What post hoc test should I use after ANOVA?

Use Tukey HSD when variances are roughly equal; it compares every pair of groups and controls the familywise error rate. If Levene's test is significant, use Games-Howell instead and report Welch's ANOVA.

What does a significant F mean in a one-way ANOVA?

It means at least one group mean differs from the others. It does not say which groups differ, which is why you follow it with post hoc tests such as Tukey HSD.

What is a good eta squared value?

Common benchmarks (Cohen, 1988) are .01 small, .06 medium, and .14 large. Report it with F and p, for example F(2, 21) = 38.85, p < .001, η² = .79.

Can I use ANOVA for Likert scale data?

For summed or averaged scale scores, ANOVA is widely accepted when scores are roughly normal. For a single Likert item, use the Kruskal-Wallis test instead.

ONE-WAY ANOVA CALCULATOR

One-way ANOVA calculator with Tukey HSD post hoc tests.

Q: Can I use ANOVA for Likert scale data?

For summed or averaged scale scores, ANOVA is widely accepted when scores are roughly normal. For a single Likert item, use the Kruskal-Wallis test instead.

Compare the means of three or more independent groups. Paste raw scores to get the full ANOVA table, F, p, eta squared, omega squared, Levene's test, Welch's ANOVA, Tukey pairwise comparisons, and an APA sentence.

GROUPS

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Write each group as "Name: value, value, …". Values can be separated by commas, spaces, or tabs, so you can paste a column from Excel or SPSS after the name.

GROUP VALUES

One-way ANOVA formulas, assumptions, post hoc tests, and APA reporting

When to use a one-way ANOVA

Use it to compare the means of three or more independent groups on one numeric outcome, for example exam scores under lecture, blended, and flipped teaching. With only two groups, use a t-test (the results are equivalent: F = t²). If the same people are measured in every condition, use a repeated-measures ANOVA instead.

Formula

F = MS_between ÷ MS_within,   MS_between = Σ nᵢ(x̄ᵢ − x̄)² ÷ (k − 1),   MS_within = Σ Σ (x − x̄ᵢ)² ÷ (N − k)

k is the number of groups and N the total sample size. A large F means the group means differ more than you would expect from the variation inside the groups. The p value comes from the F distribution with (k − 1, N − k) degrees of freedom.

Effect size: eta squared and omega squared

η² = SS_between ÷ SS_total, ω² = (SS_between − (k − 1)·MS_within) ÷ (SS_total + MS_within)

Cohen's (1988) benchmarks for η² are .01 small, .06 medium, and .14 large. ω² is less biased in small samples, so some journals prefer it. For a one-way design, partial η² equals η².

Tukey HSD post hoc test

q = |x̄ᵢ − x̄ⱼ| ÷ √[(MS_within ÷ 2)(1/nᵢ + 1/nⱼ)]

A significant F only tells you that at least one mean differs. Tukey's honestly significant difference test compares every pair while keeping the familywise error rate at 5%. p values come from the studentized range distribution; with unequal group sizes this is the Tukey-Kramer method, the same as SPSS's "Tukey" option.

Assumptions to check

Independent observations: each participant belongs to one group only
Roughly normal scores within each group (check with the Shapiro-Wilk test or Q-Q plots); ANOVA is fairly robust with 20+ per group
Equal variances (homogeneity): check Levene's test. If p < .05, report Welch's ANOVA and use Games-Howell post hoc tests
Ordinal outcome or strong skew: use the Kruskal-Wallis test with Dunn post hoc tests

How to report it (APA style)

“A one-way ANOVA showed a significant effect of teaching method on exam scores, F(2, 21) = 38.85, p < .001, η² = .79. Tukey HSD tests indicated that the flipped group (M = 83.11, SD = 2.93) scored higher than the blended group (M = 76.71, SD = 2.56), p = .001, which scored higher than the lecture group (M = 70.50, SD = 3.25), p = .002.”

References

Fisher, R. A. (1925). Statistical methods for research workers. Oliver and Boyd.
Tukey, J. W. (1949). Comparing individual means in the analysis of variance. Biometrics, 5(2), 99–114.
Kramer, C. Y. (1956). Extension of multiple range tests to group means with unequal numbers of replications. Biometrics, 12(3), 307–310.
Welch, B. L. (1951). On the comparison of several mean values: An alternative approach. Biometrika, 38(3–4), 330–336.
Brown, M. B., & Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American Statistical Association, 69(346), 364–367.

One-Way ANOVA Calculator

Compare three or more group means from raw data, with a full ANOVA source table, effect sizes, assumption checks, Tukey-Kramer pairwise comparisons, and a ready APA sentence.

Compare mean scores across three or more teaching methods, departments, or treatment groups

Find which pairs of groups differ with Tukey HSD after a significant F

Check equal variances and switch to Welch's ANOVA when needed

KEY FUNCTIONS

Full ANOVA table: SS, df, MS, F, pEta squared and omega squaredTukey HSD (Tukey-Kramer) with 95% CIs and adjusted p valuesLevene's (Brown-Forsythe) test and Welch's ANOVAAPA 7 report sentence

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