# F-Distribution / F-Test Calculator

Free F-test calculator. Compare two sample variances using the F-distribution. Get F-statistic, degrees of freedom, and p-value.

## What this calculates

Perform an F-test to compare two sample variances. Enter the variances and sample sizes to calculate the F-statistic and determine if the variances are significantly different.

## Inputs

- **Sample 1 Variance (s₁²)** — min 0.0001 — Variance of the first sample (larger variance for one-tailed).
- **Sample 1 Size (n₁)** — min 2 — Number of observations in sample 1.
- **Sample 2 Variance (s₂²)** — min 0.0001 — Variance of the second sample.
- **Sample 2 Size (n₂)** — min 2 — Number of observations in sample 2.

## Outputs

- **F-Statistic** — The ratio of the two variances (larger / smaller).
- **df₁ (numerator)** — Degrees of freedom for the numerator.
- **df₂ (denominator)** — Degrees of freedom for the denominator.
- **P-Value (One-Tailed)** — One-tailed p-value for variance ratio test.
- **P-Value (Two-Tailed)** — Two-tailed p-value for equality of variances.
- **Significant at α = 0.05?** — formatted as text — Whether the variances are significantly different.

## Details

The F-test compares two variances to determine if they are significantly different.

Formula:
F = s₁² / s₂² (larger variance in numerator)

Degrees of Freedom

- df₁ = n₁ - 1 (numerator)

- df₂ = n₂ - 1 (denominator)

Applications

- Testing equality of variances before a t-test

- ANOVA (F-statistic compares between-group to within-group variance)

- Quality control

The F-distribution is always right-skewed and defined only for positive values.

## Frequently Asked Questions

**Q: When should I use an F-test?**

A: Use an F-test to compare two sample variances, often as a preliminary test before a two-sample t-test to check if equal-variance or Welch's t-test is more appropriate. It is also the core statistic in ANOVA. The F-test assumes both populations are normally distributed.

**Q: Why is the F-statistic always greater than or equal to 1 here?**

A: By convention, the larger variance is placed in the numerator so that F ≥ 1. This simplifies looking up critical values. Some software does not rearrange the variances, in which case F can be less than 1.

**Q: What are the assumptions of the F-test?**

A: The F-test for variance equality assumes: (1) both samples come from normally distributed populations, (2) samples are independent, and (3) observations within each sample are independent. The F-test is highly sensitive to non-normality. For non-normal data, use Levene's test instead.

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Source: https://vastcalc.com/calculators/statistics/f-distribution
Category: Statistics
Last updated: 2026-04-21