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

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

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.

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.
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.

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.