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Adding more definitions

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.