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F Distribution

TL;DR

Section titled “TL;DR”
  • A continuous distribution for ratios of two independent chi-squared variables.
  • Commonly applied in ANOVA and in tests comparing variances between two samples.
  • Used to evaluate whether observed differences between groups are likely due to chance.

Definition

Section titled “Definition”

The F-distribution, also known as the Fisher-Snedecor distribution, is a continuous probability distribution that describes the ratio of two independent chi-squared random variables. It is commonly used in statistical hypothesis testing.

Explanation

Section titled “Explanation”

The F-distribution arises when forming a ratio between two independent chi-squared variates (typically scaled by their degrees of freedom). In practice, this ratio appears when comparing sample variances or when partitioning variance in Analysis of Variance (ANOVA). The computed f-statistic—the ratio formed from the variances being compared—is referenced against the F-distribution to assess whether observed differences are statistically significant or plausibly due to random variation.

Examples

Section titled “Examples”

ANOVA example

Section titled “ANOVA example”

A researcher compares average heights across three groups: men, women, and children. Heights are recorded for each individual in each group and the variance for each group is calculated. The f-statistic is computed by taking the ratio of the variance of the two groups being compared, placing the larger variance in the numerator. That f-statistic is then compared to the F-distribution to determine if the difference in variances is statistically significant.

Equality of variances example

Section titled “Equality of variances example”

A researcher compares the time spent on a task by two different groups. For each group, time measurements are collected and variances are calculated. The f-statistic is computed as the ratio of the two variances, with the larger variance in the numerator. This f-statistic is compared to the F-distribution to determine whether the variance difference between the groups is statistically significant.

Use cases

Section titled “Use cases”
  • Testing the equality of variances between two samples.
  • Analysis of Variance (ANOVA) to determine if there are significant differences between group means.
Section titled “Related terms”
  • Fisher-Snedecor distribution
  • Chi-squared distribution
  • ANOVA
  • f-statistic