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In anyquantitative science, the termsrelative change andrelative difference are used to compare twoquantities while taking into account the "sizes" of the things being compared, i.e. dividing by astandard orreference orstarting value.^[1] The comparison is expressed as aratio and is aunitlessnumber. By multiplying these ratios by 100 they can be expressed aspercentages so the termspercentage change,percent(age) difference, orrelative percentage difference are also commonly used. The terms "change" and "difference" are used interchangeably.^[2]

Relative change is often used as a quantitative indicator ofquality assurance andquality 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) calledpercent 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).

The relative change formula is not well-behaved under many conditions. Various alternative formulas, calledindicators of relative change, have been proposed in the literature. Several authors have foundlog change andlog points to be satisfactory indicators, but these have not seen widespread use.^[3]

Given two numerical quantities,v_ref andv withv_ref somereference value, theiractual change,actual difference, orabsolute change is

Δv =v −v_ref.

The termabsolute difference is sometimes also used even though the absolute value is not taken; the sign ofΔ typically is uniform, e.g. across an increasing data series. 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 value 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. The actual difference is not usually a good way to compare the numbers, in particular because it depends on the unit of measurement. For instance,1 m is the same as100 cm, but the absolute difference between2 and 1 m is 1 while the absolute difference between200 and 100 cm is 100, giving the impression of a larger difference.^[4] But even with constant units, the relative change helps judge the importance of the respective change. For example, an increase in price of$100 of a valuable is considered big if changing from$50 to 150 but rather small when changing from$10,000 to 10,100.

We can adjust the comparison to take into account the "size" of the quantities involved, by defining, for positive values ofv_ref :

$${\displaystyle \text{relative change}(v_{\text{ref}},v)=\frac{\text{actual change}}{\text{reference value}}=\frac{\mathrm{\Delta }v}{v_{\text{ref}}}=\frac{v}{v_{\text{ref}}}-1.}$$

The relative change is independent of the unit of measurement employed; for example, the relative change from2 to 1 m is−50%, the same as for200 to 100 cm. The relative change is not defined if the reference value (v_ref) is zero, and gives negative values for positive increases ifv_ref is negative, hence it is not usually defined for negative reference values either. For example, we might want to calculate the relative change of −10 to −6. The above formula gives⁠(−6) − (−10)/ −10⁠ =⁠4/ −10⁠ = −0.4, indicating a decrease, yet in fact the reading increased.

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

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 ofv_ref:

$${\displaystyle \text{Relative change}(v_{\text{ref}},v)=\frac{v-v_{\text{ref}}}{|v_{\text{ref}}|}.}$$

This still does not solve the issue when the reference is zero. It is common to instead use an indicator of relative change, and take the absolute values of bothv and${\displaystyle v_{\text{reference}}}$. Then the only problematic case is${\displaystyle v=v_{\text{reference}}=0}$, which can usually be addressed by appropriately extending the indicator. For example, for arithmetic mean this formula may be used:^[5]$${\displaystyle d_r(x,y)=\frac{|x-y|}{(|x|+|y|)/2},d_r(0,0)=0}$$

Apercentage change is a way to express a change in a variable. It represents the relative change between the old value and the new one.^[6]

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

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

More generally, ifV₁ represents the old value andV₂ the new one,$${\displaystyle \text{Percentage change}=\frac{\mathrm{\Delta }V}{V_1}=\frac{V_2-V_1}{V_1}\times 100\mathrm{\%}.}$$

Some calculators directly support this via a%CH orΔ% function.

When the variable in question is a percentage itself, it is better to talk about its change by usingpercentage points, to avoid confusion betweenrelative difference andabsolute difference.

Thepercent 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.

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

The terms "Experimental" and "Theoretical" used in the equation above are commonly replaced with similar terms. Other terms used forexperimental could be "measured," "calculated," or "actual" and another term used fortheoretical 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 thetheory of relativity) and is a newsworthy result.

The percent error equation, when rewritten by removing the absolute values, becomes:$${\displaystyle \mathrm{\%}\text{ Error}=\frac{\text{Experimental}-\text{Theoretical}}{|\text{Theoretical}|}\times 100.}$$

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

Suppose that carM costs $50,000 and carL costs $40,000. We wish to compare these costs.^[7] With respect to carL, the absolute difference is$10,000 = $50,000 − $40,000. That is, carM costs $10,000 more than carL. The relative difference is,$${\displaystyle \frac{\mathrm{\$}10,000}{\mathrm{\$}40,000}=0.25=25\mathrm{\%},}$$and we say that carM costs 25%more than carL. It is also common to express the comparison as a ratio, which in this example is,$${\displaystyle \frac{\mathrm{\$}50,000}{\mathrm{\$}40,000}=1.25=125\mathrm{\%},}$$and we say that carM costs 125%of the cost of carL.

