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**Formula of F-Test (Table of Contents)**

## What is the F-Test Formula?

F-Test is any test that uses F-distribution. F value is a value on the F distribution. Various statistical tests generate an F value. The value can be used to determine whether the test is statistically significant. In order to compare two variances, one has to calculate the ratio of the two variances, which is as under:

**F Value = Larger Sample Variance / Smaller Sample Variance = σ**

_{1}^{2 }/ σ_{2}^{2}While carrying out F-Test, we need to frame the null and alternate hypothesis. Then, we need to determine the level of significance under which the test has to be carried out. Subsequently, we have to find out the degrees of freedom of both the numerator and denominator. This will help determine the F table value. The F Value seen in the table is then compared to the calculated F Value to determine whether or not to reject the null hypothesis.

### Explanation of the F-Test Formula

Below are the steps where the F-Test formula is used to test the null hypothesis that the variances of two populations are equal:

**Step 1:** Firstly, frame the null and alternate hypothesis. The null hypothesis assumes that the variances are equal. **H _{0}: σ_{1}^{2 }= σ_{2}^{2}**. The alternate hypothesis states that the variances are unequal.

**H**. Here σ

_{1}: σ_{1}^{2}_{ ≠ }σ_{2}^{2}_{1}

^{2}and σ

_{2}

^{2}are the symbols for variances.

**Step 2:** Calculate the test statistic (F distribution).

F = σ_{1}^{2 } / σ_{2}^{2}

Where σ_{1}^{2 } is assumed to be larger sample variance and σ_{2}^{2 }is smaller sample variance

**Step 3:** Calculate the degrees of freedom. Degree of freedom (df_{1}) = n_{1 }– 1 and Degree of freedom (df_{2}) = n_{2 }– 1 where n_{1 }and n_{2 }are the sample sizes

**Step 4:** Look at the F value in the F table. For 2 tailed tests, divide the alpha by 2 for finding the right critical value. Thus, the F value is found looking at the degrees of freedom in the numerator and the denominator in the F table. Df_{1} is read across in the top row. Df_{2 }is read down the first column.

* Note:* There are different F Tables for different levels of significance. Above is the F table for alpha = .050.

**Step 5:** Compare the F statistic obtained in Step 2 with the critical value obtained in Step 4. If the F statistic is greater than the critical value at the required level of significance, we reject the null hypothesis. If the F statistic obtained in Step 2 is less than the critical value at the required level of significance, we cannot reject the null hypothesis.

### Examples of F-Test Formula

Let’s see some simple to advanced examples of F-Test Formula to understand it better.

#### F-Test Formula – Example #1

**A statistician was carrying out F-Test. He got the F statistic as 2.38. The degrees of freedom obtained by him were 8 and 3. Find out the F value from the F Table and determine whether we can reject the null hypothesis at 5% level of significance (one-tailed test).**

Solution:

We have to look for 8 and 3 degrees of freedom in the F Table. The F critical value obtained from the table is **8.845**. Since the F statistic (2.38) is lesser than the F Table Value (8.845), we cannot reject the null hypothesis.

#### F-Test Formula – Example #2

**An insurance company sells health insurance and motor insurance policies. Premiums are paid by customers for these policies. The CEO of the insurance company wonders if premiums paid by either of insurance segments (health insurance and motor insurance) are more variable as compared to another. He finds the following data for premiums paid:**

**Conduct a two-tailed F-test with a level of significance of 10%.**

4.9 (927 ratings)

Solution:

**Step 1:** Null Hypothesis H_{0}: σ_{1}^{2 }= σ_{2}^{2}

Alternate Hypothesis H_{a}: σ_{1}^{2}_{ ≠ } σ_{2}^{2 }

**Step 2:** F statistic = F Value = σ_{1}^{2 } / σ_{2}^{2 }= 200/50 = **4**

**Step 3: **df_{1} = n_{1 }– 1 = 11-1 =10

df_{2 = } n_{2 }– 1 = 51-1 = 50

**Step 4:** Since it is a two-tailed test, alpha level = 0.10/2 = 0.050. The F value from the F Table with degrees of freedom as 10 and 50 is 2.026.

