Quick answer
One-way ANOVA tests whether group means differ using F = MSB ÷ MSW, with MSB = SSB ÷ (k − 1) and MSW = SSW ÷ (N − k). Three groups of three values (1 2 3; 2 3 4; 3 4 5) give F = 3, df = 2 and 6, and p = 0.125.
Enter the groups separated by semicolons. The calculator returns the F statistic, degrees of freedom, p-value and sums of squares.
Related tools: One-Sample T-Test Calculator: One-sample t-test from summary statistics. One-Sample Z-Test Calculator: Z-test for a mean with known population standard deviation. P-Value Calculator: P-value from a z, t, chi-square or F statistic.
How to use the One-Way ANOVA Calculator
- Type groups (separate groups with ;), for example 1 2 3; 2 3 4; 3 4 5.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Groups (separate groups with ;) | Text, such as 1 2 3; 2 3 4; 3 4 5 | — |
Formula and method
Formula
F = MSB ÷ MSW, MSB = SSB ÷ (k − 1), MSW = SSW ÷ (N − k)
IEEE-754 double precision; shown to at most 6 decimal places.
p-value from the F distribution (right tail). Requires at least two groups and at least one group with more than one value.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Groups (separate groups with ;) = 1 2 3; 2 3 4; 3 4 5.
| Result | Value |
|---|---|
| F statistic | 3 |
| Degrees of freedom | 2, 6 |
| p-value | 0.125 |
| SSB | 6 |
| SSW | 6 |
The first line is the headline result (f statistic). Below it the one-Way ANOVA Calculator lists degrees of freedom, p-value, sSB and sSW, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
Group means are 2, 3 and 4 with a grand mean of 3. SSB = 3 × (1 + 0 + 1) = 6 and SSW = 6, so MSB = 3, MSW = 1 and F = 3. With p = 0.125, the means do not differ significantly at 5%.
Assumptions
ANOVA assumes normally distributed groups, similar variances and independent observations. A significant result says some means differ but not which; follow with post-hoc tests. At least 2 groups and N > k are needed.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Groups (separate groups with ;): entering "1 2 3" shows "Enter at least two groups separated by semicolons." beside that field and no result.
Choosing between the One-Way ANOVA Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [One-Sample T-Test Calculator](/statistics/t-test-calculator/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
| [One-Sample Z-Test Calculator](/statistics/z-test/) | sample mean, hypothesised mean and population σ | z statistic |
| [P-Value Calculator](/statistics/p-value/) | test statistic, statistic value and test type | p-value |
| [T Statistic Calculator](/statistics/t-statistic/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
| [Critical Value Calculator](/statistics/critical-value/) | distribution, significance level and tails | Critical value |
Background: hypothesis testing and inference
A hypothesis test compares a test statistic with what the null hypothesis predicts. The decision rule is to reject the null hypothesis when the p-value is at or below the significance level α, which you should choose before seeing the data (0.05 is conventional, not special). For a two-tailed z test the p-value is 2 × (1 − Φ(|z|)); for a one-sample t test the statistic is (x̄ − μ₀) ÷ (s ÷ √n) with n − 1 degrees of freedom.
Failing to reject the null hypothesis does not prove it true, and a statistically significant result can still be too small to matter in practice.
Assumptions and limitations
- At least 2 groups
- N > k
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for groups (separate groups with ;) to confirm this.
Frequently asked questions
When use ANOVA instead of a t-test?
For three or more groups.
What does the p-value mean?
The probability of such an F if all means were equal.
What is SSB?
The sum of squares between groups.
What does the One-Way ANOVA Calculator calculate?
One-way analysis of variance across groups. It returns F statistic, Degrees of freedom, p-value, SSB and SSW.
What formula does the One-Way ANOVA Calculator use?
F = MSB ÷ MSW, MSB = SSB ÷ (k − 1), MSW = SSW ÷ (N − k). p-value from the F distribution (right tail).
What do I need to enter in the one-Way ANOVA Calculator?
You enter Groups (separate groups with ;).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. At least 2 groups.
What are the limits of this calculator?
At least 2 groups. N > k.
What is an example of a one-Way ANOVA result?
With groups (separate groups with ;) 1 2 3; 2 3 4; 3 4 5, the f statistic is 3.
Related calculators
- One-Sample T-Test Calculator — One-sample t-test from summary statistics.
- One-Sample Z-Test Calculator — Z-test for a mean with known population standard deviation.
- P-Value Calculator — P-value from a z, t, chi-square or F statistic.
- T Statistic Calculator — t statistic from summary statistics.
- Critical Value Calculator — Critical values for z, t, chi-square and F distributions.
- Degrees of Freedom Calculator — Degrees of freedom for one- and two-sample t-tests.
- Chi-Square Statistic Calculator — Goodness-of-fit chi-square from observed and expected counts.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.