Silhouette and Cluster Validity Indices

Clustering has no ground-truth labels to score against, so cluster validity indices exist to answer "is this a good clustering?" using only the data and the assignments themselves. The most common one, the silhouette score, measures for each point how much closer it sits to its own cluster than to the nearest rival cluster.

How it's used

Because the silhouette score doesn't need to know the "correct" number of clusters in advance, it's commonly used to choose k in k-means: run the algorithm for k = 2, 3, 4, … and pick the k with the highest average silhouette score across all points, rather than relying on an arbitrary guess or the elbow method alone.

Worked example

A point sits in a cluster where its average distance to the other 9 members is 2.0 (call this $a$), and its average distance to the nearest neighboring cluster is 5.0 (call this $b$). The silhouette score is $(b - a) / \max(a, b) = (5.0 - 2.0) / 5.0 = 0.6$, a solidly positive score indicating the point is well-placed. Averaging this across every point and every candidate value of $k$ gives a simple way to compare, say, $k=3$ against $k=5$ on the same dataset without any labels at all.