Converting absolute performance ratings to inverse relative ratings

[JenniferMurphy]

This is not an Excel question per se, but there are a lot of people here with math skills and I get such good answers here, I thought I would give it a try. I hope that's OK.

I have some performance scores or ratings. They are the results of tests in various categories or components of a skill. I would like to convert them to inverse relative ratings where the lowest score gets the value "1" and all the rest get proportionately lower scores. This is in preparation for converting them to relative weights for future drills.

If all of the ratings are positive, this is easy. I just divide the minimum value by each of the scores. Here's an example:

C/R	D	E	F	G	H	I	J
4	Components	A	B	C	D
5	Scores	12	6	3	2	2	Min
6	Inverse = MinScore / Score	0.167	0.333	0.667	1.000	2.167	Sum
7	Weights = Inverse / SumInverse	8%	15%	31%	46%	100%	Sum

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The four components are A, B, C, & D. Their scores are 12, 6, 3, & 2 (Row 5). The relative inverse ratings are in Row 6. The lowest raw score (2) gets a "1" (H6). The rest get proportionately lower ratings.

The weights are in Row 7. The component with the lowest score will get the most drills (46%). The rest will get proportionately fewer.

This is all fine and good until one of the components gets a zero score. Then we run into divide by zero errors (H6), which causes the rest of the table to fail.

C/R	D	E	F	G	H	I	J
4	Components	A	B	C	D
5	Scores	12	6	3	0	0	Min
6	Inverse = MinScore / Score	0.000	0.000	0.000	#DIV/0!	#DIV/0!	Sum
7	Weights = Inverse / SumInverse	#DIV/0!	#DIV/0!	#DIV/0!	#DIV/0!	#DIV/0!	Sum

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It also fails in tests where there is a penalty for errors making a negative score possible.

C/R	D	E	F	G	H	I	J
4	Components	A	B	C	D
5	Scores	12	6	3	-1	-1	Min
6	Inverse = MinScore / Score	-0.083	-0.167	-0.333	1.000	0.417	Sum
7	Weights = Inverse / SumInverse	-20%	-40%	-80%	240%	100%	Sum

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I would appreciate any suggestions for ways to calculate reasonably good inverse relative ratings in these cases. I realize that compared to zero, all non-zero values have an undefined (infinite) ratio. I suppose I could add 1 to all scores for an approximation. Similarly, for the negative numbers, I could add a value that will bring the lowest score to 1.

Are there better ways?

 

[wideboydixon]

Hi,

I think the thing that's missing is a measure of how much you expect a drill to improve a score so any calculation is a bit of a "guess". For example, I added an extra row for an adjusted score and used that for calculation:

Book1
	D	E	F	G	H	I	J
4	Components	A	B	C	D
5	Scores	12	6	3	2	2	Min
6	Adjusted Score	11	5	2	1	1	Min Adjusted
7	Inverse = MinScore / Score	0.091	0.200	0.500	1.000	1.791	Sum
8	Weights = Inverse / SumInverse	5%	11%	28%	56%	100%	Sum
Sheet1

Cell Formulas
Range	Formula
E6	=E$5-$I$5+1
E7	=$I$6/E$6
E8	=E$7/$I$7
I5	=MIN($E5:$H5)
I6	=MIN($E6:$H6)
I7	=SUM($E7:$H7)
I8	=SUM($E8:$H8)

WBD

 

[JenniferMurphy]

Hi,

I think the thing that's missing is a measure of how much you expect a drill to improve a score so any calculation is a bit of a "guess". For example, I added an extra row for an adjusted score and used that for calculation:

It's a little spooky that you chose the term "adjusted score". But I don't understand what it has to do with my expectations for the amount of improvement for a drill. I don't have any expectations. I just want to allocate the drills proportionately and see what happens.

Anyway, after a little playing around, I came up with something similar. I decided that one (imperfect) solution would be to raise all of the scores by enough so that the lowest one equals 1. I called that the "adjusted score". Here are three examples:

Example #1: No scores < 1 -- No adjustment needed.

C/R	E	F	G	H	I	J	K
4	Components	A	B	C	D
5	Scores	12	6	3	1	1	Min
6	Adjusted Scores	12	6	3	1	1	Min
7	Inverse = MinScore / Score	0.083	0.167	0.333	1.000	1.583	Sum
8	Weights = Inverse / SumInverse	5%	11%	21%	63%	100%	Sum

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Example #2: 1 score = 0 -- all scores raised by 1

C/R	E	F	G	H	I	J	K
4	Components	A	B	C	D
5	Scores	12	6	3	0	0	Min
6	Adjusted Scores	13	7	4	1	1	Min
7	Inverse = MinScore / Score	0.077	0.143	0.250	1.000	1.470	Sum
8	Weights = Inverse / SumInverse	5%	10%	17%	68%	100%	Sum

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Example #3: 1 score < 0 -- all scores raised by 2

C/R	E	F	G	H	I	J	K
4	Components	A	B	C	D
5	Scores	12	6	3	-1	-1	Min
6	Adjusted Scores	14	8	5	1	1	Min
7	Inverse = MinScore / Score	0.071	0.125	0.200	1.000	1.396	Sum
8	Weights = Inverse / SumInverse	5%	9%	14%	72%	100%	Sum

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This is certainly not perfect, but it's adequate for my needs.