# Intuition behind the Calinski-Harabasz Index

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?

• I can add to Glen's neat answer that CH index is also called "multivariate (pseudo) F". So, it is actually Fisher's F that is used in ANOVA. ("Pseudo" because it takes not possible covariations into account.) – ttnphns May 13 '14 at 6:44

Some simple intuition: $[B(k)/(k-1)]/[W(k)/(n-k)]$ is analogous to an F-ratio in ANOVA; $B(k)$ and $W(k)$ are between- and within-cluster sums of squares for the $k$ clusters.
$B(k)$ has $k-1$ degrees of freedom, while $W(k)$ has $n-k$ degrees of freedom.
As $k$ grows, if the clusters were all actually just from the same population, $B$ should be proportional to $k-1$ and $W$ should be proportional to $n-k$.