F distribution
A noisy denominator makes a long tail
F compares two variability estimates. A denominator estimated with few degrees of freedom can be unusually small, making the ratio unusually large.
Keep the numerator degrees of freedom at 2. Improve the denominator estimate from 6 to 60 degrees of freedom and watch the upper 5% boundary.
Null F density & rejection region
The curve shows possible F values under the null hypothesis. The dashed line marks the upper 5% critical value. Values to its right make up the rejection region. This explorer shows a critical value rather than a statistic calculated from observed data.
Worked comparison
Follow the calculation
What the symbols mean
U and V are independent chi-squared variables, with df₁ and df₂ degrees of freedom. F is the ratio after dividing each by its own degrees of freedom.
Assumptions and full calculation
F is non-negative and has no upper limit. For an upper-tail variance-ratio test or ANOVA, values above the critical value lead to rejection of the null. When the numerator has fewer than two degrees of freedom, the density grows without limit near zero. The curve omits that endpoint. On a linear axis the shaded area ends at the plot edge, while the critical value uses the full distribution. On a logarithmic axis screen area is not proportional to probability; the fill is removed. The probability and critical value remain unchanged.