach Wiley, ISBN 978-0-470-45461-9
• Greenwood, P.E., Nikulin, M.S. (1996) A guide to chi-squared testing. Wiley, New York. ISBN 0-471-55779-X
• Nikulin, M.S. (1973). "Chi-squared test for normality". In: Proceedings of the International Vilnius Conference on Probability Theory and Mathematical Statistics, v.2, pp. 119–122.
• Bagdonavicius, V., Nikulin, M.S. (2011) "Chi-square goodness-of-fit test for right censored data". The International Journal of Applied Mathematics and Statistics, p. 30-50.

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