In this example the cost of carL was considered the reference value, but we could have made the choice the other way and considered the cost of carM as the reference value. The absolute difference is now−$10,000 = $40,000 − $50,000 since carL costs $10,000 less than carM. The relative difference,$${\displaystyle \frac{-\mathrm{\$}10,000}{\mathrm{\$}50,000}=-0.20=-20\mathrm{\%}}$$is also negative since carL costs 20%less than carM. The ratio form of the comparison,$${\displaystyle \frac{\mathrm{\$}40,000}{\mathrm{\$}50,000}=0.8=80\mathrm{\%}}$$says that carL costs 80%of what carM costs.

It is the use of the words "of" and "less/more than" that distinguish between ratios and relative differences.^[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. The absolute change in this situation is 1 percentage point (4% − 3%), but the relative change in the interest rate is:$${\displaystyle \frac{4\mathrm{\%}-3\mathrm{\%}}{3\mathrm{\%}}=0.333\dots =33\frac{1}{3}\mathrm{\%}.}$$

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.^[9]

The (classical) relative change above is but one of the possible measures/indicators of relative change. Anindicator of relative change fromx (initial or reference value) toy (new value)${\displaystyle R(x,y)}$ is a binary real-valued function defined for the domain of interest which satisfies the following properties:^[10]

• Appropriate sign:
• R is an increasing function ofy whenx is fixed.
• R is continuous.
• Independent of the unit of measurement: for all${\displaystyle a>0}$,${\displaystyle R(ax,ay)=R(x,y)}$.
• Normalized:${\displaystyle {\frac{d}{dy}R(1,y)|}_{y=1}=1}$

The normalization condition is motivated by the observation thatR scaled by a constant${\displaystyle c>0}$ still satisfies the other conditions besides normalization. Furthermore, due to the independence condition, everyR can be written as a single argument functionH of the ratio${\displaystyle y/x}$.^[11] The normalization condition is then that${\displaystyle H^{\prime }(1)=1}$. This implies all indicators behave like the classical one when${\displaystyle y/x}$ is close to1.

Usually the indicator of relative change is presented as the actual change Δ scaled by some function of the valuesx andy, sayf(x,y).^[2]

$${\displaystyle \text{Relative change}(x,y)=\frac{\text{Actual change}\mathrm{\Delta }}{f(x,y)}=\frac{y-x}{f(x,y)}.}$$

As with classical relative change, the general relative change is undefined iff(x,y) is zero. Various choices for the functionf(x,y) have been proposed:^[12]

Indicators of relative change^[12]

Name	${\displaystyle f(x,y)}$ where the indicator's value is${\displaystyle \frac{y-x}{f(x,y)}}$	${\displaystyle H(y/x)}$
(Classical) Relative change	x	${\displaystyle \frac{y}{x}-1}$
Reversed relative change	y	${\displaystyle 1-\frac{x}{y}}$
Arithmetic mean change	${\displaystyle \frac{1}{2}(x+y)}$	${\displaystyle \frac{\frac{y}{x}-1}{\frac{1}{2}\left(1+\frac{y}{x}\right)}}$
Geometric mean change	${\displaystyle \sqrt{xy}}$	${\displaystyle \frac{\frac{y}{x}-1}{\sqrt{xy}}}$
Harmonic mean change	${\displaystyle \frac{2}{\frac{1}{x}+\frac{1}{y}}}$	${\displaystyle \frac{\left(\frac{y}{x}-1\right)\left(1+\frac{x}{y}\right)}{2}}$
Moment mean change of orderk	${\displaystyle {\left[\frac{1}{2}(x^k+y^k)\right]}^{\frac{1}{k}}}$	${\displaystyle \frac{\frac{y}{x}-1}{{\left[\frac{1}{2}\left(1+{\left(\frac{y}{x}\right)}^k\right)\right]}^{\frac{1}{k}}}}$
Maximum mean change	${\displaystyle max(x,y)}$	${\displaystyle \frac{\frac{y}{x}-1}{max\left(1,\frac{y}{x}\right)}}$
Minimum mean change	${\displaystyle min(x,y)}$	${\displaystyle \frac{\frac{y}{x}-1}{min\left(1,\frac{y}{x}\right)}}$
Logarithmic (mean) change		${\displaystyle \ln \frac{y}{x}}$

As can be seen in the table, all but the first two indicators have, as denominator amean. One of the properties of a mean function${\displaystyle m(x,y)}$ is:^[12]${\displaystyle m(x,y)=m(y,x)}$, which means that all such indicators have a "symmetry" property that the classical relative change lacks:${\displaystyle R(x,y)=-R(y,x)}$. This agrees with intuition that a relative change fromx toy should have the same magnitude as a relative change in the opposite direction,y tox, just like the relation${\displaystyle \frac{y}{x}=\frac{1}{\frac{x}{y}}}$ suggests.

