From an F statistic to its p-value
F statistics come from ANOVA (comparing group means) and from variance-ratio tests. F is a ratio of two variance estimates, so (like chi-square) it is never negative and only large values argue against the null. The p-value is the right-tail area, and it needs two degrees of freedom: numerator (between-groups or model) and denominator (within-groups or error).
F = 4.1028 with df (2, 10): right-tailed p = 0.0500. That F is exactly the 5% critical value for those df. Swap the df and the same F means something else entirely; F has no meaning without both of its degrees of freedom attached.
Which df goes where?
- One-way ANOVA with k groups and N total observations: numerator k − 1, denominator N − k
- Variance-ratio test: numerator df from the group in the numerator, denominator df from the other
- Regression overall test: numerator = number of predictors, denominator = n − predictors − 1
Frequently asked questions
Why does F need two degrees of freedom?
F is a ratio of two independent variance estimates, and each carries its own df. The shape of the F distribution depends on both.
My F is below 1. Is that a problem?
No. F below 1 just means the numerator variance estimate came out smaller than the denominator's. That's ordinary sampling noise under the null, with a large right-tail p-value.