📊 Silhouette Score and the Elbow Method

About this simulation

K-means clustering needs to be told how many clusters, k, to look for before it can run — it will happily produce a tidy-looking partition for almost any k you give it, even a wrong one. This lab runs real k-means on a small 3D point cloud and shows the two classic tools people use to pick a sensible k: the elbow method (inertia vs. k) and the silhouette score (a per-point measure of how well-matched a point is to its own cluster vs. the next closest one).

How it works

• Inertia — the sum of squared distances from every point to its assigned centroid. It always falls as k rises, so the "elbow" (where the drop stops being dramatic) is the signal, not the lowest value.
• Silhouette score — for each point, s = (b − a) / max(a, b), where a is the mean distance to points in its own cluster and b is the mean distance to the nearest other cluster. Averaged over all points, it peaks near a genuinely good k and can even fall as k keeps rising.
• The two orange-highlighted bars in each chart track whichever k the slider is set to, so you can see both curves respond as you move it.
• The "Uniform noise" dataset has no real cluster structure — watch how flat and unconvincing both curves become, a useful warning sign in real analysis.

Controls

• Clusters (k) — how many clusters k-means is asked to find (2–9).
• Dataset — switch between clean, overlapping, and structureless point clouds.
• Point colour — colour points by their assigned cluster, or by their individual silhouette value (green = well matched, red = poorly matched or misassigned).
• Centroid links — draw a line from every point to its cluster centroid.
• ▶ Sweep k=2→9 — automatically steps through every k so you can watch both charts build.
• ↺ Regenerate points — redraw a fresh random sample from the current dataset. Drag to rotate, scroll to zoom.

Did you know?

Silhouette scores range from −1 to 1. A negative score for a point means it is, on average, closer to a neighbouring cluster than to its own — a strong hint that either the point or the whole choice of k is wrong.