Absolute difference

The absolute difference of two real numbers ${\displaystyle x}${\displaystyle x} and ${\displaystyle y}${\displaystyle y} is given by ${\displaystyle |x-y|}${\displaystyle |x-y|}, the absolute value of their difference. It describes the distance on the real line between the points corresponding to ${\displaystyle x}${\displaystyle x} and ${\displaystyle y}${\displaystyle y}, and is a special case of the L^p distance for all ${\displaystyle 1\leq p\leq \infty }${\displaystyle 1\leq p\leq \infty }. Its applications in statistics include the absolute deviation from a central tendency.

Absolute difference has the following properties:

• For ${\displaystyle x\geq 0}${\displaystyle x\geq 0}, ${\displaystyle |x-0|=x}${\displaystyle |x-0|=x} (zero is the identity element on non-negative numbers)^[1]
• For all ${\displaystyle x}${\displaystyle x}, ${\displaystyle |x-x|=0}${\displaystyle |x-x|=0} (every element is its own inverse element)^[1]
• ${\displaystyle |x-y|\geq 0}${\displaystyle |x-y|\geq 0} (non-negativity)^[2]
• ${\displaystyle |x-y|=0}${\displaystyle |x-y|=0} if and only if ${\displaystyle x=y}${\displaystyle x=y} (nonzero for distinct arguments).^[2]
• ${\displaystyle |x-y|=|y-x|}${\displaystyle |x-y|=|y-x|} (symmetry or commutativity ).^{[1] [2]}
• ${\displaystyle |x-z|\leq |x-y|+|y-z|}${\displaystyle |x-z|\leq |x-y|+|y-z|} (the triangle inequality );^{[2] [3]} equality holds if and only if ${\displaystyle x\leq y\leq z}${\displaystyle x\leq y\leq z} or ${\displaystyle x\geq y\geq z}${\displaystyle x\geq y\geq 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.^[4] It has been called "the most natural metric space",^[5] and "the most important concrete metric space".^[2] This distance generalizes in many different ways to higher dimensions, as a special case of the L^p distances for all ${\displaystyle 1\leq p\leq \infty }${\displaystyle 1\leq p\leq \infty }, including the ${\displaystyle p=1}${\displaystyle p=1} and ${\displaystyle p=2}${\displaystyle p=2} cases (taxicab geometry and Euclidean distance, respectively). It is also the one-dimensional special case of hyperbolic distance.

Instead of ${\displaystyle |x-y|}${\displaystyle |x-y|}, the absolute difference may also be expressed as $${\displaystyle \max(x,y)-\min(x,y).}$${\displaystyle \max(x,y)-\min(x,y).} Generalizing this to more than two values, in any subset ${\displaystyle S}${\displaystyle S} of the real numbers which has an infimum and a supremum, the absolute difference between any two numbers in ${\displaystyle S}${\displaystyle S} is less or equal then the absolute difference of the infimum and supremum of ${\displaystyle S}${\displaystyle S}.

The absolute difference takes non-negative integers to non-negative integers. As a binary operation that is commutative but not associative, with an identity element on the non-negative numbers, the absolute difference gives the non-negative numbers (whether real or integer) the algebraic structure of a commutative magma with identity.^[1]

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.^[6]

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.^[7]

As well as being a special case of the L^p 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.^[8]

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.

• Weisstein, Eric W. "Absolute Difference". MathWorld .

• Real numbers
• Distance

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