Last updated on July 4th, 2025
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.
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:
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 - 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 |
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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:
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:
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.
Hypotheses:
F-ratio (larger variance / smaller variance):
F = S21/ S22 = 20/10 = 2
Degrees of Freedom:
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.
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.
Set the Hypotheses
This is a left-tailed F-test for comparing variances.
Identify Given Values
Degrees of freedom:
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.
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.
Set the Hypotheses
Identify Given Values
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.
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
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!