Relative change and difference

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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 In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

and is a
unitless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate di ...
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

. By multiplying these ratios by 100 they can be expressed as
percentage In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction (mathematics), fraction of 100. It is often Denotation, denoted using the percent sign, "%", although the abbreviations "pct.", "p ...

s so the terms percentage change, percent(age) difference, or relative percentage difference are also commonly used. The terms "change" and "difference" are used interchangeably. Relative change is often used as a quantitative indicator of
quality assurance Quality assurance (QA) is the term used in both manufacturing and service industries to describe the systematic efforts taken to ensure that the product(s) delivered to customer(s) meet with the contractual and other agreed upon performance, design ...

and
quality control Quality control (QC) is a process by which entities review the quality of all factors involved in Production (economics), production. ISO 9000 defines quality control as "a part of quality management focused on fulfilling quality requirements" ...

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 and various alternative formulas, called ''indicators of relative difference'', 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, ''x'' and ''y'', with ''y'' a ''reference value'' (a theoretical/actual/correct/accepted/optimal/starting, etc. value), their ''actual change'', ''actual difference'', or ''absolute change'' . The term
absolute difference The absolute difference of two real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' mea ...

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, because it depends on the unit of measurement. For instance, 1 meter is the same as 100 centimeters, but the absolute difference between 2 m and 1 m is 1 while the absolute difference between 200 cm and 100 cm is 100, giving the impression of a larger difference. We can adjust the comparison to take into account the "size" of the quantities involved, by defining, for positive values of ''x''reference: $\text(x, x_\text) = \frac = \frac = \frac.$ The relative change is independent of the unit of measurement employed; for example, the relative change from 2 meters to 1 meter is -50%, the same as for 200 cm to 100 cm. The relative change is not defined if the reference value (''x''reference) is zero, and gives negative values for positive increases if ''x''reference is negative, hence it is not usually defined for negative reference values either. For example, we might want to calculate the relative change of a thermometer from −10 °C to −6 °C. The above formula gives , indicating a decrease, yet in fact the reading increased. Measures of relative difference are
unitless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate di ...
numbers expressed as a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. 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).

# 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 ''x''reference: $\text(x, x_\text) = \frac = \frac = \frac.$ This still does not solve the issue when the reference is zero. It is common to instead use an indicator of relative difference, and take the absolute values of both $x$ and $x_\text$. Then the only problematic case is $x=x_\text=0$, which can usually be addressed by appropriately extending the indicator. For example, for arithmetic mean this formula may be used: $d_r=\frac, d_r(0,0)=0$

# 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. $\%\text = \frac\times 100.$ 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 The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
) and is a newsworthy result. The percent error equation, when rewritten by removing the absolute values, becomes: $\%\text = \frac\times100.$ It is important to note that the two values in the
numerator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
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. 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 $\frac = 0.1 = 10\%.$ It can then be said that the worth of the house went up by 10%. More generally, if ''V''1 represents the old value and ''V''2 the new one, $\text = \frac = \frac \times100\% .$ Some calculators directly support this via a or function. When the variable in question is a percentage itself, it is better to talk about its change by using
percentage point A percentage point or percent point is the unit (measurement), unit for the Difference (mathematics), arithmetic difference between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points, but a ...
s, to avoid confusion between
relative difference In any quantitative science, the terms relative change and relative difference are used to compare two quantities Quantity or amount is a property that can exist as a Counting, multitude or Magnitude (mathematics), magnitude, which illustrate d ...
and
absolute difference The absolute difference of two real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' mea ...

.

## 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: $\frac = 0.333\ldots = 33\frac\%.$ 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.

# Examples

## Comparisons

Car ''M'' costs $50,000 and car ''L'' costs$40,000. We wish to compare these costs. With respect to car ''L'', the absolute difference is . That is, car ''M'' costs $10,000 more than car ''L''. The relative difference is, $\frac = 0.25 = 25\%,$ 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, $\frac = 1.25 = 125\%,$ 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 since car ''L'' costs$10,000 less than car ''M''. The relative difference, $\frac = -0.20 = -20\%$ is also negative since car ''L'' costs 20% ''less than'' car ''M''. The ratio form of the comparison, $\frac = 0.8 = 80\%$ 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.

# Indicators of relative difference

An indicator of relative change $C\left(x_\text,x\right)$ is a binary real-valued function defined for the domain of interest which satisfies the following properties: * Appropriate sign: $C\left(x_\text,x\right)> 0$ iff $x>x_\text$, $C\left(x_\text,x\right)= 0$ iff $x=x_\text$, $C\left(x_\text,x\right)< 0$ iff
floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an Integer (computer science), integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. ...
values in
programming language A programming language is a system of notation for writing computer program, computer programs. Most programming languages are text-based formal languages, but they may also be visual programming language, graphical. They are a kind of computer ...
s for equality with a certain tolerance. Another application is in the computation of
approximation error The approximation error in a data value is the discrepancy between an exact value and some ''approximation An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else. Etymology ...
s when the relative error of a measurement is required. # Minimum mean change: $f\left(x,y\right)=\min\left(x,y\right)$, $H\left(y/x\right) = \left(\left(y/x\right) - 1\right)/\min\left(1,y/x\right)$. This has been recommended for use in econometrics. # Logarithmic change: $f\left(x,y\right)=\begin \left(y-x\right)/\ln\left(y/x\right) & x \neq y \\ x & x = y \end$, $H\left(y/x\right)=\ln\left(y/x\right)$ Tenhunen defines a general relative change function:

# Logarithmic change

Of these indicators of relative difference, the most natural is the
natural logarithm The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...
(ln) of the ratio of the two numbers, called ''log change''. Indeed, when $\left , \frac \right , \ll 1$, the following approximation holds: $\ln\frac = \int_^\frac \approx \int_^\frac = \frac = \text$ In the same way that relative change is scaled by 100 to get percentages, $\ln\frac$ 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 '' L%''. 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.

Using log change has the advantages of additivity compared to relative change. 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: 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 $\text = \ln\left(1 + \text\right)$. By additivity, $\ln\frac + \ln\frac = 0$, and therefore additivity implies a sort of symmetry property, namely $\ln\frac = - \ln\frac$ and thus the magnitude of a change expressed in log change is the same whether ''V''0 or ''V''1 is chosen as the reference. In contrast, for relative change, $\frac \neq - \frac$, with the difference $\frac$ becoming larger as ''V''1 or ''V''0 approaches 0 while the other remains fixed. For example: Here 0+ means taking the
limit from above In calculus, a one-sided limit refers to either one of the two Limit of a function, limits of a Function (mathematics), function f(x) of a Real number, real variable x as x approaches a specified point either from the left or from the right. The ...
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 $F_\lambda\left(x,y\right)$ for any $\lambda\in\mathbb$, such that absolute change is $F_0$ and log change is $F_1$.

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Approximation error The approximation error in a data value is the discrepancy between an exact value and some ''approximation An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else. Etymology ...
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Errors and residuals in statistics In statistics and mathematical optimization, optimization, errors and residuals are two closely related and easily confused measures of the deviation (statistics), deviation of an observed value of an Elementary event, element of a Sample (stati ...
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Relative standard deviation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a Standardized (statistics), standardized measure of statistical dispersion, dispersion of a probability distribution o ...
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Logarithmic scale A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...

# References

* * * * * * {{DEFAULTSORT:Relative Change and Difference Measurement Numerical analysis Statistical ratios Subtraction