One-Way ANOVA Calculator
Run a one-way ANOVA test on up to four groups.
Calculate the F-statistic and p-value to determine if group means differ significantly.
One-Way ANOVA (Analysis of Variance)
One-way ANOVA tests whether the means of several independent groups are equal. It answers one question: “Is at least one group mean different from the others?” Three or more groups is the usual case. It works with two as well, where it gives exactly the square of the t-statistic you would get from an independent-samples t-test, and an identical p-value.
The null hypothesis (H₀): All group means are equal. The alternative hypothesis (H₁): At least one group mean differs.
How it works: partitioning variance:
| Source | Sum of Squares | df | Mean Square | F |
|---|---|---|---|---|
| Between groups | SSB | k − 1 | MSB = SSB/dfB | F = MSB/MSW |
| Within groups | SSW | N − k | MSW = SSW/dfW | |
| Total | SST | N − 1 |
Where k = number of groups, N = total observations.
Formulas:
Grand mean x̄ = (sum of all values) / N
SSB = Σᵢ nᵢ × (x̄ᵢ − x̄)² (between-group variance)
SSW = Σᵢ Σⱼ (xᵢⱼ − x̄ᵢ)² (within-group variance)
F = MSB / MSW
Worked example you can check against the calculator:
Group 1: 1, 2, 3 (mean 2) Group 2: 4, 5, 6 (mean 5) Group 3: 7, 8, 9 (mean 8)
Grand mean = 5. Each group has 3 values, so: SSB = 3(2−5)² + 3(5−5)² + 3(8−5)² = 27 + 0 + 27 = 54 SSW = 2 + 2 + 2 = 6 (each group contributes (−1)² + 0² + 1²) dfB = 3 − 1 = 2, dfW = 9 − 3 = 6 MSB = 54 / 2 = 27, MSW = 6 / 6 = 1 F = 27 / 1 = 27
Paste those three groups in and you should get F = 27.0000 with p = 0.00100.
That p-value is exact rather than approximate, which makes it a good check on the calculator. When dfB = 2 the F-distribution tail collapses to a simple power: p = [dfW / (dfW + dfB·F)]^(dfW/2) = (6/60)³ = 0.001.
Interpreting the result:
A large F and a small p suggest at least one group mean is different. The conventional threshold is p < 0.05. ANOVA is deliberately silent about which group differs, so a significant result is the start of the analysis rather than the end. Follow it with a post-hoc test such as Tukey HSD (Honestly Significant Difference) or Bonferroni.
Alongside F, this calculator reports η² (eta squared), which is SSB divided by total SS. That is the fraction of the variation explained by which group a value belongs to. It matters because F grows with sample size: with 500 observations per group, a difference too small to care about will still come out significant. η² tells you whether the effect is worth anything. Rough conventions put 0.01 at small, 0.06 at medium and 0.14 at large.
Assumptions:
- Observations are independent
- Residuals are roughly normally distributed
- Groups have similar variances (homoscedasticity)
- If the variances are clearly unequal, Welch’s ANOVA is the safer test
Entering data. Separate values with commas, spaces, tabs or line breaks, whichever your source uses. Pasting a column out of a spreadsheet works.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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