Lab 21 — Clustering: k-means & the Elbow

Session: Week 11 · Session 21 · 90 min CLO: CLO-2 Difficulty: Intermediate

Learning objectives

By the end of this lab you can:

1. Explain unsupervised learning and k-means in plain words.
2. Scale features before clustering and say why.
3. Choose k with the elbow method (and read its limits).
4. Validate clusters against known labels when available.

Problem statement

No labels: the lab wants to know whether penguins naturally form groups from body measurements alone. You must run k-means over k = 1..8, justify a k with the elbow method, and then — because species labels actually exist — check how well the discovered clusters match reality.

Dataset requirements

Seaborn built-in penguins, dropna() (333 rows). Features: bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g. Keep species aside as ground truth.

Step-by-step tasks

1. Prep: dropna; X = penguins[features]; scale with StandardScaler().fit_transform(X). Comment: why scale before k-means?
2. Elbow: for k in 1..8, fit KMeans(n_clusters=k, random_state=42, n_init=10) and record kmeans.inertia_. Plot inertia vs k (markers + line). Mark where the elbow is.
3. Choose k from the elbow; fit k-means with that k and random_state=42.
4. Visualize: sns.scatterplot(x=scaled[:, 0], y=scaled[:, 1], hue=labels, palette="Set2") plus cluster centers (kmeans.cluster_centers_ as big markers). Scatter the first two scaled features — name them in the axis labels.
5. Ground truth check: build pd.crosstab(penguins["species"], labels) (species × cluster). If clusters align with species, each species row should concentrate in one cluster. Report the crosstab and say whether the discovered groups match the 3 species.
6. Honest note: write one sentence on what the elbow method can and cannot tell you (e.g., it's a heuristic; real-world clusters are rarely this clean).

Starter code

import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.preprocessing import StandardScaler
from sklearn.cluster import KMeans

penguins = sns.load_dataset("penguins").dropna()
features = ["bill_length_mm", "bill_depth_mm",
            "flipper_length_mm", "body_mass_g"]
X = StandardScaler().fit_transform(penguins[features])

# your code here: inertia sweep, elbow plot, chosen k, fit, scatter,
#                 crosstab, honest note

Expected output

• Inertia decreases monotonically with k; the elbow is visible at k = 3 (big drop from 1→3, then flattening) — possibly a slight bend at 2.
• Chosen k = 3 (or 4, if justified) with reasoning.
• Scatter of the two most informative scaled features showing 2–3 clear blobs; centers plotted.
• Crosstab: cluster 0 ≈ Adelie, cluster 1 ≈ Chinstrap, cluster 2 ≈ Gentoo (or a near-diagonal permutation) — clusters largely match species.
• One honest sentence about the elbow's limits.

Questions

1. What does inertia_ measure, and why does it always drop as k grows?
2. Why must features be scaled before k-means, given what the algorithm computes (distances)?
3. What does n_init=10 do, and why is it needed (k-means is initialization-sensitive)?
4. Clusters match species almost perfectly here. What would it mean if they did not?
5. k-means finds spherical clusters. What kind of data would it fail on?

Challenge task

Run k-means on unscaled features and show the elbow plot side by side with the scaled one. Answer: does scaling change the elbow's location? Then compute a simple agreement score: the fraction of rows where the cluster label matches its species (after mapping clusters to species by majority vote with pd.crosstab(...).idxmax(axis=1)). Report the fraction.

Marking rubric

Criteria	Max	Notes
Prep + scaling rationale	4	scale + comment
Elbow sweep + plot	5	1..8, marked elbow
k chosen + reasoning	3	ties to plot
Cluster scatter + centers	4	labels + centers visible
Crosstab + species match verdict	4	diagonal reading
Honest note	2	limits of elbow
Answers to questions	3	Q1, Q2, Q4 correct
Challenge: unscaled comparison + agreement	5	side-by-side + fraction
Total	30