Skip to main content

F-Distribution

NIST/SEMATECH Section 1.3.6.6.5 F-Distribution

PDF

F(5, 10) PDF 0 1 2 3 4 5 6 7 x 0 0.2 0.4 0.6 f(x)

CDF

F(5, 10) CDF 0 1 2 3 4 5 6 7 x 0 0.2 0.4 0.6 0.8 1 F(x)

Interactive Explorer

PDF

CDF

Formulas

Probability Density Function

f(x)=(d1x)d1⋅d2d2(d1x+d2)d1+d2x B ⁣(d12,d22)f(x) = \frac{\sqrt{\frac{(d_1 x)^{d_1} \cdot 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)}

Cumulative Distribution Function

F(x)=Id1xd1x+d2 ⁣(d12,d22)F(x) = I_{\frac{d_1 x}{d_1 x + d_2}}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)

Properties

Mean

d2d2−2  (d2>2)\frac{d_2}{d_2 - 2} \;(d_2 > 2)

Variance

2d22(d1+d2−2)d1(d2−2)2(d2−4)  (d2>4)\frac{2d_2^2(d_1+d_2-2)}{d_1(d_2-2)^2(d_2-4)} \;(d_2 > 4)

Overview

The FF-distribution is the ratio of two scaled chi-squared variables with d1d_1 and d2d_2 degrees of freedom. It is used in analysis of variance (ANOVA) and for comparing variances of two populations.

Related Distributions

Chi-Square Distribution Student's t-Distribution Beta Distribution