Variance Confidence Interval Calculator

StatisticsLast updated: August 22, 2026

Confidence intervals built for a mean are symmetric around the estimate; those built for a variance are not. The reason is that the quantity (n−1)s²/σ² follows a chi-square distribution, and the chi-square distribution is right-skewed. As a result the upper bound of the variance sits much further away than the lower bound: the possibility that the variability is truly larger than estimated covers a wider area than the possibility that it is smaller.

Starting from raw data or from summary statistics (n and s or s²), this calculator produces the confidence interval for both σ² and σ, shows the two chi-square critical values used, and reports how asymmetric the interval is. In measurement repeatability, process control, laboratory calibration, and test development work, variability itself is the quantity of interest; in studies like these the interval for the standard deviation, not its point estimate, should be reported.

You can write the numbers separated by commas or spaces, or one per line. A comma may be used as the decimal separator (10,2).
The probability that the interval covers the true parameter.

Chi-Square Critical Values (95% Confidence, α/2 = 0.025)

The two ends of the variance interval come from dividing the same numerator (n−1)s² by two different chi-square critical values. The large gap between the left and the right critical value shows directly why the interval is asymmetric.

df = 9 (n = 10)
2.7004 and 19.0228
The right critical value is roughly 7 times the left one — the interval is strongly asymmetric
df = 14 (n = 15)
5.6287 and 26.1189
The ratio drops to about 4.6
df = 19 (n = 20)
8.9065 and 32.8523
The ratio is about 3.7
df = 24 (n = 25)
12.4012 and 39.3641
The ratio is about 3.2
df = 29 (n = 30)
16.0471 and 45.7223
The ratio is about 2.8 — the asymmetry keeps shrinking
Source of the asymmetry
The chi-square distribution is right-skewed
As the degrees of freedom grow the distribution approaches symmetry and the interval starts to look like an interval for a mean

Critical values derived from the chi-square distribution; they do not change over timeKaynak: Chi-square (χ²) distribution tables; the same values returned by the Excel CHISQ.INV and CHISQ.INV.RT functions

Confidence Interval for the Variance and the Standard Deviation

Pivotal quantity:  (n − 1)·s² / σ²  ~  χ²(n − 1)
Variance:          [ (n−1)s² / χ²_{α/2} ,  (n−1)s² / χ²_{1−α/2} ]
                   χ²_{α/2}: the LARGE critical value with right-tail area α/2
                   χ²_{1−α/2}: the SMALL critical value with right-tail area 1−α/2
Standard deviation: [ √(lower bound) ,  √(upper bound) ]
Degrees of freedom: df = n − 1
Sample variance:   s² = Σ(xᵢ − x̄)² / (n − 1)
Asymmetry ratio:   (upper − s²) / (s² − lower)

The interval is not symmetric around s² because the chi-square distribution is right-skewed and the two critical values that cut off the central area are not equally far from the center. The asymmetry shrinks as the degrees of freedom grow, but it never disappears entirely.

How to Calculate

  1. Choose the input mode: paste the raw data, or enter n together with s (or s²).
  2. In summary mode, state whether the value you entered is a standard deviation or a variance.
  3. Choose the confidence level; 95% is the standard choice in academic reporting.
  4. Always check the normality assumption: when you enter raw data, the Shapiro-Wilk result and a Q-Q plot appear in the results.
  5. Read the σ² and σ intervals together; reports usually favor the σ interval because it is in the same unit as the measurements.
  6. Compare the distances from s² to the lower and the upper bound to see how asymmetric the interval is.

Worked Examples

Interval for the standard deviation from raw measurements

The 12 measurements have a standard deviation of 0.2778 and a variance of 0.0772. With chi-square critical values of 3.8157 and 21.9200 for df = 11, the 95% interval is [0.0387; 0.2225] for σ² and [0.1968; 0.4717] for σ. The upper bound sits 0.1453 away from s² while the lower bound is only 0.0385 away: the interval is 3.78 times longer on the upper side. The Shapiro-Wilk test, with W = 0.9794; p = 0.981, supports the normality assumption.

