## Distance measurements for ordinal attributes

Many methods of partitioning use distance measures to determine the similarity or dissimilarity between any pair of objects (like Distance Measures for ordinal attributes). It is common to denote the distance between two instances x_i and x_j as: d(x_i, x_j). A valid distance measure must be symmetric and obtains its minimum value (usually zero) in the case of identical vectors. The distance measure is called a metric distance measure if it also satisfies the following properties:

When attributes are ordinal, the sequence of values is important. In such cases, attributes can be treated as numeric attributes after mapping their range to [0,1]. This mapping can be done as follows:

where z_i, n is the normalized value of attribute a_n of object i. r_i, n is this value before normalization, and M_n is the upper limit of the domain of the a_n attribute (assuming the lower limit is 1).