Many clustering methods use distance measures to determine the similarity or dissimilarity between any pair of objects. It is useful to denote the distance between two instances x_i and x_j as: d(x_i,x_j). A valid distance measure should be symmetric and obtains its minimum value (usually zero) in case of identical vectors. The distance measure is called a metric distance measure if it also satisfies the following properties:
When the attributes are ordinal, the sequence of the values is meaningful. In such cases, the attributes can be treated as numeric ones after mapping their range onto [0,1]. Such mapping may be carried out as follows:
where z_i,n is the standardized value of attribute a_n of object i. r_i,n is that value before standardization, and M_n is the upper limit of the domain of attribute a_n (assuming the lower limit is 1).