Confidence Intervals for Variance — Chi-Square Interval

Core Concepts

Confidence intervals for variance use the chi-square distribution. Unlike intervals for the mean, these intervals are asymmetric — the lower and upper bounds are not equidistant from the point estimate.

Confidence Interval for Standard Deviation

Derivation from the Sampling Distribution

Proof sketch: Standardize each observation: $Z_i = (X_i - \\mu)/\sigma \\sim \mathcal{N}(0,1)$. The sum of squared standard normals is $\sum Z_i^2 \\sim \\chi^2_n$. Decompose using Cochran's theorem: $\sum Z_i^2 = \sum(X_i - \bar{X})^2/\sigma^2 + (\bar{X} - \\mu)^2 n/\sigma^2$. The first term on the right is $(n-1)S^2/\sigma^2$ and the second is $\\chi^2_1$. By independence (since $\bar{X}$ is sufficient for $\\mu$ and $S^2$ is sufficient for $\sigma^2$ in the normal family), the first term is $\\chi^2_{n-1}$.

Worked Example: Quality Control

A quality control engineer measures the diameter of 25 ball bearings. The sample variance is $s^2 = 0.0036\\ \text{mm}^2$. Construct a 95% CI for $\sigma^2$.

Step 1: Identify parameters: $n = 25$, $s^2 = 0.0036$, $\alpha = 0.05$, $df = 24$.

Step 2: Find chi-square critical values:

Step 3: Compute the interval:

Step 4: For standard deviation, take square roots:

Sensitivity to Non-Normality

Proof sketch: For a non-normal population with kurtosis $\\kappa_4$, the statistic $(n-1)S^2/\sigma^2$ no longer follows exactly $\\chi^2_{n-1}$. A Cornish-Fisher expansion shows the leading correction is proportional to $\\kappa_4 / n$. For heavy-tailed distributions (e.g., $t_5$ with $\\kappa_4 = 6$), the true coverage can be 90% when 95% is nominal.

Python Implementation: Bootstrap Comparison

import numpy as np
from scipy import stats

np.random.seed(42)
n = 25
sigma_true = 1.0
data = np.random.normal(loc=0.0, scale=sigma_true, size=n)
s2 = np.var(data, ddof=1)

# Chi-square CI (parametric)
chi2_low = stats.chi2.ppf(0.975, df=n-1)
chi2_high = stats.chi2.ppf(0.025, df=n-1)
ci_parametric = [(n-1)*s2 / chi2_low, (n-1)*s2 / chi2_high]
print(f"Parametric CI for σ²: [{ci_parametric[0]:.4f}, {ci_parametric[1]:.4f}]")

# Bootstrap CI (non-parametric)
B = 10000
boot_vars = np.array([np.var(np.random.choice(data, size=n, replace=True), ddof=1)
                      for _ in range(B)])
ci_bootstrap = np.percentile(boot_vars, [2.5, 97.5])
print(f"Bootstrap CI for σ²:  [{ci_bootstrap[0]:.4f}, {ci_bootstrap[1]:.4f}]")

# Compare coverage (repeat 1000 times)
coverage_param = 0
coverage_boot = 0
M = 1000
for _ in range(M):
    sample = np.random.normal(0, sigma_true, n)
    sv = np.var(sample, ddof=1)
    chi2_lo = stats.chi2.ppf(0.975, n-1)
    chi2_hi = stats.chi2.ppf(0.025, n-1)
    lo_p, hi_p = (n-1)*sv/chi2_lo, (n-1)*sv/chi2_hi
    if lo_p <= sigma_true**2 <= hi_p:
        coverage_param += 1
    boot_v = np.array([np.var(np.random.choice(sample, n, replace=True), ddof=1)
                       for _ in range(1000)])
    lo_b, hi_b = np.percentile(boot_v, [2.5, 97.5])
    if lo_b <= sigma_true**2 <= hi_b:
        coverage_boot += 1
print(f"Parametric coverage: {coverage_param/M:.3f}")
print(f"Bootstrap coverage:  {coverage_boot/M:.3f}")

Key Takeaways