Math
ANOVA Calculator (One-Way)
Run a one-way analysis of variance on 2 to 10 groups. Returns the F statistic, critical F, p-value range, full ANOVA table, group means, and eta-squared effect size.
One-way ANOVA
Need 2 to 10 groups, each with at least 2 numeric values.
F statistic
31.22
df between = 2, df within = 12
Reject the null hypothesis at alpha = 0.05. At least one group mean differs from the others (p < 0.01). Effect size is large (eta squared = 0.84).
Critical F
3.89
P-value
p < 0.01
Eta squared
0.84
Grand mean
85.8
Total N
15
Groups
3
ANOVA table
| Source | SS | df | MS |
|---|---|---|---|
| Between | 451.6 | 2 | 225.8 |
| Within | 86.8 | 12 | 7.23 |
| Total | 538.4 | 14 | - |
Group summary
| Group | n | Mean | Variance |
|---|---|---|---|
| Method A | 5 | 86.4 | 9.3 |
| Method B | 5 | 78.8 | 6.7 |
| Method C | 5 | 92.2 | 5.7 |
Effect size is largebased on Cohen's benchmarks (small 0.01, medium 0.06, large 0.14).
Frequently Asked Questions about the ANOVA Calculator (One-Way)
What does a one-way ANOVA test?
One-way ANOVA tests the null hypothesis that the means of two or more groups are all equal against the alternative that at least one group mean differs. It compares the variance between group means to the variance within groups; if the between-group variance is much larger, the F statistic grows and the test rejects equal means. With only two groups, ANOVA reduces to a two-sample t-test (F = t squared).
What assumptions does ANOVA require?
Three key assumptions: (1) each group is normally distributed, (2) the groups share a common variance (homoscedasticity), and (3) observations are independent within and across groups. ANOVA is fairly robust to mild departures from normality when sample sizes are roughly equal. If variances differ noticeably, Welch's ANOVA or a non-parametric test like Kruskal-Wallis is safer.
How do I interpret the F statistic?
The F statistic is the ratio of mean square between groups to mean square within groups. An F close to 1 says between-group and within-group variability are similar, consistent with equal means. As F grows above 1, the between-group differences look larger relative to the noise inside each group. Compare your F to the critical F at your chosen alpha (or compare the p-value to alpha): if F exceeds the critical value, reject the null.
What should I do if ANOVA rejects the null hypothesis?
ANOVA only tells you at least one mean differs, not which one. Follow it with a post-hoc test that controls the family-wise error rate. Tukey's HSD (honestly significant difference) is the standard choice when sample sizes are equal and you want all pairwise comparisons. Bonferroni and Scheffe are more conservative alternatives; Dunnett works when you compare every treatment to a single control.
What does eta squared mean as an effect size?
Eta squared is the proportion of total variance explained by group membership: eta squared = SS between / SS total. A value of 0.20 means 20 percent of the variation in your outcome is associated with which group an observation belongs to. Cohen's benchmarks treat 0.01 as a small effect, 0.06 as medium, and 0.14 as large. Effect size matters even when the p-value is significant; a tiny effect in a large sample can still reach significance without being practically meaningful.