Given $CH(k) = [B(k) / W(k) ] \times [(n-k)/(k-1)]$, where $n$ = # data points $k$ = # clusters $W(k)$ = within cluster variation $B(k)$ = between cluster variation.
It is my understanding that the CH index can show the optimal number of clusters when doing k-means or hierarchical clustering; you would choose the number of clusters $k$ that maximize $CH(k)$. As $k$ increases, $B(k)$ increases, and $W(k)$ decreases.
However, can someone explain to me the intuition behind the second part of the formula, namely $[(n-k) / (k-1)]$? Isn't it too punitive for cases where $n$ is very large, since increasing $k$ by 1 will drastically decrease the whole term?