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Agglomerative hierarchical clustering starts with each observation as a singleton cluster in cluster analysis, repeatedly joins the closest pair of current clusters, and continues until one remains. The input is a dissimilarity matrix, so choosing variable scales and a scientifically sensible dissimilarity is part of the analysis.

Different linkages define closeness differently:Single-linkage clustering can connect elongated chains through nearest neighbors; complete-linkage clustering emphasizes compact clusters; average-linkage clustering averages all cross-pair distances. For Euclidean observations, Ward minimum-variance clustering instead chooses the smallest increase in within-cluster squared error,This follows by expanding squared deviations around the merged mean and using the zero sums of within-cluster residuals.

A dendrogram plots merge heights. In the lower-left sketch, two close pairs first merge at small heights, then join across a much larger separation; a horizontal cut between these heights yields two groups. The hierarchy gives nested partitions, not a uniquely established number of populations. Examine sensitivity to scaling, linkage and ties; once an agglomerative merge is made, the method does not undo it. The absence of a probability model also means that a visually attractive dendrogram is not itself a significance test.