You can think of the above observables as a set of quantities which define a given experiment. This experiment is then repeated several times, say \( m \) times. The total average is then $$ \begin{equation} \langle X_m \rangle= \frac{1}{m}\sum_{\alpha=1}^mx_{\alpha}=\frac{1}{mn}\sum_{\alpha, k} x_{\alpha,k}, \tag{8} \end{equation} $$ where the last sums end at \( m \) and \( n \). The total variance is $$ \begin{equation*} \sigma^2_m= \frac{1}{mn^2}\sum_{\alpha=1}^m(\langle x_{\alpha} \rangle-\langle X_m \rangle)^2, \end{equation*} $$ which we rewrite as $$ \begin{equation} \sigma^2_m=\frac{1}{m}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle). \tag{9} \end{equation} $$