UPGMA (unweighted pair group method with arithmetic mean) is a simple agglomerative (bottom-up) hierarchical clustering method. It also has a weighted variant, WPGMA, and they are generally attributed to Sokal and Michener.

Note that the unweighted term indicates that all distances contribute equally to each average that is computed and does not refer to the math by which it is achieved. Thus the simple averaging in WPGMA produces a weighted result and the proportional averaging in UPGMA produces an unweighted result (see the working example).

The UPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwise similarity matrix (or a dissimilarity matrix). At each step, the nearest two clusters are combined into a higher-level cluster. The distance between any two clusters ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle {\mathcal {B}}}$, each of size (i.e., cardinality) ${\displaystyle {|{\mathcal {A}}|}}$ and ${\displaystyle {|{\mathcal {B}}|}}$, is taken to be the average of all distances ${\displaystyle d(x,y)}$ between pairs of objects ${\displaystyle x}$ in ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle y}$ in ${\displaystyle {\mathcal {B}}}$, that is, the mean distance between elements of each cluster:

In other words, at each clustering step, the updated distance between the joined clusters ${\displaystyle {\mathcal {A}}\cup {\mathcal {B}}}$ and a new cluster ${\displaystyle X}$ is given by the proportional averaging of the ${\displaystyle d_{{\mathcal {A}},X}}$ and ${\displaystyle d_{{\mathcal {B}},X}}$ distances:

${\displaystyle d_{({\mathcal {A}}\cup {\mathcal {B}}),X}={\frac {|{\mathcal {A}}|\cdot d_{{\mathcal {A}},X}+|{\mathcal {B}}|\cdot d_{{\mathcal {B}},X}}{|{\mathcal {A}}|+|{\mathcal {B}}|}}}$

The UPGMA algorithm produces rooted dendrograms and requires a constant-rate assumption - that is, it assumes an ultrametric tree in which the distances from the root to every branch tip are equal. When the tips are molecular data (i.e., DNA, RNA and protein) sampled at the same time, the ultrametricity assumption becomes equivalent to assuming a molecular clock.

This working example is based on a JC69 genetic distance matrix computed from the 5S ribosomal RNA sequence alignment of five bacteria: Bacillus subtilis (${\displaystyle a}$), Bacillus stearothermophilus (${\displaystyle b}$), Lactobacillus viridescens (${\displaystyle c}$), Acholeplasma modicum (${\displaystyle d}$), and Micrococcus luteus (${\displaystyle e}$).

Let us assume that we have five elements ${\displaystyle (a,b,c,d,e)}$ and the following matrix ${\displaystyle D_{1}}$ of pairwise distances between them :