F distribution

You will learn: Interpret a ratio of scaled chi-squared variables and its degrees of freedom.

Take two independent chi-squared variables, each divided by its own degrees of freedom. Take their ratio. That ratio is F.

F=\dfrac{U_{1}/d_{1}}{U_{2}/d_{2}},\quad U_{i}\sim\chi^{2}_{d_{i}}

Mode: 0.50 Mean: 1.25 F = 1 (null reference)

Read the two independent pieces

First three constructions in the displayed sample
U₁/d₁	U₂/d₂	Their ratio F
0.78708	1.28000	0.614904
0.38761	0.78550	0.493448
0.93107	0.63197	1.47329

Each scaled chi-squared has expectation one. The expectation of their ratio need not be one: division by small independent denominators creates the right tail. If both pieces were the same draw with the same degrees of freedom, the ratio would instead always be one.

F cutoff 2.00

P(F≥2) = 0.164195. Equivalently, P(1/F≤0.50000) has this probability under F(10,5).

Quantile probability 0.95

The 95% quantile is 3.325835. Variance is 1.35417.

6 of 1500 draws are above the plot limit 6.8724; exact mass beyond it is 0.5000%. The density drawing starts at 0.000001 and clips at height 0.7907. Neither the full-tail calculation nor the sample denominator is truncated.

d₁ (numerator df) 5 d₂ (denominator df) 10 n (samples) 1500

PDF

f(x;d_{1},d_{2})=\frac{\sqrt{\dfrac{(d_{1}x)^{d_{1}}d_{2}^{d_{2}}}{(d_{1}x+d_{2})^{d_{1}+d_{2}}}}}{x\,B(d_{1}/2,d_{2}/2)}

. Equivalently the ratio of two independent chi-squareds, each divided by its degrees of freedom:

F=\dfrac{U_{1}/d_{1}}{U_{2}/d_{2}},\quad U_{i}\sim\chi^{2}_{d_{i}}

What to notice

Always non-negative, right-skewed. Ratios of positive quantities can only be positive, and the denominator’s lower tail drags the upper tail of the ratio.
d₂ controls the right tail. Small denominator degrees of freedom give heavy tails — the denominator can be very close to zero, inflating F. Increasing $d_{2}$ reduces that source of variability, but the required tail accuracy depends on the event.
F=1 is a reference value, not a universal mean. It marks equal scaled numerator and denominator values. For d₂>2, E[F]=d₂/(d₂−2), which is greater than one.
Reciprocal symmetry. Swapping $d_{1}$ and $d_{2}$ is equivalent to inverting the statistic:

$X\sim F_{d_1,d_2}\quad\Longrightarrow\quad 1/X\sim F_{d_2,d_1}$

Why it matters

Under the relevant independent normal-error assumptions, F reference distributions arise in:

ANOVA. The ratio “between-group mean square” over “within-group mean square” is F under the null of equal group means.
Nested regression models. The F statistic compares the sums of squared residuals of a restricted vs. unrestricted model.
Variance ratio tests. Directly testing $\sigma_{1}^{2}=\sigma_{2}^{2}$ .

The sampling construction determines whether F is appropriate; the appearance of a histogram does not establish the model.

E[F]=\frac{d_{2}}{d_{2}-2}\quad(d_{2}>2)

The denominator matters as much as the numerator

Suppose U₁=5 with d₁=5, and U₂=2 with d₂=10. The scaled pieces are 1 and 0.2, so their ratio is 5. If U₂ were 10 instead, the ratio would be 1. The construction table shows actual seeded pieces and preserves their independence. Copying one piece into both positions with equal degrees of freedom would produce a constant ratio, not an F distribution.

At d₁=5,d₂=10, P(F≥2)≈0.164195, and the 95th percentile is approximately 3.325835. A five-percent upper-tail rule uses that quantile. A two-sided variance comparison needs a specified two-tail convention; applying this single upper cutoff does not automatically make a two-sided test.

Reciprocal ratios swap the reference distribution

If F=(U₁/d₁)/(U₂/d₂), then 1/F reverses the pieces and follows F(d₂,d₁). Consequently an upper-tail event F≥c becomes a lower-tail event 1/F≤1/c with the degrees of freedom swapped. At d₁=d₂=2, the exact survival probability is 1/(1+c), so P(F≥3)=1/4 and the 95th percentile is 19. A reference line at one can coexist with a very long tail.

For independent normal samples of sizes n₁ and n₂, the ratio (S₁²/σ₁²)/(S₂²/σ₂²) has degrees of freedom n₁−1 and n₂−1. Under equal population variances this reduces to S₁²/S₂². ANOVA uses a different pair of mean squares and a null of equal group means; interpreting all F procedures as the same variance-ratio question loses that distinction.

Make a prediction

Two marginally chi-squared quantities share all of their random inputs. Can you still use the displayed F tail formula for their scaled ratio?

Explore the answer

Not from the marginal distributions alone. The construction needs independence. With identical quantities and equal degrees of freedom, the ratio is exactly one, providing a direct counterexample.

References

NIST: F distribution gives the ratio law, incomplete-beta probabilities, and moment conditions. NIST: comparing two variances describes the one- and two-sided testing choices. Connect the ratio to chi-squared variance inference and hypothesis-testing questions.

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