F-distribution

The F-distribution is a continuous probability distribution formed by the ratio of two independent, scaled chi-square variables. It underlies variance-ratio tests, including ANOVA and tests of regression models.

Chi-square distribution: A continuous distribution of sums of squared independent standard normal variables. The F-distribution is built from two independent chi-square variables, each divided by its degrees of freedom.
Analysis of variance: A statistical method that partitions variation to compare means across groups. ANOVA uses an F statistic to compare between-group variation with within-group variation.
Normal distribution: A continuous, symmetric probability distribution determined by its mean and variance. Squared standard normal variables generate the chi-square variables used to construct the F-distribution.
Levene's test: A test of equality of group variances based on deviations from group centers. It offers a less normality-sensitive alternative to the classical F-test for comparing variances.
Degrees of freedom: The number of independent values available to vary when estimating a statistical quantity. The numerator and denominator degrees of freedom determine the F-distribution’s shape.
Regression F-test: A test of whether a regression model or a specified group of predictors explains variation beyond a null model. Its test statistic follows an F-distribution under the null restrictions.
Independent random variables: Random variables are independent when knowledge of one does not change the probability distribution of another. Independence of the two chi-square variables is part of the standard F-distribution construction.
Bartlett's test: A test of whether several normally distributed populations have equal variances. It also tests variance equality, but uses a chi-square approximation rather than an F reference distribution.
F-test: A statistical test whose test statistic follows an F-distribution under a specified null hypothesis. An observed F statistic is assessed against this distribution to calculate a p-value.
Nested model: A statistical model is nested within another when the smaller model is obtained by restricting parameters of the larger one. An F-test can compare nested models by measuring the larger model’s reduction in residual variation.

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Chi-squared distributionCompared with: It compares variance estimates through a ratio, unlike the single sum of squares represented here.