Compare three or more means
One-way ANOVA calculator
How results are checked
Named statistical review pending. Selected automated reference cases are published on the validation page.Result on the sample data
.15η²
Purchase intent differs across the four packaging concepts (F(3, 116) = 6.71, p < .001)
Welch F = 6.79 · ω² = .12 · 120 respondents
One-way ANOVA tests whether several independent groups share the same mean. It compares how far the group means sit from each other with how much people vary inside each group: F = MSbetween / MSwithin. A significant F says at least one mean differs; Tukey's HSD then shows which pairs differ while holding the familywise error rate, and η² or ω² say how much of the variance the grouping explains. If the groups have unequal spreads, read Welch's F.
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Enter n, mean and SD for each group (add as many as you need), or switch to raw data and paste one column per group — or a group column and a value column. You can also open a CSV, Excel or SPSS file and pick the outcome and grouping variables; it is read in your browser and never uploaded.
One-way ANOVA
F(3, 116) = 6.71
The group means differ more than chance would explain (α = .05); group membership accounts for η² = 0.148 of the variance. Welch's F, which does not assume equal variances: F(3, 64.01) = 6.79, p < .001.
p < .001
Descriptive statistics
| n | Mean | SD | 95% CI of mean | |
|---|---|---|---|---|
| Concept A | 30 | 53.67 | 14.43 | [48.28, 59.06] |
| Concept B | 28 | 66.86 | 15.00 | [61.04, 72.68] |
| Concept C | 32 | 50.91 | 14.31 | [45.75, 56.07] |
| Concept D | 30 | 60.47 | 16.30 | [54.38, 66.56] |
ANOVA table
| Source | Sum of squares | df | Mean square | F | p |
|---|---|---|---|---|---|
| Between groups | 4542.086 | 3 | 1514.029 | 6.712 | p < .001 |
| Within groups | 26166.591 | 116 | 225.574 | ||
| Total | 30708.677 | 119 |
Tests that do not assume equal variances
| F | df | p | |
|---|---|---|---|
| Welch | 6.787 | 3, 64.01 | p < .001 |
| Brown–Forsythe | 6.698 | 3, 113.83 | p < .001 |
Levene's test and Kruskal–Wallis need the raw values; switch to Raw data to see them.
Effect size
| Estimate | 95% CI | |
|---|---|---|
| η² (eta squared) | 0.148 | [0.036, 0.251] |
| ω² (omega squared, less biased) | 0.125 |
Tukey HSD post-hoc comparisons
| Comparison | Mean difference | Adjusted p | 95% CI |
|---|---|---|---|
| Concept A − Concept B | -13.19 | p = .006 | [-23.48, -2.90] |
| Concept A − Concept C | 2.76 | p = .888 | [-7.19, 12.71] |
| Concept A − Concept D | -6.80 | p = .301 | [-16.91, 3.31] |
| Concept B − Concept C | 15.95 | p < .001 | [5.82, 26.08] |
| Concept B − Concept D | 6.39 | p = .372 | [-3.90, 16.68] |
| Concept C − Concept D | -9.56 | p = .064 | [-19.51, 0.39] |
Tukey–Kramer for unequal group sizes; p-values and intervals use the studentized range distribution and control the familywise error rate.
Which F should I report?
The classic F test assumes every group has the same population variance. Welch's ANOVA (1951) weights each group by n/s² and adjusts the denominator degrees of freedom, so it stays accurate when spreads differ — especially when the smallest groups have the largest variances. The Brown–Forsythe F (1974) is another robust alternative. When the spreads are similar all three agree, so reporting Welch costs little.
With raw data the calculator adds Levene's test of equal variances and the Kruskal–Wallis H test, which compares mean ranks and needs no normality assumption — useful for skewed scores or short rating scales. With two groups, the F test equals the squared Student t from the t-test calculator.
The method
Formula
F = [Σ nⱼ(x̄ⱼ − x̄)² / (k − 1)] / [Σ (nⱼ − 1)sⱼ² / (N − k)]
where
- k, N
- the number of groups and the total sample size
- nⱼ, x̄ⱼ, sⱼ
- size, mean and standard deviation of group j; x̄ is the grand mean
- η² = SS_between / SS_total
- share of variance explained by the groups (biased upward in small samples)
- ω² = (SS_between − (k − 1)MS_within) / (SS_total + MS_within)
- a less biased estimate of the same share
- Tukey HSD
- q = |x̄ᵢ − x̄ⱼ| / √(MS_within/2 · (1/nᵢ + 1/nⱼ)) referred to the studentized range distribution with k and N − k df (Tukey–Kramer for unequal n)
- Welch F
- uses weights wⱼ = nⱼ/sⱼ² and df₂ = (k² − 1) / (3 Σ (1 − wⱼ/Σw)²/(nⱼ − 1))
Worked example
The sample data is a synthetic packaging test: 120 respondents, split into four cells, each rated purchase intent (0–100) for one concept. Means range from 50.9 (Concept C) to 66.9 (Concept B).
