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 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 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.: "We suggest that this indicator should be used more extensively."

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Definition Given two numerical quantities, v_ref and v with v_ref some reference value, their actual change, actual difference, or absolute change is

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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, is the same as , but the absolute difference between is 1 while the absolute difference between is 100, giving the impression of a larger difference. But even with constant units, the relative change helps judge the importance of the respective change. For example, an increase in price of of a valuable is considered big if changing from but rather small when changing from .

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

$$\backslash text\{relative\; change\}(v\_\backslash text\{ref\},\; v)\; =\; \backslash frac\{\backslash text\{actual\; change\}\}\{\backslash text\{reference\; value\}\}\; =\; \backslash frac\{\backslash Delta\; v\}\{v\_\backslash text\{ref\}\}\; =\; \backslash frac\{v\}\{v\_\backslash text\{ref\}\}\; -\; 1.$$

The relative change is independent of the unit of measurement employed; for example, the relative change from is , the same as for . The relative change is not defined if the reference value ( v_ref) is zero, and gives negative values for positive increases if v_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 , 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).

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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 v_ref:

$$\backslash text\{Relative\; change\}(v\_\backslash text\{ref\},\; v)\; =\; \backslash frac\{v\; -\; v\_\backslash text\{ref\}\}$$

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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 and $v\_\backslash text\{reference\}$. Then the only problematic case is $v=v\_\backslash text\{reference\}=0$, which can usually be addressed by appropriately extending the indicator. For example, for arithmetic mean this formula may be used:

$$d\_r(x,y)=\backslash frac$${(|x|+|y|)/2},\ d_r(0,0)=0

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

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, $$\backslash text\{Percentage\; change\}\; =\; \backslash frac\{\backslash Delta\; V\}\{V\_1\}\; =\; \backslash frac\{V\_2\; -\; V\_1\}\{V\_1\}\; \backslash times100\backslash \%\; .$$

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 , to avoid confusion between relative difference and absolute difference.

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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.

$$\backslash \%\backslash text\{\; Error\}\; =\; \backslash frac$$

\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) and is a newsworthy result.

The percent error equation, when rewritten by removing the absolute values, becomes: $$\backslash \%\backslash text\{\; Error\}\; =\; \backslash frac\{\backslash text\{Experimental\}-\backslash text\{Theoretical\}\}$$

\times100.

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

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Examples

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Valuable assets Suppose that 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, $$\backslash frac\{\backslash \$10,000\}\{\backslash \$40,000\}\; =\; 0.25\; =\; 25\backslash \%,$$ 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, $$\backslash frac\{\backslash \$50,000\}\{\backslash \$40,000\}\; =\; 1.25\; =\; 125\backslash \%,$$ 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, $$\backslash frac\{-\backslash \$10,000\}\{\backslash \$50,000\}\; =\; -0.20\; =\; -20\backslash \%$$ is also negative since car L costs 20% less than car M. The ratio form of the comparison, $$\backslash frac\{\backslash \$40,000\}\{\backslash \$50,000\}\; =\; 0.8\; =\; 80\backslash \%$$ 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.

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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 (4% − 3%), but the relative change in the interest rate is: $$\backslash frac\{4\backslash \%\; -\; 3\backslash \%\}\{3\backslash \%\}\; =\; 0.333\backslash ldots\; =\; 33\backslash frac\{1\}\{3\}\backslash \%.$$

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.

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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 x (initial or reference value) to y (new value) $R(x,y)$ is a binary real-valued function defined for the domain of interest which satisfies the following properties:

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

The normalization condition is motivated by the observation that scaled by a constant $c>0$ still satisfies the other conditions besides normalization. Furthermore, due to the independence condition, every can be written as a single argument function of the ratio $y/x$. The normalization condition is then that $H\text{'}(1)\; =\; 1$. This implies all indicators behave like the classical one when $y/x$ is close to .

Usually the indicator of relative change is presented as the actual change Δ scaled by some function of the values x and y, say .

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

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

+ Indicators of relative change
$\backslash frac\; y\; x\; -\; 1$
$1-\backslash frac\; x\; y$
$\backslash frac\{\backslash frac\{y\}\{x\}\; -\; 1\}\{\backslash frac\{1\}\{2\}\backslash left(1\; +\; \backslash frac\{y\}\{\; x\}\backslash right)\}$
$\backslash frac\{\backslash frac\; y\; x\; -\; 1\}\{\backslash sqrt\{xy\}\}$
$\backslash frac\{\backslash left(\backslash frac\; \{y\}\; \{x\}\; -\; 1\backslash right)\backslash left(1+\backslash frac\; \{x\}\; \{y\}\backslash right)\}\{2\}$
$\backslash frac\{\backslash frac\; y\; x\; -\; 1\}\{\backslash left\backslash frac\{1\}\{2\}\backslash left(1+\backslash left(\backslash frac^\backslash frac\{1\}\{k\}\}$
$\backslash frac\{\backslash frac\; y\; x\; -\; 1\}\{\backslash max\backslash left(1,\backslash frac\; y\; x\backslash right)\}$
$\backslash frac\{\backslash frac\; y\; x\; -\; 1\}\{\backslash min\backslash left(1,\backslash frac\; y\; x\backslash right)\}$
Logarithmic (mean) change		$\backslash ln\backslash frac\{y\}\{x\}$

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: $m(x,y)=m(y,x)$, which means that all such indicators have a "symmetry" property that the classical relative change lacks: $R(x,y)=-R(y,x)$. This agrees with intuition that a relative change from x to y should have the same magnitude as a relative change in the opposite direction, y to x, just like the relation $\backslash frac\; y\; x\; =\; \backslash frac\; 1\; \backslash frac\{x\}\{y\}$ suggests.

Maximum mean change has been recommended when comparing floating point values in programming languages for equality with a certain tolerance. What's a good way to check for close enough floating-point equality Another application is in the computation of approximation errors when the relative error of a measurement is required. Minimum mean change has been recommended for use in econometrics.

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 L (reference value) to K: $$H(K,L)\; =\; \backslash begin\{cases\}$$

 \int_1^{K/L} t^{c-1} dt & \text{when } K>L \\
 -\int_{K/L}^1 t^{c-1} dt & \text{when } K
\end{cases}
     
     

which leads to
     

     $$H(K,L)\; =\; \backslash begin\{cases\}$$
 \frac{1}{c} \cdot ((K/L)^c-1) & c \neq 0 \\
 \ln(K/L) & c = 0, K > 0, L > 0
     
\end{cases}
     
     

In particular for the special cases $c=\backslash pm\; 1$,
     

     $$H(K,L)\; =\; \backslash begin\{cases\}$$
 (K-L)/K & c=-1 \\
 (K-L)/L & c=1
     
\end{cases}
     
     

 

 
 
  

  Logarithmic change
 
 

 
 
 
Of these indicators of relative change, arguably the most natural is the natural logarithm (ln) of the ratio of the two numbers (final and initial), called  log change. Indeed, when $\backslash left\; |\; \backslash frac\{V\_1\; -\; V\_0\}\{V\_0\}\; \backslash right\; |\; \backslash ll\; 1$, the following approximation holds:
     $$\backslash ln\backslash frac\{V\_1\}\{V\_0\}\; =\; \backslash int\_\{V\_0\}^\{V\_1\}\backslash frac$$

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