# Derive a formula to solve a specific task

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%

Does anyone have an approach to this problem?

• Shouldn't the minimum possible total score (assuming you simply add all $8$ variables together) be $8$ and the maximum possible score be $40$? Jul 23, 2013 at 7:27
• Yes you;re correct. Jul 23, 2013 at 23:15

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.

• Can you explain where the numbers 30 and 35 came from? Jul 24, 2013 at 0:32
• a + b + c + d + e + f - 6 / 30 scales the first set of numbers from 0 to 24/30 = 4/5 = 28/35. g + h - 2 / 35 scales the second set of numbers from 0 to 8/35. As the first section ranges from 0 to 28/35, and the second section ranges from 0 to 8/35, the first section is worth approximately 4 times (Actually 7/2) as much as the second section. Thus, when it is scaled such that the total is 1, the first section will be worth around .8, and the second section around .2. Jul 24, 2013 at 1:54
• Thanks for the reply. Is there any reason you didn't make it exactly 80/20? ((a+b+c+d+e+f-6)/30) + ((g+h-2)/40)? I'm curious why you chose 30 and 35. Thanks @qaphla Jul 24, 2013 at 22:43
• That would work too. When I was writing this, I was getting a bit confused by the numbers you were asking for in your original post, since they changed a bit, and because of that, settled for something close. 30 and 40 works better if you want exactly .8/.2. Jul 25, 2013 at 2:48
• Cool - thanks for the info. Jul 25, 2013 at 3:15