# One-way ANOVA calculator

> One-way ANOVA calculator: the F statistic, p-value, sums of squares, mean squares and η² for two or more groups, with the full ANOVA table.

Interactive version: https://www.calcopenly.com/statistics/one-way-anova-calculator
Subject: Statistics and probability calculators

One-way analysis of variance (ANOVA) tests whether two or more independent groups share the same mean. It splits the total variation into a between-group sum of squares and a within-group sum of squares, divides each by its degrees of freedom (k − 1 and N − k) to get mean squares, and takes their ratio as the F statistic. The p-value is the area of the F distribution beyond that F.

Typical uses are comparing crop yields under three fertilisers, exam scores across teaching methods or response times across product versions. The default data, the NIST e-Handbook example with three groups of five, give F = 9.59 on 2 and 12 degrees of freedom and p = 0.0032, so at α = 0.05 the three means are not all equal.

The test assumes independent observations, roughly normal data in each group and similar group variances. A significant F says that at least one mean differs, not which one.

## Inputs

- **Groups**: One group per line; separate values with spaces or commas. Groups may have different sizes.
- **Significance level α**

## Results

- p-value — main result
- Decision
- F statistic
- df between groups
- df within groups
- Sum of squares between
- Sum of squares within
- Mean square between
- Mean square within
- η² (share of variance explained)
- Critical F

## Formula

$$
F = \frac{SS_B/(k-1)}{SS_W/(N-k)},\quad SS_B = \sum_i n_i(\bar x_i - \bar x)^2,\quad SS_W = \sum_i\sum_j (x_{ij} - \bar x_i)^2
$$

## Worked examples

### NIST e-Handbook example, 3 groups of 5 (defaults)

- Groups: 6.9 5.4 5.8 4.6 4.0 / 8.3 6.8 7.8 9.2 6.5 / 8.0 10.5 8.1 6.9 9.3
- Significance level α: 0.05
- **Sum of squares between: 27.897**
- **Sum of squares within: 17.452**
- **F statistic: 9.59**
- **df between groups: 2**
- **df within groups: 12**
- **p-value: 0.003248**
- Checked against: NIST/SEMATECH e-Handbook §7.4.3.3 ANOVA table (SS 27.897 / 17.452, F = 9.59); p = (12/(12 + 2F))⁶ by A&S 26.6.4 in Python

### Unequal group sizes

- Groups: 23 25 21 22 / 28 30 27 26 29 / 24 26 25
- Significance level α: 0.05
- **F statistic: 13.5**
- **df between groups: 2**
- **df within groups: 9**
- **p-value: 0.001953**
- **η² (share of variance explained): 0.75**
- Checked against: Sums of squares with Python fractions (SSB 62.25, SSW 20.75); A&S 26.6.4 for d₁ = 2 gives p = (9/(9 + 2F))^4.5 = 4^−4.5 = 1/512

### Two groups equals the pooled t-test

- Groups: 12 15 11 14 13 16 / 17 14 18 16 19 15
- Significance level α: 0.05
- **F statistic: 7.714286**
- **df within groups: 10**
- **p-value: 0.019536**
- Checked against: F = t² for two groups; pooled t from Python fractions, two-sided p from the A&S 26.7.4 closed form with ν = 10

### Edge case: identical group means

- Groups: 1 2 3 / 2 1 3 / 3 2 1
- Significance level α: 0.05
- **F statistic: 0**
- **p-value: 1**
- **Decision: Fail to reject H₀**
- Checked against: Every group mean is 2, so SS between = 0 and F = 0

## Questions

### What does the p-value in ANOVA mean?

It is the probability of an F statistic at least as large as the one observed if every group mean were equal. With the default data p = 0.0032: equal means would produce F ≥ 9.59 in about 3 samples out of 1,000. It is not the probability that the null hypothesis is true, and a small p-value does not say how large the differences are; η² measures that.

### How do you interpret the F statistic?

F is the between-group mean square divided by the within-group mean square. Equal population means give F values near 1; larger values point to real differences. How large is large enough depends on the degrees of freedom: with 2 and 12 df the 5% critical value is 3.885, so F = 9.59 is significant at α = 0.05 and F = 3 would not be.

### What are the assumptions of one-way ANOVA?

Independent observations, a roughly normal distribution in each group, and equal population variances. The F test tolerates moderate non-normality when groups are of similar size. Moore and McCabe's rule of thumb accepts the equal-variance assumption if the largest group standard deviation is less than twice the smallest; otherwise use Welch's ANOVA, and for clearly non-normal data the Kruskal–Wallis test.

### How do you find which groups differ after ANOVA?

Run a post-hoc test. Tukey's honestly significant difference (HSD) compares every pair while holding the family-wise error rate at α; the Bonferroni method tests each of the m pairs at α/m, which is 0.05/3 ≈ 0.0167 for three groups. Separate t-tests at 0.05 on every pair push the chance of at least one false positive well above 5%, towards 1 − 0.95³ ≈ 14% for three comparisons.

### What is a good eta squared value?

η² is the between-group sum of squares divided by the total sum of squares: the share of variation explained by group membership. Cohen (1988) proposed 0.01, 0.06 and 0.14 as small, medium and large effects. The default data give 27.897/45.349 = 0.615, a very large effect. η² overstates the population effect in small samples; ω² corrects for that bias.

### How accurate is the one-way ANOVA calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 4 worked examples whose answers come from independent sources; for example, “NIST e-Handbook example, 3 groups of 5 (defaults)” is checked against NIST/SEMATECH e-Handbook §7.4.3.3 ANOVA table (SS 27.897 / 17.452, F = 9.59); p = (12/(12 + 2F))⁶ by A&S 26.6.4 in Python.

### Where does the method come from?

NIST/SEMATECH e-Handbook of Statistical Methods, §7.4.3 Are the means equal? (one-way ANOVA and worked example); Abramowitz & Stegun, Handbook of Mathematical Functions, §26.6 (F distribution).

## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods, §7.4.3 Are the means equal? (one-way ANOVA and worked example)](https://www.itl.nist.gov/div898/handbook/prc/section4/prc43.htm)
- Abramowitz & Stegun, Handbook of Mathematical Functions, §26.6 (F distribution)