F-Test in Statistics

F test is a statistical test that is used in hypothesis testing that determines whether the variances of two samples are equal or not. The article will provide detailed information on f test, f statistic, its calculation, critical value and how to use it to test hypotheses. To understand F test firstly we need to have some basic understanding of F-distribution.

F-distribution

The F-distribution is a continuous statistical distribution used to test whether two samples have the same variance. The F-Distribution has two parameters the numerator degrees of freedom (df1) and the denominator degrees of freedom (df2). Formula for F-distribution:

\text{f-value} =\frac{sample 1/df 1}{sample 2/df 2}

• The independent random variables Samples 1 and 2, have a chi-square distribution.
• The related samples' degrees of freedom are denoted by df1 and df2.

Understanding F-Test

In F test the data follows an F distribution. This test uses the F statistic to compare two variances by dividing them. An F test can either be one-tailed or two-tailed depending upon the parameters of the problem. The F value obtained after conducting an F test is used to perform the one-way ANOVA (analysis of variance) test. We can use this test when:

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

Hypothesis Testing Framework for F-test

For various hypothesis tests the F test formula is provided as follows:

1. Left Tailed Test:

Null Hypothesis: H₀ : \sigma_{1}^2 = \sigma_{2}^2
Alternate Hypothesis: H₁ : \sigma_{1}^2 < \sigma_{2}^2
Decision-Making Standard: The null hypothesis is to be rejected if the F statistic is less than the F critical value.

2. Right Tailed Test:

Null Hypothesis: H₀ : \sigma_{1}^2 = \sigma_{2}^2
Alternate Hypothesis: H₁ : \sigma_{1}^2 > \sigma_{2}^2
Decision-Making Standard: Dismiss the null hypothesis if the F test statistic is greater than the F test critical value.

3. Two Tailed Test:

Null Hypothesis: H₀ : \sigma_{1}^2 = \sigma_{2}^2
Alternate Hypothesis: H₁ : \sigma_{1}^2 \neq \sigma_{2}^2
Decision-Making Standard: When the F test statistic surpasses the F test critical value the null hypothesis is declared invalid.

F Test Statistics

The F test statistic or simply the F statistic is a value that is compared with the critical value to check if the null hypothesis should be rejected or not. The F test statistic formula is given below:

F statistic for large samples: F_{calc}=\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}} where \sigma_{1}^{2}  is the variance of the first population and \sigma_{2}^{2} is the variance of the second population.

F statistic for small samples: F_{calc}=\frac{s_{1}^{2}}{s_{2}^{2}} where s_{1}^{2} is the variance of the first sample and s_{2}^{2} is the variance of the second sample.

Steps to calculate F-Test

Step 1: Use Standard deviation (σ1) and find variance (σ2) of the data. (if not already given)

Step 2: Determine the null and alternate hypothesis.

•   H0: no difference in variances.
•   H1: difference in variances.

Step 3: Find F_calc using Equation 1 (F-value).

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 F_table value using d1 and d2 obtained in Step-4 from the F-distribution table. Take learning rate, α = 0.05 (if not given) 

Looking up the F-distribution table: 

In the F-Distribution 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

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 for calculating F-Test

Consider the following example In this we conduct a two-tailed F-Test on the following samples: 

Step 1: The statement of the hypothesis is formatted as:

• H₀: no difference in variances.
• H₁: difference in variances.

Step 2: Let's calculated the value of the variances in numerator and denominator as F-value= \frac{\sigma^2_{1}}{\sigma^2_{2}}

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

F_calc = (109.63 / 65.99) =  1.66

Step 3: Now, let's calculate the degree of freedom: Degree of freedom = sample - 1 Here we have Sample 1 = n₁ = 41 and
Sample 2 = n₂ = 21

Degree of sample 1 = d₁ = (n₁ - 1) = (41 – 1) = 40

Degree of sample 2 = d₂ = (n₂ — 1) = (21 – 1) = 20

Step 4: The usual alpha level of 0.05 is selected because the question does not specify an alpha level. The alpha level should be lowered during the test to half of its starting value. Using d₁ = 40 and d₂ = 20 in the F-Distribution table. (link here) and Take α = 0.05 as it's not given. Since it is a two-tailed F-test then:
α = 0.05/2 = 0.025

Step 5: The critical F value is found with alpha at 0.025 using the F table. For (40, 20), the critical value at alpha equal to 0.025 is 2.287. Therefore, F_table = 2.287

Step 6: Since F_calc < F_table (1.66 < 2.287):
We cannot reject null hypothesis.
∴ Variance of two populations is 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.