Using the same prescription, we can also evaluate the mean magnetization through $$ \begin{equation*} \langle {\cal M} \rangle = \sum_i^M {\cal M}_i P_i(\beta)= \frac{1}{Z}\sum_i^M {\cal M}_ie^{-\beta E_i}, \end{equation*} $$ and the corresponding variance $$ \begin{equation*} \sigma_{{\cal M}}^2=\langle {\cal M}^2 \rangle-\langle {\cal M} \rangle^2= \frac{1}{Z}\sum_{i=1}^M {\cal M}_i^2e^{-\beta E_i}- \left(\frac{1}{Z}\sum_{i=1}^M {\cal M}_ie^{-\beta E_i}\right)^2. \end{equation*} $$ This quantity defines also the susceptibility \( \chi \) $$ \begin{equation*} \chi=\frac{1}{k_BT}\left(\langle {\cal M}^2 \rangle-\langle {\cal M} \rangle^2\right). \end{equation*} $$