Note 12.4. Three properties matter in use. The algorithm converges, but only to a local minimum, and different starting points give different answers — so it should be run repeatedly from random starts. It requires \(k\) in advance, and the criterion cannot choose \(k\), since the within-cluster sum of squares decreases whenever \(k\) increases; the elbow of that curve against \(k\) is used in the same spirit, and with the same imprecision, as the scree plot of Section 4.5. And minimising a sum of squared Euclidean distances biases the result toward spherical clusters of comparable size, so genuinely elongated groups are routinely split.

Note 12.5. A clustering is a hypothesis, not a finding. Any of these algorithms will partition data that has no group structure whatever, and will do so confidently. Before interpreting clusters, check that they persist under a different distance, a different linkage or a different \(k\); that they are separated on variables not used to construct them; and that they mean something in the subject the data came from. None of these is a significance test, because none exists.