F-Test is any test that utilizes the F-Distribution table to fulfil its purpose (for eg: ANOVA). It compares the ratio of the variances of two populations and determines if they are statistically similar or not.

**We can use this test when :**

- The population is normally distributed.
- The samples are taken at random and are independent samples.

**Formulas Used**

where,FCritical F-value._{calc}=σvariance of the two samples._{1}^{2}& σ_{2}^{2}=

where,df =Degrees of freedom of the sample.nSample size._{S}=

**Steps involved: **

**Step 1:** Use Standard deviation (σ) and find variance (σ2) of the data. (if not already given)

**Step 2:** Determine the null and alternate hypothesis.

- H0 -> no difference in variances.
- Ha -> difference in variances.

**Step 3:** Find Fcalc using Eq-1.

NOTE :While calculatingF, divide the larger variance with small variance as it makes calculations easier._{calc}

**Step 4:** Find the degrees of freedom of the two samples.

**Step 5:** Find Ftable value using d1 and d2 obtained in Step-4 from the F-distribution table. (link here). Take learning rate, α = 0.05 (if not given)

**Looking up the F-distribution table: **

In the F-Distribution table (Link here), refer the table as per the given value of α in the question.

- d1 (Across) = df of the sample with numerator variance. (larger)
- d2 (Below) = df of the sample with denominator variance. (smaller)

Consider the F-Distribution table given below,

**While performing One-Tailed F-Test.**

**GIVEN : **

α = 0.05

d_{1 }= 2

d_{2 }= 3

d_{2} /d_{1} | 1 | 2 | . . |
---|---|---|---|

1 | 161.4 | 199.5 | . . |

2 | 18.51 | 19.00 | . . |

3 | 10.13 | 9.55 | . . |

: | : | : | . . |

Then, **F _{table} = 9.55**

**Step 6:** Interpret the results using *Fcalc* and *Ftable*.

**Interpreting the results:**

If Fcalc < Ftable :Cannot reject null hypothesis. ∴ Variance of two populations are similar.If Fcalc > Ftable :Reject null hypothesis. ∴ Variance of two populations are not similar.

**Example Problem (Step by Step)**

Consider the following example,

Conduct a two-tailed F-Test on the following samples:

Sample 1 | Sample 2 | |
---|---|---|

σ | 10.47 | 8.12 |

n | 41 | 21 |

**Step 1:**

**σ**(10.47)_{1}^{2}=^{2}= 109.63**σ**(8.12)_{2}^{2}=^{2}= 65.99

**Step 2:**

**H**no difference in variances._{0}:**H**difference in variances._{a}:

**Step 3:****F _{calc} = **(109.63 / 65.99) =

**1.66**

**Step 4:****d _{1} = (n_{1} – 1) =** (41 – 1) = 40

**d**= (21 – 1) = 20

_{2}= (n_{2}— 1)Step 5 -Usingdand_{1}= 40din the F-Distribution table. (link here) Take_{2}= 20α = 0.05as it's not given. Since it is a two-tailed F-test,α = 0.05/2= 0.025Therefore,F_{table}= 2.287

Step 6 -Since F_{calc}< F_{table}(1.66 < 2.287): We cannot reject null hypothesis. ∴ Variance of two populations are similar to each other.

F-Test is the most often used when comparing statistical models that have been fitted to a data set to identify the model that best fits the population. Researchers usually use it when they want to test whether two independent samples have been drawn from a normal population with the same variability. For any doubt/query, comment below.