In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a standard or reference or starting value.
A special case of percent change (relative change expressed as a percentage) called percent error occurs in measuring situations where the reference value is the accepted or actual value (perhaps theoretically determined) and the value being compared to it is experimentally determined (by measurement).
Various alternative formulas, called indicators of relative change, have been proposed in the literature.
[3] Given two numerical quantities, vref and v with vref some reference value, their actual change, actual difference, or absolute change is The term absolute difference is sometimes also used even though the absolute value is not taken; the sign of Δ typically is uniform, e.g. across an increasing data series.
The actual difference is not usually a good way to compare the numbers, in particular because it depends on the unit of measurement.
We can adjust the comparison to take into account the "size" of the quantities involved, by defining, for positive values of vref:
The above formula gives (−6) − (−10)/ −10 = 4/ −10 = −0.4, indicating a decrease, yet in fact the reading increased.
Measures of relative change are unitless numbers expressed as a fraction.
The domain restriction of relative change to positive numbers often poses a constraint.
To avoid this problem it is common to take the absolute value, so that the relative change formula works correctly for all nonzero values of vref:
It is common to instead use an indicator of relative change, and take the absolute values of both v and
Some calculators directly support this via a %CH or Δ% function.
When the variable in question is a percentage itself, it is better to talk about its change by using percentage points, to avoid confusion between relative difference and absolute difference.
The percent error is a special case of the percentage form of relative change calculated from the absolute change between the experimental (measured) and theoretical (accepted) values, and dividing by the theoretical (accepted) value.
Although it is common practice to use the absolute value version of relative change when discussing percent error, in some situations, it can be beneficial to remove the absolute values to provide more information about the result.
For example, experimentally calculating the speed of light and coming up with a negative percent error says that the experimental value is a velocity that is less than the speed of light.
This is a big difference from getting a positive percent error, which means the experimental value is a velocity that is greater than the speed of light (violating the theory of relativity) and is a newsworthy result.
The percent error equation, when rewritten by removing the absolute values, becomes:
[8] If a bank were to raise the interest rate on a savings account from 3% to 4%, the statement that "the interest rate was increased by 1%" would be incorrect and misleading.
is a binary real-valued function defined for the domain of interest which satisfies the following properties:[10] The normalization condition is motivated by the observation that R scaled by a constant
Furthermore, due to the independence condition, every R can be written as a single argument function H of the ratio
Various choices for the function f(x, y) have been proposed:[12] As can be seen in the table, all but the first two indicators have, as denominator a mean.
, which means that all such indicators have a "symmetry" property that the classical relative change lacks:
Maximum mean change has been recommended when comparing floating point values in programming languages for equality with a certain tolerance.
[citation needed] Minimum mean change has been recommended for use in econometrics.
Tenhunen defines a general relative difference function from L (reference value) to K:[16]
Of these indicators of relative change, the most natural arguably is the natural logarithm (ln) of the ratio of the two numbers (final and initial), called log change.
[17] Log points are equivalent to the unit centinepers (cNp) when measured for root-power quantities.
[18][19] This quantity has also been referred to as a log percentage and denoted L%.
This approximation property does not hold for other choices of logarithm base, which introduce a scaling factor due to the derivative not being 1.