Calculators

ANOVA Calculator

One-way analysis of variance on 3 or more groups — with the full ANOVA summary table, exact F-distribution p-value, critical value, a shaded F-curve diagram and a plain-language significance verdict.

ANOVA explained, step by step

Analysis of variance compares three or more group means with a single F test that controls the false-positive rate. Each card below covers one piece of the calculation this tool runs.

1. What ANOVA tests — equality of three or more group means

One-way ANOVA asks whether the population means of k independent groups are all equal. It's an omnibus test: a significant F says at least one group differs somewhere, not which pair. With k = 2 groups it reduces exactly to the squared t-statistic.

H₀: μ₁ = μ₂
= …
= μₖ  ·  H₁: at least one μᵢ differs
Example
Given
3 teaching methods, 5 scores each
Substitute
H₀: μA = μB
= μC
Answer
test
one F
not 3 t-tests

2. Why not run many t-tests — the family-wise error problem

Every t-test at α = 0.05 has a 5% false-positive risk. With k = 5 groups you'd run 10 pairwise tests and the chance of at least one false alarm balloons to ~40%. A single ANOVA controls the overall error rate.

family-wise error ≈ 1 − (1 − α)^m   where m = k(k−1)/2
k = 33 tests · ≈ 14%
k = 46 tests · ≈ 26%
k = 510 tests · ≈ 40%
k = 615 tests · ≈ 54%
Example
Given
k = 5
α = 0.05
Substitute
1 − 0.95¹⁰
Answer
≈ 40% false-positive risk

3. Partitioning variance — SS, MS and the F ratio

Total variability splits into between-group (differences among group means) and within-group (random scatter around each group's mean). Dividing by degrees of freedom turns each SS into a variance estimate; their ratio is F.

SSbetween = Σ nᵢ (x̄ᵢ − x̄)²
SSwithin = Σ Σ (xᵢⱼ − x̄ᵢ)²
F = MSbetween / MSwithin
knumber of groups·Ntotal observations·x̄grand mean
Example
Given
SSb = 90
SSw = 30
k = 3
N = 15
Substitute
F = (90/2) / (30/12)
Answer
F = 18 · p ≈ 0.00025

4. After a significant F — post-hoc tests and effect size

A significant F says only that some pair differs. Use Tukey's HSD (equal n), Bonferroni, or Scheffé to identify which pairs. Report an effect size — η² = SS_between / SS_total — so readers see the magnitude, not just the p-value. Check assumptions: independence, roughly normal residuals, similar variances (use Welch's ANOVA if not).

η² = SSbetween / SStotal
significant F → run Tukey / Bonferroni
report η² or ω² alongside p
unequal variances → Welch's ANOVA
non-normal, small n → Kruskal–Wallis
Example
Given
F(2,12) = 18
SSb = 90
SSt = 120
Substitute
η² = 90 / 120
Answer
η² = 0.75 (large effect)

Features of this calculator

  • Enter unlimited groups with any (unequal) sample sizes.
  • Accepts messy paste from spreadsheets — commas, tabs, spaces, newlines all work.
  • Full ANOVA table: SS, df, MS, F and exact p-value from the F CDF.
  • Critical F at your α with a plain-language reject / fail-to-reject verdict.
  • Shaded F-distribution diagram highlighting the observed statistic and rejection region.
  • Effect size η² = SS_between / SS_total reported alongside the F.
  • Grand mean and per-group means shown for quick sanity checks.
  • Copy a text summary or download the results as a PNG.

Frequently asked questions

+When should I use one-way ANOVA?
When you're comparing means of three or more independent groups on a single numeric outcome. With just two groups, use an independent-samples t-test — the F test in one-way ANOVA reduces to t² in that case.
+What is the difference between one-way and two-way ANOVA?
One-way ANOVA has one categorical factor. Two-way ANOVA has two factors and can also test their interaction (e.g. does the effect of teaching method depend on school type?). This calculator handles the one-way case.
+What if my groups have unequal sample sizes?
One-way ANOVA still works with unbalanced groups — the formulas already weight each group by its own n. Unequal sizes do, however, make the test more sensitive to violations of the equal-variance assumption.
+What if the equal-variance assumption fails?
Use Welch's ANOVA, which corrects the degrees of freedom for unequal variances. With non-normal data and small samples, consider the Kruskal–Wallis non-parametric test.
+How do I report ANOVA results?
APA style: F(df_between, df_within) = value, p = value, plus an effect size such as η² or ω². Example: F(2, 12) = 18.00, p < .001, η² = .75.
+Which post-hoc test should I use?
Tukey's HSD is the default for pairwise comparisons when sample sizes are equal or roughly so. Bonferroni is more conservative and easy to apply. Scheffé is very conservative but flexible for complex contrasts.

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