F distribution | Vose Software

# F distribution

Format: F(n1, n2)

The F distribution (sometimes known as the Fisher-Snedecor distribution (Sir Ronald Aylmer Fisher (1890-1962), George Waddell Snedecor (1882 - 1974)) 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 n1 and n2 degrees of freedom as follows:

F(n1,n2)= (ChiSq(n1)/n1) / (ChiSq(n2)/n2)                  (1)

Examples of the F distribution are given below:

## Uses

Equation 1 shows the relationship between the ChiSq and F distributions. From this relationship you can see that one can compare any two independently estimated parameters, whose estimates follow ChiSq distributions, using the F distribution.

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. The uncertainty about a Normally distributed population's variance can be described as follows when the population mean is unknown:

Thus, if we wish to compare the variances of two populations A, B where we have nA and nB data points respectively and sample variances sA2 ,sB2  we have:

So, for example, we may wish to know what confidence we have that  sA2 > sB2 - the condition that:

or

The VoseFProb function in ModelRisk can give us this directly:

From Equation (1) you'll readily appreciate the identity F(n1,n2) = 1/ F(n2,n1): it is just switching the numerator and denominator around.

## ModelRisk functions added to Microsoft Excel for the F distribution

VoseF generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseFObject constructs a distribution object for this distribution.

VoseFProb returns the probability density or cumulative distribution function for this distribution.

VoseFProb10 returns the log10 of the probability density or cumulative distribution function.

VoseFFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseFFitObject constructs a distribution object of this distribution fitted to data.

VoseFFitP returns the parameters of this distribution fitted to data.