The ANOVA gives F(3, 116) = 6.71, p < .001; Welch's F(3, 64.0) = 6.79, p < .001 agrees, and Levene's test (p = .941) shows similar spreads. The concept explains η² = .15 of the variance (95% CI [.04, .25]; ω² = .12).
Tukey's HSD finds 2 of 6 pairs different at α = .05: Concept A vs Concept B (p = .006); Concept B vs Concept C (p < .001). The rank-based Kruskal–Wallis test agrees: H(3) = 17.17, p < .001.
Using it well
When to use it
Use it when
- Use: Comparing a numeric outcome across three or more independent groups (concepts, segments, regions, ad cells).
- Use: Each respondent belongs to exactly one group.
- Use: Roughly normal scores within groups, or groups of about 20 or more where the means are stable.
- Use: You want one overall test plus controlled pairwise comparisons instead of many separate t tests.
Not the right tool when
- Avoid: The same people rated every concept (a within-subjects design) — use a repeated-measures model.
- Avoid: Two or more factors at once (e.g. concept × region) — use a two-way ANOVA or a general linear model.
- Avoid: A categorical outcome — use the chi-square test.
- Avoid: Weighted survey data or complex samples — these formulas assume simple random sampling.
How to interpret it
Rule of thumb
A significant F (p < 0.05) means the group means are not all equal; it does not say which ones differ — that is what Tukey's HSD is for. Its adjusted p-values and intervals already account for every pairwise comparison. For effect size, η² around .01, .06 and .14 are often read as small, medium and large (Cohen's convention); ω² is the less biased choice for small samples. If Welch's F and the classic F disagree, trust Welch.
Questions
Frequently asked questions
Can I run an ANOVA from means and standard deviations?
Yes. The F test, Welch's and Brown–Forsythe's F, eta squared, omega squared and Tukey's HSD depend only on each group's n, mean and SD, so summary statistics give exactly the same results as raw data. Levene's test and Kruskal–Wallis need the individual values and appear only in raw-data mode.
What is the difference between ANOVA and Welch's ANOVA?
Classic ANOVA pools the within-group variances and assumes they are equal. Welch's ANOVA weights each group by n/s² and uses approximate degrees of freedom, so it remains accurate when variances differ. With equal spreads they agree closely, so Welch's version is a safe default.
Why use Tukey's HSD instead of several t tests?
Each extra t test adds another chance of a false positive: with four groups there are six pairs, and the chance of at least one false 'significant' result at α = .05 is far above 5%. Tukey's HSD uses the studentized range distribution so that the whole family of comparisons keeps the 5% error rate.
What do eta squared and omega squared mean?
Both estimate the share of outcome variance explained by group membership. Eta squared is SS_between / SS_total and is biased upward in small samples; omega squared corrects that bias and is usually a little smaller. The calculator gives eta squared a confidence interval from the noncentral F distribution.
When should I use Kruskal–Wallis instead of ANOVA?
When scores are strongly skewed, have outliers, or are short ordinal ratings in small groups. Kruskal–Wallis compares the groups' mean ranks, so it does not rely on normality. It answers a slightly different question (whether one group tends to score higher), so report it as a check alongside, or instead of, the ANOVA.
Sources and next steps
References & further reading
- [1]Welch, B. L. (1951). On the comparison of several mean values: an alternative approach. Biometrika, 38(3–4), 330–336.
- [2]Brown, M. B., & Forsythe, A. B. (1974). The small sample behavior of some statistics which test the equality of several means. Technometrics, 16(1), 129–132.
- [3]Tukey, J. W. (1949). Comparing individual means in the analysis of variance. Biometrics, 5(2), 99–114.
- [4]Kramer, C. Y. (1956). Extension of multiple range tests to group means with unequal numbers of replications. Biometrics, 12(3), 307–310.
- [5]Kruskal, W. H., & Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47(260), 583–621.
- [6]SciPy — scipy.stats.tukey_hsddocs.scipy.org
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