The total sample variance over the \( mn \) measurements is defined as
$$ \sigma^2=\frac{1}{mn}\sum_{i=1}^{m} \sum_{j=1}^{n}\left(x_{ij}-\overline{X}\right)^2. $$We have from the equation for \( \sigma_m^2 \)
$$ \overline{x}_i-\overline{X}=\frac{1}{n}\sum_{j=1}^{n}\left(x_{i}-\overline{X}\right), $$and introducing the centered value \( \tilde{x}_{ij}=x_{ij}-\overline{X} \), we can rewrite \( \sigma_m^2 \) as
$$ \sigma^2_{m}=\frac{1}{m}\sum_{i} \left( \overline{x}_{i}-\overline{X}\right)^2=\frac{1}{m}\sum_{i=1}^{m}\left[ \frac{i}{n}\sum_{j=1}^{n}\tilde{x}_{ij}\right]^2. $$