ANOVA Calculator
One-way ANOVA, F-statistic, p-value & ANOVA table
Null hypothesis
H₀: μ₁ = μ₂ = μ₃
All 3 population means are equal.
Alternative hypothesis
H₁: at least one population mean differs
Not that every mean differs from every other.
- F-statistic
- 28.731
- Degrees of freedom
- 2, 12
- df₁ = k − 1, df₂ = N − k
- p-value
- 2.66e-5
- Upper tail of the F distribution
- Critical F (α = 0.05)
- 3.8853
- Eta squared (η²)
- 0.8272
- SSB / SST, share of sample variation between groups
- Omega squared (ω²)
- 0.7871
- Less biased estimate for the population
- Grand mean
- 76.5333
- Groups and observations
- k = 3, N = 15
Significance and effect size answer different questions. The p-value is about evidence against H₀; η² and ω² describe how large the differences between means are relative to the total variation. Values around 0.01, 0.06 and 0.14 are often called small, medium and large, but these are rough conventions that depend on the field. ANOVA shows evidence of differences, not their cause.
ANOVA table
| Source | Sum of squares | df | Mean square | F | p-value |
|---|---|---|---|---|---|
| Between groups (SSB, MSB) | 277.7333 | 2 | 138.8667 | 28.731 | 2.66e-5 |
| Within groups (SSW, MSW) | 58 | 12 | 4.8333 | — | — |
| Total (SST) | 335.7333 | 14 | — | — | — |
Group summary
| Group | n | Mean | Std. deviation | Minimum | Maximum |
|---|---|---|---|---|---|
| Group A | 5 | 75 | 2.2361 | 72 | 78 |
| Group B | 5 | 82.4 | 2.4083 | 79 | 85 |
| Group C | 5 | 72.2 | 1.9235 | 70 | 75 |
Visualizing the groups
Individual observations, group means and the grand mean
Group means with 95% confidence intervals
Total variation = between-group + within-group variation
F compares the two parts after dividing each by its degrees of freedom: F = MSB / MSW. When group means are far apart relative to the spread inside groups, F is large.
Where F falls on the F distribution
Step-by-step calculation
Step 1: Group means
Group A: x̄1 = 375 / 5 = 75
Group B: x̄2 = 412 / 5 = 82.4
Group C: x̄3 = 361 / 5 = 72.2
Step 2: Grand mean
The mean of all N observations together.
x̄ = Σx / N = 1148 / 15 = 76.5333
Step 3: Between-group sum of squares
How far each group mean is from the grand mean, weighted by group size.
SSB = Σ nᵢ(x̄ᵢ − x̄)²
SSB = 5(75 − 76.5333)² + 5(82.4 − 76.5333)² + 5(72.2 − 76.5333)²
SSB = 277.7333
Step 4: Within-group sum of squares
How far each observation is from its own group mean, added over all groups.
Group A: Σ(x − 75)² = 20
Group B: Σ(x − 82.4)² = 23.2
Group C: Σ(x − 72.2)² = 14.8
SSW = 58
Step 5: Total sum of squares
SST = SSB + SSW = 277.7333 + 58 = 335.7333
Check: Σ(x − x̄)² over all observations = 335.7333
Step 6: Degrees of freedom
df between = k − 1 = 3 − 1 = 2
df within = N − k = 15 − 3 = 12
df total = N − 1 = 14
Step 7: Mean squares
MSB = SSB / df between = 277.7333 / 2 = 138.8667
MSW = SSW / df within = 58 / 12 = 4.8333
Step 8: F-statistic
F = MSB / MSW = 138.8667 / 4.8333 = 28.731
Step 9: p-value
The upper-tail area of the F distribution with 2 and 12 degrees of freedom beyond F = 28.731.
p = 2.66 × 10⁻⁵
p < α = 0.05: reject H₀
Explore: what makes F large?
