Clustering Validation Metrics · Silhouette, Davies-Bouldin, and Calinski-Harabasz

Evaluating clustering is fundamentally harder than evaluating classification — there are no ground-truth labels. Internal validation metrics assess clustering quality using only the data and the cluster assignments, without external labels.

Silhouette Score

Measures how similar a point is to its own cluster versus the nearest other cluster. Range: [-1, 1]. Higher = better.

$$ s(i) = \frac{b(i) - a(i)}{\max(a(i), b(i))} $$

where $a(i)$ = mean distance to points in its own cluster, $b(i)$ = mean distance to points in the nearest other cluster.

Best for: Spherical, well-separated clusters (k-means output). Penalises DBSCAN-style arbitrary-shape clusters because peripheral points in elongated clusters have high distance to dense centre points.

Davies-Bouldin Index

Average similarity between each cluster and its most similar other cluster, where similarity is the ratio of within-cluster scatter to between-cluster separation. Range: [0, ∞). Lower = better.

Best for: Balanced cluster sizes. Penalises one large + several small clusters.

Calinski-Harabasz Index (Variance Ratio Criterion)

Ratio of between-cluster dispersion to within-cluster dispersion. Higher = better.

$$ CH = \frac{\text{tr}(B_k)}{\text{tr}(W_k)} \times \frac{N-k}{k-1} $$

Best for: Gaussian-distributed clusters with equal covariance. Heavily favours equal-sized, spherical clusters.

Which Metric When?

• Silhouette: k-means, hierarchical clustering · Arbitrary-shaped, varying-density
• Davies-Bouldin: Comparing across k values · Density-based clustering
• Calinski-Harabasz: Well-separated Gaussians · Outliers, non-spherical

Key insight: These metrics share the implicit assumption of spherical, roughly-equal-sized clusters. Using them to tune DBSCAN parameters (ε, minPts) is misleading — DBSCAN’s strength is finding arbitrary-shaped clusters, and these metrics penalise that strength. For density-based clustering, visual inspection, stability analysis, or DBCV (Density-Based Clustering Validation) are more appropriate — but no metric is a complete substitute for understanding your data.