Relative change and difference

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. The comparison is expressed as a ratio and is a unitless number. By multiplying these ratios by 100 they can be expressed as percentages so the terms percentage change, percent(age) difference, or relative percentage difference are also commonly used. The distinction between "change" and "difference" depends on whether or not one of the quantities being compared is considered a standard or reference or starting value. When this occurs, the term relative change (with respect to the reference value) is used and otherwise the term relative difference is preferred. Relative difference is often used as a quantitative indicator of quality assurance and quality control for repeated measurements where the outcomes are expected to be the same. 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).

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 x_reference:

{\text{Relative change}}(x,x_{\text{reference}})={\frac {\text{Actual change}}{x_{\text{reference}}}}={\frac {\Delta }{x_{\text{reference}}}}={\frac {x-x_{\text{reference}}}{x_{\text{reference}}}}.

The relative change is not defined if the reference value (x_reference) 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 x_reference is positive, and reverses this behavior if x_reference 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 x_reference:

{\text{Relative change}}(x,x_{\text{reference}})={\frac {\text{Actual change}}{|x_{\text{reference}}|}}={\frac {\Delta }{|x_{\text{reference}}|}}={\frac {x-x_{\text{reference}}}{|x_{\text{reference}}|}}.

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).^[1]

{\text{Relative difference}}(x,y)={\frac {\text{Absolute difference}}{|f(x,y)|}}={\frac {|\Delta |}{|f(x,y)|}}=\left|{\frac {x-y}{f(x,y)}}\right|.

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.

Formulae

Measures of relative difference are unitless numbers expressed as a fraction. Corresponding values of percent difference would be obtained by multiplying these values by 100 (and appending the % sign to indicate that the value is a percentage).

One way to define the relative difference of two numbers is to take their absolute difference divided by the maximum absolute value of the two numbers.

d_{r}={\frac {|x-y|}{\max(|x|,|y|)}}

if at least one of the values does not equal zero. This approach is especially useful when comparing floating point values in programming languages for equality with a certain tolerance.^[2] Another application is in the computation of approximation errors when the relative error of a measurement is required.

Another way to define the relative difference of two numbers is to take their absolute difference divided by some functional value of the two numbers, for example, the absolute value of their arithmetic mean:

d_{r}={\frac {|x-y|}{\left({\frac {|x+y|}{2}}\right)}}\,.

This approach is often used when the two numbers reflect a change in some single underlying entity.^{[citation needed]} A problem with the above approach arises when the functional value is zero. In this example, if x and y have the same magnitude but opposite sign, then

{\frac {|x+y|}{2}}=0,

which causes division by 0. So it may be better to replace the denominator with the average of the absolute values of x and y:^{[citation needed]}

d_{r}={\frac {|x-y|}{\left({\frac {|x|+|y|}{2}}\right)}}\,.

Percent error

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.

\%{\text{ Error}}={\frac {|{\text{Experimental}}-{\text{Theoretical}}|}{|{\text{Theoretical}}|}}\times 100.

The terms "Experimental" and "Theoretical" used in the equation above are commonly replaced with similar terms. Other terms used for experimental could be "measured," "calculated," or "actual" and another term used for theoretical could be "accepted." Experimental value is what has been derived by use of calculation and/or measurement and is having its accuracy tested against the theoretical value, a value that is accepted by the scientific community or a value that could be seen as a goal for a successful result.

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. Thus, if an experimental value is less than the theoretical value, the percent error will be negative. This negative result provides additional information about the experimental 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:

\%{\text{ Error}}={\frac {{\text{Experimental}}-{\text{Theoretical}}}{|{\text{Theoretical}}|}}\times 100.

It is important to note that the two values in the numerator do not commute. Therefore, it is vital to preserve the order as above: subtract the theoretical value from the experimental value and not vice versa.

Percentage change

A percentage change is a way to express a change in a variable. It represents the relative change between the old value and the new one.^[3]

For example, if a house is worth $100,000 today and the year after its value goes up to $110,000, the percentage change of its value can be expressed as

{\frac {110000-100000}{100000}}=0.1=10\%.

It can then be said that the worth of the house went up by 10%.

More generally, if V₁ represents the old value and V₂ the new one,

{\text{Percentage change}}={\frac {\Delta V}{V_{1}}}={\frac {V_{2}-V_{1}}{V_{1}}}\times 100\%.

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.

Example of percentages of percentages

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 ambiguous. The absolute change in this situation is 1 percentage point (4% − 3%), but the relative change in the interest rate is:

{\frac {4\%-3\%}{3\%}}=0.333\ldots =33{\frac {1}{3}}\%.

In general, the term "percentage point(s)" indicates an absolute change or difference of percentages, while the percent sign or the word "percentage" refers to the relative change or difference.^[4]

Examples

Comparisons

Car M costs $50,000 and car L costs $40,000. We wish to compare these costs.^[5] With respect to car L, the absolute difference is $10,000 = $50,000 − $40,000. That is, car M costs $10,000 more than car L. The relative difference is,

{\frac {\$10,000}{\$40,000}}=0.25=25\%,

and we say that car M costs 25% more than car L. It is also common to express the comparison as a ratio, which in this example is,

{\frac {\$50,000}{\$40,000}}=1.25=125\%,

and we say that car M costs 125% of the cost of car L.

In this example the cost of car L was considered the reference value, but we could have made the choice the other way and considered the cost of car M as the reference value. The absolute difference is now −$10,000 = $40,000 − $50,000 since car L costs $10,000 less than car M. The relative difference,

{\frac {-\$10,000}{\$50,000}}=-0.20=-20\%

is also negative since car L costs 20% less than car M. The ratio form of the comparison,

{\frac {\$40,000}{\$50,000}}=0.8=80\%

says that car L costs 80% of what car M costs.

It is the use of the words "of" and "less/more than" that distinguish between ratios and relative differences.^[6]

Logarithmic scale

Change in a quantity can also be expressed as the natural logarithm (ln) of the ratio of the two numbers, called log change.^[1] Indeed, when $\left|{\frac {V_{1}-V_{0}}{V_{0}}}\right|\ll 1$ , the following approximation holds:

\ln {\frac {V_{1}}{V_{0}}}=\int _{V_{0}}^{V_{1}}{\frac {{\mathrm {d} }V}{V}}\approx \int _{V_{0}}^{V_{1}}{\frac {{\mathrm {d} }V}{V_{0}}}={\frac {V_{1}-V_{0}}{V_{0}}}={\text{relative change}}

In the same way that relative change is scaled by 100 to get percentages, $\ln {\frac {V_{1}}{V_{0}}}$ can be scaled by 100 to get what is commonly called log points.^[7] Log points are equivalent to the unit centinepers (cNp) when measured for root-power quantities.^[8]^[9] This quantity has also been referred to as a log percentage and denoted L%.^[1] Since the derivative of the natural log at 1 is 1, log points are approximately equal to percentage difference for small differences – for example an increase of 1% equals an increase of 0.995 cNp, and a 5% increase gives a 4.88 cNp increase. This approximation property does not hold for other choices of logarithm base, which introduce a scaling factor due to the derivative not being 1. Log points can thus be used as a replacement for percentage differences.^[10]^[8]

Using log change has the advantages of additivity compared to relative change.^[1]^[8]

Additivity

When using log change, the total change after a series of changes equals the sum of the changes. With percent, summing the changes is only an approximation, with larger error for larger changes.^[8] For example:

Log change 0 (cNp)	Log change 1 (cNp)	Total log change (cNp)	Relative change 0 (%)	Relative change 1 (%)	Total relative change (%)
10	5	15	10	5	15.5
10	−5	5	10	−5	4.5
10	10	20	10	10	21
10	−10	0	10	−10	−1
50	50	100	50	50	125
50	−50	0	50	−50	−25

Note that in the above table, since relative change 0 (respectively relative change 1) has the same numerical value as log change 0 (respectively log change 1), it does not correspond to the same variation. The conversion between relative and log changes may be computed as ${\text{log change}}=\ln(1+{\text{relative change}})$ .

By additivity, $\ln {\frac {V_{1}}{V_{0}}}+\ln {\frac {V_{0}}{V_{1}}}=0$ , and therefore additivity implies a sort of symmetry property, namely $\ln {\frac {V_{1}}{V_{0}}}=-\ln {\frac {V_{0}}{V_{1}}}$ and thus the magnitude of a change expressed in log change is the same whether V₀ or V₁ is chosen as the reference.^[8] In contrast, for relative change, ${\frac {V_{1}-V_{0}}{V_{0}}}\neq -{\frac {V_{0}-V_{1}}{V_{1}}}$ , with the difference ${\frac {(V_{1}-V_{0})^{2}}{V_{0}V_{1}}}$ becoming larger as V₁ or V₀ approaches 0 while the other remains fixed. For example:

V₀	V₁	Log change (cNp)	Relative change (%)
10	9	−10.5	−10.0
9	10	+10.5	+11.1
10	1	−230	−90
1	10	+230	+900
10	0⁺	−∞	−100
0⁺	10	+∞	+∞

Here 0⁺ means taking the limit from above towards 0.

