What is the difference between a t-test and a one-way ANOVA?

A t-test compares the means of exactly two groups. One-way ANOVA generalises it to three or more groups in a single test. Running many t-tests instead inflates the chance of a false positive; ANOVA controls that by testing all groups at once. For two groups, ANOVA and the two-sided t-test give the same p-value (F equals t²).

How do you calculate the F-statistic in one-way ANOVA?

Split the total variation into a between-groups sum of squares (SSB = Σ nᵢ(x̄ᵢ − x̄)²) and a within-groups sum of squares (SSW = ΣΣ(xᵢⱼ − x̄ᵢ)²). Divide each by its degrees of freedom to get mean squares MSB = SSB/(k − 1) and MSW = SSW/(N − k). The F-statistic is F = MSB / MSW.

What does a statistically significant ANOVA result mean?

It means at least one group mean differs from the others — the p-value is below your significance level α (usually 0.05). ANOVA does not tell you which groups differ. To find the specific pairs, follow a significant ANOVA with a post-hoc test such as Tukey's HSD or Bonferroni-corrected t-tests.

What is eta-squared (η²) effect size in ANOVA?

Eta-squared is the share of total variation explained by group membership: η² = SSB / SST. It ranges from 0 to 1. Rough benchmarks (Cohen) are 0.01 small, 0.06 medium, 0.14 large. Omega-squared (ω²) is a less biased version that adjusts for sample size and can be slightly negative when the effect is near zero.

When should I use ANOVA instead of running multiple t-tests?

Whenever you compare three or more group means. With k groups there are k(k − 1)/2 possible pairwise t-tests — five groups need ten. Each test carries its own false-positive risk, so the combined error rate climbs well above 5%. One omnibus ANOVA keeps the overall error rate at α, then post-hoc tests explore the specifics.

How many values does each group need?

At least two per group, and at least two groups. More is better: small groups give low within-group degrees of freedom (N − k) and a weak test. This calculator ignores empty group boxes, so you can leave spare boxes blank. Group sizes do not have to be equal — unbalanced designs are handled correctly.

Is my data uploaded anywhere?

No. Every calculation runs in your browser with JavaScript. Nothing you paste is sent to a server, logged, or stored. You can disconnect from the internet after the page loads and the calculator still works.

How is the p-value computed — is it exact?

Yes. The p-value is the upper-tail area of the F distribution, p = I_{d2/(d2+d1·F)}(d2/2, d1/2), evaluated through the regularised incomplete beta function rather than a rounded printed table. Any F and degrees of freedom resolve exactly, matching a standard F table to at least three significant figures.

Statistics · Education

One-Way ANOVA Calculator

Compare the means of three or more groups from raw data. Paste your groups and get the full ANOVA table — sum of squares, degrees of freedom, mean squares, the F-statistic, an exact p-value and effect size (η², ω²) — with a plain-English verdict, computed entirely in your browser.

By Induwara Ashinsana— Executive Director, Ryzera TechnologiesUpdated Jul 8, 2026

One-way ANOVA

NIST formulas · cross-checked

3 values

Numbers separated by commas, spaces, or new lines.

3 values

Numbers separated by commas, spaces, or new lines.

3 values

Numbers separated by commas, spaces, or new lines.

3 of 10 groups

Try an example

Significance level (α)

Any value from 0.0001 to 0.5. 0.05 is the usual default.

Reject H₀ — at least one group mean differs (p = 0.0080 < α = 0.05).

F(2, 6)

p-value (upper tail)

0.0080

Critical F at α = 0.05

5.1433

F exceeds F* → significant

Effect size η²

0.8

ω² = 0.7097

ANOVA table

Source	SS	df	MS	F	p-value
Between groups	24	2	12	12	0.0080
Within groups (error)	6	6	1	—	—
Total	30	8	—	—	—

Per-group summary

Group	n	Mean	Sample SD	Variance
Method A	3	5	1	1
Method B	3	7	1	1
Method C	3	9	1	1
Overall (grand)	9	7	—	—

Cross-check: the total sum of squares from the table identity (SSB + SSW = 30) and from the direct definition ΣΣ(x − x̄)² = 30 agree to floating-point precision.

One-way ANOVA assumes independent observations, roughly normal groups, and similar variances across groups (homogeneity). If variances differ widely, prefer Welch's ANOVA; if the data are ranks or heavily skewed, use the non-parametric Kruskal–Wallis test. A significant F tells you the group means are not all equal — run a post-hoc test (Tukey HSD) to find which pairs differ.

