F(*n*1,*n*2) - no Crystal Ball distribution, = FINV(Uniform(0,1), *n*1,*n*2)

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. It is derived from the ratio of two normalized chi-squared distributions with v1 and v2 degrees of freedom as follows:

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

Examples of the F distribution are given below:

#### 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 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:

= FINV(Uniform(0,1), *n*1,*n*2) (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 distribution, i.e.:

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:

=Pearson6(*n*1/2, *n*2/2, 1) * *n*2/*n*1 (4)

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(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)