Chi-squared test

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Chi-squared distribution, showing X2 on the x-axis and P-value on the y-axis.

A chi-squared test, also referred to as chi-square test or χ² 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. The chi-square (I) test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. Do the number of individuals or objects that fall in each category differ significantly from the number you would expect? Is this difference between the expected and observed due to sampling error, or is it a real difference?

Examples of chi-squared tests[edit]

The following are examples of chi-squared tests where the chi-squared distribution is approximately valid:

Pearson's chi-squared test[edit]

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 χ², see Fisher's exact test).

Yates's correction for continuity[edit]

Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution. This assumption is not quite correct, and introduces some error.

To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction for continuity that adjusts the formula for Pearson's chi-squared test by subtracting 0.5 from the difference between each observed value and its expected value in a 2 × 2 contingency table.[1] This reduces the chi-squared value obtained and thus increases its p-value.

Other tests[edit]

Exact chi-squared distribution[edit]

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-Square Test Requirements[edit]

1. Quantitative data. 2. One or more categories. 3. Independent observations. 4. Adequate sample size (at least 10). 5. Simple random sample. 6. Data in frequency form. 7. All observations must be used.

Chi-squared test for variance in a normal population[edit]

If a sample of size n is taken from a population having a normal distribution, then there is a 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[edit]


  1. ^ Yates, F (1934). "Contingency table involving small numbers and the χ2 test". Supplement to the Journal of the Royal Statistical Society 1(2): 217–235. JSTOR 2983604
  • Weisstein, Eric W., "Chi-Squared Test", MathWorld.
  • 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.[full citation needed]