Hierarchical Clustering & Dendrogram
Agglomerative clustering starts with every point as its own cluster and repeatedly merges the two closest clusters until one remains; the whole merge history is a tree called a dendrogram. Scatter up to 60 points from a preset (three blobs, two moons, uniform) or by clicking, choose a linkage — single, complete, average or Ward — and play: the left canvas shows points recolouring under growing cluster hulls as merges happen one by one, while the right canvas builds the dendrogram bottom-up, each junction drawn at its merge distance. Drag the cut-height slider to slice the dendrogram with a dashed line and recolour the points into the clusters that cut produces — one tree answers every choice of cluster count. Try single linkage on the moons preset to see chaining follow the curved shapes.
Runs 100% in your browser — models are trained and computed locally on your device.
Read the full guide to this tool
Notes
- Linkage defines cluster distance: single takes the minimum pairwise distance, complete the maximum, average the mean, and Ward merges the pair that increases total within-cluster variance the least.
- All four linkages update distances after a merge with one Lance–Williams formula d(i∪j, k) = αᵢd(i,k) + αⱼd(j,k) + βd(i,j) + γ|d(i,k) − d(j,k)|, only the coefficients differ.
- Single linkage chains: it happily connects long thin shapes like the moons, but one noisy bridge point can fuse two genuine clusters. Complete and Ward prefer compact round clusters instead.
- The naive algorithm used here is O(n³) time and O(n²) memory — fine for 60 points, hopeless for a million; that is why k-means scales better.
- Runs 100% in your browser — models are trained and computed locally on your device.
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