Comparing Two Groups: Student's t, Welch's t, Mann-Whitney, and Paired Tests
Choosing between parametric mean comparisons and non-parametric rank tests across independent and paired clinical designs.
In this lesson you will learn
- 1Distinguish Independent-Samples designs (two separate groups) from Paired-Samples designs (same participant measured twice).
- 2Apply Shapiro-Wilk normality and Levene's homogeneity checks to choose between Student's t-test, Welch's t-test, and Mann-Whitney U.
- 3Interpret Cohen's d and rank-biserial correlation alongside p-values in APA 7th prose.
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Two-Group Comparison Framework & Effect Sizes
This section defines the four core two-group tests—Student's t, Welch's t, Mann-Whitney U, and Paired t / Wilcoxon—alongside Cohen's d effect size benchmarks.
Running an independent t-test on paired before/after data discards within-subject correlation and sacrifices substantial statistical power.
Comparing two conditions is the most frequent analytical task in clinical trials. Selecting the defensible test requires checking three structural questions: (1) Are observations independent or paired? (2) Is the outcome normally distributed? (3) Are group variances equal?
| Test / Metric | When Valid | Summary & Effect Size |
|---|---|---|
| Student's t-Test | 2 independent groups, normal distribution in each arm, equal variances (Levene's p > .05). | Mean (SD) | Cohen's d (0.2 small, 0.5 medium, 0.8 large). |
| Welch's t-Test | 2 independent groups, normal distribution, unequal variances (Levene's p < .05). | Mean (SD) with Welch-adjusted df | Cohen's d. |
| Mann-Whitney U | 2 independent groups, skewed continuous or ordinal outcome. | Median [IQR] | Rank-biserial correlation (r_rb). |
| Paired t / Wilcoxon | Same participant measured twice (Pre vs Post): Paired t if differences are normal; Wilcoxon Signed-Rank if skewed. | Mean (SD) of differences or Median [IQR] | Cohen's d_z. |
Clinical Example
An antihypertensive RCT compares systolic blood pressure reduction between Drug A (n = 45, variance = 14.4) and Standard Care (n = 45, variance = 68.9); Levene's test yields p = .004.
Interpretation: Because both arms are normal but Levene's test shows unequal variances (p = .004 < .05), Welch's t-test is reported with decimal-adjusted degrees of freedom (df = 59.4).
Common Pitfall
Reading the 'Equal variances assumed' row in SPSS t-test output when Levene's test significance column is less than .05.
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