F = MSB / MSW = 13.5 / 0.7733 = 17.4569, p = 0.0001, η² = 0.6995
Pulling the groups apart increases between-group variation (SSB) and F; adding spread increases within-group variation (SSW) and shrinks F, even though the means stay where they are.
One-way ANOVA formulas
x̄ = Σx / N
SSB = Σ nᵢ(x̄ᵢ − x̄)²
SSW = Σᵢ Σⱼ (xᵢⱼ − x̄ᵢ)²
SST = SSB + SSW = Σ(xᵢⱼ − x̄)²
df₁ = k − 1, df₂ = N − k
MSB = SSB / df₁, MSW = SSW / df₂
F = MSB / MSW
η² = SSB / SST
ω² = (SSB − df₁·MSW) / (SST + MSW)
k = number of groups, nᵢ = size of group i, N = total observations, xᵢⱼ = observation j in group i, x̄ᵢ = mean of group i, x̄ = grand mean.
Next step: post-hoc comparisons
ANOVA tells you that at least one group mean differs, but not which specific groups differ. This calculator performs the overall one-way ANOVA. Use a post-hoc multiple-comparison procedure to compare pairs of groups: Tukey’s HSD when variances are similar, or Games-Howell when they are not.
Avoid running many ordinary t-tests between every pair of groups: with 4 groups there are 6 pairs, and without a correction the chance of at least one false positive grows well above α.
Comparing just two groups? The T-Test Calculator gives the same answer as classical ANOVA (F = t²) plus a confidence interval for the difference. Summarize a single group with the Standard Deviation Calculator, estimate one mean with the Confidence Interval Calculator, or, for counts in categories instead of measurements, use the Chi-Square Calculator.
An ANOVA calculator tests whether the means of several groups are likely to differ, using a one-way analysis of variance. It splits the total variation in the data into variation between the group means and variation within the groups, and compares them with the F-statistic.
Enter the observations for each group and a significance level to get the F-statistic, degrees of freedom, p-value and decision, the full ANOVA table, a summary of each group, the effect sizes eta squared and omega squared, charts of the data and of the variance decomposition, and every calculation step. Choose Welch's ANOVA when the group variances are clearly unequal.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Exam scores for three teaching methods (hypothetical) | F(2, 12) = 28.73, p ≈ 0.00003 | Group means 75, 82.4 and 72.2; grand mean 76.53. SSB = 5(75 − 76.53)² + 5(82.4 − 76.53)² + 5(72.2 − 76.53)² = 277.73; SSW = 20 + 23.2 + 14.8 = 58. MSB = 277.73 / 2 = 138.87, MSW = 58 / 12 = 4.83, F = 28.73 with df (2, 12). p ≈ 0.00003 < 0.05, so reject H₀: at least one method’s mean score differs. η² ≈ 0.83. A post-hoc test would show which methods differ. |
| Tensile strength from four production processes (hypothetical) | F(3, 19) = 19.84, p < 0.00001 | Unequal group sizes (6, 5, 7, 5; N = 23). Means 48.52, 50.58, 49.10 and 48.14. SSB = 17.57, SSW = 5.61, MSB = 5.86, MSW = 0.295, F = 19.84 with df (3, 19). Reject H₀: mean strength is not the same for all four processes, and η² ≈ 0.76. Process 2 looks highest, but only a post-hoc comparison can show which processes differ, and only a controlled study design would show the process causes the difference. |
| Plant height with three fertilizers: no significant difference (hypothetical) | F(2, 15) = 0.57, p ≈ 0.58 | Means 21.5, 22.5 and 21.5 with a within-group SD of about 1.9. SSB = 4 and SSW = 52.5, so F = 2 / 3.5 = 0.57 and p ≈ 0.58: fail to reject H₀. The data do not show a difference, which is not the same as proving the fertilizers are equal. ω² is slightly negative here, usually reported as 0. |
What Is One-Way ANOVA?
One-way analysis of variance (ANOVA) compares the means of two or more independent groups defined by one factor, such as three teaching methods or four production processes. The null hypothesis is that all the population means are equal, H₀: μ₁ = μ₂ = … = μₖ. The alternative is that at least one population mean differs. It does not claim that every mean is different from every other.
