Confidence Interval for Ratio of Two Variances (F-Distribution)
Use this calculator for confidence intervals comparing population variability/consistency between two groups using the F-distribution (ratio of variances).
When to Use
- Comparing population variances between two groups
- Testing homogeneity of variance assumption
- Quality control: comparing consistency between machines/processes
- Variability comparison: which process is more stable?
How to Use
Step 1: Paste first group data (comma-separated values)
Step 2: Paste second group data (comma-separated values)
Step 3: Select confidence level (typically 95%)
Step 4: Click “Calculate”
| Confidence interval for Variances | ||
|---|---|---|
| Sample 1 | Sample 2 | |
| Enter Data (Separated by comma ,) | ||
| Confidence Level ($1-\alpha$) | ||
| Results | ||
| Sample sd: | ||
| Degrees of Freedoms: | ||
| F Critical values : | ||
| Confidence Limit | ||
Formula
$$CI = \left[\frac{s_1^2}{s_2^2} \times \frac{1}{F_{\alpha/2, n_1-1, n_2-1}}, \frac{s_1^2}{s_2^2} \times F_{\alpha/2, n_2-1, n_1-1}\right]$$
Where:
- $s_1^2, s_2^2$ = sample variances
- $F_{\alpha/2}$ = F-critical values
Interpreting Results
If 95% CI includes 1:
- Cannot conclude variances differ
- Variances are likely equal
If 95% CI doesn’t include 1:
- Variances likely differ significantly
- One group has greater variability
Worked Example
Scenario: Comparing precision of two machines. Machine A (n=15): s₁²=25. Machine B (n=15): s₂²=16.
Solution:
- Ratio: s₁²/s₂² = 25/16 = 1.5625
- With df=(14,14) and α=0.05
- CI ≈ [0.54, 4.54]
Interpretation: We’re 95% confident Machine A has 0.54 to 4.54 times the variance of Machine B. Since this includes 1, we cannot definitively say variances differ.
Related: Levene’s Test, Tutorial