One-Way ANOVA Calculator

Test whether the means of two or more independent groups differ on a continuous outcome. Enter raw observations or summary statistics, run classic or Welch one-way ANOVA, review the ANOVA table and effect sizes, then explore pairwise post-hoc comparisons when the omnibus test is significant.

Data input

Analysis method

Welch ANOVA does not assume equal population variances. Prefer it when group sizes or variances differ substantially.

Significance level (α)

Custom α (0–1)

0.00 0.50

Please enter a valid value

Groups

Raw mode: paste numbers separated by commas, spaces, or new lines. Summary mode: n, mean, and SD per group. Unequal sample sizes are supported.

Post-hoc method

The omnibus F-test does not identify which groups differ. Pairwise post-hoc tests control the familywise error rate across comparisons. Classic ANOVA pairs with Tukey; Welch pairs with Games-Howell.

Calculations run in your browser. Pasted or typed data is not uploaded to the server.

One-way ANOVA

F = 80.8226

One-way ANOVA found a statistically significant difference among the group means, F(2, 12) = 80.8226, p = 1.09e-7.

Method

One-way ANOVA

F statistic

80.8226

p-value

< 0.000001

df (between, within)

2, 12

F critical

3.8853

Decision

Reject H₀

η² (eta squared)

0.9309

ω² (omega squared)

0.9141

Cohen's f

—

APA-style

F(2, 12) = 80.82, p < .001, η² = 0.931

Hypotheses

H₀: H₀: All population means are equal (μ₁ = μ₂ = … = μₖ).

H₁: H₁: At least one population mean differs.

ANOVA table

Source	SS	df	MS	F	p

Group descriptives

Group	n	Mean	SD	SE	95% CI

Post-hoc pairwise comparisons

Pair	Diff	SE	Adj. p	95% CI	Sig.

Assumption diagnostics

Homogeneity (Brown–Forsythe)

—

Residual normality (Shapiro–Wilk)

—

Residual summary

—

Potential outliers (1.5×IQR)

—

Brown–Forsythe (median-centered Levene) tests equality of variances. A small p-value suggests unequal variances — consider Welch ANOVA. Shapiro–Wilk is applied to pooled residuals (y − group mean). A small p-value does not by itself invalidate ANOVA; use residual plots and subject-matter judgment.

Independence

Independence of observations cannot be verified from the numbers alone. Confirm your sampling design supports independent groups.

Group means

Box plots

Residual normal Q–Q

F distribution

Calculation steps

One-way ANOVA (SSR default example)
k = 3, N = 15
SS_between = 668.1333, SS_within = 49.6
F = 80.8226, p = 1.09e-7

How to Use the One-Way ANOVA Calculator

Choose raw data or summary statistics, enter at least two independent groups, run classic or Welch ANOVA, review the table and effect sizes, and interpret post-hoc comparisons correctly.

1. Choose the input mode: Pick Raw data if you have individual observations per group — this enables the full diagnostic set (box plots, Q-Q plot, Shapiro-Wilk, Brown-Forsythe). Pick Summary statistics if you only have n, mean, and SD per group — the ANOVA table, F, p, effect sizes, and post-hoc comparisons still work, but raw-data diagnostics are unavailable.
2. Enter the group data: In raw mode, enter each group's observations in the textarea — numbers can be separated by commas, spaces, or new lines. In summary mode, enter n, mean, and SD per group. Add or remove groups with the buttons under the group list. Up to 15 groups are supported. Unequal sample sizes are handled automatically. Click Load example to populate one of the built-in datasets.
3. Choose the analysis method: Pick Classic one-way ANOVA for the standard equal-variance F test. Pick Welch ANOVA when group variances or sizes differ substantially, or when Brown-Forsythe on the results panel flags unequal variances. Welch does not assume equal variances and uses adjusted degrees of freedom.
4. Set α and the post-hoc method: Choose the significance level from 0.10, 0.05, 0.01, or a custom value between 0 and 1. Choose a post-hoc method: Tukey HSD for classic ANOVA with equal variances, Games-Howell for Welch or unequal variances, or Bonferroni, Holm, or Šidák for a more conservative multiplicity adjustment. Default pairing is automatic — classic pairs with Tukey, Welch pairs with Games-Howell.
5. Read the ANOVA table and effect sizes: The ANOVA table shows SS, df, MS, F, and p for between groups, within groups, and total. The SS check line confirms that SS_Between + SS_Within equals SS_Total. Check eta squared, omega squared, and Cohen's f to judge practical magnitude. The group descriptives table shows n, mean, SD, SE, and 95% CI per group.
6. Review diagnostics and interpret: Review Brown-Forsythe for homogeneity of variance, Shapiro-Wilk on the residuals, the box plots for outliers, and the Q-Q plot for residual normality. If the omnibus test is significant, look at the post-hoc comparisons to identify which pairs differ. Use the APA-style result line as the basis for reporting, and add the substantive interpretation in the context of the research question.

What Is a One-Way ANOVA?

A one-way ANOVA tests whether the means of three or more independent groups differ on a continuous outcome. One categorical factor, one continuous response, independent observations. It answers an omnibus question: is there evidence of a difference somewhere among the group means? It does not identify which groups differ — that is a post-hoc question.

Use ANOVA instead of running multiple t-tests because each additional t-test inflates the chance of at least one false positive. A single ANOVA holds the familywise error rate at the chosen α.

Hypotheses and Formula

H₀: μ₁ = μ₂ = … = μₖ (all population means are equal).

H₁: At least one population mean differs.

