F-Distribution

Comparing variances between two or more samples.

The "Ratio of Variances" Distribution

The F-distribution is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the Analysis of Variance (ANOVA) and the F-test to compare two variances.

It describes the ratio of two independent chi-squared variables, each divided by their respective degrees of freedom. In practical terms, if you take two samples from normal populations, the ratio of their sample variances follows an F-distribution. This is why it's the cornerstone for checking if the volatility (variance) of two different stocks or trading strategies is significantly different.

The Formula

The probability density function (PDF) is defined by its two parameters: the degrees of freedom of the numerator (

d_1

) and the denominator (

d_2

f(x; d_1, d_2) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1 + d_2}}}}{x B(\frac{d_1}{2}, \frac{d_2}{2})}

$d_1$ is the degrees of freedom for the numerator variance.
$d_2$ is the degrees of freedom for the denominator variance.
$B$ is the Beta function.

Interactive F-Distribution

Adjust the degrees of freedom to see how they influence the shape of the distribution. Notice how it becomes more bell-shaped as the degrees of freedom increase.

Numerator Degrees of Freedom (d₁): 5

Denominator Degrees of Freedom (d₂): 10