A one-way ANOVA (analysis of variance) tests whether the means of three or more independent groups differ. It answers one question, "is there any difference among these group means?", without inflating the false-positive rate the way a long series of t-tests would.
This guide explains the logic of ANOVA, its assumptions, a full worked example, how to follow up a significant result with post hoc tests, and how to report it in APA 7 style.
In this guide
Why not just run several t-tests?
With three groups you would need three t-tests (A vs B, A vs C, B vs C). If each uses α = .05, the chance of at least one false positive across the three is about 1 − 0.95³ ≈ 14%. With five groups (10 comparisons) it is about 40%. ANOVA tests all groups at once with a single F-test, keeping the overall error rate at 5%. Post hoc tests then find which groups differ, with corrections built in.
When to use a one-way ANOVA
- One continuous outcome (exam score, satisfaction scale, reaction time).
- One categorical factor with three or more independent groups (three teaching methods, four age bands, five regions).
- Different participants in each group.
If the same people are measured several times, use a repeated measures ANOVA. If you have two factors (say, method and gender), use a two-way ANOVA. If you want to adjust for a covariate such as pre-test score, use an ANCOVA.
How ANOVA works (in one paragraph)
ANOVA splits the total variability in the outcome into two parts: variability between groups (how far the group means are from the grand mean) and variability within groups (how far individuals are from their own group mean). The F-ratio compares them:
F = mean square between ÷ mean square within
If the groups really have the same population mean, F should be close to 1. A large F means the group means are further apart than within-group noise would explain.
Assumptions
- Independent observations, guaranteed by the design.
- Normally distributed residuals in each group. ANOVA is robust to moderate non-normality when groups are of similar size and have 20 or more participants each. Check with a Q-Q plot or the Shapiro-Wilk test.
- Homogeneity of variance. Check with Levene's test. If it is significant, report Welch's F instead of the classic F.
- No extreme outliers within groups.
If your data are ordinal or badly skewed with small groups, use the Kruskal-Wallis test, the nonparametric alternative to one-way ANOVA. It comes with Dunn's post hoc tests.
Worked example
A school compares end-of-term maths scores for three teaching methods, with 20 students in each group.
| Method | n | Mean | SD |
|---|---|---|---|
| Lecture | 20 | 70.2 | 8.4 |
| Flipped classroom | 20 | 75.8 | 9.1 |
| Problem-based | 20 | 78.1 | 8.7 |
Step 1: Check equal variances
The SDs are similar (8.4 to 9.1), and Levene's test is not significant, so the classic F-test is appropriate. (Welch's F gives the same conclusion here: F(2, 37.96) = 4.46, p = .018.)
Step 2: Read the ANOVA table
| Source | SS | df | MS | F | p |
|---|---|---|---|---|---|
| Between groups | 660.4 | 2 | 330.2 | 4.32 | .018 |
| Within groups | 4352.1 | 57 | 76.4 | ||
| Total | 5012.5 | 59 |
F(2, 57) = 4.32, p = .018. Because p < .05, at least one group mean differs from the others.
Step 3: Effect size
- Eta squared (η²) = SS between ÷ SS total = 660.4 ÷ 5012.5 = .13. Teaching method accounts for about 13% of the variance in scores in this sample.
- Omega squared (ω²) = .10. It is less biased than η², especially in small samples, and many methodologists prefer it.
Common rough benchmarks for η² are .01 (small), .06 (medium) and .14 (large).
Step 4: Post hoc tests
The F-test says something differs, not what. The Tukey HSD test compares every pair of groups while controlling the familywise error rate:
| Comparison | Mean difference | p (Tukey) |
|---|---|---|
| Flipped − Lecture | 5.6 | .115 |
| Problem-based − Lecture | 7.9 | .016 |
| Problem-based − Flipped | 2.3 | .685 |
Only the problem-based vs lecture difference is significant. The flipped classroom sits in between and does not differ significantly from either.
Which post hoc test should I use?
| Situation | Recommended test |
|---|---|
| Equal variances, all pairwise comparisons | Tukey HSD |
| Unequal variances or very unequal group sizes | Games-Howell |
| A few planned comparisons only | Bonferroni or Holm-corrected t-tests |
| Every group vs one control group | Dunnett's test |
The free one-way ANOVA calculator gives Tukey HSD comparisons automatically, along with Levene's test, Welch's F, η² and ω².
How to report a one-way ANOVA in APA 7
A one-way ANOVA showed that maths scores differed by teaching method, F(2, 57) = 4.32, p = .018, η² = .13. Tukey post hoc tests indicated that students taught with problem-based learning (M = 78.1, SD = 8.7) scored significantly higher than students taught by lecture (M = 70.2, SD = 8.4), p = .016. The flipped-classroom group (M = 75.8, SD = 9.1) did not differ significantly from either group.
Checklist for your write-up:
- Means and SDs for every group (in text or a table).
- F with both degrees of freedom, the exact p-value and an effect size.
- Which post hoc test you used, and its results.
- If Levene's test was significant, say you used Welch's F and Games-Howell comparisons.
Common mistakes
- Running post hoc tests after a non-significant F and reporting the one comparison that "worked". Unless the comparisons were planned in advance, stop at the non-significant F.
- Ignoring unequal variances when group sizes differ. That combination makes the classic F-test unreliable.
- Reporting only p-values. Always add η² or ω².
- Using ANOVA for repeated measurements of the same people. That needs a repeated measures ANOVA.
Frequently asked questions
What does the F value mean in ANOVA?
F is the ratio of the variance between group means to the variance within groups. A value near 1 suggests no difference between groups; larger values suggest the group means differ more than chance would explain.
What is a good eta squared value?
As rough benchmarks, η² of .01 is small, .06 is medium and .14 is large. Interpret it in the context of your field and outcome rather than relying only on these labels.
Can I run ANOVA with unequal group sizes?
Yes. Unequal group sizes are fine, but combined with unequal variances they make the classic F-test unreliable. Check Levene's test and use Welch's F with Games-Howell comparisons if needed.
What should I do if Levene's test is significant?
Report Welch's F instead of the standard F, and use the Games-Howell post hoc test, which does not assume equal variances.
What is the nonparametric alternative to one-way ANOVA?
The Kruskal-Wallis H test. It compares the rank distributions of three or more independent groups and is followed by Dunn's post hoc tests.