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

• Properties

• Applications

• References

• External links

Absolute difference has the following properties:

• For x ≥ ≥ 0 {\displaystyle x\geq 0} , | x − − 0 | = x {\displaystyle |x-0|=x} (zero is the identity element on non-negative numbers)cite-ref-taldas-1-0[1]

• For all x {\displaystyle x} , | x − − x | = 0 {\displaystyle |x-x|=0} (every element is its own inverse element)cite-ref-taldas-1-1[1]

• | x − − y | ≥ ≥ 0 {\displaystyle |x-y|\geq 0} (non-negativity)cite-ref-kubrusly-2-0[2]

• | x − − y | = 0 {\displaystyle |x-y|=0} if and only if x = y {\displaystyle x=y} (nonzero for distinct arguments).cite-ref-kubrusly-2-1[2]

• | x − − y | = | y − − x | {\displaystyle |x-y|=|y-x|} (symmetry or commutativity).cite-ref-taldas-1-2[1]cite-ref-kubrusly-2-2[2]

• | x − − z | ≤ ≤ | x − − y | + | y − − z | {\displaystyle |x-z|\leq |x-y|+|y-z|} (the triangle inequality);cite-ref-kubrusly-2-3[2]cite-ref-3[3] equality holds if and only if x ≤ ≤ y ≤ ≤ z {\displaystyle x\leq y\leq z} or x ≥ ≥ y ≥ ≥ 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.cite-ref-4[4] It has been called "the most natural metric space",cite-ref-footnotekhamsikirk201114-5-0[5] and "the most important concrete metric space".cite-ref-kubrusly-2-4[2] This distance generalizes in many different ways to higher dimensions, as a special case of the Lp distances for all 1 ≤ ≤ p ≤ ≤ ∞ ∞ {\displaystyle 1\leq p\leq \infty } , including the p = 1 {\displaystyle p=1} and 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 | x − − y | {\displaystyle |x-y|} , the absolute difference may also be expressed as max ( x , y ) − − min ( x , y ) . {\displaystyle \max(x,y)-\min(x,y).} Generalizing this to more than two values, in any subset S {\displaystyle S} of the real numbers which has an infimum and a supremum, the absolute difference between any two numbers in S {\displaystyle S} is less or equal then the absolute difference of the infimum and supremum of 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.cite-ref-taldas-1-3[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.cite-ref-6[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.cite-ref-7[7]

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.cite-ref-8[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.

cite-note-taldas-11. ↑ citereftalukdardas1996Talukdar, D.; Das, N. R. (July 1996). "80.33 Measuring associativity in a groupoid of natural numbers". The Mathematical Gazette. 80 (488): 401–404. doi:10.2307/3619592. JSTOR 3619592.

cite-note-kubrusly-22. ↑ citerefkubrusly2001Kubrusly, Carlos S. (2001). Elements of Operator Theory. Boston: Birkhäuser. p. 86. doi:10.1007/978-1-4757-3328-0. ISBN 9781475733280.

cite-note-33. ↑ citerefkhamsikirk2011Khamsi, Mohamed A.; Kirk, William A. (2011). "1.3 The triangle inequality in R {\displaystyle \mathbb {R} } ". An Introduction to Metric Spaces and Fixed Point Theory. John Wiley & Sons. pp. 7–8. ISBN 9781118031322.

cite-note-44. ↑ citerefgeorgievzennir2019Georgiev, Svetlin G.; Zennir, Khaled (2019). Functional Analysis with Applications. Walter de Gruyter GmbH. p. 25. ISBN 9783110657722.

cite-note-footnotekhamsikirk201114-55. ↑ Khamsi & Kirk (2011), p. 14.

cite-note-66. ↑ citerefrebashier2014Reba, Marilyn A.; Shier, Douglas R. (2014). Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking. CRC Press. p. 463. ISBN 9781482297935.

cite-note-77. ↑ citerefgolomb1972Golomb, Solomon W. (1972). "How to number a graph". In Read, Ronald C. (ed.). Graph Theory and Computing. Academic Press. pp. 23–37. doi:10.1016/B978-1-4832-3187-7.50008-8. MR 0340107.

cite-note-88. ↑ citerefwebb2003Webb, Andrew R. (2003). Statistical Pattern Recognition (2nd ed.). John Wiley & Sons. p. 421. ISBN 9780470854785.

• reference-mathworld-absolute-differenceciterefweissteinWeisstein, Eric W. "Absolute Difference". MathWorld.