# F statistic

The F statistic is the mean square between groups divided by the mean square within groups. A mean square is a sum of squares divided by its degrees of freedom. A large F means the group means differ more than chance explains.

Statistics (university) · Part 07: Comparing categories and many groups · Card 5 of 8 · Last updated 2026-10-19

## Analogy

A signal-to-noise ratio. The group differences are the signal, the spread within groups is the noise.

## Example

**Computing F**

k = 3 groups, N = 9
MS between = 24 / 2 = 12
MS within = 6 / 6 = 1
F = 12 / 1 = 12 with df (2, 6)
Critical 5.14; p = 0.008

## In practice

ANOVA output is usually shown as a table with sums of squares, degrees of freedom, mean squares and F.

## Practice

### Flashcards

- **What is the F statistic?** Mean square between groups divided by mean square within groups.

### Interview questions

**Why use ANOVA instead of several t tests, and what does the F statistic compare?**

- Many t tests inflate the false alarm rate
- ANOVA tests all means at once
- F compares between-group and within-group variation
- A large F suggests real differences
- A post-hoc test finds which groups differ

## Questions people ask

### What does an F value near 1 mean?

It means the variation between group means is about the same as the variation expected from noise within groups. That gives no evidence of real differences. Values well above 1, relative to the critical value, suggest the groups truly differ.

### What assumptions does ANOVA need?

Observations should be independent, each group roughly normal, and the groups should have similar variances. ANOVA is fairly robust to mild departures, especially with equal group sizes. Strong skew or very unequal spread calls for a rank-based alternative such as the Kruskal-Wallis test, which compares groups without assuming a normal shape.

## Related concepts

- [One-way ANOVA](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-7/statistics-university-part-7-concept-4)
- [Post-hoc comparisons](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-7/statistics-university-part-7-concept-6)

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