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The traditional formula for calculating a percentage change is as follows:

  • Going from 45 to 430: $$ \frac{430 - 45}{45} \times 100 = 856\% $$
  • Going from 430 to 45: $$ \frac{45- 430}{430} \times 100 = -90\% $$

The values are correct, but they're not intuitive: looking at the percentages, a 90% decrease seems a lot less in magnitude than an 856% increase.

Is there is a related formula that gives more intuitive results, such that the two answers have different signs but the same magnitude?

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2 Answers 2

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So after quite a lot of research, I ended up on calculating price changes in microeconomics. This excellent video from KhanAcademy discusses my problem exactly, but for prices instead of quantities.

The formula used in economics for calculating a percentage change is as follows:

(For a quantity going from 45 to 430)
$$ \frac{430 - 45}{Avg(430,45)} \times 100 = 162\% $$

(For a quantity going from 430 to 45)
$$ \frac{45 - 430}{Avg(45,430)} \times 100 = -162\% $$

The value of the percentage for all the changes will range from -200% to 200%, and it's reflective too.

That's one solution. Feel free to post your own.

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    $\begingroup$ The use of the arithmetic mean as the denominator -- that is, as the standard against which you compare the actual change -- is more or less arbitrary. If all you care about is getting percent changes that are equal in magnitude but opposite in sign, you can also use the geometric mean, or indeed any symmetric real-valued function of two variables. Which standard you use depends on what you want to do with the comparison. $\endgroup$
    – symplectomorphic
    Commented Mar 29, 2018 at 15:28
  • $\begingroup$ Yeah that makes sense now. Not a math guy, so it took me a while to understand it's all about keeping the denominator constant lol. $\endgroup$
    – Sulaiman
    Commented Mar 29, 2018 at 17:52
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  1. The relative change (percentage change) from $x_1$ to $x_2,$ where $x_1\ne0,$ is $$\frac{x_2-x_1}{|x_1|}.\tag1$$ For example, the relative change from $\,-100\,$ to $\,-70\;$ is $\,30\%.$
  2. The relative difference (percentage difference) between $x$ and $y,$ scaled by $f(x,y),$ where $f(x,y)\ne0,$ is $$\left|\frac{x-y}{f(x,y)}\right|.\tag2$$ For example, the relative difference between $\,-100\,$ and $\,-70,$ scaled by their arithmetic mean, is $\,35.3\%,$ while their relative difference, scaled by their geometric mean, is $\,35.9\%.$

Technically, the formula that you want is neither $(1)$ nor $(2)$ nor $$\frac{x-y}{f(x,y)},\tag3\\f(x)=\operatorname{ArithmeticMean}(x,y)$$ (as suggested in your self-Answer), but rather $$\frac{x-y}{|f(x,y)|}.\tag4$$

Expressions $(3)$ and $(4)$ coincide when $f(x,y)$ is positive.

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