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· Co-Founder, 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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