**Step 5: **Since F statistic (4) is more than the table value obtained (2.026), we reject the null hypothesis.

#### F-Test Formula – Example #3

**A bank has a Head Office in Delhi and a branch at Mumbai. There are long customer queues at one office, while customer queues are short at the other office. The Operations Manager of the bank wonders if the customers at one branch are more variable than the number of customers at another branch. A research study of customers is carried out by him.**

**The variance of Delhi Head Office customers is 31 and that for Mumbai branch is 20. The sample size for Delhi Head Office is 11 and that for Mumbai branch is 21. Carry out a two-tailed F-test with a level of significance of 10%.**

Solution:

** Step 1:** Null Hypothesis H_{0}: σ_{1}^{2 }= σ_{2}^{2}

Alternate Hypothesis H_{a}: σ_{1}^{2}_{ ≠ } σ_{2}^{2}

**Step 2:** F statistic = F Value = σ_{1}^{2 } / σ_{2}^{2 }= 31/20 = **1.55**

**Step 3: **df_{1} = n_{1 }– 1 = 11-1 = 10

df_{2 = } n_{2 }– 1 = 21-1 = 20

**Step 4: **Since it is a two-tailed test, alpha level = 0.10/2 = 0.05. The F value from the F Table with degrees of freedom as 10 and 20 is 2.348.

**Step 5:** Since F statistic (1.55) is lesser than the table value obtained (2.348), we cannot reject the null hypothesis.

### Relevance and Uses

F-Test can be used in a wide variety of settings. F-Test is used to test the hypothesis that the variances of two populations are equal. Secondly, it is used for testing the hypothesis that the means of given populations which are normally distributed, having the same standard deviation, are equal. Thirdly, it is used to test the hypothesis that a proposed regression model fits the data well.

### F-Test Formula in Excel (with Excel Template)

Workers in an organization are paid daily wages. The CEO of the organization is concerned about the variability in wages between males and females in the organization. Below is the data are taken from a sample of males and females.

**Conduct a one-tailed F test at 5% level of significance.**

Solution:

**Step 1:** H_{0}: σ_{1}^{2 }= σ_{2}^{2}, H_{1}: σ_{1}^{2}_{ ≠ } σ_{2}^{2}

**Step 2:** Click on Data Tab > Data Analysis in Excel.

**Step 3:** The below-mentioned window will appear. Select F-Test Two-Sample for Variances and then click on OK.

**Step 4:** Click on the Variable 1 range box and select the range A2: A8. Click on the Variable 2 range box and select the range B2: B7. Click A10 in the output range. Select 0.05 as alpha as a level of significance is 5%. Then click on OK.

The values for F statistic and F table value will be displayed along with other data.

**Step 4:** From the above table we can see F statistic (8.296) is greater than F critical one-tail (4.95), so we will reject the null hypothesis.

**Note 1:** Variance of variable 1 has to be higher than the variance of variable 2. Otherwise, calculations made by Excel will be wrong. If not, then swap the data.

**Note 2:** If the Data analysis button is not available in Excel, go to File > Options. Under Add-ins, select Analysis ToolPak and click on the Go button. Check Analysis Tool Pack and click on OK.

**Note 3:** There is a formula in Excel to calculate the F table value. Its syntax is:

You can download this F Test Formula Excel Template from here – F-Test Formula Excel Template

### Recommended Articles

This has been a guide to F-Test Formula. Here we discuss how to perform F-Test to determine whether or not to reject the null hypothesis along with examples and downloadable excel template. You can learn more about statistical modeling from the following articles –

- Examples of ISNULL Function in VBA
- Sample Size Formula
- Hypergeometric Distribution Formula | Definition
- Median Formula Statistics
- T-TEST Excel
- P-Value Calculation
- Weibull Distribution Excel

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