I have a specific problem.
I have 8 different variables a, b, c, d, e, f, g, h. Each of these variables has a score out of 5, where 1 is bad and 5 is good. So a max score of 45 and a min of 0.
Of these variables I can influence the scores for 6 a, b, c, d, e, f. I can not influence the scores for 2 g, h.
I need to derive a formula which has the following approximate properties:
- To represent a positive collective rating where every score is good I want a score of ~100%
- To represent a rating where the influencable variables are bad I want a score of ~0%
- To represent where the influencable variables can be improved, but the variables out of control are good, I want ~50%
- To represent where the influenceable variables are good, but the set variables are bad, I want ~100%
Or with less text:
- influenceable = good & non-influencable = good => 100%
- influenceable = good & non-influencable = bad => 80%
- influenceable = bad & non-influencable = good => 40%
- influenceable = bad & non-influencable = bad => 0%
Does anyone have an approach to this problem?
Very roughly, the following is a reasonable start.
$\frac{35}{36}(\frac{a + b + c + d + e + f - 6}{30} + \frac{g + h - 2}{35})$.
At (5,5,5,5,5,5,5,5), this gives 1 (100%).
At (5,5,5,5,5,5,1,1), it gives .78 (roughly).
At (1,1,1,1,1,1,5,5), it gives .22 (roughly).
At (1,1,1,1,1,1,1,1), it gives 0.
It's hard to come up with a simple one that gives ~80% to the first six good and last two bad and 40% to the reverse, since that means that simply making the first six good should bring the value up by .8, and making the last two good should bring the value up by .4, giving a total of 1.2 for making them all good, unless another term is added to make the increase less.