The F statistic compares between-group variation to within-group variation:

SS_Between = Σ nᵢ(x̄ᵢ − x̄)²
SS_Within = Σ Σ (xᵢⱼ − x̄ᵢ)²
SS_Total = SS_Between + SS_Within
df_Between = k − 1, df_Within = N − k
MS_Between = SS_Between ÷ df_Between
MS_Within = SS_Within ÷ df_Within
F = MS_Between ÷ MS_Within

A large F means the group means are spread further apart than within-group noise would explain by chance. The p-value is the upper-tail probability of the F distribution with (df_Between, df_Within).

The ANOVA Table

Three rows — between groups, within groups, total — and columns for SS, df, MS, F, and p. Only the between row has an F and p. The within row provides the error term. The total row verifies that SS_Between + SS_Within equals SS_Total.

Assumptions

Independence — observations are independent within and across groups. This is a design property and cannot be verified from the numbers alone.
Normality — residuals within groups are approximately normal. Matters most for small samples.
Homogeneity of variance — group variances are approximately equal. Matters most when group sizes are unequal.
No extreme outliers — a single extreme value can distort F. Flag and investigate; never auto-remove.

A non-significant Shapiro-Wilk p-value does not prove normality. Use the Q-Q plot, box plots, and subject-matter judgement together.

Welch ANOVA

Welch's one-way ANOVA does not assume equal variances. It uses weights wᵢ = nᵢ ÷ sᵢ² and adjusts both numerator and denominator degrees of freedom. Prefer Welch when group sizes or variances differ substantially, or when Brown-Forsythe flags unequal variances. Classical F and Welch F are separate calculations — the calculator reports which one was run.

Post-Hoc Comparisons

When the omnibus test is significant, post-hoc tests identify which pairs differ. Tukey HSD (Tukey–Kramer for unequal n) controls the familywise error rate and is the standard choice after a classic ANOVA with equal variances. Games-Howell is the corresponding choice after Welch ANOVA or when variances are unequal. Bonferroni, Holm, and Šidák are alternative multiplicity adjustments.

Default pairing: classic ANOVA → Tukey; Welch ANOVA → Games-Howell. Other adjustments are available as secondary options.

Effect Sizes

η² (eta squared) = SS_Between ÷ SS_Total. Proportion of total variance explained by the factor.
ω² (omega squared) = (SS_Between − df_Between × MS_Within) ÷ (SS_Total + MS_Within). Less biased than η².
Cohen's f = √(η² ÷ (1 − η²)). Conventional guidelines: ~0.10 small, ~0.25 medium, ~0.40 large. Guidelines, not laws.

Worked Example

Three groups, five observations each. Group A: 12, 15, 14, 18, 16. Group B: 21, 19, 24, 22, 20. Group C: 31, 29, 34, 32, 30.

Group means: 15, 21.2, 31.2. Grand mean: 22.47. SS_Between ≈ 665.7, SS_Within ≈ 43.2, SS_Total ≈ 708.9. df_Between = 2, df_Within = 12. MS_Between ≈ 332.9, MS_Within ≈ 3.6. F ≈ 92.5, p < 0.001. η² ≈ 0.94 — the factor explains most of the variance. Tukey HSD would flag all three pairs as significantly different.

Common Mistakes and Reporting

Reading the omnibus F as telling you which groups differ. It does not. That is what post-hoc tests are for.
Running multiple t-tests instead of ANOVA. This inflates the familywise error rate.
Treating p > α as proof of equal means. It means the data did not provide sufficient evidence at the chosen α.
Ignoring the variance assumption. If variances differ substantially, classic F is not the right test.
Reporting p without an effect size. Statistical significance and practical magnitude are separate questions.

APA format: F(2, 12) = 92.50, p < .001, η² = .94. Report the design, group sizes, the omnibus result, effect size, and any significant post-hoc comparisons separately.

Frequently Asked Questions

Q: When should I use one-way ANOVA instead of multiple t-tests?

Use ANOVA when comparing three or more groups. Running multiple t-tests inflates the familywise error rate — with three groups you would need three t-tests, and the chance of at least one false positive grows quickly. A single ANOVA holds the overall error rate at α. When the omnibus test is significant, post-hoc tests identify which pairs differ while keeping the error rate controlled.

Q: What is the difference between one-way and two-way ANOVA?

One-way ANOVA has a single categorical factor and tests whether its level means differ. Two-way ANOVA has two factors and also tests the interaction between them — whether the effect of one factor depends on the level of the other. The two share the same F and p-value framework, but two-way adds an interaction term and uses different degrees of freedom. Use two-way when you have two factors and want to know whether they interact.

Q: Can one-way ANOVA handle unequal sample sizes?

Yes. Unequal group sizes are fully supported. The sums of squares weight each group mean by its sample size, and Tukey–Kramer (the unequal-n variant of Tukey HSD) is used for post-hoc comparisons. When group sizes are both unequal and variances differ substantially, Welch ANOVA with Games-Howell is the more appropriate pairing.

Q: What is the difference between Tukey HSD and Games-Howell?

Tukey HSD (Tukey–Kramer for unequal n) assumes equal variances and is the standard post-hoc after a classic ANOVA. Games-Howell does not assume equal variances and is the corresponding post-hoc after Welch ANOVA. Both control the familywise error rate across pairwise comparisons. Default pairing: classic → Tukey, Welch → Games-Howell.

Q: Can I calculate one-way ANOVA from summary statistics?

Yes. Enter n, mean, and SD for each group. The calculator computes SS_Between from the group means and the grand mean, and SS_Within from the group SDs via SS_Within = Σ (nᵢ − 1)sᵢ². The ANOVA table, F, p, effect sizes, and Tukey post-hoc all work from summary input. Diagnostics that need individual observations — box plots, Q-Q plot, Shapiro-Wilk, Brown-Forsythe — are not available in summary mode.