Let us say I have a scaling score from 1 to 10, in increments of 1. So:

1

2

3

4

5

6

7

8

9

10

And for each of those scores I have a population of 10.

10

10

10

10

10

10

10

10

10

10

When I multiply each population with each score, and then divide the sum of that product with the sum of the populations (the weighting values), I get a weighted value of 5.5, which I believe for my purposes skews the result too high (by a value of .5). To adjust this 5.5 to 5.0, I assume I have to include a 0 value on the 1-10 scale but I’m not sure how that would then adjust the result to 5, as I would simply be multiplying a score of 0 with a population of 0. Am I thinking about this correctly (albeit in an unsophisticated way) and what can I do to adjust the score accordingly? Or should I not adjust the score—that is, is 5.5 the correct way of averaging this and my inclination or intuition is wrong?

I’ll note further that I’m actually looking at a scale from 0 to 100, the scores of which I’m given in deciled increment ranges of 5—so 0.0-5.0, 5.01-10…95.01-100.0. Each score, like 0.0-5.01, has a corresponding population value. My inclination has been to change these to a scale of 2.5, 7.5…97.5, and then calculate the weighted average accordingly but that may be quite wrong; the only other alternative I can think of would be to calculate the weighted average with a scoring system of 5, 10...100, but that seems to skew the result high, as I outlined in the example above. Any advice here would be welcome. Thank you.