More about the confidence interval for the population variance

A confidence interval is an statistical concept that refers to an interval that has the property that we are confident at a certain specified confidence level that the population parameter, in this case, the population standard deviation, is contained by it. For the case the population standard deviation (\(\sigma^2\)), the following expression is used:

\[ CI(\text{Variance}) = \displaystyle \left( \frac{(n-1) s^2}{\chi^2_{\alpha/2,n-1}}, \frac{(n-1) s^2}{\chi^2_{1-\alpha/2,n-1} } \right) \]

where the critical values corresponds to critical values associated to the Chi-Square distribution. The critical values for the given \(\alpha\) and \(df\) degrees of freedom are \(\chi_L^2 = \chi^2_{1-\alpha/2,n-1}\) and \(\chi_U^2 = \chi^2_{\alpha/2,n-1}\).

Assumptions that need to be met

Most people do not bother to check assumptions and they will rush to use the above expression to calculate the confidence interval for the variance, or confidence interval calculator above, with no regards. But in reality, you ensure that the sample comes from an at least approximately normally distributed population, in order to guarantee the validity of the interval obtained.

There is also the case when instead of dealing with one population variance, what you need is to deal with the ratio of two population variances, in which case you will use this calculator for the ratio of variances .

You may be interested in computing other confidence intervals. For example, you can use this confidence interval for the mean , or this confidence interval for variance when mean is known , or you can also this confidence interval for mean regression responses .