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Statistics LibreTexts

11: The Chi-Square Distribution

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  • A chi-squared 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.

    • 11.1: Prelude to The Chi-Square Distribution
      You will now study a new distribution, one that is used to determine the answers to such questions. This distribution is called the chi-square distribution.
    • 11.2: Facts About the Chi-Square Distribution
      The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.
    • 11.3: Goodness-of-Fit Test
      In this type of hypothesis test, you determine whether the data "fit" a particular distribution or not. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test (meaning the distribution for the hypothesis test is chi-square) to determine if there is a fit or not. The null and the alternative hypotheses for this test may be written in sentences or may be stated as equations or inequalities.
    • 11.4: Test of Independence
      Tests of independence involve using a contingency table of observed (data) values. The test statistic for a test of independence is similar to that of a goodness-of-fit test.
    • 11.5: Test for Homogeneity
      The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to draw a conclusion about whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence.
    • 11.6: Comparison of the Chi-Square Tests
      You have seen the Chi-square test statistic used in three different circumstances. The following bulleted list is a summary that will help you decide which Chi-square test is the appropriate one to use.
    • 11.7: Test of a Single Variance (Not Included in the Course)
      A test of a single variance assumes that the underlying distribution is normal. The null and alternative hypotheses are stated in terms of the population variance (or population standard deviation). A test of a single variance may be right-tailed, left-tailed, or two-tailed
    • 11.8: Lab 1: Chi-Square Goodness-of-Fit (Worksheet)
      A statistics Worksheet: The student will evaluate data collected to determine if they fit either the uniform or exponential distributions.
    • 11.9: Lab 2: Chi-Square Test of Independence (Worksheet)
      A statistics Worksheet: The student will evaluate if there is a significant relationship between favorite type of snack and gender.
    • 11.E: The Chi-Square Distribution (Exercises)
      These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.

    Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at