One-way ANOVA calculator
Enter your values. The answer and the full worked solution update as you type.
Result
| Source | SS | df | MS | F | p |
|---|---|---|---|---|---|
| Between groups | 156.1 | 2 | 78.07 | 20.91 | 0.0001228 |
| Within groups | 44.8 | 12 | 3.733 | ||
| Total | 200.9 | 14 |
Compare the means of two or more groups with a one-way analysis of variance. Enter the values of each group in its own field. The calculator gives the full ANOVA table (sums of squares, degrees of freedom, mean squares and F), the exact p-value from the F distribution, the critical value and the decision, plus η² as a measure of effect size.
Step-by-step solution
Hypotheses
Group means
Sums of squares
Mean squares and F
p-value and decision
Key formulas
- Between groups
- SSB = Σ ni(x̄i − x̄)², dfB = k − 1
- Within groups
- SSW = Σ (ni − 1)si², dfW = N − k
- F statistic
- F = MSB/MSW = (SSB/dfB)/(SSW/dfW)
- Decision
- reject H₀ if F > Fcrit(dfB, dfW)
How to do a one-way ANOVA
- State H₀: all group means are equal, and H₁: at least one mean differs.
- Compute each group mean and the grand mean of all values.
- Between groups: SSB = Σ nᵢ(x̄ᵢ − x̄)², with k − 1 degrees of freedom. Within groups: SSW = ΣΣ(x − x̄ᵢ)², with N − k degrees of freedom.
- Mean squares: MSB = SSB/(k − 1) and MSW = SSW/(N − k). Then F = MSB/MSW.
- Find the p-value P(F > F_observed) and reject H₀ if p ≤ α.
Questions and answers
What does a significant ANOVA tell me?
Only that not all group means are equal. To find which groups differ, follow up with a post-hoc test such as Tukey’s HSD or with pairwise t-tests with a Bonferroni correction.
What are the assumptions?
Independent observations, roughly normal data within each group and similar variances in the groups. With equal group sizes ANOVA is fairly robust; with very unequal variances use Welch’s ANOVA.
Is ANOVA with two groups the same as a t-test?
Yes. With two groups, F equals t² from the pooled two-sample t-test and the p-values are identical.
What is η²?
Eta squared, SSB/SST, is the fraction of the total variation explained by the groups. Rough guidelines are 0.01 small, 0.06 medium and 0.14 large.
Practise it yourself
Practise Hypothesis tests & confidence intervals
A calculator gives the answer. In the exam you need the method: practise it with new problems, hints and worked solutions.