Definitions
Given two numerical quantities, x and y, their difference, Δ = x - y, can be called their actual difference. When y is a reference value (a theoretical/actual/correct/accepted/optimal/starting, etc. value; the value that x is being compared to) then Δ is called their actual change. When there is no reference value, the sign of Δ has little meaning in the comparison of the two values since it doesn't matter which of the two values is written first, so one often works with |Δ| = |x - y|, the absolute difference instead of Δ, in these situations. Even when there is a reference value, if it doesn't matter whether the compared value is larger or smaller than the reference value, the absolute difference can be considered in place of the actual change.
The absolute difference between two values is not always a good way to compare the numbers. For instance, the absolute difference of 1 between 6 and 5 is more significant than the same absolute difference between 100,000,001 and 100,000,000. We can adjust the comparison to take into account the "size" of the quantities involved, by defining, for positive values of xreference:
The relative change is not defined if the reference value (xreference) is zero.
For values greater than the reference value, the relative change should be a positive number and for values that are smaller, the relative change should be negative. The formula given above behaves in this way only if xreference is positive, and reverses this behavior if xreference is negative. For example, if we are calibrating a thermometer which reads -6° C when it should read -10° C, this formula for relative change (which would be called relative error in this application) gives ((-6) - (-10)) / (-10) = 4/-10 = -0.4, yet the reading is too high. To fix this problem we alter the definition of relative change so that it works correctly for all nonzero values of xreference:
If the relationship of the value with respect to the reference value (that is, larger or smaller) does not matter in a particular application, the absolute difference may be used in place of the actual change in the above formula to produce a value for the relative change which is always non-negative.
Defining relative difference is not as easy as defining relative change since there is no "correct" value to scale the absolute difference with. As a result, there are many options for how to define relative difference and which one is used depends on what the comparison is being used for. In general we can say that the absolute difference |Δ| is being scaled by some function of the values x and y, say f(x,y).
As with relative change, the relative difference is undefined if f(x,y) is zero.
Several common choices for the function f(x, y) would be:
- max (|x|,|y|),
- max (x, y),
- min (|x|, |y|),
- min (x, y),
- (x + y)/2, and
- (|x| + |y|)/2.
Read more about this topic: Relative Change And Difference
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