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Comparing means or medians across two or more groups -- from a simple t-test to factorial and repeated-measures designs.
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Unpaired (Independent) T TestCompares means between two independent groups of different subjects. Requires a continuous, approximately normally distributed outcome variable.Paired T TestCompares means from the same subjects at two time points, or from individually matched pairs. More statistically powerful than the unpaired t test for the same sample size.Mann-Whitney U TestNon-parametric test for two independent groups. Tests whether one group tends to rank higher than the other -- no normality assumption required.Wilcoxon Signed-Rank TestNon-parametric equivalent of the paired t test. Tests whether the median difference between paired measurements is significantly different from zero.Kruskal-Wallis TestNon-parametric equivalent of one-way ANOVA for 3+ independent groups.Friedman TestNon-parametric equivalent of repeated-measures ANOVA for 3+ related (repeated) measurements that are non-normal or ordinal.One-Way ANOVACompares means across 3 or more independent groups simultaneously, controlling the Type I error rate at 5% -- unlike running multiple t tests.Two-Way ANOVAExtends one-way ANOVA to two categorical factors simultaneously, testing both main effects and the interaction between them.Repeated Measures ANOVACompares means from the same subjects across 3 or more time points or conditions, accounting for within-subject correlation.ANCOVACombines ANOVA and regression to compare group means while statistically adjusting for one or more continuous covariates.MANOVATests whether two or more groups differ on a combination of multiple continuous outcome variables simultaneously.Post-Hoc TestsPairwise comparisons after a significant ANOVA or Kruskal-Wallis result, controlling the family-wise error rate (Tukey HSD or Dunn's test).
Related Articles
ANOVA vs T Test: How to Choose the Correct Statistical TestThe number-of-groups rule with medical examples and a decision flowchart.Paired vs Unpaired T Test: Which One Should You Use?Full guide with clinical examples and FAQ.Mann-Whitney U Test vs Independent t-Test: When Parametric Assumptions FailChoosing between parametric and non-parametric group comparisons.Tukey vs Bonferroni Post Hoc Tests After ANOVAComplete guide to pairwise comparisons after a significant ANOVA.Parametric vs Non-Parametric Tests in Medical ResearchHow to decide which family of tests fits your data.
Medical Research Examples
Unpaired T Test: A trial compares systolic BP reduction with Drug A (n=25) vs. placebo (n=25): mean difference=8.2 mmHg (95% CI 3.1-13.3), t(48)=3.24, p=0.002.
Mann-Whitney U Test: Pain scores (0-10, non-normal) compared between two analgesic groups (n=20 each): median 3 vs. 5, U=120, p=0.03.
One-Way ANOVA: Three analgesic groups (n=20 each) compared on pain relief: F(2,57)=12.3, p<0.001, η²=0.30 -- Tukey HSD is needed to locate the specific pairwise differences.
Two-Way ANOVA: Systolic BP reduction across two drugs and two doses (n=12/cell): Drug×Dose interaction F(1,44)=7.3, p=0.010 -- Drug A shows greater benefit at high dose, Drug B is similar at both doses.
Repeated Measures ANOVA: Knee OA pain (VAS) at Baseline, Week 4, Week 8, and Week 12 in 12 patients: F(3,33)=24.6, p<0.001, η²p=0.69; treatment effect plateaus by Week 8.
MANOVA: Three MS treatments compared on 4 outcome measures in 90 patients: Wilks Λ=0.61, F(8,168)=4.83, p<0.001 -- a pattern of difference missed by separate ANOVAs.
Common Mistakes
- Running multiple pairwise t-tests across 3+ groups instead of one ANOVA, inflating the Type I error rate.
- Using a t-test or ANOVA on clearly non-normal or ordinal data instead of the non-parametric equivalent.
- Treating paired (within-subject) data as if it were independent, which wastes statistical power and can bias results.
- Reporting a significant ANOVA without follow-up post-hoc tests to identify which groups actually differ.
- Ignoring a significant interaction in Two-Way ANOVA and interpreting the main effects in isolation.
- Not checking the sphericity assumption before interpreting repeated-measures ANOVA results.
Reporting Recommendations
Journals expect more than a p-value. For each test, report the effect estimate and its 95% confidence interval alongside the p-value:
Unpaired (Independent) T TestMean difference, 95% CI, t statistic, df, p-value, Cohen’s d
Paired T TestMean difference of pairs, 95% CI, t statistic, p-value, Cohen’s d
Mann-Whitney U TestMedians, IQR, U statistic, exact p-value, effect size r = Z/√N
Wilcoxon Signed-Rank TestMedian difference, W statistic, exact p-value, effect size r
Kruskal-Wallis TestH statistic, df, p-value, effect size epsilon-squared (ε²), group medians
Friedman TestChi-square (Friedman) statistic, df, p-value, Kendall's W (effect size)
One-Way ANOVAF statistic, df, p-value, eta-squared (η²), group means ± SD
Two-Way ANOVAF statistic, df, p-value, and partial eta-squared for each main effect and the interaction
Repeated Measures ANOVAF statistic, df, p-value, partial eta-squared, pairwise comparisons with Bonferroni correction
ANCOVAAdjusted F statistic, df, p-value, partial eta-squared, adjusted group means (estimated marginal means)
MANOVAWilks' Lambda, approximate F, df, p-value, multivariate eta-squared, followed by protected univariate ANOVAs per outcome
Post-Hoc TestsAdjusted p-value for each pairwise comparison, mean/rank difference, 95% CI
Recommended Learning Order
- Unpaired / Paired T Test — the simplest case: comparing exactly 2 groups
- Mann-Whitney U / Wilcoxon — non-parametric alternatives when data is not normally distributed
- One-Way ANOVA — extend to 3 or more independent groups
- Post-Hoc Tests — find out which specific groups differ after a significant ANOVA
- Two-Way ANOVA / ANCOVA — add a second factor or adjust for a covariate
- MANOVA — most advanced: multiple correlated outcomes at once
Frequently Compared Tests
One-Way ANOVA vs. Tukey vs. Bonferroni
ANOVA gives one omnibus test across all groups; Tukey HSD and Bonferroni are post-hoc corrections used afterward to find which specific pairs differ.
Frequently Asked Questions
When do I use a t-test vs. ANOVA?
Use a t-test for exactly 2 groups. For 3 or more groups, use ANOVA -- running multiple t-tests instead inflates the false-positive rate.
What's the difference between paired and unpaired tests?
Paired tests compare the same subjects measured twice (or matched pairs); unpaired tests compare two completely independent groups of different subjects.
My data isn't normally distributed -- what do I use instead?
Use the non-parametric equivalent: Mann-Whitney U instead of unpaired t-test, Wilcoxon instead of paired t-test, Kruskal-Wallis instead of ANOVA, Friedman instead of repeated-measures ANOVA.
My ANOVA is significant -- what's next?
A significant ANOVA only tells you that at least one group differs. Run a post-hoc test (Tukey HSD after ANOVA, Dunn's after Kruskal-Wallis) to identify which specific pairs differ.
When do I need Two-Way ANOVA instead of One-Way?
When you have two categorical grouping factors (e.g. drug and dose) and want to test each factor's effect plus whether they interact.
When should I use ANCOVA instead of ANOVA?
When you have a continuous covariate (often a baseline measurement) that you want to statistically adjust for, increasing power and correcting for baseline imbalance.