Relative change and difference

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:

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

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:

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

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]

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

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.

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.

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

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:

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

This approach is often used when the two numbers reflect a change in some single underlying entity. 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

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

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:

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

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.

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

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:

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

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.

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

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

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,

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

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.

Change in a quantity can also be expressed logarithmically. Using the natural logarithm (ln) and normalization with a factor of 100, as done for percent aligns with the definition for percentage change for very small changes (called "log change" in the tables below):

${\displaystyle D=100\cdot \ln {\frac {V_{2}}{V_{1}}}\approx 100\cdot {\frac {V_{2}-V_{1}}{V_{1}}}={\text{Percentage change}}{\text{ when }}\left|{\frac {V_{2}-V_{1}}{V_{1}}}\right|<<1}$

Using a logarithmic scale has advantages. First, the magnitude of the change expressed in this way is the same whether V₁ or V₂ is chosen as the reference, since ${\displaystyle \ln {\frac {V_{2}}{V_{1}}}=-1\cdot \ln {\frac {V_{1}}{V_{2}}}}$. In contrast, ${\displaystyle {\frac {V_{2}-V_{1}}{V_{1}}}\approx -1\cdot {\frac {V_{1}-V_{2}}{V_{2}}}}$, with the approximation error becoming more significant as V₂ and V₁ diverge. For example:

Another advantage is that the total change after a series of changes equals the sum of the changes when expressed logarithmically. With percent, summing the changes is only an approximation, with larger error for larger changes. For example:

• 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 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