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CI for Population Variance: Measuring Dispersion

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The Formal Theorem

Let X1,X2,…,Xn X_1, X_2, \dots, X_n be a random sample from a normal distribution N(μ,σ2) N(\mu, \sigma^2) . Let S2=1n−1∑i=1n(Xi−Xˉ)2 S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 be the sample variance. The statistic (n−1)S2σ2 \frac{(n-1)S^2}{\sigma^2} follows a chi-squared distribution with n−1 n-1 degrees of freedom, denoted by χn−12 \chi^2_{n-1} . For a confidence level of 1−α 1 - \alpha , the (1−α) (1 - \alpha) confidence interval for σ2 \sigma^2 is given by:
[(n−1)S2χα/2,n−12,(n−1)S2χ1−α/2,n−12] \left[ \frac{(n-1)S^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n-1)S^2}{\chi^2_{1-\alpha/2, n-1}} \right]

Analytical Intuition.

Imagine you are calibrating the precision of a high-end lens manufacturing line. You don't just care about the average focal length (the mean); you care deeply about the 'spread' or consistency—the variance. If your machine drifts, the product quality collapses. We treat the sample variance S2 S^2 as our best estimate, but we know it's a volatile snapshot. To quantify our uncertainty, we look at the distribution of the sum of squared deviations. Because squares of normal variables aggregate into the Chi-squared distribution, we create a 'trap' for the true variance σ2 \sigma^2 . We select two boundary values from the χn−12 \chi^2_{n-1} distribution—the lower tail and the upper tail—to capture the true population variance with 1−α 1 - \alpha confidence. Unlike the symmetric intervals we use for means, this interval is inherently asymmetric because the Chi-squared distribution itself is skewed, reflecting the physical reality that it is easier to underestimate variance than to overestimate it in small samples.
CAUTION

Institutional Warning.

Students often struggle because the interval is asymmetric and the critical values are ordered 'flipped' (the larger Chi-squared value creates the smaller denominator for the lower bound). Always remember that dividing by a larger number yields a smaller result; thus, the larger quantile belongs in the lower bound.

Academic Inquiries.

01

Why is the Chi-squared distribution used instead of the Z or t-distributions?

The Z and t-distributions describe the sampling distribution of the mean. Because variance involves squared deviations from the mean, its sampling distribution follows the Chi-squared distribution by definition of the sum of squared standard normal variables.

02

Is this confidence interval robust to non-normal data?

No. The Chi-squared distribution relies heavily on the normality assumption. If the population distribution has heavy tails (kurtosis), the interval will be highly unreliable.

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 Population Variance: Measuring Dispersion: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/statistical-inference-i/ci-for-population-variance--measuring-dispersion

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