I need to figure out how to create an equation that gives a cricket player a ranking score based on their batting and bowling averages.
For those who don't know cricket, basically, the higher your batting average the better you are and the lower your bowling average the better you are.
For batting I can just use:
A batting average of 50 has a rating of 100, 40 = 80, 30 = 60, 20 = 40, 10 = 20. So for this I can use average * 2. That's cool.
But then I need a bowling equation that will give a player a rating
For bowling, I want a bowler with an average of 20 to have a rating of 100, 30 = 80, 40 = 50, 50 = 20.
Batsman can have an average over 50 and bowlers can have an average under 20 and the averages can range from 0 - 100 in both bowling and batting so the equation needs to work within this range.
Hopefully this question makes sense (been a long while since I've tried to figure something like this out).
Hope you can help.
For each of the characteristics, you can use a bounded linear rating: $$bat(x) = \max\{l_{bat},\min\{2x,L_{bat}\}\},$$ $$bow(x) = \max\{l_{bow},\min\{170-3x,L_{bow}\}\},$$ where $l_{bat},l_{bow}$ and $L_{bat},L_{bow}$ are the lower and higher possible values for each of the ratings, respectively. Note that the bowling rating does not fit exactly with the data you are giving (as @Pieter21 pointed out, yours is not linear) but this could be a starting point for you.
The combined rating could be something like: $$rank(x) = \omega_1 bat(x) + \omega_2 bow(x) + \omega_0, $$ where $\omega_1$ and $\omega_2$ are the weights you assign to each of the characteristics. You just need to choose them, according to $l$ and $L$ values, so that the maximum is 100 and the minimum $0$. For instance, for $l_{bat}=l_{bow}=20$, $L_{bat}=L_{bow}=100$ and equal weight for each of the characteristics ($\omega_1=\omega_2=5/8$ and $\omega_0=-25$), you have the following:
Ratings of batting (blue) and bowling (orange) with respect to average:
Ranking with respect to the averages of batting and bowling: