Why is it ok to estimate the population variance with the sample variance formula (i.e. the unbiased estimate of the variance). I understand how we can build a confidence interval using the central limit theorem and the following formula . But, the unbiased estimate of the variance means that the estimate corresponds to the population variance in the long run (and the sd is biased, but nearly correct). Replacing s with σ in the formula above is just that, not the value of s squared in the long run. So, why is it ok to use it?