Let $A$ be a set of values $\{a_1,a_2,a_3,a_4,a_5\}$ where $a_1 = 2$, $a_2 = 1$, $a_3 = 4$, $a_4 = 1$ and $a_5 = 2$, so, the $avg(A) = 2$.
I'm looking for a normalization where the values below the average would be increased while the values over the average would be reduced, with the condition that the normalized version of a greater number would also be greater than the normalized version of a smaller one.
I could just tranform to the logarithmic scale but the reduction is too much. My attempt was: $\frac{a_i}{log_{avg(A)}a_i}$. The problem is that it does not work if the value of $a_i$ is 1, for example. Also, $\frac{2}{log_{2}2} = 2$ while $\frac{3}{log_{2}3} = 1.89$, violating the condition.
Is there any (good) way to achieve what I stated in the 2nd paragraph?
You can compute $a_{avg}$, the average of all your numbers. Then replace $a_i$ with $a_{avg} + \frac 1n (a_i-a_{avg})$. Choose $n$ to your liking. The average will not change, nor will values that start at the average.