F Test Calculator for Equality of Variances

How to Use the F Test Calculator

This F test calculator helps you test whether the variances of two independent populations are significantly different. The calculator accepts both raw or summary data.

To use the calculator:

1. Select the alternative hypothesis that matches your research question.
2. Enter the significance level, α.
3. Choose Summary Data or Raw Data.
4. Enter the required information for Sample 1 and Sample 2.
5. Click Calculate.

The calculator will instantly return the test statistic, p value, critical value, and tell you whether the equality of variance assumption is met if the test is two-tailed. It will also provide a clear, step-by-step solution showing how the test was performed using the six steps of hypothesis testing.

Tips.

• Use the Summary Data option if you know the sample standard deviation/variance and sample size for each group
• Select the Raw Data option whenever you have the individual observations for both groups. For raw data, you can copy and paste values directly from Excel, Google Sheets, or text documents. You can also manually enter the values separated by commas, spaces, tabs, or line breaks.
• The two datasets do not need to contain the same number of observations. However, each sample must contain at least two values and have a variance greater than zero.

What Is an F Test for Equality of Variances?

An F test for equality of variances is a hypothesis test used to determine whether two independent populations have the same variance.

The null hypothesis states that the two population variances are equal. In symbols, we want to test the null hypothesis: H₀: σ₁² = σ₂²

The alternative hypothesis can be either of these:

• Two-tailed Test: H₁: σ₁² ≠ σ₂². This is the alternative hypothesis when you’re testing for equality of variance assumption to help you determine whether to use a Welch’s t test or a two-sample t-test (assuming equal variances).
• Right-tailed Test: H₁: σ₁² > σ₂²
• Left-tailed Test: H₁: σ₁² < σ₂²

This F-test for equality of variance test is also known as:

• Two-sample F test
• F test for two variances
• Variance ratio test
• F test for homogeneity of variance
• F test for equality of two population variances

Using the F Test to Check the Equal-Variance Assumption

The pooled two sample t test calculator assumes that the two populations have equal variances. This F test calculator can help assess whether the sample provides evidence against this assumption.

For a two-tailed test:

• If p ≤ α, we reject the null hypothesis (H₀) and conclude that the the population variances differ. In other words, the equality of variance assumption is not met.
• If p > α, we fail to reject the null hypothesis (H₀). There is insufficient evidence that the population variances differ. In other words, the equality of variance assumption is met.

Tip. A non-significant result does not prove that the variances are exactly equal. It only means the test has not detected a statistically significant difference between them and the equality of variance assumption can be assumed.

Note. If the equality of variance assumption is not met, the Welch’s t test calculator is generally more appropriate for comparing two population means.

F Test Assumptions

The following assumptions must be met before using an F-test for equality of variances:

• The two samples must be independent
• Observations within each sample should be independent
• Both populations should be approximately normally distributed.
• The dependent variable should be measured on a continuous scale.

Tip. If the normality assumption is not met, Levene’s test or the Brown–Forsythe test is usually more robust than the classical variance-ratio test.

F Test (Variance-ratio Test) Formula

The F test for equality of variance formula is: F = s₁²/s₂²

Where:

• F is the variance-ratio test statistic
• s₁² is the variance of Sample 1
• s₂² is the variance of Sample 2

Note. The F-test for equality of variance is also known as the Variance-ratio test.

The degrees of freedom formula for this test are:

• df₁ = n₁-1
• df₂ = n₂-1

Where:

• df₁ is the numerator degrees of freedom
• df₂ is the denominator degrees of freedom
• n₁ is the size of Sample 1
• n₂ is the size of Sample 2

The calculator always places Sample 1 in the numerator and Sample 2 in the denominator. Therefore, an F statistic smaller than 1 is possible and completely valid.

Important. You should not automatically move the larger variance to the numerator when using a directional test because doing so changes the direction of the hypothesis.

How to Perform an F Test

A complete F test follows these six hypothesis steps:

1. State the null and alternative hypotheses.
2. State the significance level.
3. Calculate the F statistic and degrees of freedom.
4. Find the p-value and critical value.
5. Make the statistical decision.
6. Write the conclusion in the context of the question.

The examples below show how to apply these steps using summary statistics and raw data.

Example 1. Summary Data

A healthcare data analyst at an urban medical center is evaluating operational efficiencies under two separate, independent outpatient scheduling systems. Prior to conducting a pooled-variance two-sample t-test to compare the mean patient wait times, the analyst must verify the assumption of homoscedasticity.

Summary metrics extracted from the electronic health record (EHR) database are provided in the table below:

Assuming both underlying populations are normally distributed, evaluate whether there is a statistically significant difference between the two population variances at the 5% significance level.

