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F-Test

  • Last Updated : 26 Nov, 2020

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

$F_{c a l c}=\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}$
where, 
Fcalc = Critical F-value.
σ12 & σ22 = variance of the two samples.

$d f=n_{S}-1$



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 Fcalc, 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

. .

1

161.4199.5. .

2

18.5119.00. .

3

10.139.55. .

:



:

:

. .

Then, Ftable = 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 1Sample 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.
  • Ha: difference in variances.

Step 3:
Fcalc = (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. 

Attention reader! Don’t stop learning now. Get hold of all the important Machine Learning Concepts with the Machine Learning Foundation Course at a student-friendly price and become industry ready.

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