Sample size calculator for the variance (CI)

Explanation

The confidence interval for the variance \(\sigma^2\) uses the chi-square distribution and has the form:

\( \left[\frac{(n-1)s^2}{\chi^2_{1-\alpha/2,\,n-1}},\;\; \frac{(n-1)s^2}{\chi^2_{\alpha/2,\,n-1}}\right] \)

Unlike CIs for means or proportions, the absolute width of the CI for the variance depends on the unknown value of \(\sigma^2\), which makes it impossible to express precision in absolute units without knowing it beforehand. The natural solution is to specify precision in relative terms: the U/L ratio between the upper and lower limits of the CI.

This ratio depends only on the quantiles of the chi-square distribution, and therefore only on the sample size \(n\) and the confidence level. No assumption about \(\sigma^2\) is needed.

Formula for the U/L ratio

\( \frac{U}{L} = \frac{\chi^2_{1-\alpha/2,\,n-1}}{\chi^2_{\alpha/2,\,n-1}} \)

For a given confidence level and a desired ratio \(R\), the calculator searches for the smallest \(n\) such that \(\chi^2_{1-\alpha/2,\,n-1}/\chi^2_{\alpha/2,\,n-1} \leq R\). There is no closed-form formula: the result is obtained by an iterative search over \(n\).

• R close to 1: very narrow CI → enormous sample.
• R = 2: the upper limit doubles the lower limit → reasonable sample (≈ 65 observations at 95%).
• R = 3: the upper limit triples the lower limit → much smaller sample (≈ 28 observations at 95%).

Interpreting the U/L ratio

A ratio \(R = 2.5\) means the upper limit of the CI can be up to 2.5 times larger than the lower limit. For example, if \(s^2 = 4.0\), the CI could go from 2.5 to 6.25 (a variability of 2.5 to 1 between the extremes). The smaller \(R\) is, the narrower and more informative the CI, but the more observations are needed.

Quick setup

• Ratio R: R between 2 and 4 is common in quality control and process studies. R very close to 1 requires prohibitively large samples.
• Confidence level: 95% is the standard. With 99% the required n is significantly larger for the same R.
• Normality: make sure the data are compatible with normality; the chi-square CI is very sensitive to this assumption.
• Expected losses: divide n by (1 − dropout rate) to get the number of subjects to recruit.

Worked example

A metrology laboratory wants to estimate the variance of a measurement instrument's error to validate its calibration. It is established that the 95% CI must have a U/L ratio no greater than \(R = 2.5\) (the upper limit cannot be more than 2.5 times the lower limit).

The calculator searches for the smallest \(n\) such that \(\chi^2_{0.975,\,n-1} / \chi^2_{0.025,\,n-1} \leq 2.5\). For \(n = 39\) (df = 38): \(\chi^2_{0.975,38} \approx 56.90\) and \(\chi^2_{0.025,38} \approx 22.88\), ratio \(\approx 2.49 \leq 2.5\). For \(n = 38\): ratio \(\approx 2.52 > 2.5\). Result: \(n = 39\) observations.

With those 39 measurements, if \(s = 0.12\) mm (variance \(s^2 = 0.0144\) mm²), the 95% CI would be approximately... more precisely, using the exact quantiles: \([(38 \times 0.0144)/56.90,\; (38 \times 0.0144)/22.88] = [0.0096,\; 0.0239]\) mm². The actual U/L ratio is \(0.0239/0.0096 \approx 2.49\).

Sensitivity analysis: if the laboratory accepts \(R = 3\) (the upper limit can triple the lower limit), the sample shrinks to \(n = 28\). With \(R = 2\) the sample rises to \(n \approx 65\). The choice of \(R\) has a very large impact on the required n.

Model assumptions

• The variable of interest follows a normal distribution. The chi-square CI for the variance is very sensitive to non-normality, more so than any CI for the mean.
• Observations are independent of each other.
• The target U/L ratio must be greater than 1; the iterative search starts at \(n = 2\).

Common uses

• Statistical process control (SPC): estimating the process variance for control charts.
• Validation of measurement instruments and reproducibility studies.
• Design of experiments where variability is the main variable of interest.
• Auditing of production processes where dispersion needs to be characterized.

How to interpret the result

The value \(n\) guarantees that the confidence interval for \(\sigma^2\) based on the chi-square distribution will have a ratio between the upper and lower limits no greater than the specified value \(R\), at the chosen confidence level. This ensures relative precision: the upper limit of the CI will not exceed \(R\) times the lower limit. Always round up. If you expect losses or exclusions during data collection, divide \(n\) by \((1 - \text{dropout rate})\); with a 10% dropout rate, recruit \(\lceil n / 0.90 \rceil\) units.

The most critical and delicate assumption of this method is the normality of the data: the chi-square CI for \(\sigma^2\) is very sensitive to deviations from normality, much more so than the CI for the mean. With skewed or heavy-tailed distributions, the actual CI can have coverage well below the nominal level (e.g., a 95% CI may cover the true \(\sigma^2\) only 85% of the time). Before applying this result, check normality with the Shapiro-Wilk test or with Q-Q plots. Sensitivity to \(R\): requiring a ratio \(R = 2\) (narrow CI) requires many more subjects than \(R = 4\) (wide CI); increasing \(R\) by 50% can cut \(n\) in half.

If the calculated \(n\) is not feasible, the alternatives are: (1) accept a larger ratio \(R\) (wider CI), (2) reduce the confidence level (e.g., from 99% to 95%), or (3) if the data are not normal, use bootstrap methods to build the CI for \(\sigma^2\) without relying on the chi-square distribution. Once the data are collected, build the actual CI with the CI calculator for the variance; if you also want to test \(\sigma^2\) against a reference value, use the sample size calculator for a one-variance test.

References and further reading

• Zar, J. H. (2010). Biostatistical Analysis (5th ed.). Pearson. — sample size calculation for variance.
• Montgomery, D. C. (2019). Introduction to Statistical Quality Control (8th ed.). Wiley. — applications in SPC.
• Wikipedia (en): Chi-squared distribution — properties and percentiles.

Frequently asked questions

• Why do I need such a large sample for a narrow variance CI? Because the chi-square distribution converges slowly to symmetry. With few degrees of freedom, the extreme quantiles are far apart, producing very asymmetric and wide CIs.
• What happens if the data aren't normal? The chi-square CI can have actual coverage very different from the nominal confidence level. First apply a normality test (Shapiro-Wilk) and, if the data aren't normal, consider transformations or bootstrap methods.
• Can I use this result if I want to estimate σ instead of σ²? Yes. The CI for σ is obtained by taking the square root of each limit of the σ² CI, and the U/L ratio is the same.