K-Means Clustering & The Elbow Method

Centroids, inertia, Voronoi partitions, and silhouette scores

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K-Means is an unsupervised clustering algorithm that partitions unlabeled data points into K distinct groups by iteratively updating centroid positions.

The K-Means Convergence Algorithm

1Initialize K cluster centroids randomly (or via k-means++).

2Assignment Step: Assign each data point to its nearest centroid using Euclidean distance.

3Update Step: Recalculate centroids as the arithmetic mean of all points assigned to that cluster.

4Repeat until centroids stop moving (convergence).

The Elbow Method: Plotting Inertia (within-cluster sum of squares) vs K reveals an "elbow" point representing the optimal cluster count.

Customer Segmentation with K-Means

Try It Yourself »

from sklearn.cluster import KMeans
import numpy as np

# Customer purchase data: [annual_spend, visit_frequency]
X = np.array([
    [12000, 45], [14000, 50], [11500, 48], # High value
    [1200, 6],   [1500, 8],   [900, 5],     # Low value
    [5000, 22],  [6200, 25],  [4800, 20]    # Moderate
])

kmeans = KMeans(n_clusters=3, init="k-means++", random_state=42)
clusters = kmeans.fit_predict(X)

print("Assigned cluster IDs:", clusters)
print("Centroid coordinates:
", np.round(kmeans.cluster_centers_, 1))

Groups customers into natural demographic clusters without labels.

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Test Yourself With Exercise

+15 XP

In K-Means clustering, what visual heuristic is commonly used to choose the optimal number of clusters K?

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