F-Statistic Calculator Formula
Understand the math behind the f-statistic calculator. Each variable explained with a worked example.
Formulas Used
F-Statistic
f_stat = s1_sq / s2_sqdf1 (numerator)
df1 = n1 - 1df2 (denominator)
df2 = n2 - 1SD Ratio
variance_ratio = sqrt(s1_sq / s2_sq)Variables
| Variable | Description | Default |
|---|---|---|
s1_sq | Sample Variance 1 (larger) | 25 |
s2_sq | Sample Variance 2 (smaller) | 16 |
n1 | Sample Size 1 | 20 |
n2 | Sample Size 2 | 25 |
How It Works
How to Calculate the F-Statistic
Formula
F = s1^2 / s2^2
The F-statistic is the ratio of two sample variances, with the larger variance in the numerator by convention. Under the null hypothesis of equal population variances, F follows an F-distribution with df1 = n1-1 and df2 = n2-1. F-values near 1 suggest equal variances.
Worked Example
Group 1: variance = 25, n = 20. Group 2: variance = 16, n = 25.
- 01F = 25 / 16 = 1.5625
- 02df1 = 20 - 1 = 19
- 03df2 = 25 - 1 = 24
- 04SD ratio = sqrt(1.5625) = 1.25
- 05Compare F = 1.5625 to F-distribution(19, 24)
Frequently Asked Questions
Why put the larger variance in the numerator?
By convention, this ensures F >= 1 and simplifies looking up critical values. Some procedures test both tails, in which case the order matters less.
What does F = 1 mean?
F = 1 means the two sample variances are identical, providing no evidence that the population variances differ.
How is the F-test used in ANOVA?
In ANOVA, the F-statistic is the ratio of between-group variance to within-group variance. A large F indicates that group means differ more than expected from random variation alone.
Ready to run the numbers?
Open F-Statistic Calculator