F Statistic Calculator

Variance of Group 1:

Variance of Group 2:

Sample Size of Group 1:

Sample Size of Group 2:

Significance Level (α):

In statistics, the F-statistic is crucial for comparing variances between two groups and forms the basis for tests like the F-test and ANOVA. It measures how much the variability between groups differs relative to the variability within groups. This helps decide whether group variances differ significantly.

Manually calculating the F-statistic, critical value, and p-value can be complex without proper tools. Our F Statistic Calculator simplifies this by taking group variances and sample sizes as inputs and outputs the F-statistic, degrees of freedom, critical value at a chosen significance level, approximate p-value, and a clear decision about the null hypothesis.

This tool is perfect for students, researchers, and anyone conducting variance tests or hypothesis testing.

How to Use the F Statistic Calculator

Step 1: Enter Variance of Each Group

Input the sample variance for Group 1 and Group 2.
Both must be positive numbers.

Step 2: Enter Sample Sizes

Enter the sample size for Group 1 and Group 2.
Each must be an integer greater than or equal to 2.

Step 3: Select Significance Level (α)

Choose the significance level from the dropdown:
- 0.01 (99% confidence)
- 0.025 (97.5% confidence)
- 0.05 (95% confidence) — default
- 0.10 (90% confidence)

Step 4: Calculate

Click the Calculate button to view:

F-Statistic calculated as variance1 ÷ variance2.
Degrees of Freedom for numerator and denominator (sample size – 1).
Critical F Value at the selected significance level.
Approximate P-Value for the test.
Decision on null hypothesis whether variances differ significantly.

Example Usage

Suppose you have two groups:

Variance of Group 1 = 25.0
Variance of Group 2 = 10.0
Sample size Group 1 = 15
Sample size Group 2 = 18
Significance level α = 0.05

After entering these values and clicking Calculate, the calculator shows:

F-Statistic = 2.50
Degrees of Freedom: df1 = 14, df2 = 17
Critical F Value ≈ 2.55
P-Value ≈ 0.0530
Decision: Fail to reject null hypothesis (variances are not significantly different)

Since the F-statistic is less than the critical value and the p-value is greater than α, we conclude the variances are not significantly different at 95% confidence.

How This Calculator Works (Briefly)

The calculator computes the F-statistic by dividing the first variance by the second. Degrees of freedom are derived from sample sizes minus one. It then uses advanced mathematical functions (incomplete beta function, logarithmic gamma) to compute:

The critical F value based on significance level and degrees of freedom.
The cumulative distribution function (CDF) for the F-statistic to approximate the p-value.

Using these, it guides you whether to reject or fail to reject the null hypothesis.

Tips for Accurate Use

Ensure variances are positive numbers (variances can’t be zero or negative).
Sample sizes must be integers ≥ 2.
Pick the correct α level based on your confidence requirements.
Remember the F-statistic is variance1 ÷ variance2 — you can reverse groups if needed but interpret results accordingly.
Use the p-value and decision to understand the significance of your test.

15 Frequently Asked Questions (FAQs)

1. What is the F-statistic?
It is the ratio of two variances used to test if they are significantly different.

2. Why do we need degrees of freedom?
Degrees of freedom account for sample size and affect the shape of the F-distribution.

3. How is the F-statistic calculated?
By dividing the variance of Group 1 by the variance of Group 2.

4. What does the critical F value represent?
It’s the cutoff point from the F-distribution that determines statistical significance.

5. What is the p-value?
The probability of obtaining the observed F-statistic or more extreme under the null hypothesis.

6. How do I interpret the decision?
If the F-statistic > critical value, reject the null hypothesis; otherwise, fail to reject it.

7. Can variances be zero?
No, variances must be positive.

8. What happens if sample sizes are small?
Smaller samples affect degrees of freedom and the shape of the F-distribution, which influences critical values.

9. Is this test one-tailed or two-tailed?
This calculator assumes a right-tailed test typical for F-tests.

10. What if I switch the groups in variance1 and variance2?
The F-statistic and interpretation will change because the ratio is order-dependent.

11. Can I use this for ANOVA?
ANOVA uses F-statistics but typically compares multiple groups. This calculator is for two-group variance comparison.

12. Why approximate p-values?
Exact p-values require complex calculations; this tool uses numerical methods for a close approximation.

13. What is significance level α?
It’s the threshold probability for rejecting the null hypothesis (commonly 0.05).

14. What does “fail to reject null hypothesis” mean?
It means there is not enough evidence to say variances differ significantly.

15. Can I use this calculator offline?
You need a browser to run the JavaScript, so it can work offline if saved locally.

Conclusion

The F Statistic Calculator is an essential tool for quickly computing the F-statistic, critical value, and approximate p-value for comparing variances between two groups. It simplifies hypothesis testing and aids in making clear decisions, ideal for students, teachers, researchers, and analysts.

Try it now to streamline your statistical analysis with ease and confidence!