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title: "F distribution"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575229/F%20distribution"
format: markdown
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F(*n*1,*n*2) - no Crystal Ball distribution, = FINV(Uniform(0,1), *n*1,*n*2)

[F distribution equations](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26576144/)

 

 

The F distribution (sometimes known as the Fisher(1)–Snedecor(2) distribution, and taking Fisher's initial) is commonly used in a variety of [statistical tests](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575342/). It is derived from the ratio of two normalized [chi-squared](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575221/) distributions with v1 and v2 degrees of freedom as follows:


  (1)

F(*n*1,*n*2)= (ChiSquared(*n*1)/*n*1)/(ChiSquared(*n*2)/*n*2)                 

 

Examples of the F distribution are given below:

 

![image](media://53121606-ce77-4cb6-9e58-979346bccba3)

 

 

#### Uses

The most common use of the F distribution you'll see in statistics text books is to compare the variance between two (assumed Normally distributed) populations. From a risk analysis perspective, it is very infrequent that we would wish to model the *ratio* of two estimated variances (which is essentially the F-test in this circumstance) so the F distribution is not particularly useful to us.

 

#### Generation

The F distribution is not directly available with Crystal Ball, but can be produced in two ways. The first is with Equation 1 (above) using two Chi Squared distributions, but you will lose the benefits of [Latin Hypercube sampling](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575293/) because two distributions are being used to create the random samples.

 

The second method uses the FINV( ) function in Microsoft Excel which returns the inverse of the F distribution, as follows:

 

(2)

= FINV(Uniform(0,1), *n*1,*n*2)                                       

 

This will generate Latin Hypercube samples (provided the Crystal Ball settings have that option selected) from the F(*n*1,*n*2) distribution.

 

The un-normalized version of equation (1) is a particular case of the [Pearson Type VI](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575250/) distribution, i.e.:


(3)

Pearson6(*n*1/2, *n*2/2, 1) = ChiSquared(*n*1)/ChiSquared(*n*2)         (3)

 

Equation (3) gives yet another (the third) way of generating the F distribution using Crystal Ball, as follows:

 

(4)

=Pearson6(*n*1/2, *n*2/2, 1) * *n*2/*n*1                                 

 

Just as equation (2), equation (4) keeps the advantages of Latin Hypercube sampling.

 

#### Comments

From Equation (1) you'll readily appreciate the identity F(*n*1,*n*2) = 1/ F(*n*2,*n*1): it is just switching the numerator and denominator around.

 

 

The Excel function [FDIST](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575570/)(x, *n*1,*n*2) returns the cumulative probability from an F(*n*1,*n*2) distribution

 

 

(1) Sir Ronald Aylmer Fisher (1890-1962)

(2) George Waddell Snedecor (1882–1974)

 

 

 

 

 

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