Interactive module guide

F Distribution Calculator & Guide

This F distribution calculator lets you calculate right-tail probabilities and critical values used in ANOVA and variance-ratio testing.

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The F distribution is a continuous probability model on nonnegative values for the ratio of two independent chi-square variables that have each been divided by their degrees of freedom.

Its parameters are the numerator degrees of freedom d₁ and the denominator degrees of freedom d₂, and it serves as the reference distribution for one-way ANOVA and for tests that compare two variances.

Use this page to examine the density, cumulative probability, tail areas, and quantiles implied by a chosen pair of degrees of freedom; do not treat the page as a test, because it evaluates a reference model rather than analyzing data.

Is this the right module for my question?

Use it when

Use an F model as the reference distribution for a ratio of two independent variance estimates computed from normally distributed samples. Under equal population variances, the ratio s₁²/s₂² follows F(d₁, d₂) with each degrees-of-freedom value equal to its sample size minus one, and the explorer shows how large that ratio can plausibly become by chance alone.
It is also the null reference for the one-way ANOVA statistic, the between-group mean square divided by the within-group mean square. Real group differences inflate the numerator but not the denominator, so evidence against the null hypothesis accumulates in the upper tail, and the upper-tail quantiles displayed here are the critical values printed in ANOVA tables.
Choose this page when the question asks for a tail probability, an interval probability, or a quantile of a specified F(d₁, d₂) model, for example locating the upper 5% critical value for stated degrees of freedom or checking a p-value reported elsewhere against the same reference.

Choose another method when

Do not apply an F reference to a statistic that is not a ratio of independent, appropriately scaled chi-square quantities. Variance ratios from markedly non-normal populations are the leading example: the sampling distribution of a sample variance is very sensitive to heavy tails, so the nominal F reference can be seriously misleading even with moderate samples.
Avoid the model when the two variability estimates are not independent, for instance variances computed from overlapping observations or from paired measurements on the same units. Dependence between numerator and denominator changes the sampling distribution, and the F curve drawn here no longer describes the ratio.
Do not use this page as a substitute for the test itself. It does not compute an F statistic from raw data, does not correct for multiple comparisons, and does not verify assumptions; it only describes what the reference model implies for a chosen d₁ and d₂.

Interactive tool

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Example preview

ANOVA-style F critical value

Inputs: F with d₁=5 and d₂=10; find the 95th percentile.
Representative result: Q(0.95) ≈ 3.3258; observed ratios beyond it fall in the upper 5% tail.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

2:59 min

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Read the complete transcript

The F Distribution. Why can dividing two variance estimates create a ratio with such a long right tail? Divide two independent chi-square variables by their degrees of freedom, then take their ratio. Both scaled components are nonnegative, so every F variable also lives from zero rightward. The first degrees of freedom belongs to the numerator and controls behavior near zero. The second belongs to the denominator, whose instability controls the heavy upper tail. Invert the ratio and the degrees of freedom swap, creating an exact reciprocal identity.

At finite degrees of freedom the density is right-skewed, especially with small denominator degrees. When d one exceeds two, the mode follows a closed formula below one. The mean exists only above denominator degrees two and then remains greater than one. Finite variance requires denominator degrees above four; otherwise exact probabilities still remain well defined. Skewness needs denominator degrees above six and can be very large near that threshold. Excess kurtosis needs denominator degrees above eight, revealing even stricter tail-moment requirements. As both degrees grow, the distribution concentrates more tightly around a ratio of one. Density height still is not probability; every probability requires area over a stated event.

Now consider an F reference with numerator degrees five and denominator degrees ten. An upper five-percent boundary is the quantile at cumulative probability zero point nine five. For F five ten, that exact critical value is approximately three point three two five eight. The survival area beyond three point three two five eight is therefore exactly five percent. Its mean is one point two five, variance about one point three five four. The mode is one half and median about zero point nine three two, showing right skew. The swapped lower quantile multiplies this upper quantile to one, confirming the reciprocal identity.

