Absolute Difference

An Absolute Difference between two numbers is the Absolute Value of the subtraction between the two, i.e. [math]\displaystyle{ |A-B| }[/math] .

• Example(s):
  • [math]\displaystyle{ |3-5|= |-5|=5 }[/math]
• Counter-Example(s):
  • Absolute Deviation.
  • Relative Difference

See: Absolute Value, Number, Mean Deviation.

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References

2016

• (Wikipedia, 2016) ⇒ http://www.wikiwand.com/en/Absolute_difference Retrieved 2016-07-10
  • The absolute difference of two real numbers x, 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. It is a special case of the L^p distance for all 1 ≤ p ≤ ∞ and is the standard metric used for both the set of rational numbers Q and their completion, the set of real numbers R.

    As with any metric, the metric properties hold:

• |x − y| ≥ 0, since absolute value is always non-negative.
• |x − y| = 0   if and only if   x = y.
• |x − y| = |y − x|     (symmetry or commutativity).
• |x − z| ≤ |x − y| + |y − z|     (triangle inequality); in the case of the absolute ::difference, equality holds if and only if x ≤ y ≤ z.

By contrast, simple subtraction is not non-negative or commutative, but it does obey the second and fourth properties above, since x − y = 0 if and only if x = y, and x − z = (x − y) + (y − z).

The absolute difference is used to define other quantities including the relative difference, the L¹ norm used in taxicab geometry, and graceful labelings in graph theory.

When it is desirable to avoid the absolute value function – for example because it is expensive to compute, or because its derivative is not continuous – it can sometimes be eliminated by the identity

|x − y| < |z − w| if and only if (x − y)² < (z − w)².

This follows since |x − y|² = (x − y)² and squaring is monotonic on the nonnegative reals.

• (Eric W. Weisstein, 2016) ⇒ Weisstein, Eric W. “Absolute Difference.” From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/AbsoluteDifference.html Retrieved 2016-07-10
  • The absolute difference of two numbers [math]\displaystyle{ n_1 }[/math] and [math]\displaystyle{ n_2 }[/math] is [math]\displaystyle{ |n_1-n_2| }[/math], where the minus sign denotes subtraction and |x| denotes the absolute value.

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised.” Oxford University Press. ISBN:0199541450

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