External measures can be useful for examining whether the structure of the clusters match to some predefined classification of the instances.

## Mutual information based measure

The mutual informationcriterion can be used as an external measure for clustering. The measure for m instances clustered using C={C_1, . . . , C_g} and referring to the target attribute y whose domain is dom(y) ={c_1, . . . , c_k} is defined as follows:

where m_l,h indicate the number of instances that are in cluster C_l and also in class c_h. m.,h denotes the total number of instances in the class c_h. Similarly, m_l,. indicates the number of instances in cluster C_l.

MI is combined with entropy in the Normalized mutal information:

MI is combined with entropy in the Adjusted mutal information:

## Precision-recall measure

The precision-recall measure from information retrieval can be used as an external measure for evaluating clusters.The cluster is viewed as the results of a query for a specific class. Precision is the fraction of correctly retrieved instances, while recall is the fraction of correctly retrieved instances out of all matching instances. A combined F-measure can be useful for evaluating a clustering structure.

## Rand index

The Rand index is a simple criterion used to compare an induced clustering structure (C1) with a given clustering structure (C2). Let a be the number of pairs of instances that are assigned to the same cluster in C1 and in the same cluster in C2; b be the number of pairs of instances that are in the same cluster in C1, but not in the same cluster in C2; c be the number of pairs of instances that are in the same cluster in C2, but not in the same cluster in C1; and d be the number of pairs of instances that are assigned to different clusters in C1 and C2. The quantities a and d can be interpreted as agreements, and b and c as disagreements. The Rand index is defined as:

The Rand index lies between 0 and 1. When the two partitions agree perfectly, the Rand index is 1.

A problem with the Rand index is that its expected value of two random clustering does not take a constant value (such as zero). Hubert and Arabie in 1985 suggest an adjusted Rand index that overcomes this disadvantage.