F(α=0.10, ν₁, ν₂) — upper 10% tail
F(α=0.05, ν₁, ν₂) — upper 5% tail
F(α=0.025, ν₁, ν₂) — upper 2.5% tail
F(α=0.01, ν₁, ν₂) — upper 1% tail
P(F ≤ f, ν₁, ν₂) for f from 0.0 to 15.9
How to use these tables
F Distribution (Snedecor)
If X₁ ~ χ²(ν₁) and X₂ ~ χ²(ν₂) are independent, then F = (X₁/ν₁) / (X₂/ν₂) follows an F distribution with ν₁ and ν₂ degrees of freedom. Its density function is:
\( f(f;\nu_1,\nu_2) = \dfrac{\sqrt{\dfrac{(\nu_1 f)^{\nu_1}\,\nu_2^{\nu_2}}{(\nu_1 f+\nu_2)^{\nu_1+\nu_2}}}}{f\,B\!\left(\tfrac{\nu_1}{2},\tfrac{\nu_2}{2}\right)}, \quad f > 0 \)
The mean is ν₂/(ν₂ − 2) for ν₂ > 2. It has positive skew.
Critical values
The cell in row ν₂ and column ν₁ shows the quantile F such that P(F > f) = α. For an ANOVA at the 5% level with ν₁ = 3 groups and ν₂ = 20 residuals, look up the α = 0.05 table → F = 3.098.
- For equality-of-variances tests (two-sided), use α/2 in the table and reject if F > F(α/2) or F < 1/F(α/2).
- As ν₁ → ∞ and ν₂ → ∞, the F distribution tends to 1.