Confidence Interval for Variance Calculator

A confidence interval for the ratio of two variances gives a range of plausible values for σ²1/σ²2, the true ratio of two population variances. Because we only observe sample variances, we use the F-distribution to construct this interval. A 95% CI means that if we repeated sampling many times, 95% of the constructed intervals would contain the true ratio.

When two independent samples are drawn from normal populations, the ratio of their sample variances divided by the ratio of their population variances follows an F-distribution. This property makes the F-distribution the natural tool for constructing confidence intervals and hypothesis tests involving variance ratios.

The two-sided CI is: [ (S²1/S²2) / F_{α/2}(df1, df2) , (S²1/S²2) × F_{α/2}(df2, df1) ], where df1 = n1 − 1, df2 = n2 − 1, and F_{α/2} is the upper critical value of the F-distribution at significance level α/2. α equals 1 minus the confidence level.

If 1 falls within the confidence interval for σ²1/σ²2, there is no statistically significant evidence that the two population variances differ at your chosen confidence level. If the entire interval is above or below 1, it suggests the variances are significantly different.

Use a two-sided interval when you want to detect differences in either direction. Choose a lower-bound (one-sided) interval when you only care that σ²1/σ²2 exceeds some minimum value. Choose an upper-bound interval when you want to establish a maximum on how large the ratio can be. One-sided intervals provide a tighter bound in one direction.

There is no strict minimum, but larger samples produce narrower (more precise) confidence intervals. With very small samples (n < 10), the interval can be extremely wide and the normality assumption becomes harder to verify. Most practitioners recommend at least 20–30 observations per group for reasonably tight intervals.

Yes. The F-distribution-based confidence interval for variance ratios assumes that both groups come from normally distributed populations. If your data are heavily skewed or contain many outliers, the resulting interval may be unreliable. In such cases, consider data transformations or non-parametric alternatives.

Sample variance (S²) is computed from observed data using n − 1 in the denominator (Bessel's correction) and estimates the unknown population variance (σ²). The confidence interval uses sample variances to infer a plausible range for the true population variance ratio, which is what researchers ultimately care about.