Absolute difference

Absolute difference has the following properties: • For x\ge 0, |x-0|=x (zero is the identity element on non-negative numbers) • |x-y|= 0 if and only if x=y (nonzero for distinct arguments). equality holds if and only if x\le y\le z or x\ge y\ge z. Because it is non-negative, nonzero for distinct arguments, symmetric, and obeys the triangle inequality, the real numbers form a metric space with the absolute difference as its distance, the familiar measure of distance along a line. It has been called "the most natural metric space", and "the most important concrete metric space". ==Applications==

The absolute difference is used to define the relative difference, the absolute difference between a given value and a reference value divided by the reference value itself. In the theory of graceful labelings in graph theory, vertices are labeled by natural numbers and edges are labeled by the absolute difference of the numbers at their two vertices. A labeling of this type is graceful when the edge labels are distinct and consecutive from 1 to the number of edges. As well as being a special case of the Lp distances, absolute difference can be used to define Chebyshev distance (L∞), in which the distance between points is the maximum or supremum of the absolute differences of their coordinates. In statistics, the absolute deviation of a sampled number from a central tendency is its absolute difference from the center, the average absolute deviation is the average of the absolute deviations of a collection of samples, and least absolute deviations is a method for robust statistics based on minimizing the average absolute deviation. == References ==