One-Way ANOVA Calculator
F statistic and p value for comparing two to six group means, with the full table.
What it does
Analysis of variance answers one question: are these group means really different, or could this spread have happened by chance inside the groups? It compares the variation between group means against the variation within the groups themselves.
The tool above takes two to six lists of values. It computes the between-group and within-group sums of squares from the grand mean, divides them by their degrees of freedom into mean squares, and forms the F ratio with its p value.
The defaults are the three-group example from the NIST engineering statistics handbook: F of 9.59 on 2 and 12 degrees of freedom with a p value of about 0.0032, comfortably significant at the 0.05 level.
The layout follows the NIST handbook pages for the ANOVA table: total variation splits into a between part with k - 1 degrees of freedom and a within part with N - k, and F is the ratio of the two mean squares, read against the F distribution.
How to use
- Enter each group's values, separated by commas or spaces.
- Fill at least three value lists; the first three groups are required.
- Leave optional groups blank if you have fewer than six.
- Press Calculate for F, p, and the mean squares.
- Read the conclusion line, then follow up with pairwise tests.
Formula
- k
- Number of groups.
- N
- Total number of values across all groups.
- SSB
- Between-group sum of squares, how far group means sit from the grand mean.
- SSW
- Within-group sum of squares, the pooled noise inside the groups.
- F
- The variance ratio; large F means the spread between groups dwarfs the noise.
The p value comes from the F distribution with k - 1 and N - k degrees of freedom, computed here from the regularized incomplete beta function. A within-group variance of zero makes F undefined, which is why identical values inside every group are rejected.
Worked example
Example: the NIST handbook three-group data
Groups: 6.9, 5.4, 5.8, 4.6, 4.0 and 8.3, 6.8, 7.8, 9.2, 6.5 and 8.0, 10.5, 8.1, 6.9, 9.3.
- Grand mean: 108.1 / 15 = 7.207.
- Between SS: 27.897 over 2 df, so MST = 13.949.
- Within SS: 17.452 over 12 df, so MSE = 1.454.
- F = 13.949 / 1.454 = 9.59.
- p = 0.0032, below 0.05, so reject the equal means hypothesis.
Reference table
| Piece | Value here | Meaning |
|---|---|---|
| Between MS | 13.949 | Signal: how far apart the group means are |
| Within MS | 1.454 | Noise: typical scatter inside groups |
| F | 9.59 | Signal divided by noise |
| p value | 0.0032 | Chance of an F this large under equal means |
| Conclusion | Reject | At least one mean differs |
Limits
- Assumes roughly normal data within groups.
- Assumes similar spread across groups.
- Groups must be independent of each other.
- Says means differ somewhere, not where.
Common mistakes
- Running ANOVA on paired before-and-after data.
- Reading a big F as proof of a big effect.
- Comparing groups with wildly unequal variances.
- Skipping pairwise tests after a significant F.
- Letting one outlier drive the whole between-group signal.
Try it yourself
Quick checks. Answers sit tucked under each question.
Three groups, F of 9.59, 12 df within?
p about 0.0032; significant at 0.05.
What if every group holds identical values?
No within-group variation, so F is undefined and the tool asks for spread.
Two groups only?
ANOVA on two groups and a two-sample t test agree; F equals t squared.
Questions
When do I use ANOVA?
When you compare the means of three or more independent groups in one pass and want to control the false positive rate.
What does the F statistic mean?
The ratio of between-group variance to within-group variance; values near 1 suggest the groups behave alike.
What is a small enough p value?
The usual cutoff is 0.05; the p value is the chance of seeing this F if all means were truly equal.
What comes after a significant result?
Pairwise comparisons such as Tukey's test to find which means differ, with adjustment for multiple testing.
Does ANOVA need equal group sizes?
No; the formulas here handle unequal sizes, though very lopsided groups are less robust to unequal variances.
Which value goes in each list?
One row per subject's measurement; each group holds the raw values, not means.