## Quick Reference

A measure of the distance between points in multidimensional space (also called a metric). Distance measures are used with techniques such as cluster analysis and multidimensional scaling. The two measures most commonly used are Euclidean distance and the city-block metric.

Euclidean distance is the straight-line distance between two points, also called the *l*_{2}**-metric**. In *d* dimensions, if the positions of *P* and *Q* are given by the coordinates (*p*_{1}, *p*_{2},…, *p** _{d}*) and (

*q*

_{1},

*q*

_{2},…,

*q*

*) then the Euclidean distance between*

_{d}*P*and

*Q*is given by . The city-block metric in two dimensions measures the distance between two points in a city if, for example, the only directions in which one could travel were north-south and east-west. It is also called the

*l*

_{1}**-metric**or Manhattan distance. In

*d*dimensions the city-block distance between

*P*and

*Q*is given by . A generalization of the previous measures is the Minkowski distance where

*k*is a positive integer. This is also called the

*l*-metric.

_{k}The Chebyshev distance (the *l*_{∞}-metric) is the largest difference over the *d* dimensions , Two other measures that have been proposed are the Canberra distance, given by . and, in the context of counts of organisms, the Bray-Curtis distance (also called the Sorensen distance):

**Distance measure.** These are the two simplest of an infinite number of possible measures of distance.

*Subjects:*
Probability and Statistics.