Finally, the Normal distribution requires a numerical value for \(\sigma\text{,}\) the population’s standard deviation. It can be shown that the maximum likelihood estimator for \(\sigma^2\) is the variance v found in chapter one. However, you may remember that at that time we always adjusted this value somewhat using the formula \(s^2 = \frac{n}{n-1} v\) which increased the variance slightly. To uncover why you would not use the maximum likelihood estimator v requires you to look up the idea of "bias". As it turns out, v is maximum likelihood but exhibits mathematical bias whereas \(s^2\) is slightly suboptimal with respect to likelihood but exhibits no bias. Therefore, for estimating the unknown population variance \(\sigma^2\) you can use sample variance

and similarly sample standard deviation

to approximate the theoretical standard deviation.