What is the F-Distribution?

The F-distribution, also known Fisher-Snedecor distribution, is extensively used to test for equality of variances from two normal populations. F-distribution got its name after R.A. Fisher, who initially developed this concept in the 1920s. It is a probability distribution of an F-Statistic.

F-distribution probability density curve, positively skewed and bounded at zero on the left

The F-distribution is generally skewed and also related to a Chi-Squared distribution. Additionally, the f-distribution is the ratio of the X1 random chi-square variable with degrees of freedom ϑ1 and the X2 random chi-square variable with degrees of freedom ϑ2. In other words, each Chi-Square random variable has been divided by its degrees of freedom.

Formula defining the F-distribution as the ratio of two chi-square random variables, each divided by its own degrees of freedom

The shape of the distribution depends on the degrees of freedom of numerator ϑ1 and denominator ϑ2.  

Family of F-distribution curves at different combinations of numerator and denominator degrees of freedom, showing the shape flattening and becoming less skewed as degrees of freedom increase

What are the properties of an F-distribution?

F-distribution curve labeled with its key properties: positively skewed, bounded at zero, and shape determined by numerator and denominator degrees of freedom
  • The F-distribution curve is positively skewed towards the right with a range of 0 and ∞.
  • The value of F is always positive or zero. No negative values.
  • The shape of the distribution depends on the degrees of freedom of numerator ϑ1 and denominator ϑ2.  
  • From the above graph, it is clear that the degree of skewness decreased with an increase in the degrees of freedom of the numerator and denominator.
  • The f-distribution curve can never be symmetrical; if degrees of freedom increase it will be more similar to the symmetrical.

When would you use the F-distribution?

The F-test compares more than one level of independent variable with multiple groups, which uses the F distribution. This is generally used in ANOVA calculations. Always use F-distribution for F-test to compare more than two groups.

Example: In a manufacturing unit, torque values are key parameters in terminal squeeze welding. To check the significant effect of various torque (nm) values of the squeeze welding, an operator set up trials of 5nm, 8nm, 10nm, and 12nm of terminals in four randomly selected batches of 30. ANOVA can determine whether the means of these four trials are different. ANOVA uses F-tests to statistically test the equality of means.

Assumptions of F-distribution

  • Assumes both populations are normally distributed.
  • Both populations are independent of each other.
  • The larger sample variance always goes in the numerator to make the right-tailed test, and the right-tailed tests are always easy to calculate.

F-distribution Videos

Once you understand the F-distribution, see the F-Test page for how it’s actually used to compare variances, including the F-statistic calculation and worked examples for right-tailed, two-tailed, and left-tailed tests. For the full mathematical treatment and critical value tables, NIST’s Engineering Statistics Handbook is a reliable reference.

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