ANOVA Calculator

One-way ANOVA · F-statistic · group charts

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Group 1
Group 2
Group 3

Result

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Enter group data and click Calculate

Compare means of multiple groups with one-way ANOVA.

Box Plots & F-Distribution

Python Compiler

What Is ANOVA?

Analysis of Variance (ANOVA) is a statistical method that tests whether the means of three or more groups are significantly different. Instead of running multiple t-tests, ANOVA compares the variance between groups to the variance within groups using a single F-test.

Group 1 Group 2 Group 3 x̄
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Compare Multiple Groups

Test whether 3 or more group means differ significantly in a single test, avoiding multiple comparison problems.

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F-Statistic

The ratio of between-group variance to within-group variance. A large F indicates the group means are spread apart more than expected by chance.

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Beyond T-Tests

Running multiple t-tests inflates Type I error. ANOVA controls the overall error rate while testing all groups simultaneously.

ANOVA Table Explained

The ANOVA table summarizes the decomposition of total variance into between-group and within-group components.

SourceSSdfMSF
BetweenSSBk − 1MSBF = MSB / MSW
WithinSSWN − kMSW
TotalSSTN − 1

Key: k = number of groups, N = total number of observations. SST = SSB + SSW. The F-statistic follows an F-distribution with (k−1, N−k) degrees of freedom.

Key Formulas

Sum of Squares Between (SSB):  SSB = ∑ ni(x̄i − x̄)2
Sum of Squares Within (SSW):  SSW = ∑∑ (xij − x̄i)2
Mean Squares:  MSB = SSB / (k − 1)     MSW = SSW / (N − k)
F-Statistic:  F = MSB / MSW
Worked Example: Three groups: A = {4, 5, 6}, B = {8, 9, 7}, C = {5, 6, 4}. Grand mean x̄ = 6.0.
x̄A = 5, x̄B = 8, x̄C = 5
SSB = 3(5−6)² + 3(8−6)² + 3(5−6)² = 3 + 12 + 3 = 18
SSW = (1+0+1) + (0+1+1) + (1+0+1) = 6
MSB = 18 / 2 = 9,   MSW = 6 / 6 = 1
F = 9 / 1 = 9.0,   df = (2, 6),   p = 0.0156

Effect Size: Eta-Squared

Eta-squared (η²) measures how much of the total variance in the data is explained by group membership. It complements the p-value by indicating practical significance.

η² = SSB / SST  —  proportion of variance explained by between-group differences
0.01
Small effect
0.06
Medium effect
0.14
Large effect
SSB SSW Between (η²) Within

Assumptions & Alternatives

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Independence

Observations must be independent within and across groups. Random sampling or random assignment satisfies this.

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Normality

Data within each group should be approximately normally distributed. ANOVA is robust to moderate violations, especially with larger samples.

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Homogeneity of Variances

Group variances should be roughly equal. Use Levene’s test to check. If violated, consider Welch ANOVA.

Alternatives & Post-Hoc Tests

Non-parametric:
Kruskal-Wallis test (no normality assumption)
Unequal variances:
Welch ANOVA (robust to heteroscedasticity)
Pairwise comparisons:
Tukey HSD (controls family-wise error rate)
Conservative correction:
Bonferroni (α / number of comparisons)

Frequently Asked Questions

One-way ANOVA tests whether the means of three or more independent groups are significantly different. It compares between-group variance to within-group variance using the F-statistic. If the result is significant, at least one group mean differs from the others.
ANOVA assumes independence of observations, normal distribution within each group, and homogeneity of variances across groups. ANOVA is robust to moderate violations of normality, especially with larger sample sizes.
A significant ANOVA tells you at least one group differs, but not which ones. Use post-hoc tests like Tukey HSD to find specific pairwise differences while controlling the family-wise error rate.
Eta-squared measures the proportion of total variance explained by group membership. Guidelines: 0.01 is a small effect, 0.06 is medium, and 0.14 is large. It helps assess practical significance beyond the p-value.
One-way ANOVA has one independent variable (factor). Two-way ANOVA has two factors and can test for interaction effects between them. This calculator performs one-way ANOVA.
If normality is violated, consider the Kruskal-Wallis non-parametric test. If variances are unequal, use Welch ANOVA. For small samples, consider bootstrapping or exact permutation tests.

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