Instead of comparing the groups two at a time, ANOVA asks one question: are the group means further apart than the variation inside the groups would lead you to expect by chance?
What Does the F-Statistic Mean?
F = MSB / MSW, the ratio of between-group variation to within-group variation, each divided by its degrees of freedom. If all the population means were equal, both mean squares would estimate the same random variation and F would be around 1. Group means that are far apart relative to the spread inside the groups make MSB large and F large.
F is never negative. The F distribution has two degrees of freedom, df₁ = k − 1 for the numerator and df₂ = N − k for the denominator, and ANOVA uses its upper tail: only large values of F count as evidence against H₀.
How to Calculate One-Way ANOVA
1. Calculate each group mean x̄ᵢ and the grand mean x̄ of all N observations.
2. SSB = Σ nᵢ(x̄ᵢ − x̄)², the between-group sum of squares.
3. SSW = sum of (x − group mean)² within every group.
4. SST = SSB + SSW, which also equals Σ(x − x̄)² over all observations.
5. df₁ = k − 1 and df₂ = N − k.
6. MSB = SSB / df₁ and MSW = SSW / df₂.
7. F = MSB / MSW, and the p-value is the area of the F(df₁, df₂) distribution above F.
For the exam-score example (Group A 72, 75, 78, 74, 76; Group B 81, 84, 79, 83, 85; Group C 70, 73, 71, 75, 72): the means are 75, 82.4 and 72.2, the grand mean is 76.53, SSB = 277.73, SSW = 58, df = 2 and 12, MSB = 138.87, MSW = 4.83, F = 28.73 and p ≈ 0.00003.
How to Interpret ANOVA Results
The p-value is the probability of an F-statistic at least as large as the one observed if all the population means were equal and the model assumptions held. It is not the probability that the null hypothesis is true.
If p < α, reject H₀: the data provide evidence that at least one group mean differs. ANOVA does not tell you which groups differ; that needs a post-hoc test.
If p ≥ α, fail to reject H₀: the data do not provide sufficient evidence to conclude that the means differ. This does not prove the means are equal; a small sample may not detect a real difference.
Look at the effect size as well. Eta squared, η² = SSB / SST, is the proportion of the total sample variation associated with differences between the group means; omega squared, ω², is a less biased estimate of the same idea for the population. A significant result can have a small effect, and a non-significant result does not prove the effect is zero.
ANOVA Assumptions
Independence: observations should be independent within and between groups, which depends on how the data were collected. Repeated measurements on the same subjects need a repeated-measures design instead.
Quantitative outcome: the dependent variable is numerical.
Approximately normal residuals: the deviations from each group mean should be roughly normal. Mild departures matter little, especially with balanced group sizes, but strong skew or outliers can distort the results. ANOVA does not require perfectly normal raw data.
Homogeneity of variance: classical one-way ANOVA assumes the groups have roughly equal population variances. When variances clearly differ, particularly with unequal group sizes, Welch's ANOVA is the safer choice.
No automated check can guarantee the assumptions hold; look at the data and think about the study design.
ANOVA vs Welch's ANOVA
Classical ANOVA pools the within-group variation into one estimate, MSW, which is appropriate when the groups have similar variances. Welch's ANOVA weights each group by nᵢ / sᵢ², so noisier groups count for less, and adjusts the denominator degrees of freedom. It is designed to be more robust when variances differ, at little cost when they do not. The calculator lets you choose either, and never switches automatically.
ANOVA vs T-Test and MANOVA
A t-test compares one mean with a reference value, or two group means. One-way ANOVA compares two or more group means in one test, and is most useful for three or more. With exactly two groups, classical ANOVA and the pooled two-sample t-test are equivalent (F = t²), and Welch's ANOVA matches Welch's t-test. For three or more groups, running every pairwise t-test inflates the chance of a false positive.
