For an item i, I have a low price estimation L, an high price estimation H and the price realized P. I need to figure out some kind of numeric indicator showing the performance of an item in regards to its estimations. Examples: - an item that has P=100 when L=40 and H=90 should have a very good score because it outperforms H - an item with P=80 when L=40 and H=90 should have a good score because P is close to 90
- an item with P=65 when L=40 and H=90 should have an average score because P is right at the mean
- an item with P=45 when L=40 and H=90 should have a low score because P is close to L
- an item with P=30 when L=40 and H=90 should have a very low score because P underperforms L.
What formula would you apply to compute such a score ?
A function of the form: $$ S = \frac{P - L}{H - L} $$ Is very simple to calculate and plot, as well as continuous and (affine) linear; its plot against $P$ is just a straight line. Bad scores are negative, average scores are in the range $[0, 1]$, while good scores are above $1$. In particular, the score of the mean is always $0.5$.
If you want a different value (maybe you want the average range to be from $0$ to $\alpha$) it is sufficient to multiply the right hand side by $\alpha$.
The very first formula I mentioned is: $$ S = \frac PH + \frac PL $$ This function happens to be a linear function of $P$, always positive, increasing in $P$ and continuous. However, the mean has score $1 + (\frac H{2L} + \frac L{2H})$ so comparing items belonging to different price ranges will be more difficult.