Maximum mean change has been recommended when comparingfloating point values inprogramming languages forequality with a certain tolerance.^[13] Another application is in the computation ofapproximation errors when the relative error of a measurement is required.^{[citation needed]} Minimum mean change has been recommended for use ineconometrics.^[14][15] Logarithmic change has been recommended as a general-purpose replacement for relative change and is discussed more below.

Tenhunen defines a general relative difference function fromL (reference value) toK:^[16]

which leads to

In particular for the special cases${\displaystyle c=\pm 1}$,

Of these indicators of relative change, arguably the most natural is thenatural logarithm (ln) of the ratio of the two numbers (final and initial), calledlog change.^[2] Indeed, when${\displaystyle \left|\frac{V_1-V_0}{V_0}\right|\ll 1}$, the following approximation holds:$${\displaystyle \ln \frac{V_1}{V_0}={\int }_{V_0}^{V_1}\frac{\mathrm{d}V}{V}\approx {\int }_{V_0}^{V_1}\frac{\mathrm{d}V}{V_0}=\frac{V_1-V_0}{V_0}=\text{classical relative change}}$$

In the same way that relative change is scaled by 100 to get percentages,${\displaystyle \ln \frac{V_1}{V_0}}$ can be scaled by 100 to get what is commonly calledlog points.^[17] Log points are equivalent to the unitcentinepers (cNp) when measured for root-power quantities.^[18][19] This quantity has also been referred to as a log percentage and denoted L%.^[2]Since the derivative of the natural log at 1 is 1, log points are approximately equal to percent change 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. Thisapproximation 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 percent change.^[20][18]

Using log change has the advantages of additivity compared to relative change.^[2][18] Specifically, 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.^[18] For example:

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

By additivity,${\displaystyle \ln \frac{V_1}{V_0}+\ln \frac{V_0}{V_1}=0}$, and therefore additivity implies a sort of symmetry property, namely${\displaystyle \ln \frac{V_1}{V_0}=-\ln \frac{V_0}{V_1}}$ and thus themagnitude of a change expressed in log change is the same whetherV₀ orV₁ is chosen as the reference.^[18] In contrast, for relative change,${\displaystyle \frac{V_1-V_0}{V_0}\ne -\frac{V_0-V_1}{V_1}}$, with the difference${\displaystyle \frac{(V_1-V_0{)}^2}{V_0{V}_1}}$ becoming larger asV₁ orV₀ approaches 0 while the other remains fixed. For example:

Here 0⁺ means taking thelimit from above towards 0.

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${\displaystyle F_{\lambda }(x,y)}$ for any${\displaystyle \lambda \in \mathbb{R}}$, such that absolute change is${\displaystyle F_0}$ and log change is${\displaystyle F_1}$.^[21]

• Approximation error
• Errors and residuals in statistics
• Relative standard deviation
• Logarithmic scale

• 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 fromthe original(PDF) on 2010-06-15. Retrieved2010-05-05.
• "Percent Difference – Percent Error"(PDF). Illinois State University, Dept of Physics. 2004-07-20. Archived fromthe original(PDF) on 2019-07-13. Retrieved2010-05-05.
• Törnqvist, Leo; Vartia, Pentti; Vartia, Yrjö (1985),"How Should Relative Changes Be Measured?"(PDF),The American Statistician,39 (1):43–46,doi:10.2307/2683905,JSTOR 2683905
• Tenhunen, Lauri (1990).The CES and par production techniques, income distribution and the neoclassical theory of production (PhD). A. Vol. 290. University of Tampere.
• Vartia, Yrjö O. (1976).Relative changes and index numbers(PDF). ETLA A 4. Helsinki: Research Institute of the Finnish Economy.ISBN 951-9205-24-1. Retrieved20 November 2022.

• Measurement
• Numerical analysis
• Ratios
• Subtraction
• Dimensionless quantities

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