123 LearnersLast updated on July 22nd, 2025
F-Test
In statistics, the F-test is used to compare the variances of two samples when evaluating hypotheses. This test checks whether the data come from populations with equal variability. The F-test is used in quality checks, education, finance, and many other sectors to compare variations. In this article, we will explore the F-test and its applications in detail.
What is the F-test in Math?
In mathematics, the F-test is a statistical test used to compare two variances. We use this test to determine if the difference between the variances is statistically significant. The criteria for performing an F-test are that the data should come from a population that follows an F-distribution, and the samples compared should be independent of each other. When conducting the hypothesis test, if the findings of the F-test are statistically significant, the null hypothesis can be rejected; otherwise, it cannot.
The F-tests possess certain properties that make it unique:
- F tests are based on the ratio of two sample variances and degrees of freedom:
F = S12/ S22. (F ≥ 1).
Where:
S12 is the variance of sample 1
S22 is the variance of sample 2
- The F-value is based on the data, that is it can be greater than, less than, or equal to 1.
- Since variances cannot be negative, the F-value will always be a positive number.
- The test assumes that the populations being compared are normally distributed. If the assumption is not met by the test, the results of the test may be inaccurate.
- F-tests are primarily one-tailed, as they check whether one variance is greater than the other. However, they can also be two-tailed in some cases.
What is the Formula for the F-test?
The F-test is a hypothesis testing tool used to verify if variances are equal. The F-test formula used in various hypothesis tests
Left-Tailed Test:
Null Hypothesis: H0: σ21= σ22
Alternate Hypothesis: : H1 : σ21 < σ22
Decision Criteria: If the F-statistic is less than the critical value, then reject the null hypothesis
Right-Tailed test:
Null Hypothesis: H0: σ21= σ22
Alternate Hypothesis: H1 : σ21 > σ22
Decision Criteria: Reject H0 if the calculated F-value is greater than the right critical value.
Two-Tailed test:
Null Hypothesis: H0: σ21= σ22
Alternate Hypothesis: H1 : σ21 ≠ σ22
Decision Criteria: If the F-test statistic > f test critical value, then the null hypothesis is rejected.
F-test statistic
The F-test statistic for larger samples is mathematically represented as: F = σ21/ σ22
Where:
σ21= variance of the first population
σ22 = variance of the second population
For small samples:
F = S21/ S22
Where:
S21= variance of the first sample
S22= variance of the second sample
F-Test vs. T-Test
| F - Tests | T - Tests |
| To assess the equality of variances between two populations | To check the difference in means between two groups |
| Use the F distribution table | Use the students t-distribution table |
| It is used primarily for variances | Used in testing the means |
| Formula used: F = S21/ S22 |
|
Common Mistakes and How to Avoid Them in F-Test
The F test is an important hypothesis test in statistics. However, students often face some difficulties in understanding the concept. We will now look at a few common mistakes and their solutions:
Mistake 1
Confusion between Variances and Means
One common mistake is that students confuse the variances with the means.
For example, If the mean of dataset X is 6 and dataset Y is 5.5, and their variances are 7 and 6, they wrongly compare the means instead of the variances. Always keep in mind that F-tests are applied for variances and not for means. So, variances should be compared even before the application of the formula.
Mistake 2
Overlooking the F-distribution Table
Sometimes they assume the final step is to find the F-value and may forget to compare the F-value to the critical value. After finding the F-value, ensure that you compare it with the critical value from the F-distribution table.
For example, if the value we calculate for f is 3.1 and the critical value is 3.5, we fail to reject
Mistake 3
Incorrect Division
Some students mistakenly divide the smaller variance by the larger variance, which may lead to wrong conclusions when comparing with the critical value. Ensure you divide the larger variance by the smaller variance to determine the F value.
For example, if variances S21= 15 and S22=3, then always calculate F = Larger Variance / Smaller Variance
= 15/ 3 = 5
Mistake 4
Incorrect Hypothesis Application
There is an incorrect assumption among some students that the null hypothesis means the variances are different. Always remember that in an F-test, the null hypothesis considers the two variances to be the same. For example,
H₀: σ12 = σ22 (The variances are equal)
H₁: σ12 ≠ σ22 (The variances are not equal)
Mistake 5
Not Checking the Sample Independence
Without understanding the criteria, students may use related samples for determining the F test. Always use independent samples for the F-test. If the samples are related, a paired t-test should be used instead.
