F Distribution Calculator

Compute the probability density, cumulative probability, upper tail p value, and mean of the F distribution with degrees of freedom d1 and d2.

Input

Calculate the probability density, cumulative distribution, upper tail probability (p value), and mean of the F distribution with degrees of freedom d1 and d2. Enter an x of 0 or greater.

Value of x

A real number of 0 or greater

Degrees of freedom d1

Numerator degrees of freedom (positive)

Degrees of freedom d2

Denominator degrees of freedom (positive)

Result

Upper tail probability P that X exceeds 2 | d1=5, d2=10

16.419495 %

Probability of a value greater than x (p value)

Probability density PDF

0.16200574

Cumulative distribution CDF (lower tail)

0.83580505

Upper tail probability (p value)

0.16419495

Mean

1.25

Probability density PDF curve

Cumulative distribution CDF curve

How it works

The cumulative distribution (lower tail) is CDF(x)=I_z(d1/2, d2/2), where z=d1 x/(d1 x+d2) and I is the regularized incomplete beta function.
The upper tail probability (p value) equals 1 minus the CDF and represents the chance of a value greater than x.
The probability density PDF takes the square root of the product of (d1 x) to the power d1 and d2 to the power d2, divided by (d1 x+d2) to the power (d1+d2), then divides by x times the beta function B(d1/2, d2/2).
The mean equals d2/(d2-2) when d2 is greater than 2, and is undefined when d2 is 2 or less.
Degrees of freedom d1 and d2 must be positive, and x must be 0 or greater.

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