Davies–Bouldin Index Calculator

The Davies–Bouldin Index (DBI) measures how well a set of clusters is separated relative to how spread out each one is. Like the silhouette score it is an internal validity index — it needs no ground-truth labels, so it is a standard way to judge a k-means, hierarchical, or DBSCAN result and to compare different choices of k. It was defined by David Davies and Donald Bouldin in 1979 and is what scikit-learn returns from davies_bouldin_score.

For a clustering with k clusters (k ≥ 2), the tool computes:

1. Centroid cᵢ. The coordinate-wise mean of the points in cluster i.
2. Dispersion Sᵢ. The mean Euclidean distance from each point in cluster i to its centroid: Sᵢ = (1 / |Cᵢ|) · Σ ‖x − cᵢ‖. This is scikit-learn's p = q = 1 choice. Smaller means a tighter cluster.
3. Separation Mᵢⱼ. The Euclidean distance between the centroids of clusters i and j: Mᵢⱼ = ‖cᵢ − cⱼ‖. Larger means the clusters sit further apart.
4. Similarity ratio Rᵢⱼ. Rᵢⱼ = (Sᵢ + Sⱼ) / Mᵢⱼ. It is high when two clusters are both loose and close — that is, when they overlap.
5. Worst rival Dᵢ. Dᵢ = max over j ≠ i of Rᵢⱼ. Each cluster is judged by its most similar competing cluster.

The index is the plain average of Dᵢ over all clusters: DB = (1 / k) · Σ Dᵢ. Distance is Euclidean throughout, matching scikit-learn's default. The minimum is 0 — reached only by perfectly tight, perfectly separated clusters — and there is no fixed upper bound, so lower is betterand DBI values are only comparable across clusterings of the same data. Two edge cases are guarded exactly as scikit-learn does: if every dispersion is zero or every centroid coincides, the index is reported as 0; and if two distinct clusters happen to share a centroid, that pair's ratio is treated as 0 rather than a division by zero. As a credibility check the tool also recomputes the index a second way — an independent reference pass that rebuilds every centroid, dispersion, and centroid distance from scratch — and confirms the two agree.

• scikit-learn — sklearn.metrics.davies_bouldin_score (the average-Euclidean-distance dispersion, the mean-over-clusters form, the 0-floor, and the coincident-centroid handling this tool matches)
• Davies, D.L. & Bouldin, D.W. (1979) — A Cluster Separation Measure, IEEE TPAMI (the original definition of Rᵢⱼ, Dᵢ, and the index)

The formulas on this page were last cross-checked against these sources on 2026-07-12. The Davies–Bouldin Index is a stable mathematical definition, so this tool needs no rate or schedule updates — only the worked examples are periodically re-reconciled against scikit-learn.