Sources cited

How it works

One-way ANOVA (analysis of variance) tests one null hypothesis: that every group is drawn from a population with the same mean. It answers this by comparing two independent estimates of the underlying variance — one built from how far the group means sit from the overall mean, the other from the spread inside each group. The method follows the NIST/SEMATECH e-Handbook of Statistical Methods §7.4.3.

Let there be k groups; group i has nᵢ observations and total N = Σ nᵢ. The calculator carries out these steps:

Means. Compute each group mean x̄ᵢ and the grand mean x̄ over all N values.
Between-groups sum of squares. SSB = Σ nᵢ (x̄ᵢ − x̄)² with df = k − 1. This grows when the group means are far apart.
Within-groups (error) sum of squares. SSW = ΣΣ (xᵢⱼ − x̄ᵢ)² with df = N − k. This is the noise inside the groups.
Mean squares. MSB = SSB / (k − 1) and MSW = SSW / (N − k). Each is a variance estimate; under the null they estimate the same quantity.
F-statistic. F = MSB / MSW, referenced to the F distribution with d1 = k − 1 and d2 = N − k degrees of freedom. A large F means the between-group signal beats the within-group noise.
p-value. The upper-tail probability p = I_{d2/(d2+d1·F)}(d2/2, d1/2), computed exactly through the regularised incomplete beta function (Abramowitz & Stegun §26.6.2) — the same numerical routine that powers this site's t-test tool. The critical value F* at your α is found by inverting the same relation, so it matches a printed F table.
Effect sizes. η² = SSB / SST is the fraction of variation explained by group membership; ω² = (SSB − (k − 1)·MSW) / (SST + MSW) is the less biased version. The identity SST = SSB + SSW is checked both ways and shown as a credibility line under the results.

ANOVA rests on three assumptions: independent observations, approximately normal groups, and roughly equal variances. When variances differ sharply, Welch's ANOVA is safer; for ranked or heavily skewed data, the non-parametric Kruskal–Wallis test applies instead.

Worked examples

Three equal-sized groups — F = 12.00, p = 0.008

A = 4, 5, 6 (x̄ = 5) · B = 6, 7, 8 (x̄ = 7) · C = 8, 9, 10 (x̄ = 9)
N = 9, k = 3, grand mean = 7
SSB = 3(5−7)² + 3(7−7)² + 3(9−7)² = 12 + 0 + 12 = 24, df = 2
SSW: each group deviations (−1, 0, 1) → 2 per group × 3 = 6, df = 6
MSB = 24/2 = 12, MSW = 6/6 = 1 → F = 12.00 on (2, 6) df
p = I_0.2(3, 1) = 0.2³ = 0.008 → reject H₀ at α = 0.05
η² = 24/30 = 0.80 (a large effect)

Unequal group sizes — exam scores by method, F = 17.43, p ≈ 0.001

A (n=4) = 82, 84, 85, 89 (x̄ = 85) · B (n=3) = 78, 80, 82 (x̄ = 80) · C (n=5) = 88, 90, 92, 94, 96 (x̄ = 92)
N = 12, k = 3, grand mean = 1040/12 = 86.667
SSB = 4(85−86.667)² + 3(80−86.667)² + 5(92−86.667)² = 286.667, df = 2
SSW = 26 + 8 + 40 = 74, df = 9
MSB = 143.333, MSW = 8.222 → F = 17.43 on (2, 9) df
x = 9/(9 + 2·17.43) = 0.2052; p = x^4.5 ≈ 0.0008 → reject H₀
η² = 286.667/360.667 = 0.79

Edge case — two identical groups, F = 0, p = 1.000

A = 1, 2, 3 · B = 1, 2, 3 (both means = 2)
N = 6, k = 2, grand mean = 2
SSB = 3(2−2)² + 3(2−2)² = 0, df = 1
SSW = 2 + 2 = 4, df = 4 → MSB = 0, MSW = 1
F = 0/1 = 0 → p = 1.000, η² = 0
Verdict: fail to reject H₀ — the groups are indistinguishable

Frequently asked questions

Sources & references

The formulas on this page were last cross-checked against the NIST e-Handbook on 2026-07-08. Both worked examples reconcile to the calculator to at least three significant figures, and the ANOVA-table identity SST = SSB + SSW is verified in the results panel.

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