**The Fisher-Snedecor test makes it possible to compare the variance of 2 samples.**

## Introduction

Created in 1920 by the work of Ronald Aylmer Fisher (British biologist and mathematician) and based on the distribution table of George Waddel Snedecor (American Statistician), the Fisher Snedecor test is a parametric approach to study the equality of variances of 2 samples.

This variance test is a good alternative to the Chi2 test when the samples are small (less than 30 individuals).

Note that this test is not robust and that is very sensitive to the respect of the normality of the data. So, we will prefer the Brown Forsythe test.

**Detail on Fisher’s Law**

The principle of this law is based on the representation of the division of variable variances, in other words, to divide a Chi2 law by another Chi2 law.

It calculates the ratio between a **Signal **and** Noise **(causing the second name of the Fischer Snedecor’s law: the signal-to-noise ratio). Hence, it is used for variance comparisons (

**Fisher test…).**

**ANOVA**,Fisher’s test compares the Variance of 2 subpopulations. He confronts the two following hypotheses :

**H0 : ****σ _{1}^{2} = σ_{2}^{2}**

**H1 : σ _{1}^{2} ≠ σ_{2}^{2}**

## Step 2 : Practical Value

The practical value is based on the ratio of the 2 variances that are compared. We consider that if this ratio is very different from 1, then the Variances are different. We will simply note that S1 > S2, because the ratio must necessarily be > 1.

Two cases of calculation is presented.

### We know the Variance of the population

In this case, the simplest but also the most rare in practice, we simply perform the report Variances.

### We do not know the Variance of the population

The most common case in practice, we do our calculations on the basis of the sample we did, but without knowing the Variance of the population*.

*We estimate them punctually via the formula :

## Step 3 : Critical value

The critical value follows a Fisher-Snedecor law, and is calculated as follows :

- dof
_{1}= Size of the sample with the largest variance – 1 - dof
_{2}= Size of the sample with the smallest variance – 1 - Risk level: This depends on the direction of the test. For a bilateral test, we will have 1-α/2 and α/2, for a right unilateral test, we will have 1 – α and for a unilateral left α.

The value is identified either with the Fisher table or with the Excel spreadsheet via the formula : FINV.(probability; n_{1} – 1; n_{2} – 1)

## Step 4 : p-Value

The p-Value follows a Fisher-Snedecor law and is calculated under excel via the formula :

**FDIST.(Practical value ; dof _{1}; dof_{2})**

## Step 5 : Interpretation

Test direction | Result | Statistical conclusion | Practical Conclusion |
---|---|---|---|

Bilateral | Practical value >Critical value α/2 and Practical value < Critical value 1 - α/2 | We reject H0 | The 2 samples are different at the given level of risk α. |

Unilateral right | Practical value > Critical value 1 - α | We reject H0 | The sample with the greatest variability shows a significant difference in the given level of risk α. |

Unilateral left | Practical value < Critical value α | We reject H0 | The sample with the smallest variability shows a significant difference in the given level of risk. |

Result | Statistical Conclusion | Practical Conclusion |
---|---|---|

p-value > α | We retain H0 | Our 2 data series are identical or similar with a risk of being wrong with p-value% |

p-Value < α | We reject H0 | Our 2 series of data are statistically different with a risk of being wrong with p-value% |

## Source

F. Dress (2007) – Les probabilités et la statistique

A. Grous (2013) – Eléments d’analyse de la fiabilité et du contrôle qualité

M. E. Tremblay, P. Avallée, M. El Haj Tirari (2011) – Pratiques et méthodes de sondage