Q: Vote "0-10" dignity. 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.

A: 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 $9.99$.
