F-Test Formula

F-Test Formula (Table of Contents)

• Formula
• Examples

What is F-Test Formula?

F-test is a statistical test which helps us in finding whether two population sets which have a normal distribution of their data points have the same standard deviation or variances. But the first and foremost thing to perform F-test is that the data sets should have a normal distribution. This is applied to F distribution under the null hypothesis. F-test is a very crucial part of the Analysis of Variance (ANOVA) and is calculated by taking ratios of two variances of two different data sets. As we know that variances give us the information about the dispersion of the data points. F-test is also used in various tests like regression analysis, the Chow test, etc.

Formula FOR F-Test:

There is no simple formula for F-Test but it is a series of steps which we need to follow:

Step 1: To perform an F-Test, first we have to define the null hypothesis and alternative hypothesis. These are given by:-

• H0 (Null Hypothesis): Variance of 1^st data set = Variance of a 2^nd data set
• Ha: Variance of 1^st data set < Variance of 2^nd data set (for a lower one-tailed test)
• Ha: Variance of 1^st data set > Variance of a 2^nd data set (for an upper one-tailed test)
• Ha: Variance of 1^st data set ≠ Variance of a 2^nd data set (for a two-tailed test)

Step 2: Next thing we have to do is that we need to find out the level of significance and then determine the degrees of freedom of both the numerator and denominator. This helps us in determining their critical values. Degree of freedom is sample size -1.

Step 3: F-Test Formula:

Step 4: Find the F critical value from F table taking a degree of freedom and level of significance.

Step 5: Compare these two values and if a critical value is smaller than the F value, you can reject the null hypothesis.

Examples of F-Test Formula (With Excel Template)

Let’s take an example to understand the calculation of F-Test in a better manner.

F-Test Formula – Example #1

Let’s say we have two data sets A & B which contains different data points. Perform F-Test to determine whether we can reject the null hypothesis at a 1% level of significance.

Data Sets:

Solution:

Null Hypothesis: Variance of A = Variance of B

Degree of Freedom is calculated as

Degree of Freedom

• For A = 10 – 1 = 9
• For B = 20 – 1 =19

Variation is calculated as:

• Variance of A = 1385.61
• Variance of B = 521.22

F Value is calculated using the formula given below

F Value = Variance of 1^st Data Set / Variance of 2^nd Data Set

• F Value = 1385.61 / 521.22
• F Value = 2.6584

F-Table:

So F critical value = 3.5225

Since F critical is greater than the F value, we cannot reject the null hypothesis.

F-Test Formula – Example #2

Suppose that you are working in a research company and want to the level of carbon oxide emission happening from 2 different brands of cigarettes and whether they are significantly different or not. In your analysis, you have collected the following information:

Solution:

Degree of Freedom is calculated as

Degree of Freedom

• For XYZ = 11 – 1 = 10
• For ABC = 10 – 1= 9

Variation is calculated as:

• Variance of XYZ = 1.2^2 = 1.44
• Variance of ABC= 1.1^2 = 1.21

• F Value = 1.44 / 1.21
• F Value = 1.19

F Critical Value = 3.137

Since the F critical > F value, the null hypothesis cannot be rejected.

Explanation

In the examples above, we have seen the application of F-Test and how it is performed. But there is a set of assumption we need to take care before performing F-Test otherwise we will not get required results:

• First thing is that we need to always place the higher variance value numerator while calculating the F value. So if F = V1 / V2, V1 should be > V2
• If we want to perform 2 tail test, we need to divide the level of significance by 2 and that will the correct level to find the critical value
• We only use variance is the F value calculation and if we are given with standard deviations, as in example 2, they must be squared to find the variance.
• Both the samples should be independent of each other and sample size should be less than 30
• Population sets out of which the samples are drawn out must be normally distributed

These are the key parameters/assumption which should be taken care of while performing F-Test.

Relevance and Use of F-Test Formula

F-Test, as discussed above, helps us to check for the equality of the two population variances. So when we have two independent samples which are drawn from a normal population and we want to check whether or not they have the same variability, we use F-test. F-test also has great relevance in regression analysis and also for testing the significance of R². So in a nutshell, F-Test is a very important tool in statistics if we want to compare the variation of 2 or more data sets. But one should keep all the assumptions in mind before performing this test.

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