# 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.
Ïƒ12 & Ïƒ22 = variance of the two samples.


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

Steps involved:

Step 1: Use Standard deviation (Ïƒ) and find variance (Ïƒ2) 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, Î± = 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 Î± 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 :
Î± = 0.05
d1 = 2
d2 = 3

Then, Ftable = 9.55

Step 6: Interpret the results using Fcalc and Ftable.

Interpreting the results:

If Fcalc < Ftable :
Cannot reject null hypothesis.
âˆ´ Variance of two populations are similar.

If Fcalc > Ftable :
Reject null hypothesis.
âˆ´ 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:

Step 1:

• Ïƒ12 = (10.47)2 = 109.63
• Ïƒ22 = (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 â€“ 1) = 40
d2 = (n2 â€” 1) = (21 â€“ 1) = 20

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

Step 6 - Since Fcalc < Ftable (1.66 < 2.287):
We cannot reject null hypothesis.
âˆ´ 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.

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