ANOVA Calculator
Paste two or more groups of numbers and the calculator runs a one-way ANOVA to test whether the group means differ significantly. It returns the F-statistic, degrees of freedom, p-value, and eta-squared effect size — the standard outputs any stats paper or lab report would cite.
How to run a one-way ANOVA
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1
Enter group data
Paste each group as a comma- or newline-separated list. Minimum two groups; at least 3-5 observations per group is recommended.
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2
Check assumptions
One-way ANOVA assumes normally-distributed residuals and roughly equal variance across groups. Flag if your groups have wildly different spreads.
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3
Read the F-statistic
F is the ratio of between-group variance to within-group variance. Larger F means stronger evidence of group differences.
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4
Interpret the p-value
Below your chosen alpha (usually 0.05), reject the null hypothesis that all group means are equal. ANOVA does not tell you which groups differ — use a post-hoc test for that.
The ANOVA table
| Source | SS (sum of squares) | df | MS (mean square) | F |
|---|---|---|---|---|
| Between groups | SSB | k - 1 | MSB = SSB/(k-1) | MSB/MSW |
| Within groups | SSW | N - k | MSW = SSW/(N-k) | |
| Total | SST = SSB + SSW | N - 1 |
Where k = number of groups, N = total observations.
F-distribution critical values (alpha = 0.05)
| df1 \ df2 | 10 | 20 | 30 | 60 | 120 |
|---|---|---|---|---|---|
| 2 | 4.10 | 3.49 | 3.32 | 3.15 | 3.07 |
| 3 | 3.71 | 3.10 | 2.92 | 2.76 | 2.68 |
| 4 | 3.48 | 2.87 | 2.69 | 2.53 | 2.45 |
| 5 | 3.33 | 2.71 | 2.53 | 2.37 | 2.29 |
If your computed F exceeds the table value for your df1 (= k-1) and df2 (= N-k), reject the null at p < 0.05.
Assumptions to verify before citing ANOVA
- Independence of observations within and between groups.
- Normality of residuals (Shapiro-Wilk test, or visual: Q-Q plot).
- Homogeneity of variance (Levene’s test, or rule of thumb: largest SD is less than 2× smallest SD).
If normality fails: Kruskal-Wallis test is the non-parametric alternative. If homoscedasticity fails: Welch’s ANOVA handles unequal variances.
After a significant ANOVA: post-hoc tests
One-way ANOVA tells you some groups differ but not which. Follow up with:
- Tukey HSD — conservative, controls family-wise error rate.
- Bonferroni — simple adjustment: α / number of comparisons.
- Scheffé — flexible but low power; good for exploratory analysis.
- Dunnett — only compares each treatment to a control group.
Effect size
A significant p-value says “there is a difference.” Effect size says “how big.” Report eta-squared (η²) = SSB / SST. Rough guide: 0.01 small, 0.06 medium, 0.14 large.
Frequently Asked Questions
If you have three or more groups. Running multiple t-tests inflates the family-wise Type I error rate (α of 0.05 across three pairwise tests becomes roughly 0.14). ANOVA keeps the overall alpha at 0.05.
One-way has a single grouping factor (e.g. treatment type). Two-way has two factors (e.g. treatment × sex) and can test main effects plus interaction. This calculator handles the one-way case.
Statistically, no — they are almost identical. The 0.05 threshold is a convention, not a physical constant. Report the exact p-value and effect size so readers can judge, rather than treating 0.05 as a sharp cut-off.
Small samples produce unstable F values. A huge F on n=3 per group is suggestive but should be replicated. Report confidence intervals around group means alongside the F.
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