Relative change

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. 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 terms "change" and "difference" are used interchangeably.Template:Sfn

Relative change 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).

The relative change formula is not well-behaved under many conditions. Various alternative formulas, called indicators of relative change, have been proposed in the literature. Several authors have found log change and log points to be satisfactory indicators, but these have not seen widespread use.

Definition

Given two numerical quantities, Template:Math and Template:Mvar with Template:Math some reference value, their actual change, actual difference, or absolute change is $$\mathrm{\Delta }v=v-v_{\mathrm{r}\mathrm{e}\mathrm{f}}.$$ 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. 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, Template:Val is the same as Template:Val, but the absolute difference between Template:Val is 1 while the absolute difference between Template:Val is 100, giving the impression of a larger difference.Template:Sfn But even with constant units, the relative change helps judge the importance of the respective change. For example, an increase in price of Template:Val of a valuable is considered big if changing from Template:Val but rather small when changing from Template:Val.

We can adjust the comparison to take into account the "size" of the quantities involved, by defining, for positive values of Template:Math: $$\text{relative change}(v_{\mathrm{r}\mathrm{e}\mathrm{f}},v)=\frac{\text{actual change}}{\text{reference value}}=\frac{\mathrm{\Delta }v}{v_{\mathrm{r}\mathrm{e}\mathrm{f}}}=\frac{v}{v_{\mathrm{r}\mathrm{e}\mathrm{f}}}-1.$$

The relative change is independent of the unit of measurement employed; for example, the relative change from Template:Val is Template:Val, the same as for Template:Val. The relative change is not defined if the reference value (Template:Math) is zero, and gives negative values for positive increases if Template:Math is negative, hence it is not usually defined for negative reference values either. For example, we might want to calculate the relative change of Template:Val. The above formula gives Template:Math, indicating a decrease, yet in fact the reading increased.

Measures of relative change are unitless numbers expressed as a fraction. 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).

Domain

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 Template:Tmath: $$\text{Relative change}(v_{\mathrm{r}\mathrm{e}\mathrm{f}},v)=\frac{v-v_{\mathrm{r}\mathrm{e}\mathrm{f}}}{|v_{\mathrm{r}\mathrm{e}\mathrm{f}}|}.$$

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 both Template:Tmath and Template:Tmath. Then the only problematic case is Template:Tmath, which can usually be addressed by appropriately extending the indicator. For example, for arithmetic mean this formula may be used: $$d_r(x,y)=\frac{|x-y|}{(|x|+|y|)/2},d_r(0,0)=0.$$

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.

For example, if a house is worth Template:Val today and the year after its value goes up to Template:Val, the percentage change of its value can be expressed as $$\frac{110,000-100,000}{100,000}=0.1=10\mathrm{\%}.$$ It can then be said that the worth of the house went up by 10%.

More generally, if Template:Tmath represents the old value and Template:Tmath the new value, $$\text{Percentage change}=\frac{\mathrm{\Delta }V}{V_1}=\frac{V_2-V_1}{V_1}\times 100\mathrm{\%}.$$

Some calculators directly support this via a Template:Kbd or Template:Kbd 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.

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

The terms "Experimental" and "Theoretical" used in the equation above are commonly replaced with similar terms. Other terms used 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: $$\mathrm{\%}\text{ Error}=\frac{\text{Experimental}-\text{Theoretical}}{|\text{Theoretical}|}\times 100.$$

The two values in the numerator do not commute. It is vital to preserve the order as above: subtract the theoretical value from the experimental value and not vice versa.

Examples

Valuable assets

Suppose that car Template:Mvar costs Template:Val and car Template:Mvar costs Template:Val. We wish to compare these costs.Template:Sfn With respect to car Template:Mvar, the absolute difference is Template:Nowrap. That is, car Template:Mvar costs Template:Val more than car Template:Mvar. The relative difference is, $$\frac{\mathrm{\$}10,000}{\mathrm{\$}40,000}=0.25=25\mathrm{\%},$$ and we say that car Template:Mvar costs 25% Template:Em car Template:Mvar. It is also common to express the comparison as a ratio, which in this example is, $$\frac{\mathrm{\$}50,000}{\mathrm{\$}40,000}=1.25=125\mathrm{\%},$$ and we say that car Template:Mvar costs 125% Template:Em the cost of car Template:Mvar.

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

It is the use of the words "of" and "less/more than" that distinguish between ratios and relative differences.Template:Sfn

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 incorrect and misleading. The absolute change in this situation is 1 percentage point (Template:Nowrap), but the relative change in the interest rate is: $$\frac{4\mathrm{\%}-3\mathrm{\%}}{3\mathrm{\%}}=0.333\dots =33\frac{1}{3}\mathrm{\%}.$$

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.Template:Sfn

Indicators of relative change

The (classical) relative change above is but one of the possible measures/indicators of relative change. An indicator of relative change from Template:Tmath (initial or reference value) to Template:Tmath (new value) $R(x,y)$ is a binary real-valued function defined for the domain of interest which satisfies the following properties:Template:Sfn

• Appropriate sign: $\{\begin{array}{cc}R(x,y)>0& \text{iff }y>x\\ R(x,y)=0& \text{iff }y=x\\ R(x,y)<0& \text{iff }y<x\end{array}.$
• Template:Mvar is an increasing function of Template:Mvar when Template:Mvar is fixed.
• Template:Mvar is continuous.
• Independent of the unit of measurement: for all Template:Tmath, Template:Tmath.
• Normalized: ${\frac{d}{dy}R(1,y)|}_{y=1}=1$

The normalization condition is motivated by the observation that Template:Tmath scaled by a constant $c>0$ still satisfies the other conditions besides normalization. Furthermore, due to the independence condition, every Template:Tmath can be written as a single argument function Template:Tmath of the ratio Template:Tmath.Template:Sfn The normalization condition is then that Template:Tmath. This implies all indicators behave like the classical one when Template:Tmath is close to Template:Math.

