Distribution F

The F-distribution is a fundamental concept in the fields of statistics and machine learning, used to compare variances between populations and assess whether the difference observed between two groups is due to chance or to significant factors.  

This distribution is named after the letter «F» in honour of Sir Ronald Fisher, a British statistician who developed it in the 1920s.

What is the F-distribution?

The F-distribution, also known as the Fisher–Snedecor distribution, is a fundamental tool for analysing variability in data sets.

Imagine two data sets, each with its own characteristics and behaviour. The F-distribution allows us to compare the variability of these data sets – that is, how widely the values are spread around their mean.  

Mathematically, it is defined as the distribution of the ratio of two variables random variables, both with chi-squared distributions, scaled by their respective degrees of freedom.

What is the F-distribution used for?

Its main function lies in hypothesis testing, specifically the Snedecor F-test, which is used to determine whether the variability in one set of data is significantly greater than that in another.  

This test is widely used in various fields, such as:

• Analysis of variance (ANOVA): It allows you to compare the variability between groups in an experiment, determining whether the observed differences are attributable to random factors or to intrinsic characteristics of the groups.
• Linear regression: It assesses the significance of the regression coefficients in a model, determining whether these coefficients have a real impact on the dependent variable.
• Design of experiments: It helps to select the appropriate sample size for an experiment, ensuring that there is sufficient statistical power to detect significant differences.

The F-value in statistics

The F-value, which is the result of the Snedecor F-test, is expressed as F = Variance 1 / Variance 2, where Variance 1 and Variance 2 correspond to the variances of the two data sets being compared.

• High F values: They point out that the variability in the first dataset is greater than that in the second, which could suggest the existence of significant differences between the two.
• Low F values: They suggest that the variability in both data sets is similar, which does not indicate any significant differences.

The F-score in machine learning

In machine learning, the F-distribution also plays an important role in the selection of models. Algorithms such as linear regression or feature selection use the F-score to assess the relevance of predictor variables, discarding those that do not provide significant information to the model.

Practical examples of the F-distribution

Let’s imagine a study comparing the academic performance of two groups of students: one that was taught using an innovative teaching method and another that followed the traditional method.  

The F-distribution can be used to determine whether the innovative method had a significantly positive impact on pupils’ performance.

Another example could be a machine learning model that predicts house prices. In this case, the F-distribution could be used to select the most relevant features – such as size, location or the number of bedrooms – in order to improve the model’s accuracy.