Theabsolute difference of tworeal numbers${\displaystyle x}$ and${\displaystyle y}$ is given by${\displaystyle |x-y|}$, theabsolute value of theirdifference. It describes the distance on thereal line between the points corresponding to${\displaystyle x}$ and${\displaystyle y}$, and is a special case of theL^p distance for all${\displaystyle 1\le p\le \mathrm{\infty }}$. Its applications in statistics include theabsolute deviation from acentral tendency.

Absolute difference has the following properties:

• For${\displaystyle x\ge 0}$,${\displaystyle |x-0|=x}$ (zero is theidentity element on non-negative numbers)^[1]
• For all${\displaystyle x}$,${\displaystyle |x-x|=0}$ (every element is its owninverse element)^[1]
• ${\displaystyle |x-y|\ge 0}$ (non-negativity)^[2]
• ${\displaystyle |x-y|=0}$ if and only if${\displaystyle x=y}$ (nonzero for distinct arguments).^[2]
• ${\displaystyle |x-y|=|y-x|}$ (symmetry orcommutativity).^[1][2]
• ${\displaystyle |x-z|\le |x-y|+|y-z|}$ (thetriangle inequality);^[2][3] equality holds if and only if${\displaystyle x\le y\le z}$ or${\displaystyle x\ge y\ge z}$.

Because it is non-negative, nonzero for distinct arguments, symmetric, and obeys the triangle inequality, the real numbers form ametric 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 theL^p distances for all${\displaystyle 1\le p\le \mathrm{\infty }}$, including the${\displaystyle p=1}$ and${\displaystyle p=2}$ cases (taxicab geometry andEuclidean distance, respectively). It is also the one-dimensional special case ofhyperbolic distance.

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

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 acommutative magma with identity.^[1]

The absolute difference is used to define therelative difference, the absolute difference between a given value and a reference value divided by the reference value itself.^[6]

In the theory ofgraceful labelings ingraph theory, vertices are labeled bynatural 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 defineChebyshev distance (L^∞), in which the distance between points is the maximum or supremum of the absolute differences of their coordinates.^[8]

In statistics, theabsolute deviation of a sampled number from acentral tendency is its absolute difference from the center, theaverage absolute deviation is the average of the absolute deviations of a collection of samples, andleast absolute deviations is a method forrobust statistics based on minimizing the average absolute deviation.

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

• Real numbers
• Distance

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