# Finding an accurate rating between 1 and 10 based on numerous data points

I'm going to provide a fictitious setup that aligns with my mathematical needs. I am a company that is measuring the bounciness of balls and providing two 1-10 rating to each ball based on how it compares to all other balls. The first rating is based on more bounces being good. The second rating is based on less bounces being good.

Is there a way I can take all data points (number of ball bounces) and find an accurate 1-10 rating (for each ball) for both scenarios?

Thank you for your time and help! (If I used incorrect tags, I apologize)

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Let $M$ be the maximum and $m$ be the minimum number of bounces among all the balls. Then the rating is just a matter of rescaling $m \le x \le M$ to $1 \le y \le 10$. The following function does this: $$f(x) = 1+\left \lfloor 10 \frac{x-m}{M+1-m} \right \rfloor.$$ Then $f$ will map a ball's number of bounces, $x$, to an integer between 1 and 10, inclusive, with the property that if $x_1>x_2$, then $f(x_1) \ge f(x_2)$. More bounces results in a higher rating.
To get a lower rating with more bounces, just subtract $f(x)$ from 11: $$g(x) = 10-\left \lfloor 10 \frac{x-m}{M+1-m} \right \rfloor.$$
You are not using the functions I gave correctly. The symbol $\lfloor z \rfloor$ means the greatest integer less than or equal to z. For example, $\lfloor 5.2 \rfloor=5$. The function $f(x)$ I gave, with $m=3$ and $M=15$, gives $f(3)=1, f(9)=5,$ and $f(15)=10$. –  Matthew Conroy Jun 22 '12 at 1:11