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

Properties

Absolute difference has the following properties:

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 Lp distances for all 1\le p\le\infty, including the p=1 and p=2 cases (taxicab geometry and Euclidean distance, respectively). It is also the one-dimensional special case of hyperbolic distance.

Instead of |x-y|, the absolute difference may also be expressed as

\max(x,y)-\min(x,y).

Generalizing this to more than two values, in any subset S of the real numbers which has an infimum and a supremum, the absolute difference between any two numbers in S is less or equal than the absolute difference of the infimum and supremum of 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]

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.[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 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.[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.

References

  1. ^ Talukdar, 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
  2. ^ Kubrusly, Carlos S. (2001). Elements of Operator Theory. Boston: Birkhäuser. p. 86. doi:10.1007/978-1-4757-3328-0. ISBN 9781475733280.
  3. ^ Khamsi, Mohamed A. & Kirk, William A. (2011). "1.3 The triangle inequality in \R". An Introduction to Metric Spaces and Fixed Point Theory. John Wiley & Sons. pp. 7–8. ISBN 9781118031322.
  4. ^ Georgiev, Svetlin G. & Zennir, Khaled (2019). Functional Analysis with Applications. Walter de Gruyter GmbH. p. 25. ISBN 9783110657722.
  5. ^ Khamsi & Kirk (2011), p. 14.
  6. ^ Reba, Marilyn A. & Shier, Douglas R. (2014). Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking. CRC Press. p. 463. ISBN 9781482297935.
  7. ^ Golomb, Solomon W. (1972). "How to number a graph". Graph Theory and Computing. Read, Ronald C. (ed.). Academic Press. pp. 23–37. doi:10.1016/B978-1-4832-3187-7.50008-8. MR 340107.
  8. ^ Webb, Andrew R. (2003). Statistical Pattern Recognition. 2nd ed. John Wiley & Sons. p. 421. ISBN 9780470854785.