Confidence Interval For Variance Calculator

Work out confidence interval for variance instantly with clear inputs, formula shown and shareable results.

About Confidence Interval For Variance Calculator

Because (n − 1)s²/σ² follows a chi-square distribution with n − 1 degrees of freedom, the interval for σ² is ((n−1)s²/χ²_upper, (n−1)s²/χ²_lower). Note the interval is asymmetric.

Formula

Variance interval

((n−1)s² / χ²₁₋α/₂, (n−1)s² / χ²α/₂) with n − 1 degrees of freedom

Worked example

With the default values below, Confidence Interval For Variance Calculator returns Lower bound for the variance of 14.458641. Change any input above and every figure here is recalculated in your browser.

Inputs used

Sample varianceminimum 0: 25
Sample sizeminimum 2: 20
Confidence levelaccepts 50–99.99 %: 95 %

Results

Lower bound for the variance: 14.458641
Upper bound for the variance: 53.331738
Lower bound for σ: 3.802452
Upper bound for σ: 7.302858

Step by step

1Values used
2Variance interval
3Lower bound for the variance= 14.458641
4Upper bound for the variance= 53.331738
5Lower bound for σ= 3.802452
6Upper bound for σ= 7.302858

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For s² = 25 with n = 20 at 95%?

The chi-square bounds on 19 df are about 32.85 and 8.91, giving a variance interval of roughly 14.46 to 53.31.

Why is the interval asymmetric?

The chi-square distribution is right-skewed, so the two tail cut-offs are not symmetric about the sample variance.