ANOVA Calculator
One-way analysis of variance from each group’s size, mean and standard deviation: the F statistic, p-value, sums of squares and the ANOVA table.
Result
F statistic: 9.5911
Exact arithmetic on the summaries. The p-value assumes independent, roughly normal groups with similar spreads.
- P-value
- 0.003248
- Compared with α
- p = 0.003248 is at or below α = 5%, so the result is statistically significant at that level.
- Degrees of freedom between groups
- 2
- Degrees of freedom within groups
- 12
- Sum of squares between groups
- 27.897333
- Sum of squares within groups
- 17.45208
- Total sum of squares
- 45.349413
- Mean square between groups
- 13.948667
- Mean square within groups
- 1.45434
- Grand mean
- 7.206667
- Eta squared, share of variation between groups
- 0.615164
https://calcgallery.com/statistics/anova-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the ANOVA Calculator
- Choose how many groups you are comparing, from two to six.
- For each group, enter its size, mean and sample standard deviation.
- Read F and its p-value, with the full ANOVA table of sums of squares, degrees of freedom and mean squares below.
The ANOVA table
grand mean x̄ = Σ nᵢ x̄ᵢ ÷ Nbetween SSB = Σ nᵢ (x̄ᵢ − x̄)² df = k − 1 MSB = SSB ÷ (k − 1)within SSW = Σ (nᵢ − 1) sᵢ² df = N − k MSW = SSW ÷ (N − k)F = MSB ÷ MSW, p = area to the right of F under the F distributionk is the number of groups and N the total number of observations.
Assumptions
The groups are independent samples, each roughly normal, with similar standard deviations. When the spreads differ a lot, a Welch-type ANOVA, which this page does not run, is the safer choice. If every group has a standard deviation of zero there is no within-group variation, F does not exist, and the page says so.
What the diagram shows
Each group’s mean as a dot with a bar reaching one standard deviation either side, and the grand mean as a dashed line. The further the dots sit from the line compared with the length of the bars, the larger F.
With your numbers
- Grand mean = Σ n × mean ÷ N = 7.206667
- SSB = Σ n × (mean − grand mean)² = 27.897333, with 2 degrees of freedom
- SSW = Σ (n − 1) × SD² = 17.45208, with 12 degrees of freedom
- F = (27.897333 ÷ 2) ÷ (17.45208 ÷ 12) = 9.5911
Quick reference
Exact arithmetic on the summaries. The p-value assumes independent, roughly normal groups with similar spreads.
| Example | F statistic |
|---|---|
| Three groups of 5, means 5.34, 7.72, 8.56 | 9.5911 |
| Two groups: 20 (4, 12) and 23 (6, 30) | 2.5293 |
| Four groups of 8 to 15 | 1.4049 |
| Three groups with equal means | 0 |
| Five groups of 20 | 2.2344 |
| Six groups of 6 | 5.5202 |
Common questions about the ANOVA Calculator
- What does a one-way ANOVA test?
- Whether the means of several independent groups could all be equal. It compares how far the group means spread around the grand mean with how much the values spread inside each group. If the means differ by much more than the within-group scatter would explain, F is large and p is small.
- Why can I run it from means and standard deviations?
- The between-group sum of squares needs only each group’s size and mean, and the within-group sum of squares is Σ (n − 1) × SD², which is exactly what a sample standard deviation encodes. The result is identical to running the ANOVA on the raw data, as long as the SDs are sample SDs with n − 1 in the denominator.
- A significant F says the means differ. Which ones?
- ANOVA does not say. A small p-value means at least one mean differs from the others; finding which needs a follow-up comparison such as Tukey’s honestly significant difference, which this page does not run. Comparing every pair with separate t-tests inflates the chance of a false positive.
- How does this relate to a t-test?
- With two groups, a one-way ANOVA is the pooled two-sample t-test: F equals t squared and the p-values match. ANOVA extends that comparison to three or more groups in one test.
- What is eta squared?
- The share of the total variation that lies between the groups, SSB ÷ SST. An η² of 0.6 means 60% of the variation in the data goes with which group a value is in. The page reports the number and leaves its interpretation to you and your field.