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Confidence Interval for Variance Calculator

Calculate the confidence interval for a population variance and standard deviation from sample data.

Result

CI for Variance
39.018 to 124.001
CI for Standard Deviation
6.246 to 11.136

This interval uses the chi-square distribution (via the Wilson-Hilferty approximation) and, unlike a confidence interval for the mean, is not symmetric around the sample variance.

About the CI for Variance

The Confidence Interval for Variance Calculator estimates a plausible range for a population's true variance and standard deviation, based on a single sample's standard deviation and size. Enter the sample standard deviation, sample size, and confidence level, and it returns intervals for both variance and standard deviation. This is useful whenever the goal is to understand how much uncertainty surrounds a measured spread, not just a measured average.

How It Works

The calculator converts the sample standard deviation into variance by squaring it, then uses the chi-square distribution, approximated via the Wilson-Hilferty method, to find two cutoff values tied to the sample's degrees of freedom and the chosen confidence level. Dividing degrees of freedom times the sample variance by each of those cutoffs produces the lower and upper bounds for the population variance; taking the square root of each bound converts the interval into standard deviation units.

df = n - 1; variance = sd²; lower bound = (df × variance) / χ²(1 - α/2, df); upper bound = (df × variance) / χ²(α/2, df); where α = 1 - confidence level, and χ² values come from the chi-square distribution's inverse CDF (Wilson-Hilferty approximation).

Formula & Methodology

With a sample of size n, degrees of freedom equal n minus 1. Square the sample standard deviation to get the sample variance. Find the chi-square value that cuts off the top α/2 of the distribution at those degrees of freedom, used in the lower bound's denominator, and the value that cuts off the bottom α/2, used in the upper bound's denominator - the upper-tail cutoff produces the lower variance bound, and vice versa, because chi-square appears in the denominator. Multiply degrees of freedom by the sample variance and divide by each cutoff to get the two variance bounds, then take square roots for the standard deviation bounds.

Examples

Manufacturing tolerance check

Sample standard deviation of 8 from a sample of 25 units, at 95% confidence. The calculator returns a variance interval of roughly 39.0 to 124.0 and a standard deviation interval of roughly 6.25 to 11.14, noticeably wider on the upper side than the lower side.

Small pilot sample

Sample standard deviation of 5 from a sample of just 10 units, at 90% confidence. The variance interval widens to roughly 13.3 to 67.9 (standard deviation interval about 3.65 to 8.24), showing how much less precise a variance estimate becomes with a small sample.

Advantages

  • Extends confidence interval logic beyond the mean to variability itself, which mean-only intervals don't cover.
  • Automatically applies the correct asymmetric chi-square-based bounds instead of an inappropriate symmetric approximation.
  • Returns both variance and standard deviation intervals in one calculation, since practitioners often think in standard deviation units.

Common Mistakes

  • Assuming the interval is symmetric around the sample variance the way a mean's confidence interval is, and misreading the asymmetric result.
  • Using a small, non-random sample and treating the resulting wide interval as a tight, reliable estimate of population variability.
  • Confusing the standard deviation interval bounds with a simple square root of the variance point estimate, rather than the separately computed bounds.

Edge Cases to Watch For

  • Sample standard deviation must be positive and sample size must exceed 1 so degrees of freedom is at least 1; otherwise the calculator returns an error.
  • Unlike the confidence interval for a mean, this interval is not symmetric around the sample variance - the chi-square distribution is skewed, especially at small sample sizes, so the upper bound sits proportionally farther from the point estimate than the lower bound.
  • The chi-square quantiles come from the Wilson-Hilferty approximation rather than an exact chi-square table lookup, which is very accurate for typical degrees of freedom but remains an approximation.

Common Use Cases

  • Manufacturing and quality engineers assessing whether measured process variability is precisely estimated or still uncertain given sample size.
  • Analysts reporting uncertainty around a volatility or spread estimate, not just the estimate itself.
  • Students and researchers verifying chi-square-based interval calculations by hand against a reference tool.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

Why isn't this confidence interval symmetric like the one for a mean?

The chi-square distribution, which underlies variance estimation, is skewed rather than symmetric - especially with smaller sample sizes. That skew carries through to the confidence interval, which is why the upper and lower bounds aren't equally spaced from the sample variance.

Related calculators

Confidence Interval CalculatorCalculate the confidence interval for a sample mean.Sample Size CalculatorCalculate the minimum sample size needed for a survey or study at a given confidence level and margin of error.Margin of Error CalculatorCalculate the margin of error for a survey result given sample size and confidence level.Correlation Coefficient CalculatorCalculate the Pearson correlation coefficient between two paired sets of numbers.

Conclusion

The Confidence Interval for Variance Calculator makes clear that variability estimates carry their own uncertainty, distinct from and often less precise than an equivalent mean estimate. Because the interval is asymmetric, it should be reported as computed rather than approximated as evenly spread around the sample variance.