Confidence Interval for Ratio of Variances

Basic Concepts

Property 1: If we draw two independent samples of size n₁ and n₂ with sample variances s₁² and s₂² from two normal populations with corresponding population variances σ₁² and σ₂², then

where F_+crit is the right critical value of F(n₂-1, n₁-1) and  F_–crit is the left critical value; i.e. F_+crit = F.INV.RT(α/2, n₂-1, n₁-1) and F_–crit = F.INV(α/2, n₂-1, n₁-1).

Proof: By Property 7 of Chi-square Distribution

By Definition 1 of F Distribution

from which it follows that

or equivalently

which means that

where F′_+crit is the right critical value of F(n₁-1, n₂-1) and  F′_–crit is the left critical value. This completes the proof.

Conclusion

Since, F.INV(α, df₁, df₂) = 1/F.INV.RT(α, df₂, df₁), we can express the confidence interval by

We can now use this property to calculate the confidence interval for the ratio of variances.

Example

Example 1: Calculate a 95% confidence interval for the ratio of (population) variances based on the data from Example 1 of Two Sample Hypothesis Testing of the Variance.

The calculation of the confidence interval is shown in Figure 1.

Figure 1 – Confidence interval for variance ratio

A ratio of 1 means that the two population variances are equal. Since 1 is in the confidence interval (.597, 6.204), once again we have a non-significant result; i.e. we cannot conclude that the two population variances are different.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

PennState (2023) Confidence intervals for variances. Introduction to Mathematical Statistics
https://online.stat.psu.edu/stat415/book/export/html/810

Srivatsan, R. (2020) Confidence interval for the ratio of population variances. CountBio
http://www.countbio.com/web_pages/left_object/R_for_biology/R_biostatistics_part-1/confidence_interval_variance_ratio.html