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I have this problem very difficult for me to solve.

I have a company and some employees.

Each of them expresses an opinion on some apples of varying quality.

The vote can be:

0 1 2 3 4 5 6 7 8 9 10

For example, the first two apples are rated in this way:

1 apple = 2 employees vote for 0, 3 employees vote for 1, 100 employees vote for 10.

2 apple = 10 employees vote for 0, 10 employees vote for 5, 120 employees vote for 10.

The weighted average of these results is:

1 apple = Weighted average = 9.55.

2 apple = Weighted average = 8.92.

So far so good.

But there is a problem.

In the first apple I would like that the result of the weighted average is a little less democratic and could indicates "10" (or more than "9.55"), given the smallness of quantitative votes assigned to "0" and "1" (2 votes for the vote "0" and 3 votes for the vote "1").

I need a mathematical system that automatically detects the distance between the maximum value (100 in this case) and minimum values ​​(2 and 3, in this case) and assign to the highest value a "majority bonus".

Is that clear? I do not think can I explain it better than that.

In other words, I would like that the single unit that composes the sum of the votes for the lower grades (in this case: single unit [1] that composes the sum "2" for the vote "0" and the sum "3" for the vote "1") have less dignity of the single unit (in this case, the single [1] that composes the sum "100" for the vote "10") that composes the sum of the votes for the higher vote.

In the event of a tie between high grades then that's great "simple" weighted average.

I hope for your help , I do not know where to turn, my brain is crying.

ginolin
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1 Answers1

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A very (very) simple thing you could do would be to raise the vote counts to the power $\alpha>0$, where $\alpha > 1$ would be 'less democratic' and $\alpha < 1$ would be 'more democratic.'

For example, $\alpha =2$ in your first example gives new vote counts of $4, 9,$ and $10000$, which gives a weighted average of $9.99$.

Ross B.
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  • You've centered my problem and you solved it. A great man. Thank you very much. But if I wanted something more "difficult" and not a simple one? – ginolin Dec 03 '13 at 00:41
  • In general, you could apply any monotonic function to transform the vote counts, with strictly increasing or decreasing functions (such as $\exp(x)$ or $\ln(x-1)$) used to skew the relative importance of large values. If you have prior information on what the counts should/might be, you could take a Bayesian approach by defining prior probabilities on vote counts and computing the posterior probability from the data. One way to do this is http://en.wikipedia.org/wiki/Additive_smoothing, though, as you may imagine, this reduces the effect of abnormally large counts ('more democratic'). – Ross B. Dec 03 '13 at 13:33
  • Yes, I've studied a lot of the additive smoothing and Bayes theory, which, as you well say, do not in this my case, beacuse i need less democratic functions. Thank you anyway. – ginolin Dec 03 '13 at 17:24