Usually the indicator of relative change is presented as the actual change Template:Math scaled by some function of the values Template:Mvar and Template:Mvar, say Template:Math:Template:Sfn $$\text{Relative change}(x,y)=\frac{\text{Actual change}\mathrm{\Delta }}{f(x,y)}=\frac{y-x}{f(x,y)}.$$

As with classical relative change, the general relative change is undefined if Template:Math is zero. Various choices for the function Template:Math have been proposed:Template:Sfn

Indicators of relative changeTemplate:Sfn

Name	$f(x,y)$ where the indicator's value is $\frac{y-x}{f(x,y)}$	$H(y/x)$
(Classical) Relative change	Template:Mvar	$\frac{y}{x}-1$
Reversed relative change	Template:Mvar	$1-\frac{x}{y}$
Arithmetic mean change	$\frac{1}{2}(x+y)$	$\frac{\frac{y}{x}-1}{\frac{1}{2}(1+\frac{y}{x})}$
Geometric mean change	$\sqrt{xy}$	$\frac{\frac{y}{x}-1}{\sqrt{xy}}$
Harmonic mean change	$\frac{2}{\frac{1}{x}+\frac{1}{y}}$	$\frac{(\frac{y}{x}-1)(1+\frac{x}{y})}{2}$
Moment mean change of order Template:Mvar	${[\frac{1}{2}(x^k+y^k)]}^{\frac{1}{k}}$	$\frac{\frac{y}{x}-1}{{\left[\frac{1}{2}(1+{\left(\frac{y}{x}\right)}^k)\right]}^{\frac{1}{k}}}$
Maximum mean change	$max(x,y)$	$\frac{\frac{y}{x}-1}{max(1,\frac{y}{x})}$
Minimum mean change	$min(x,y)$	$\frac{\frac{y}{x}-1}{min(1,\frac{y}{x})}$
Logarithmic (mean) change	$\{\begin{array}{cc}\frac{y-x}{\ln \frac{y}{x}}& x\ne y\\ x& x=y\end{array}$	$\ln \frac{y}{x}$

As can be seen in the table, all but the first two indicators have, as denominator a mean. One of the properties of a mean function $m(x,y)$ is:Template:Sfn Template:Tmath, which is to say that all such indicators have a "symmetry" property that the classical relative change lacks: Template:Tmath. This agrees with intuition that a relative change from Template:Tmath to Template:Tmath should have the same magnitude as a relative change in the opposite direction, Template:Tmath to Template:Tmath, just like the relation $\frac{y}{x}=\frac{1}{x/y}$ suggests.

Maximum mean change has been recommended when comparing floating point values in programming languages for equality with a certain tolerance. Another application is in the computation of approximation errors when the relative error of a measurement is required.Template:Cn Minimum mean change has been recommended for use in econometrics.Template:Sfn 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 from Template:Mvar (reference value) to Template:Mvar:Template:Sfn $$H(K,L)=\{\begin{array}{cc}{\int }_1^{K/L}{t}^{c-1}dt& \text{when }K>L\\ -{\int }_{K/L}^1{t}^{c-1}dt& \text{when }K<L\end{array}$$

which leads to $$H(K,L)=\{\begin{array}{cc}\frac{1}{c}\cdot ((K/L{)}^c-1)& c\ne 0\\ \ln (K/L)& c=0,K>0,L>0\end{array}$$

In particular for the special cases Template:Math, $$H(K,L)=\{\begin{array}{cc}(K-L)/K& c=-1\\ (K-L)/L& c=1\end{array}$$

Logarithmic change

Of these indicators of relative change, arguably the most natural is the natural logarithm (Template:Tmath) of the ratio of the two numbers (final and initial), called log change.Template:Sfn Indeed, when Template:Tmath, the following approximation holds: $$\begin{array}{cc}\ln \frac{V_1}{V_0}& ={\int }_{V_0}^{V_1}\frac{\mathrm{d}V}{V}\approx {\int }_{V_0}^{V_1}\frac{\mathrm{d}V}{V_0}\\ & =\frac{V_1-V_0}{V_0},\end{array}$$ which is the classical relative change formula.

Because Template:Tmath, it follows that, when the previous condition holds, the difference between (natural) logarithms is approximately equal to the percentange 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. Log points are equivalent to the unit centinepers (cNp) when measured for root-power quantities. This quantity has also been referred to as a log percentage and denoted Template:Mvar.Template:Sfn Since the derivative of the natural log at 1 is 1, log points are approximately equal to percent change for small differencesTemplate:Snd for example an increase of 1% equals an increase of Template:Val, and a 5% increase gives a Template:Val 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 percent change.

Additivity

Using log change has the advantages of additivity compared to relative change.Template:Sfn 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. 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 Template:Tmath.

By additivity, Template:Tmath, 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 Template:Tmath or Template:Tmath is chosen as the reference. In contrast, for relative change, Template:Tmath, with the difference $\frac{(V_1-V_0{)}^2}{V_0{V}_1}$ becoming larger as Template:Tmath or Template:Tmath approaches Template:Math 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 Template:Tmath, such that absolute change is $F_0$ and log change is Template:Tmath.

See also

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

References

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