95% confidence interval for the variance (σ²): [0.0387; 0.2225] · 95% confidence interval for the standard deviation (σ): [0.1968; 0.4717] · Sample variance (s²): 0.0772

Variance interval from summary statistics

For n = 25 and s = 3.2 the variance is 10.24, df = 24, and the chi-square critical values are 12.4012 and 39.3641. The 95% confidence interval is [6.2433; 19.8175] for σ² and [2.4987; 4.4517] for σ. Even though the point estimate of the standard deviation is 3.2, the population value could plausibly lie anywhere between 2.50 and 4.45; the asymmetry ratio is 2.40.

95% confidence interval for the variance (σ²): [6.2433; 19.8175] · 95% confidence interval for the standard deviation (σ): [2.4987; 4.4517] · Sample variance (s²): 10.2400

How wide the interval gets in a small sample

With n = 10 and s² = 25 (s = 5), the 99% chi-square critical values for df = 9 are 1.7349 and 23.5894. The interval is [9.5382; 129.6880] for σ² and [3.0884; 11.3881] for σ: with 10 observations you cannot rule out any standard deviation between 3.09 and 11.39. With an asymmetry ratio of 6.77, the interval stretches extremely far on the upper side.

99% confidence interval for the variance (σ²): [9.5382; 129.6880] · 99% confidence interval for the standard deviation (σ): [3.0884; 11.3881] · Sample variance (s²): 25.0000

Frequently Asked Questions

Why is the confidence interval for a variance not symmetric?
Because the interval is not built in the "estimate ± margin of error" form used for means. The quantity (n−1)s²/σ² follows a chi-square distribution, which is right-skewed; the two critical values that cut off the central area are not equally far from the center of the distribution. Dividing the fixed numerator (n−1)s² by those two different values naturally produces an asymmetric interval whose upper bound is always further from the estimate.
Is the normality assumption really that critical?
Yes. Intervals for a mean are robust to departures from normality thanks to the central limit theorem; the variance interval is not. The chi-square approach rests directly on the normality of the data, and the true coverage departs seriously from the intended level, especially when the kurtosis differs from normal. Enlarging the sample does not solve this problem.
What should I use if normality does not hold?
The most practical option is a bootstrap confidence interval: samples are drawn from the data with replacement over and over, the variance is computed in each one, and the percentiles of the resulting distribution are used as the interval. Alternatively you can apply a transformation that normalizes the data (logarithm, square root), in which case the results are interpreted on the transformed scale.
Does taking the square root of the σ² interval give the correct σ interval?
Yes. The square root is a monotonically increasing transformation, so the square root of an interval with 95% coverage for σ² also has 95% coverage for σ. The same does not hold for the point estimate: s is not an unbiased estimator of σ (s² is the unbiased one), so the standard deviation is somewhat underestimated in small samples.
How do I compute this interval in Excel, SPSS, and R?
In Excel the lower bound is =(n-1)*VAR.S(data)/CHISQ.INV.RT(0.025, n-1) and the upper bound uses CHISQ.INV.RT(0.975, n-1) in the denominator. SPSS does not provide this interval in its standard menus; it requires syntax or a custom macro. In R, (n-1)*var(x)/qchisq(c(0.975, 0.025), n-1) returns the lower and the upper bound in a single line.
How does the sample size affect the interval?
In two ways: the interval narrows and the asymmetry shrinks. At n = 10 the ratio between the right and the left chi-square critical value is about 7, while at n = 30 it falls to about 2.8. Even so, variance intervals are always wider than intervals for a mean; estimating variability precisely takes more observations than estimating a mean.
Can this calculator be used to compare the variances of two groups?
No. It estimates the σ² of a single group. For the ratio of two variances you need an interval based on the F distribution, and to test the equality of variances you should use Levene's or Bartlett's test. Levene's test is far more robust to departures from normality than Bartlett's and is usually preferred in practice.