Chi-Square Goodness-of-Fit Test

Use this test when you have a single categorical variable and want to test whether the observed proportions match a theoretical distribution. For example, testing whether a die is fair (each face equally likely), or whether the distribution of blood types in your sample matches the expected population proportions.

• The variable is categorical with two or more categories.
• Observations are independent.
• Expected frequency in each category is at least 5.
• The hypothesised proportions are specified before examining the data.

• If the p-value is less than alpha, the observed distribution differs significantly from the expected distribution.
• Examine which categories have the largest discrepancies between observed and expected counts to understand where the deviation occurs.
• A non-significant result means you cannot reject the hypothesised distribution, but it does not prove the distribution is correct.

• The test requires specifying expected proportions a priori. Fitting proportions from the data and then testing them invalidates the test.
• Categories with very small expected counts (< 5) can inflate the chi-square statistic. Combine categories if necessary.
• The test is sensitive to sample size. Very large samples can detect trivial departures from the expected distribution.

1. Specify the expected proportion for each category under the null hypothesis.
2. Multiply each expected proportion by the total sample size to get expected frequencies.
3. Compute chi-square = sum of (observed - expected)^2 / expected across all categories.
4. Compare to the chi-square distribution with (k-1) degrees of freedom, where k is the number of categories.