The Snedecor’s \(F\)-distribution or the Fisher-Snedecor distribution (after Sir Ronald A. Fisher and George W. Snedecor), or short the \(F\)-distribution, is a continuous probability distribution with the range \([0, +\infty)\). The \(F\)-distribution depends on two parameters denoted \(v_1\) and \(v_2\) (Lovric 2011). In statistical applications \(v_1\) and \(v_2\) are positive integers.

Let \(Y_1\) and \(Y_2\) be two independent random variables distributed as chi-square, with \(v_1\) and \(v_2\) degrees of freedom, respectively. Then the distribution of the ratio \(Z\),

\[Z = \frac{Y_1/v_1}{Y_2/v_2},\]

is called the \(F\)-distribution with \(v_1\) and \(v_2\) degrees of freedom. The \(F\)-distribution is often referred to as the distribution of the variance ratio (Lovric 2011).

A \(F\)-distribution has two numbers of degrees of freedom, \(v_1\) and \(v_2\), determining its shape. The first number of degrees of freedom, \(v_1\), is called the degrees of freedom of the numerator and the second, \(v_2\), the degrees of freedom of the denominator.

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Basic Properties of \(F\)-curves (Weiss 2010):

• The total area under a \(F\)-curve equals 1.
  
• A \(F\)-curve starts at 0 on the horizontal axis and extends indefinitely to the right, approaching, but never touching, the horizontal axis as it does so.
• A \(F\)-curve is right skewed.

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Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.