In practical situations however, a sample is always of finite size. Let that size be \( n \). The expectation value of a sample \( \alpha \), the sample mean, is then defined as follows $$ \begin{equation*} \langle x_{\alpha} \rangle \equiv \frac{1}{n}\sum_{k=1}^n x_{\alpha,k}. \end{equation*} $$ The sample variance is: $$ \begin{equation*} \mathrm{Var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_{\alpha,k} - \langle x_{\alpha} \rangle)^2, \end{equation*} $$ with its square root being the standard deviation of the sample.