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CI for Ratio of Variances: Relative Spread

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The Formal Theorem

Let X1,…,Xn1∼N(μ1,σ12) X_1, \dots, X_{n_1} \sim N(\mu_1, \sigma_1^2) and Y1,…,Yn2∼N(μ2,σ22) Y_1, \dots, Y_{n_2} \sim N(\mu_2, \sigma_2^2) be independent random samples. Let S12 S_1^2 and S22 S_2^2 be the respective unbiased sample variances. The ratio F=S12/σ12S22/σ22 F = \frac{S_1^2 / \sigma_1^2}{S_2^2 / \sigma_2^2} follows an F-distribution with (n1−1,n2−1) (n_1 - 1, n_2 - 1) degrees of freedom. A 100(1−α)% 100(1-\alpha)\% confidence interval for the ratio of variances σ12σ22 \frac{\sigma_1^2}{\sigma_2^2} is given by:
[S12S22⋅1Fα/2,n1−1,n2−1,S12S22⋅Fα/2,n2−1,n1−1] \left[ \frac{S_1^2}{S_2^2} \cdot \frac{1}{F_{\alpha/2, n_1-1, n_2-1}}, \quad \frac{S_1^2}{S_2^2} \cdot F_{\alpha/2, n_2-1, n_1-1} \right]

Analytical Intuition.

Imagine two separate manufacturing lines, each pulsating with its own internal 'rhythm' of variation, defined by σ12 \sigma_1^2 and σ22 \sigma_2^2 . We aren't interested in the absolute magnitude of their spread, but rather the 'relative spread'—a dimensionless ratio that tells us which process is more volatile. To capture this, we deploy the Fisher-Snedecor F-distribution, the mathematical bridge between two independent chi-squared landscapes. By taking the ratio of our sample variances S12/S22 S_1^2/S_2^2 , we construct a test statistic that pivots on the true ratio σ12/σ22 \sigma_1^2/\sigma_2^2 . If the interval contains 1, the evidence is insufficient to suggest that the variances differ; if the interval sits entirely above 1, we have definitive proof that the first process is significantly noisier. This interval acts as a 'zoom lens,' sharpening our focus on the inherent instability ratio, allowing us to quantify precision drift across disparate experimental environments with rigorous probabilistic confidence.
CAUTION

Institutional Warning.

Students frequently invert the F-critical values, confusing the degrees of freedom ordering (n1−1,n2−1) (n_1-1, n_2-1) versus (n2−1,n1−1) (n_2-1, n_1-1) . Always remember: the numerator degrees of freedom of the statistic must match the sample size of the numerator variance.

Academic Inquiries.

01

Why is this interval only valid for Normal distributions?

The derivation relies explicitly on the Cochran's Theorem property that (n−1)S2/σ2∼χn−12 (n-1)S^2/\sigma^2 \sim \chi_{n-1}^2 , which holds strictly for Normal data. Non-normal data causes this test to be extremely sensitive to kurtosis.

02

How does this relate to the F-test for equality of variances?

The confidence interval provides a bilateral test; if the value '1' is excluded from the interval at level α \alpha , the null hypothesis H0:σ12=σ22 H_0: \sigma_1^2 = \sigma_2^2 is rejected at the same significance level.

Standardized References.

  • Definitive Institutional SourceCasella, G., & Berger, R. L., Statistical Inference.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). CI for Ratio of Variances: Relative Spread: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/statistical-inference-i/ci-for-ratio-of-variances--relative-spread

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