I can't do basic math, but my competence is in other things...Help?

March 14, 2013 7:54 AM

We want to assess a list of vendors in two domains: Accessibility and Cost. We have two tools that we are using to essentially score them in each of these domains separately. The Accessibility assessment tool totals 110 points; the Cost tool totals 90 points. We want a final score for each vendor that combines the domains and is out of 100. We want the Accessibility score to "count" about 20% more than the Cost score in the final because it is more important to us. How do we do this?

Were both domain scores out of 100 this would be easy: we would simply multiply Accessibility by 0.55 & Cost by 0.45.

Some folks are proposing that because there are already about 20% more points in the Accessibility section, we simply add both together and average. (The two sections add to 200 points.) I know from brute arithmetic that this does not work exactly in the middle of the range (although it does at each end), but I'm having trouble articulating why not. There are more points to gain or lose in the Accessibility section, but does that translate to a greater weight being placed on Accessibility?

I would like to find a solution that "weights" Acc. 20% more than Cost.
I would like to understand why the proposed average does or does not do that.
I would like the solution to be as transparent as possible so that vendors can understand the results even if they are as challenged as I am. (Assume they will be, this is not about cost or access and the vendors are not business people.)

Can you help?

posted by OmieWise to science & nature (10 answers total)

Make it a two step process: scale both domains to out of 100, then apply your weighting.

1.10*Accessibility score = A
0.90*Cost score = C

0.60*A + 0.40*C = Score
posted by dobi at 8:01 AM on March 14

(Accessibility)/110*60+(Cost)/90*40 = Total Score out of 100.

Or use 55/45 as the multipliers if that's better, depending on what you mean by 20% more.

That converts each score to a percent and then the multipliers weight them how you want them weighted.
posted by brainmouse at 8:02 AM on March 14 [1 favorite]

I assume that just because Accessibility is out of 110 and Cost is only out of 90, that does not mean that Accessibility is "inherently" more valuable than Cost.

So, first, reduce both scores to the same scale. The range from 0 (no points) to 1 (every possible point) works as well as anything: divide A-score by 110 and C-score by 90.

Now, give preferential weighting to C-scores: multiply them by 1.2. A-scores will range from 0 to 1 and C-scores will range from 0 to 1.2. The total score ranges from 0 to 2.2.

Add up your rescaled A-scores and C-scores. Divide by 2.2 (to get back to 0-1 for combined score) and multiply by 100 (to get percentage out of a decimal between 0-1).

Did I understand correctly?
posted by Nomyte at 8:03 AM on March 14

I would use ((6a / 110) + (5c / 90)) / 11 for a composite percentage score. By saying '20% more', you would want each a-point to count as 1.2 (or 6/5) c-points. This accomplishes that exactly.

You can simplify it to (54a + 55c) / 10890 if you'd prefer.

* Extra parentheses are for clarity.
posted by jmfitch at 8:18 AM on March 14

If you want it weighted 55% Accessibility 45% cost, all you need to do is divide by two and add the scores together. 110/2 = 55, 90/2= 45.
posted by Zalzidrax at 8:22 AM on March 14 [3 favorites]

Clarification: 'a-point' and 'c-point' expressed as a percentage. Each actual a-point would be of slightly less value than each c-point, as evidenced by the smaller factor in the simplified equation.
posted by jmfitch at 8:27 AM on March 14

Seems to me that they're already weighted that way, you just need to add them together and divide by 2.

eg. If a vendor were to score max points on both, accessibility would make up 55% of the total score and cost would be 45%, which is the exact same weighting you're proposing to apply if both were scored out of 100. (and works out exactly the same regardless of actual scores, accessibility scores up to 55% of the total and cost up to 45%)
posted by missmagenta at 8:27 AM on March 14

Were both domain scores out of 100 this would be easy: we would simply multiply Accessibility by 0.55 & Cost by 0.45.

Making them both be out of the 100 is easy. Divide each score by the max then multiply by 100 (or multiply by max/100) - just like if you were calculating a percentage. If you then multiply accessibility by 0.55 and cost by 0.45 the range of scores for accessibility is 0-55 and cost is 0-45 which is exactly the same as dividing each of the current scores by 2.

Convert the current scores to both be out of 100 by dividing accessibility by 1.1 and cost by 0.9
eg. a/1.1 + c/0.9 = score
now you want to weight them
(a/1.1)*0.55 + (c/0.9)*0.45 = score

0.5a+0.5c = score
posted by missmagenta at 8:46 AM on March 14

By "20% more", do you mean Accessibility is worth 60% of the total and Cost is worth 40% of the total? Or that the extra points you get for a perfect A score should be 20% of the points you'd get for a perfect C score (that would be 55% and 45%)? Mathematically, you get ALMOST the same thing, but it's conceptually clearer to say "Accessibility is 60% of what we care about", and I think closer to what you are trying to communicate.

But even if you decide to go with a 55%/45% split, stick to a formula that uses percentage weights. It's easier to show a vendor "Accessibility is 55% of the score" than to say "we added the scores and divided by two, and because of the points awarded to each category Accessibility is 55% of the total." Also, it's easier to adjust total number of points and percentage weights on future projects without re-figuring the formula.

Having said that, I'm with brainmouse. What brainmouse is doing is converting each score to a scale of 100 (that's what the /110 and /90 are doing), and then multiplying by a weight to get the final score out of 100 (that's what the *60 and *40 are doing).

If you were going to present this to a vendor (say on a report or score card), you could say:
Accessibility Score: __________ out of 110 points = _____ * 100 = _____ %
Cost Score: __________ out of 90 points = _____ * 100 = _____ %

Total Weighted Score: A * .6 + C * .4 = _____ %


TL;DR - what brainmouse said.
posted by rakaidan at 9:13 AM on March 14

Right now, (acc + cost)/2, using the 110/90 scoring, produces exactly the same results as if you normalized the scores to 100/100, and then weighted them 55/45.

If you normalized to 100/100 and then weighted them 60/40, you'd get slightly different results, but in terms of determining rankings between vendors, the differences would only be significant for vendors that had wildly disparate cost and accessibility ranking--like if they're 80 on one and 5 on the other.
posted by adamrice at 10:00 AM on March 14