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Chi-Square Tests

The χ2 Goodness-of-Fit

This test, also referred to as the χ2 test for a single sample, evaluates whether the observed cell frequencies are different from the expected cell frequencies.


  • The test requires data in a single data column and treats the data as categorical (see the example below).

    • Example: The histogram illustrated here represents the observed frequencies of failure in six different hardware systems for a given period of time. The χ2 goodness-of-fit test evaluates whether there is a statistical difference with respect to the number of failures taken place in these systems that can be differentiated from a random process with equal probability for failure in each system.

    The test summary table consists of columns for observed frequencies, expected frequencies, observed minus expected frequencies, squared differences and cell χ2. The test results include the overall χ2 (summation of all cells chi-square), the critical chi-square χ2(0.05) value, and p value.

The Single-Sample χ2 Test for a Population Variance

This test evaluates whether a sample of n subjects come from a parent population in which the variance equals a specific hypothesized value.

  • The method requires a single numeric column of data.
  • The test results include the estimated χ2, the critical two-tailed and one-tailed χ2 0.05 and χ2 0.01 values.