Confidence Interval for Variance Calculator

It uses the chi-square distribution and the sample degrees of freedom, so it is appropriate for independent observations drawn from an approximately normal population. Enter the variance computed with an n minus one denominator, not a population variance computed with n. The result includes both interval limits and the critical values used in the calculation.

Enter the three quantities consistently

Start with the sample variance, the number of observations, and a confidence level written as a decimal. The sample variance must be the usual unbiased estimate whose sum of squared deviations is divided by n minus one. If your software labels alternatives as sample and population variance, choose sample variance. Size means the count of independent observations used to obtain that variance, not the degrees of freedom; the calculator subtracts one automatically. A 95 percent confidence level is entered as 0.95, while 0.90 and 0.99 produce 90 percent and 99 percent intervals. Variance cannot be negative, although zero is mathematically accepted and produces an interval whose endpoints are both zero. A sample must contain at least two observations because a one-observation sample has no usable sample variance or positive degrees of freedom. Keep the measurement scale in mind: if observations are measured in centimeters, the variance and both returned bounds are in square centimeters. The calculator does not convert units or accept a standard deviation in place of variance; square a sample standard deviation first if that is the statistic you have.

Understand the chi-square calculation

For normally distributed independent observations, the quantity formed by multiplying the sample variance by n minus one and dividing by the true population variance follows a chi-square distribution with n minus one degrees of freedom. The calculator splits the probability outside the requested confidence level equally between the two tails. It finds the lower and upper chi-square critical values for those tail probabilities, then reverses their positions when forming the variance bounds: the upper critical value divides the lower bound, and the lower critical value divides the upper bound. This reversal is easy to miss in a manual calculation and is a common source of incorrect intervals. The returned critical values make the work auditable. The algorithm evaluates the regularized incomplete gamma function and uses a bounded numerical search for each quantile, without network calls, random sampling, or time-dependent behavior. Because the chi-square distribution is asymmetric, the resulting interval is generally not centered on the sample variance. Smaller samples often produce especially wide and visibly skewed intervals, while increasing the sample size usually narrows the range.

Interpret the interval and check assumptions

A confidence interval describes the long-run performance of the procedure, not a probability assigned to a fixed population variance after observing the data. At a 95 percent confidence level, repeated independent samples analyzed in the same way would produce intervals that contain the true variance about 95 percent of the time under the model assumptions. The strongest assumption is population normality: unlike many procedures for a mean, the classical chi-square interval for variance can be sensitive to skewness, heavy tails, outliers, dependence, or a mixture of populations. Review a plot of the observations and the sampling process before treating the bounds as definitive. Also distinguish variance from standard deviation. If you need an interval for population standard deviation, take the nonnegative square root of each variance endpoint; do not apply the chi-square formula directly to the sample standard deviation. Compare intervals only when variables use compatible units and were collected under comparable conditions. For automated use, the API returns the same deterministic object as the browser calculator and costs $0.002 per request, which makes it suitable for reports, quality-control workflows, and reproducible analysis pipelines.