How to Choose K in K-Means (Elbow Method + Silhouette Score)

How to choose K in K‑Means (Elbow Method + Silhouette Score)

Key concepts and definitions

• K-means requires you to specify K (number of clusters) before running it; choosing K is critical to meaningful results.
• Inertia: a measure of cluster tightness — the sum of squared distances from each point to its cluster centroid. Lower inertia means tighter clusters.
• Silhouette score: evaluates per-point clustering quality by comparing the average distance to points in the same cluster versus the nearest other cluster. Scores range from -1 to 1:
  • Near +1: point is well inside its cluster.
  • Around 0: point lies near a boundary between clusters.
  • Negative: point is likely assigned to the wrong cluster.
• Distance is the basis for the data’s structure; evaluation metrics (inertia, silhouette) are used to verify that structure.

Silhouette score is a per-point, normalized measure; inertia summarizes cluster compactness across the dataset.

Methods / workflow (tutorial-style)

1. Elbow method (visual guide)

   • Run K-means for multiple K values (e.g., 1, 2, 3, … up to some reasonable limit).
   • Compute inertia for each K and plot inertia versus K.
   • Look for the “elbow”: a bend where the inertia curve sharply flattens. That K often indicates diminishing returns from adding more clusters.
   • Limitation: the elbow can be ambiguous or absent, so the method is mainly useful to narrow a range of candidate K values.

2. Silhouette score (verification)

   • For candidate K values (for example, those suggested by the elbow), compute the average silhouette score across all points.
   • The K with the highest average silhouette is typically the most meaningful choice.
   • Use the silhouette score to confirm or reject choices suggested by the elbow method.

Practical recommendation

• First use the elbow method to narrow down a reasonable range of K.
• Then use the silhouette score to select and confirm the best K within that range.
• Combine visual inspection (elbow) and quantitative evaluation (silhouette) to avoid over- or under-clustering.

Main speaker / sources

• Video narrator / presenter (unnamed)
• Discussed methods: K-means algorithm, Elbow method (inertia), Silhouette score (cluster quality)