Why is the homogeneity of variance assumption not required for a related-samples t-test?

Why is the homogeneity of variance assumption not required for a related-samples t-test?

• A. The test compares two independent groups with equal variances.
• B. Variance across groups is irrelevant because the test ignores variance.
• C. The test analyzes the difference scores within each pair, not two separate groups.
• D. The test uses a nonparametric approach.

Answer: (C)

In a related-samples (paired) t-test, the focus is on the difference within each pair. For every subject you compute a difference Δ = value before - value after, then test whether the average of these differences is zero. Since there’s only one set of differences, there aren’t two independent groups with their own variances to compare. The test statistic uses the standard deviation of the difference scores and the standard error of the mean difference, not a pooled variance across two separate groups. Thus, the homogeneity of variances assumption across groups isn’t part of this test. What matters instead is that the distribution of the difference scores (or the sampling distribution of the mean difference) is approximately normal. If normality is a concern, nonparametric alternatives like the Wilcoxon signed-rank test can be used.

In a related-samples (paired) t-test, the focus is on the difference within each pair. For every subject you compute a difference Δ = value before - value after, then test whether the average of these differences is zero. Since there’s only one set of differences, there aren’t two independent groups with their own variances to compare. The test statistic uses the standard deviation of the difference scores and the standard error of the mean difference, not a pooled variance across two separate groups. Thus, the homogeneity of variances assumption across groups isn’t part of this test. What matters instead is that the distribution of the difference scores (or the sampling distribution of the mean difference) is approximately normal. If normality is a concern, nonparametric alternatives like the Wilcoxon signed-rank test can be used.