Solution

Because the analyst is interested in a difference in either direction, this is a two-tailed F test. To perform the test manually, follow these steps:

Step 1: State the Hypotheses

The hypotheses are:

• H₀: σ₁² = σ₂²
• H₁: σ₁² ≠ σ₂²

Step 2: State the Significance Level

From the question, we want test the hypothesis at 5% significance level. Thus, the significance level is α = 0.05

Step 3: Calculate the test Statistic

By definition, the F-test statistic formula is: F = s₁²/s₂²

Substituting the values into the formula gives:

F = 8.4²/6.9²

=70.56/47.61

=1.4820

Step 4: Find the P-Value and Critical Values

Before finding the p-value, we need to compute the correct degrees of freedom.

The numerator degrees of freedom, df₁ = n₁-1

= 22-1

= 21

The denominator degrees of freedom, df₂ = n₂-1

=20-1

=19

Next, we can use the p-value from F calculator to find the two-tailed p-value with F = 1.4820, df₁ = 21, and df₂ = 19:. This gives: p-value = 0.392739

To find the two-tailed critical value for the test, we use the F-table and look up for F_0.05/2, 21, 19 = 2.493 . Alternatively, we can use the F-critical value calculator to get similar results.

Step 5: Make the Decision

P value approach: Since the p-value (0.392739) is greater than the 0.05 significance level, we fail to reject the null hypothesis.

Critical value Approach: Since the test statistic (1.4820) is less than the critical value (2.493), we fail to reject the null hypothesis.

Step 6: Write the Conclusion

At the 0.05 significance level, there is insufficient evidence to conclude that the two population variances are different. Therefore, the equal-variance assumption is met. Therefore, the analyst may consider using the pooled two sample t test rather than the Welch’s t test.

Verify Example 1 Using the Calculator

To verify the result using the F-test calculator, follow these steps:

1. Select H₁: σ₁² ≠ σ₂².
2. Enter α = 0.05.
3. Select Summary Data.
4. Enter s₁ = 8.4 and n₁ = 22.
5. Enter s₂ = 6.9 and n₂ = 20.
6. Click Calculate.

The calculator will return approximately:

• F statistic: F(21, 19) = 1.4820
• P-value: 0.392739
• Critical values: 0.4094 and 2.4930
• Decision: Fail to reject H₀

Example 2: Raw Data

A quality-control analyst wants to compare the variability of measurements produced by two independent, manufacturing machines. A random sample of measurements from each machine yields the following data:

• Machine 1: 42, 55, 49, 61, 37, 58, 46, 63, 40, 54, 35, 60
• Machine 2: 48, 50, 52, 49, 51, 53, 47, 50, 52, 49, 51, 48

Assuming the data from both populations are approximately normally distributed, which of the following is the most appropriate test statistic to determine if the population variances are different at the α = 0.05 significance level?

Solution

To perform the test by hand, follow these steps:

Step 1: Hypotheses

Because the analyst wants to detect a difference in either direction, this is a two-tailed test.

Therefore, the hypotheses are:

• H₀: σ₁² = σ₂²
• H₁: σ₁² ≠ σ₂²

Step 2: Significance Level

We want to test the hypothesis at 0.05 significance level. Hence, α = 0.05

Step 3: Test statistic

Since have the raw data in this case, we can either calculate the sample standard deviation or variance for each group and manually count the number of observations in each of these groups.

Using the variance calculator, we get:

• The sample variance of Group 1, S₁²= 97.2727
• The sample variance of Group 2, S₂²= 3.4545

Alternatively, you can use the standard deviation calculator and later square the values to get the variance.

Recall. The test statistic formula for the variance ratio test is: F = s₁²/s₂²

Substituting the values into the formula gives:

F = 97.2727/3.4545

=28.1583

Step 4: P-Value and Critical Values

To find the p-value and critical values, we need the degrees of freedom. From the question, we know that the two groups have the same number of observations. That is, n₁ = n₂ = 12.

Therefore:

• The numerator degrees of freedom, df₁ = 12-1 = 11
• The denominator degrees of freedom, df₂ = 12-1 = 11

Now, using the p value from F statistic calculator, the two-tailed p-value for the test is 0.000004.

To find the critical value for the test, you can either use an F distribution table and look up for F_0.05/2, 11, 11 or use the critical F value calculator when alpha = 0.025. The critical value for the test will be 3.4737

Step 5: Decision

P-value method. Since the p-value (0.000004) is less than the 0.05 significance level, we reject the null hypothesis

Critical value method. Since the test statistic (28.1583) is greater than the critical value (3.4737), we reject the null hypothesis.

Step 6: Conclusion

At the 0.05 significance level, there is sufficient evidence to conclude that the two population variances are different.

The homogeneity-of-variance assumption is therefore not supported. When comparing the two population means, Welch’s t test would generally be more appropriate than the pooled independent samples t test.

Verify Example 2 Using the Calculator

To verify the result:

1. Select H₁: σ₁² ≠ σ₂².
2. Enter α = 0.05.
3. Select Raw Data.
4. Enter the Machine 1 observations in the Sample 1 field.
5. Enter the Machine 2 observations in the Sample 2 field.
6. Click Calculate.

The calculator will return approximately:

• F statistic: F(11, 11) = 28.1579
• P-value: 0.000004
• Critical values: 0.2879 and 3.4737
• Decision: Reject H₀

Frequently Asked Questions