Are the results of my chi-square test invalid?

If the expected counts (also called expected frequencies) for the cells are very small, the results of the test may not be valid. If one or more categories have expected counts that are too low, you can combine them with adjacent categories to achieve the minimum required expected count. You can also use Fisher's exact test, which is accurate for all sample sizes. To perform Fisher's exact test, choose Stat > Tables > Cross Tabulation and Chi-Square and click Other Stats. Use the following guidelines to determine when you can trust the results.

Note

Fisher's exact test is only available for 2x2 contingency tables.

Are the results of my chi-square test of association invalid?

If either variable has only 2 or 3 levels, then you can trust the results if either of the following is true:
  • All cells have expected counts of at least 3.
  • All cells have expected counts of at least 2, and 50% or fewer of the cells have expected counts of less than 5.
If both variables have 4 to 6 levels, then you can trust the results if either of the following is true:
  • All cells have expected counts of at least 2.
  • All cells have expected counts of at least 1, and 50% or fewer of the cells have expected counts of less than 5.
Note

Minitab does not display the p-value when any expected count is less than 1 because the results can be invalid.

Are the results of my chi-square goodness-of-fit test invalid?

You can trust the results when either of the following is true:
  • All cells have expected counts of at least 2.5.
  • All cells have expected counts of at least 1.25, and 50% or fewer of the cells have expected counts of less than 5.

The second assumption is necessary because the distribution of counts under the null hypothesis is multinomial, and the normal distribution can be used to approximate the multinomial distribution if the sample size is sufficiently large and the probability parameters aren’t too small. You can use the Central Limit Theorem to show that the multinomial distribution converges to the normal distribution as the sample size approaches infinity. However, there is no easy way to show mathematically how and when the convergence fails. Guidelines like the second assumption ensure that the approximations that you use are reasonably accurate.

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