Percentage with negative numbers

Question

I have several competitors in an economic market, each one with a score of a strength indicator. Scores can vary in [-inf,inf]. For a subset of competitors, I need to calculate their relative importance.

With all positive scores (like in CASE A), I've been calculating relative importance using the following formula (like in percentages):

CASE A

Score 1 = 1 -- > r = 0.1111

Score 2 = 2 --> r = 0.2222

Score 3 = 6 --> r = 0.6667

How can I replicate my intent in CASE B, where I have some negative scores? Here, the above-mentioned formula does not reach my goal anymore.

CASE B

Score 1 = 1 -- > r = ?

Score 2 = 2 --> r = ?

Score 3 = 6 --> r = ?

Score 4 = -100 --> r = ?

Answer 1

Let's stick to your notation and define the score of the $i$th competitor by $x_i$ and the relative importance by $r_i$.

The definition $r_i=\frac{x_i}{\sum_{j=1}^n |x_j|}$ has the following drawbacks:

1. The relative importance can be negative if $x_i<0$. This is not easy to interpret. Let's say you have two competitors with relative importance $r_i=0.1$ and $r_j=-0.1$. How would you compare these numbers?
2. Large negative score strongly influences the resulting values of the relative importance which is counterintuitive.

One possible option would be to define the relative importance using an exponential function: $$r_i=\frac{\alpha^{x_i}}{\sum_{j=1}^n \alpha^{x_j}},$$ with the base $\alpha>1$. This formulation has an advantage of penalizing very small (negative) scores. However, the relative importance grows nonlinearly with $x_i$, i.e., $x_i=2x_j$ does not imply $r_i=2r_j$.

Answer 2

Try to normalize your data with Min Max function.