Uniqueness and extensions

The log change is the unique two-variable function that is additive, and whose linearization matches relative change. There is a family of additive difference functions $F_{\lambda }(x,y)$ for any $\lambda \in \mathbb {R}$ , such that absolute change is $F_{0}$ and log change is $F_{1}$ .^[11]

Notes

^ ^a ^b ^c ^d Törnqvist, Vartia & Vartia 1985.
^ What's a good way to check for close enough floating-point equality
^ Kazmi, Kumail (March 26, 2021). "Percentage Increase Calculator". Smadent - Best Educational Website of Pakistan. Smadent Publishing. Retrieved March 26, 2021.
^ Bennett & Briggs 2005, p. 141
^ Bennett & Briggs 2005, pp. 137–139
^ Bennett & Briggs 2005, p.140
^ Békés, Gábor; Kézdi, Gábor (6 May 2021). Data Analysis for Business, Economics, and Policy. Cambridge University Press. p. 203. ISBN 978-1-108-48301-8.
^ ^a ^b ^c ^d ^e Karjus, Andres; Blythe, Richard A.; Kirby, Simon; Smith, Kenny (10 February 2020). "Quantifying the dynamics of topical fluctuations in language". Language Dynamics and Change. 10 (1). Section A.3.1. doi:10.1163/22105832-01001200. S2CID 46928080.
^ Roe, John; deForest, Russ; Jamshidi, Sara (26 April 2018). Mathematics for Sustainability. Springer. p. 190. doi:10.1007/978-3-319-76660-7_4. ISBN 978-3-319-76660-7.
^ Doyle, Patrick (2016-08-24). "The Case for a Logarithmic Performance Metric". Vena Solutions.
^ Brauen, Silvan; Erpf, Philipp; Wasem, Micha (2020). "On Absolute and Relative Change" (PDF). SSRN Electronic Journal. arXiv:2011.14807. doi:10.2139/ssrn.3739890. S2CID 227228720.

References

Bennett, Jeffrey; Briggs, William (2005), Using and Understanding Mathematics: A Quantitative Reasoning Approach (3rd ed.), Boston: Pearson, ISBN 0-321-22773-5
"Understanding Measurement and Graphing" (PDF). North Carolina State University. 2008-08-20. Archived from the original (PDF) on 2010-06-15. Retrieved 2010-05-05.
"Percent Difference – Percent Error" (PDF). Illinois State University, Dept of Physics. 2004-07-20. Retrieved 2010-05-05.
Törnqvist, Leo; Vartia, Pentti; Vartia, Yrjö (1985), "How Should Relative Changes Be Measured?", The American Statistician, 39 (1): 43–46, doi:10.2307/2683905, JSTOR 2683905

[FOOTNOTETörnqvistVartiaVartia1985-1] Törnqvist, Vartia & Vartia 1985.

[2] What's a good way to check for close enough floating-point equality

[3] Kazmi, Kumail (March 26, 2021). "Percentage Increase Calculator". Smadent - Best Educational Website of Pakistan. Smadent Publishing. Retrieved March 26, 2021.

[4] Bennett & Briggs 2005, p. 141

[5] Bennett & Briggs 2005, pp. 137–139

[6] Bennett & Briggs 2005, p.140

[7] Békés, Gábor; Kézdi, Gábor (6 May 2021). Data Analysis for Business, Economics, and Policy. Cambridge University Press. p. 203. ISBN 978-1-108-48301-8.

[Blythe2020-8] Karjus, Andres; Blythe, Richard A.; Kirby, Simon; Smith, Kenny (10 February 2020). "Quantifying the dynamics of topical fluctuations in language". Language Dynamics and Change. 10 (1). Section A.3.1. doi:10.1163/22105832-01001200. S2CID 46928080.

[9] Roe, John; deForest, Russ; Jamshidi, Sara (26 April 2018). Mathematics for Sustainability. Springer. p. 190. doi:10.1007/978-3-319-76660-7_4. ISBN 978-3-319-76660-7.

[10] Doyle, Patrick (2016-08-24). "The Case for a Logarithmic Performance Metric". Vena Solutions.

[11] Brauen, Silvan; Erpf, Philipp; Wasem, Micha (2020). "On Absolute and Relative Change" (PDF). SSRN Electronic Journal. arXiv:2011.14807. doi:10.2139/ssrn.3739890. S2CID 227228720.

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