Interpret clustering procedures outputs.

A regional auto book contains 8,400 policies. K-means with K=4 returns four centroids and four cluster sizes. Your job: read those centroids, the within-cluster sum of squares, and a dendrogram well enough to tell the pricing committee which segments are real.

What clustering returns. Unsupervised methods take $n$ observations on $p$ features and produce group labels. Nothing in the algorithm tells you the labels are meaningful. Interpretation is the entire exam skill.

K-means picks centroids ${\bar{x}}_k$ and assignments $C_k$ that minimize total within-cluster variation:

$$W=\sum _{k=1}^K\sum _{i\in C_k}\Vert x_i-{\bar{x}}_k{\Vert }^2$$

Reading the output. A K-means report lists, for each cluster: size $n_k$, centroid ${\bar{x}}_k$ on every feature, and $W_k$. Three reads:

• Compare each centroid to the grand mean feature by feature. On standardized data a centroid at +1.8 is "1.8 SD above the grand mean" on that feature.
• Compare $n_k$ across clusters. One micro-cluster of 8 against three clusters of 2,000+ usually signals outliers, not a real segment.

Common mistakes

• Picking K by minimum $W$. $W$ hits zero at K = n. The correct rule is the elbow, silhouette, or gap statistic. Choosing K = 10 because it had the smallest $W$ is the canonical wrong answer.
• Ignoring scale. Clustering raw salary (range 0 to $200,000) with raw age (range 18 to 90) makes age effectively invisible. Always standardize when units differ.
• Reading horizontal position on a dendrogram. Adjacent leaves are not necessarily similar. Only fusion height matters. Two leaves sharing a parent at height 0.2 are close; two that touch at the bottom but merge only at height 8.0 are far apart.

Bottom line

• K-means output = K centroids (cluster means on each feature), cluster assignments, and within-cluster sum of squares $W_k$. Total $W={\sum }_k{W}_k$ decreases monotonically in K.
• Interpret each centroid against the grand mean feature by feature; on standardized data a centroid at +2.0 SD means that cluster sits two SD above average there.
• Hierarchical output = a dendrogram. Y-axis is fusion height (linkage distance at which clusters merged). Cut horizontally at height $h$ to extract clusters.
• Read a dendrogram by fusion height, not horizontal position; find the largest vertical gap and cut there. A cut crosses one branch per resulting cluster.

Exam shortcut

When a problem reports K-means output as centroids on standardized features, read each centroid as standard deviations above or below the grand mean and label segments by their largest absolute coordinate. When the prompt gives a dendrogram with explicit merge heights, list successive differences and cut above the largest one.

The full lesson (about 2,026 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C8

Browse all free MAS-II lessons or jump into free MAS-II practice questions.