# F Distribution

The F Distribution (also  called Snedecor’s F, Fisher’s F or Fisher–Snedecor distribution) is a probability distribution of the F Statistic. In other words, it’s a distribution of all possible values of the F statistic. The distribution is important to understand when dealing with ANOVA (Analysis of Variance) tests, as it helps researchers test whether samples have the same variance and decide whether or not their results are statistically significant. To better understand this concept, let’s take a closer look at the F Distribution and how it works.

## What is the F Distribution?

The F distribution is an asymmetric distribution that is used in Analysis of Variance (ANOVA) tests. It has a minimum value of zero; there is no maximum value. The higher the f-value after that point, the lower the curve becomes.

The F Distribution is a family of several individual distributions with different shapes which can help indicate whether or not statistical significance exists between two or more groups being compared.

The shape of the density curve depends on the degrees of freedom in the numerator (dfn) and denominator (dfd). These depend upon your sample characteristics.

For example, in a one-way ANOVA between-groups,

• dfn = a – 1
• dfd = N – a

where

• a is the number of groups
• n is the total number of subjects in the experiment.

## F distribution properties

The F distribution is the distribution of

where S1 and S2 are independent random variables with chi-squared distributions and d1, d2 are their respective degrees of freedom.

The F distribution and chi-square distribution are very similar [2]:

• The chi square distribution arises in hypothesis tests for independence of two random variables. Specifically, concerning whether a discrete random variable follows a specified distribution.
• The F distribution arises in tests of hypotheses to measure whether two population variances are equal and whether three or more population means are equal.

The probability density function (PDF) is given by

where Β is the beta function.

Where I is the regularized incomplete beta function.

Values for the F distribution can be found in F-tables. However, most statistical software will calculate the values for you during hypothesis testing.

## How does it work?

In order for the F-distribution to be useful in ANOVA testing, one must first understand what an “f-statistic” is. An f-statistic measures how much variation within each group versus how much variation between groups exists in a set of data being analyzed. In other words, if two groups have very similar scores on average but also have large variations within each group, then that would indicate that there isn’t much difference between them and thus no significant statistical difference can be determined from those results. On the other hand, if two groups have vastly different scores on average but small variations within each group, then that would indicate that there may be some kind of significant statistical difference between them and thus further investigation should be done to determine exactly what that difference might be.

In order for researchers to determine whether or not their results are statistically significant based on their data analysis findings, they use an “F-test” which calculates an “F-Statistic” from their data set using specific formulas and then compares this number to an “F Distribution” which contains all possible values for an F statistic under various conditions (i.e., different numbers of groups being compared). If their calculated f statistic falls outside of this range then they can conclude that there is a statistically significant difference between their groups and further investigation should be done in order to determine why this might be happening (i.e., what caused these differences). This type of analysis helps researchers better understand their data by providing them with valuable insights they may not have seen before just by looking at raw numbers or charts alone.

## References

[1] IkamusumeFan, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[2] Chi-Square Tests and F-Tests Retrieved Jan 1 2023 from: https://bluebox.creighton.edu/demo/modules/en-saylor/content/saylordotorg.github.io/text_introductory-statistics/s15-chi-square-tests-and-f-tests.html

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