$$ x_{ki} - \overline{x} = (x_{ki} - \overline{x}_k) + ( \overline{x}_k - \overline{x} ) $$ also
$$ (x_{ki} - \overline{x})^2 = (x_{ki} - \overline{x}_k)^2 + 2 (x_{ki} - \overline{x}_k) ( \overline{x}_k - \overline{x} ) + ( \overline{x}_k - \overline{x} )^2 $$ also
$$ \sum_{i=1}^{H_k}(x_{ki} - \overline{x})^2 = \sum_{i=1}^{H_k}(x_{ki} - \overline{x}_k)^2 + \sum_{i=1}^{H_k}( \overline{x}_k - \overline{x} )^2 = \sum_{i=1}^{H_k}(x_{ki} - \overline{x}_k)^2 + H_k( \overline{x}_k - \overline{x} )^2 $$ weil gilt
$$ \sum_{i=1}^{H_k} 2 (x_{ki} - \overline{x}_k) ( \overline{x}_k - \overline{x} ) = 2 ( \overline{x}_k - \overline{x} ) \sum_{i=1}^{H_k} (x_{ki} - \overline{x}_k) = 0 $$ also
$$ s^2 = \frac{1}{n-1} \sum_{k=1}^K \sum_{i=1}^{H_k} (x_{ki} - \overline{x})^2 = \frac{1}{n-1} \sum_{k=1}^K \sum_{i=1}^{H_k} (x_{ki} - \overline{x}_k)^2 + \sum_{k=1}^K \frac{H_k}{n-1}( \overline{x}_k - \overline{x} )^2 $$ und wegen
$$ \sum_{i=1}^{H_k} (x_{ki} - \overline{x}_k)^2 = (H_k - 1) s_k^2 $$ folgt
$$ s^2 = \sum_{k=1}^K \frac{H_k - 1}{n-1}s_k^2 + \sum_{k=1}^K \frac{H_k}{n-1}( \overline{x}_k - \overline{x} )^2 $$
