How should one explain the conventional formula for relative variation of signed variables? The conventional formula for the relative variation between two signed values is $\displaystyle \frac{b-a}{|a|}$. Unfortunately, I could not find any clear justification for it, especially when considering examples like this one: according to the conventional formula, the relative variation between -10 and +20 is +300%, while the relative variation between -20 and +20 is +200%. This is puzzling, because the absolute variation for the former is +30 while the absolute variation for the latter is +40 and both datasets end at the same +20 value, yet the relative variation for the former is greater than the relative variation for the latter.
How can one justify this counter-intuitive scenario?
To better understand what is going on, it usually helps to think of the relative growth between two values of different signs as being composed of two separate parts: the relative growth from the first value to zero, plus the relative growth from zero to the second value. For example, the relative growth between $-10$ and $20$ is equal to the relative growth between $-10$ and $0$, plus the relative growth between $0$ and $20$.
The relative growth between $-10$ and $0$ is $\displaystyle \frac{0-(-10)}{|-10|} = \frac{10}{10} = 1 = 100\%$ according to our previous formula.
In fact, the relative growth between any negative value and $0$ is always equal to $100\%$, which actually makes sense when one comes to think about it. But things become a bit more tricky when we need to compute the relative growth between $0$ and a positive value, like $20$ in our previous example. There, we cannot simply use our formula, because it would lead to a division by zero. Instead, we have to compute this relative growth in relation to the previous one, by computing the ratio of two absolute growths.
For example, when going from $-10$ to $20$, we gain $10$ in absolute terms between $-10$ and $0$, then $20$ in absolute terms again between $0$ and $20$. Therefore, we gained twice as much going from $0$ to $20$ than we did when going from $-10$ to $0$. And as we have seen earlier, the relative growth between $-10$ and $0$ is $100\%$, therefore the relative growth between $0$ and $20$ should be twice that amount, or $200\%$, and the relative growth between $-10$ and $20$ should be the sum of our two relative growths, or $100\% + 200\% = 300\%%$, which matches what our adjusted formula gave us in the first place. In other words, we found two ways of getting to the same result, but this should not come as a surprise if you look at the original equation.
Indeed, if $a$ is negative, we can rewrite $\displaystyle \frac{b-a}{|a|}$ as $\displaystyle \frac{b-a}{-a}$. Then, we can split the fraction as $\displaystyle \frac{b}{-a} + \frac{-a}{-a}$ and further simplify it into $\displaystyle \frac{b}{-a} + 1$, or even $\displaystyle 1 + \frac{b}{-a}$. This addition has two addends, $1$ and $\displaystyle \frac{b}{-a}$. The former is the relative growth from the first value to zero (always equal to $1$), and the latter is the ratio of the absolute growth between the first value and $0$ and the absolute growth between $0$ and the second value, which we can write as $\displaystyle \frac{b - 0}{0 -a}$.
When we look at the formula through that angle, we realize that a relative growth between two values of different signs is only affected by the ratio between the two values $\displaystyle \frac{b}{-a}$, unlike the relative growth between two values of the same sign, which is the ratio between the difference of the two values and the first value $\displaystyle \frac{b-a}{a}$. But this is where the adjusted formula becomes misleading, because we should not look at the former as a ratio between two values. Instead, we should look at it as a ratio between two absolute growths $\displaystyle \frac{b - 0}{0 -a}$.
Make no mistake though: this is not a trivial arithmetic rewriting. Instead, by writing $\displaystyle \frac{b}{-a}$ as $\displaystyle \frac{b - 0}{0 -a}$, we clearly communicate the fact that our fraction is a ratio between two absolute growths, instead of a ratio between two values. And this is what explains that the relative growth between $-10$ and $20$ ($300\%$) is greater than the relative growth between $-20$ and $20$ ($200\%$).
As explained earlier, to grow from $-10$ to $20$, you first need to grow from $-10$ to $0$, then grow from $0$ to $20$, and the first growth component to $0$ is always equal to $100\%$. But when you grow from $0$ to $20$, while this absolute growth ($20$) is the same whether you started from $-10$ or $-20$, the relative growth is twice as much if you started from $-10$ than if you started from $-20$.
In other words, what the relative growth of values of different signs really measures is the relative growth after you have crossed the zero line, because the relative growth before you cross the zero line is always the same ($100\%$).
A quantity, $a$, is changed by $b-a$. What fraction of $a$ does that change constitute?
The answer is $$\dfrac{b-a}{a}$$
This is called the relative change because its value is computed &quot;realtive to&quot; $a$.
This is not a useful measure when $a$ is close to $0$.
Wikipedia: approximation error