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.
where, Fcalc = Critical F-value. σ12 & σ22 = variance of the two samples.
where, df = Degrees of freedom of the sample. nS = Sample size.
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 : 1 2 1 2 3 : : :
α = 0.05
d1 = 2
d2 = 3
d2 /d1 . . 161.4 199.5 . . 18.51 19.00 . . 10.13 9.55 . . . .
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: σ 10.47 8.12 n 41 21
Sample 1 Sample 2
- σ12 = (10.47)2 = 109.63
- σ22 = (8.12)2 = 65.99
- H0: no difference in variances.
- Ha: difference in variances.
Fcalc = (109.63 / 65.99) = 1.66
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.
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