ANOVA uses an upper tail because between-group signal inflates the numerator mean square. Two-sided variance tests instead need separate lower and upper boundaries, never mirrored around one. Keep numerator and denominator degrees attached to their positions, because reversing them changes the distribution. Variance-ratio inference requires independent samples and is especially sensitive to departures from normality. The explorer supplies reference areas and quantiles; it does not calculate a statistic from data. Report d one, d two, the exact event, and whether the result is area or quantile. Explore F tails, reciprocal quantiles, and changing degrees of freedom free at Distri Scope dot com.

How to read the result

Read the PDF panel as relative concentration on x ≥ 0: for d₁ > 2 the curve rises to a single mode below 1 and then decays along a long right tail, and only area under the curve corresponds to probability. In the Key Statistics panel, expect the mean above the median whenever the mean exists, a direct display of right skew; at small d₂ the standard deviation can be large or shown as a dash even though every probability and quantile remains perfectly well defined.

When reporting a result, state the event with both degrees of freedom in fixed order, because F(d₁, d₂) and F(d₂, d₁) are different distributions. A small upper-tail area says that ratios this large are uncommon under the reference model; it does not by itself establish that group means differ or that variances are unequal, since that conclusion also depends on the design, the assumptions, and a test procedure applied to actual data.

Inputs and parameter meaning

Set the numerator degrees of freedom d₁ with the first slider, from 1 to 50 with a default of 5. In a two-sample variance ratio d₁ is the numerator sample size minus one, and in one-way ANOVA it is the number of groups minus one. Small d₁ controls behavior near zero: for d₁ < 2 the density is unbounded as x approaches zero, at d₁ = 2 the mode is exactly zero, and for d₁ > 2 the mode is ((d₁−2)/d₁)·(d₂/(d₂+2)), which always sits below 1.

Set the denominator degrees of freedom d₂ with the second slider, from 1 to 50 with a default of 10. This parameter governs the right tail and which moments exist: the mean requires d₂ > 2 and equals d₂/(d₂−2), the variance requires d₂ > 4, skewness requires d₂ > 6, and excess kurtosis requires d₂ > 8. The Key Statistics panel renders any undefined quantity as a dash, which is a mathematical statement about the model, not a computational failure.

The point calculator accepts a value x on the support x ≥ 0 and reports the density f(x), the cumulative probability P(X ≤ x), and the survival probability P(X > x). The interval tool gives P(a < X ≤ b), and the quantile lookup inverts the CDF to return x = Q(p). Parameter settings can be shared by URL, and the charts can be exported as PNG images or CSV values for records.

Formula and available parameters

Density or probability mass

f(x) = \frac{\sqrt{\dfrac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1+d_2}}}}{x\, B\!\left(\frac{d_1}{2}, \frac{d_2}{2}\right)}, \quad x > 0

Mean

\mu = \frac{d_2}{d_2 - 2} \quad (d_2 > 2)

Variance

\sigma^2 = \frac{2 d_2^2 (d_1 + d_2 - 2)}{d_1 (d_2-2)^2 (d_2-4)} \quad (d_2 > 4)

Numerator df (d₁): interactive range 1 to 50; default 5.
Denominator df (d₂): interactive range 1 to 50; default 10.

What the method is doing

Construction explains the shape. If U~χ²(d₁) and V~χ²(d₂) are independent, then X = (U/d₁)/(V/d₂) follows F(d₁, d₂). Each scaled chi-square has expectation 1, yet the ratio is not centered at 1: averaging over the reciprocal of a volatile denominator inflates the expectation, so the mean is d₂/(d₂−2), which exceeds 1 whenever it exists, even when the two underlying variances are exactly equal. Small d₂ makes the denominator unstable and produces the heavy right tail visible at low slider settings.

Quantiles are strongly asymmetric. Swapping numerator and denominator inverts the variable, so if X follows F(d₁, d₂) then 1/X follows F(d₂, d₁), which yields the reciprocal identity F(α; d₁, d₂) = 1/F(1−α; d₂, d₁) for quantiles. Printed tables historically listed only upper-tail critical values and relied on this identity for lower tails; the quantile lookup here returns any probability level directly, but the identity remains a useful cross-check, and two-sided variance tests genuinely need both tails.

