ANOVA Calculator

StatisticsLast updated: August 17, 2026

Analysis of variance (ANOVA) is a parametric hypothesis test that compares the means of more than two independent groups in a single test. Comparing three groups with pairwise t-tests inflates the Type I error rate from 5% to roughly 14%; a one-way ANOVA compares all the groups at once and keeps that error at 5%. The null hypothesis says that all group means are equal, the alternative that at least one group differs.

Paste the data of one group per line and this calculator returns the F statistic, the degrees of freedom, the p-value, and the eta squared (η²) effect size; it also produces a group summary table and a chart of the group means. For comparisons such as the effect of three teaching methods on achievement or the spending of four income groups, it hands thesis and paper authors a ready-made APA-style results sentence.

Write all the values of one group on each line, separated by commas or spaces. At least 2 groups and at least 2 values per group are required; the groups do not have to be the same size.

Formula

SS_between = Σ nⱼ (x̄ⱼ − x̄_grand)²
SS_within = Σ Σ (xᵢⱼ − x̄ⱼ)²
F = MS_between / MS_within = (SS_between / (k − 1)) / (SS_within / (N − k))
η² = SS_between / SS_total

k is the number of groups and N the total number of observations. When H₀ is true, F fluctuates around 1; as the groups diverge, F grows and the p-value is read from the right tail of the F(k−1, N−k) distribution. When the result is significant, post-hoc tests (Tukey HSD and the like) show which pairs differ.

How to Calculate

  1. Paste the data of each group on its own line, separating the values with commas or spaces.
  2. Make sure there are at least 2 groups and at least 2 values in every group; the group sizes may differ.
  3. Read the F value, the degrees of freedom, the p-value, and the eta squared effect size in the results section.
  4. Use the group summary table and the means chart to see which group is high and which is low.
  5. If p < 0.05, run a post-hoc test (such as Tukey HSD) to find which groups differ, and copy the APA sentence into your report.

Worked Examples

Three groups, a strong difference

For three groups of five with means of 5.00, 7.00, and 9.00, the between-groups sum of squares is 40 and the within-groups sum of squares is 6. The result: F(2, 12) = 40.00; p < 0.001; η² = 0.870 — a very strong and significant difference in which 87% of the variance is explained by group membership.

F statistic: 40.00 · Degrees of freedom — between groups (df₁): 2 · Degrees of freedom — within groups (df₂): 12

Three teaching methods (8 students each)

The score means of the students taught with the three methods are 59.50, 67.50, and 62.50. The ANOVA gives F(2, 21) = 17.37; p < 0.001; η² = 0.623: there is a large and significant difference between the methods, and a post-hoc test should be used to find which pairs differ.

F statistic: 17.37 · Degrees of freedom — between groups (df₁): 2 · Degrees of freedom — within groups (df₂): 21

Two groups, a difference that is not significant

For two groups of six with means of 23.50 and 24.50, the result is F(1, 10) = 0.86; p = 0.376; η² = 0.079: the difference is not statistically significant. In a two-group comparison F = t², so an independent samples t-test returns exactly the same p-value.

F statistic: 0.86 · Degrees of freedom — between groups (df₁): 1 · Degrees of freedom — within groups (df₂): 10

Frequently Asked Questions

Can ANOVA be used with two groups?
Yes, it works mathematically: a one-way ANOVA on two groups produces exactly the same p-value as an independent samples t-test, because F = t². Even so, the convention in the literature is to report a t-test for a two-group comparison; the t-test presents the direction of the difference and its confidence interval more directly.
What is a post-hoc test and when is it needed?
A significant ANOVA tells you only that at least two group means differ; it does not tell you which ones. Post-hoc (multiple comparison) tests compare all pairs of groups while controlling the Type I error rate. If you found p < 0.05 and have three or more groups, you need to run a post-hoc test such as Tukey HSD.
Is it a problem if the group sizes are unequal?
ANOVA works with unbalanced designs (unequal n) as well; differing group sizes are not a problem in themselves. However, unequal n combined with a violation of homogeneity of variance can make the F test misleading. If Levene's test supports homogeneity, the results are reliable; if homogeneity fails, Welch's ANOVA is preferable.
What are the assumptions of ANOVA?
There are three core assumptions: independence of observations (each participant belongs to only one group), approximate normality of the dependent variable within each group, and homogeneity of the group variances. With n ≥ 30 per group, ANOVA is quite robust to departures from normality; for severe violations the nonparametric Kruskal-Wallis H test is used.
How is eta squared (η²) interpreted?
Eta squared is a ratio between 0 and 1 showing what proportion of the total variance in the dependent variable is explained by group membership. The common thresholds are 0.01 for a small, 0.06 for a medium, and 0.14 for a large effect. Because the p-value is strongly affected by sample size, ANOVA results should always be reported together with eta squared.
What should I do if the ANOVA is not significant?
A finding of p > 0.05 does not prove that the groups are equal; it only shows that the data at hand does not provide enough evidence of a difference. Report the result as it stands, together with the descriptive statistics and eta squared; no post-hoc test is run after a non-significant ANOVA. If the sample is small, note in your limitations that statistical power may have been low.