Confidence intervals for sample standard deviations are calculated using upper and lower critical values from a chi-square distribution with an appropriate degree of freedom (typically, n-1).

For a 100(1-a)% confidence interval for s, two critical values are needed: one from the upper end of the distribution and one from the lower end. The lower confidence bound is given by:

And the upper bound is given by

Where

and

are critical values from a chi-square distribution with n-1 degrees of freedom, is the sample variance, and n is the sample size.