Worked example: the upper 5% critical value for a variance ratio

Suppose two production lines are compared for consistency. A sample of 6 units from line A and 11 units from line B yield variance estimates s₁² and s₂², so the ratio s₁²/s₂² is referred to F(5, 10) under the hypothesis of equal population variances and normally distributed measurements. The analyst wants the upper 5% critical value and a sense of the shape of the reference distribution.

1Set d₁ = 5 and d₂ = 10 with the sliders, then use the quantile lookup at p = 0.95. The explorer returns Q(0.95) ≈ 3.3258, so P(F > 3.3258) ≈ 0.05: under equal variances, only about one ratio in twenty would exceed 3.3258 by chance.
2Read the Key Statistics panel. The mean is 10/8 = 1.25 rather than 1, the median of about 0.9319 lies below the mean, and the skewness of about 3.867 confirms a strongly right-skewed shape, even though both variance estimates target the same quantity. A ratio modestly above 1 is entirely unremarkable here.
3Confirm internal consistency with the point calculator by evaluating the survival probability at x = 3.3258, which returns approximately 0.05, matching the quantile result. Then drag d₂ toward smaller values while keeping d₁ = 5 and watch the right tail thicken and the upper critical value climb: with few denominator degrees of freedom, much larger ratios arise by chance alone.

Interpretation

Under the F(5, 10) reference model, a variance ratio above about 3.33 would occur roughly 5% of the time when the population variances are truly equal, so such a value is unusual but not impossible. The calculation assumes independent samples from normal populations, and comparisons of variances are known to be sensitive to departures from normality. This page neither computes the ratio from data nor checks those assumptions; carry the observed statistic and its degrees of freedom to the F-test for variances in the hypothesis test calculator for a complete procedure with stated assumptions.

Common mistakes

Reversing the degrees of freedom is the most common error. F(d₁, d₂) and F(d₂, d₁) are different distributions with different critical values, so keep the numerator degrees of freedom first and attached to the numerator of your ratio. When in doubt, the reciprocal identity shows exactly how the two orderings correspond.
Do not build two-sided critical regions symmetric about 1, and do not read a ratio slightly above 1 as evidence of unequal variances. The distribution is right-skewed, the mean exceeds 1 whenever it exists, and lower critical values must come from the lower quantile, obtained directly here or through the reciprocal relationship, not by mirroring the upper value around 1.
Do not interpret a dash in the Key Statistics panel as a software error, and do not summarize small-d₂ settings by moments alone. When d₂ ≤ 2 the mean does not exist and when d₂ ≤ 4 the variance does not exist, yet the CDF, tail areas, and quantiles remain exact; use those directly instead of moment-based rules of thumb.

Limits and independent validation

This explorer visualizes a parameterized reference model. It does not run F-tests or one-way ANOVA, does not estimate degrees of freedom from data, and does not fit distributions to observations; the hypothesis test calculator performs the tests, while this page supplies the distributional reasoning behind their critical values and p-values.

Both sliders stop at 50 degrees of freedom, so reference curves for larger designs cannot be displayed, although the visible trend continues: as both d₁ and d₂ grow, the distribution concentrates ever more tightly around 1. Extreme tail probabilities at very small d₂ describe a genuinely heavy-tailed model, but consequential decisions resting on remote tails deserve independent numerical confirmation.

Before using the result

Confirm the pedigree of the statistic before reading any tail area: two independent samples, plausibly normal populations, variance estimates computed with the correct denominators, and degrees-of-freedom bookkeeping that matches the design, sample size minus one for each variance, and groups minus one with total minus groups for one-way ANOVA.
Cross-check one quantile with the reciprocal identity: look up Q(0.95) at your settings, swap the two slider values, look up Q(0.05), and verify that the two results multiply to approximately 1. Record d₁, d₂, the probability level, and the exact event, and share the parameter URL so another person can reproduce the numbers.

F-test for two variances →Run the actual variance-ratio test on data, using both tails of the F reference explored here.one-way ANOVA →Apply the upper-tail F reference to compare several group means with a computed test statistic.

← Chi-Square Distribution Bernoulli Distribution →

See every option in the Probability Distribution Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.