F-Test

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, 
Fcalc = Critical F-value.
σ12 & σ22 = variance of the two samples.

where,
df = Degrees of freedom of the sample. 
nS = Sample size.

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 calculating F_calc, divide the larger variance with small variance as it makes calculations easier.

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
d₁ = 2
d₂ = 3

d2 /d1	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:

• σ₁² = (10.47)² = 109.63 
• σ₂² = (8.12)² = 65.99

Step 2:

• H₀: no difference in variances.
• H_a: difference in variances.

Step 3:
F_calc = (109.63 / 65.99) =  1.66

Step 4:
d₁ = (n₁ – 1) = (41 – 1) = 40
d₂ = (n₂ — 1) = (21 – 1) = 20

Step 5 - Using d1 = 40 and d2 = 20 in the F-Distribution table. (link here)
      Take α = 0.05 as it's not given.
      Since it is a two-tailed F-test, 
      α = 0.05/2 
       = 0.025     
     Therefore, Ftable = 2.287

Step 6 - Since Fcalc < Ftable (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.