We have
$$ \begin{equation}\tag{4} T = W(C) + B(C) = \frac{1}{2}\sum_{i=1}^n \sum_{i'=1}^nd(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k} \Big(\sum_{C(i') = k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) + \sum_{C(i')\neq k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})\Big). \end{equation} $$This is a quantity that is conserved throughout the \( k \)-means algorithm. It can be thought of as the total amount of information in the data, and it is composed of the aforementioned within-cluster scatter and the between-cluster scatter \( B(C) \). In methods such as principle component analysis the total scatter is not conserved.