For example, if two types of test results of the same students are the samples, then use a paired t-test. If the results are from students of two different schools, use the F-test.
Real-life applications of the F-test
F-tests are extensively used in various fields to compare variances, such as in finance, research, and more. Here are a few real-life applications of the F-test:
- F-tests are utilized in schools to compare the consistency or variation in student performance scores across different schools.
- Athletes use these tests to assess the effectiveness of different training methods. For example, when two groups use different training routines, the F-test helps us identify which one yields more reliable results.
- These tests can help investors in comparing the variability of returns between two different investment plans.
- F-tests are used for quality-checking purposes in the product manufacturing field. For example, this test determines whether two machines that create the same product are equally consistent.
- Doctors can check the effectiveness of different drugs or medicines by comparing their variances. For example, if two medicines are proven to cure an allergy, F-tests help in identifying the more effective one.
Solved Examples of F-Test
Problem 1
A researcher wants to compare the variability in test scores of two different classes. Class A: Variance = 20, Sample Size = 15 Class B: Variance = 10, Sample Size = 12 Test at the 5% significance level whether the variances are significantly different.
There is no significant difference between the variances.
Explanation
Hypotheses:
- H0 : σ12 = σ22 (The variances are equal)
- H1 : σ12 ≠ σ22
F-ratio (larger variance / smaller variance):
F = S21/ S22 = 20/10 = 2
Degrees of Freedom:
- df1 = n1 −1 = 15 −1=14
- df2 = n2 −1=12−1=11
Critical Value:
Using F-table at α = 0.05, df1 = 14, df2 = 11
Critical value F0.025,14,11 ≈ 3.29
Decision:
Since 2 < 3.29, we fail to reject the null hypothesis.
Problem 2
The variances of pizza delivery times in two different cities are being compared. City 1: Sample size = 28, Variance = 38 City 2: Sample size = 25, Variance = 83 At a significance level of 0.05, test whether the variance in delivery times for City 1 is significantly less than that of City 2.
There is sufficient evidence at the 5% significance level to conclude that the delivery times in City 1 have significantly lower variance compared to City 2.
So we reject the null hypothesis.
Explanation
Set the Hypotheses
- Null hypothesis H0: σ21 = σ22
- Alternative hypothesis H1: σ21< σ22
This is a left-tailed F-test for comparing variances.
Identify Given Values
- n1 = 28, S12 = 38 (City 1)
- n2 = 25, S22 = 83 (City 2)
Degrees of freedom:
- df1= n1−1=27
- df2= n2 −1=24
Compute the F-statistic
F= S21/ S22 = 38/83 ≈ 0.4578
Determine the Critical Value
Since this is a left-tailed test at α=0.05, we compute:
F0.95,27,24 = 1/ F0.05,24,27 = 1/1.93 ≈ 0.5181
Step 5: Decision
Compare the calculated F-value with the critical value:
0.4578< 0.5181
Since the F-statistic falls in the rejection region, we reject the null hypothesis.
Problem 3
A tech company is testing the battery life consistency of two different brands of wireless earbuds. From Brand X, 36 samples were taken, and the variance in battery life was found to be 95 hours². From Brand Y, 24 samples were taken, and the variance was 60 hours². At a 0.05 significance level, test whether there is a significant difference in the variances of battery life between the two brands.
Fail to reject the null hypothesis.
Explanation
Set the Hypotheses
- H0 : σ21=σ22
- H1: σ12 ≠ σ22 (two-tailed test)
Identify Given Values
- n1=36, s12 = 95(Brand X)
- n2=24, s22 = 60 (Brand Y)
- Degrees of freedom:
df1=35, df2 =23
Compute the F-statistic
F = S21/ S22 = 95/60 ≈1.5833
Determine the Critical Value
This is a two-tailed test, so divide α=0.05α=0.05 into two tails:
α/2 = 0.025
From the F-table:
F0.025,35,23 ≈ 2.24(approx)
Also check the lower bound:
F0.975,35,23 = 1/F0.025,23,35 ≈ 1/2.30 ≈ 0.4348
Decision
0.4348< F =1.5833< 2.24
Since the F-value falls within the acceptance region, we fail to reject H0.
There is not enough evidence at the 5% level to conclude that the variances of battery life between Brand X and Brand Y are significantly different.
FAQs on the F Test
1.What is an F test?
2.Can we use the F-test for small samples?
3.What is the formula for calculating the F test?
4.What are the common applications of the F Test?
5.What do you mean by the critical value in the F Test?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
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