Fstatistikken

Fstatistikken, commonly referred to as the F-statistic, is a statistical measure used to assess whether a statistical model explains a significant portion of the observed variability. It is central to analysis of variance (ANOVA) and linear regression, where it tests hypotheses about the overall significance of a model or the equality of means across groups.

The statistic is defined as F = MS_between / MS_within, where MS_between is the mean square attributable to the model (between-group variation) and MS_within is the mean square of residual error. In a typical one-way ANOVA with k groups and n observations, df_between = k − 1 and df_within = n − k. More generally, F = (SS_model / df_model) / (SS_error / df_error). Under the null hypothesis that the model has no explanatory power, F follows an F-distribution with df_model and df_error degrees of freedom.

A small p-value against the F distribution leads to rejection of the null hypothesis, indicating that the model explains a significant portion of the variance. Assumptions include independence of observations, normally distributed residuals, and homoscedasticity (equal variances).

F-statistics are widely used to compare nested models and to perform overall tests of regression or group differences. Limitations include sensitivity to deviations from assumptions and reduced interpretability when models are non-nested or effects are small. The F-distribution was introduced in the early 20th century by Ronald A. Fisher.