ANOVA analyses one quantitative outcome. MANOVA (multivariate ANOVA) analyses several related outcomes jointly, such as height and weight together.
What to Do After a Significant ANOVA
A significant ANOVA tells you that at least one group mean differs, but not which pairs of groups differ. To find out, use a post-hoc multiple-comparison procedure that controls the overall error rate: Tukey's HSD is common when variances are similar, and Games-Howell is often used when they are not. This calculator performs the overall one-way ANOVA only.
ANOVA can identify evidence of differences among group means, but it does not by itself establish causation. Whether the differences can be attributed to the groups themselves depends on the study design, for example random assignment to groups.
How to Use the ANOVA Calculator
- Enter the observations for each group, separated by commas, spaces or new lines. Rename groups, and add or remove groups as needed; each group needs at least 2 values.
- Choose the significance level α (0.05 by default) and the test: classical one-way ANOVA, or Welch’s ANOVA if the group variances look clearly unequal.
- Read the F-statistic, degrees of freedom, p-value and decision (reject or fail to reject H₀), with the hypotheses tested and the effect sizes η² and ω².
- Check the ANOVA table, the group summary and the charts of the raw data and group means.
- Follow the step-by-step calculation, and remember that a significant ANOVA needs a post-hoc test to show which groups differ.
Frequently Asked Questions
What is an ANOVA calculator?
A tool that runs an analysis of variance on groups of observations and returns the F-statistic, degrees of freedom, p-value and ANOVA table, so you can test whether the group means differ.
What is one-way ANOVA?
A test of whether the means of two or more independent groups, defined by one factor, are all equal. H₀ says all population means are equal; H₁ says at least one differs.
What does the F-statistic mean in ANOVA?
F = MSB / MSW, the ratio of variation between group means to variation within groups. Around 1 is what you expect if the means are equal; larger values are evidence that at least one mean differs.
How do I calculate the ANOVA F-statistic?
Compute SSB = Σ nᵢ(x̄ᵢ − x̄)² and SSW = the sum of squared deviations from each group mean, divide them by k − 1 and N − k to get MSB and MSW, then F = MSB / MSW.
What does the ANOVA p-value mean?
The probability of an F at least as large as yours if all the population means were equal and the assumptions held. It is not the probability that H₀ is true.
What does a significant ANOVA result mean?
At your significance level, the data provide evidence that at least one population mean differs from the others. It does not say which, or how large the difference is.
Does ANOVA tell you which groups are different?
No. Use a post-hoc multiple-comparison test such as Tukey’s HSD, or Games-Howell when variances are unequal, to compare pairs of groups.
What is the difference between ANOVA and a t-test?
A t-test compares one mean with a value or two means; ANOVA compares two or more means in one test. With two groups, classical ANOVA gives F = t² and the same p-value as the pooled t-test.
What are the assumptions of one-way ANOVA?
Independent observations, a quantitative outcome, roughly normal residuals and, for classical ANOVA, similar group variances. Welch’s ANOVA drops the equal-variance assumption.
What is the difference between ANOVA and Welch’s ANOVA?
Classical ANOVA assumes equal population variances and pools them; Welch’s ANOVA weights each group by n / s² and adjusts the degrees of freedom, so it stays reliable when variances differ.
What are degrees of freedom in ANOVA?
df between = k − 1 (groups minus 1), df within = N − k (observations minus groups), and df total = N − 1. They are the numerator and denominator degrees of freedom of the F distribution.
What is eta squared?
η² = SSB / SST, the proportion of the total sample variation associated with differences between the group means. It describes effect size, separately from the p-value.
Can ANOVA prove causation?
No. ANOVA shows evidence that group means differ. Whether the grouping caused the difference depends on the study design, for example random assignment.
What should I do after a significant ANOVA?
Run a post-hoc test such as Tukey’s HSD to find which pairs of groups differ, report the effect size, and check the assumptions. Avoid many uncorrected pairwise t-tests.
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Last updated: September 27, 2026.