Intuitive formulae for percentage increase and percentage decrease

Question

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?

Answer 1

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.

Answer 2

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.