Word maps, polynomial maps and image ratios

If A is a finite group (or a finite ring) and ω is a word map (or a polynomial map), we define the quantity |ω(A)|/|A| as the image ratio of ω on A and will be denoted by μ(ω,A). In this article, we investigate the set R(ω)=\μ(ω,A) : A is a finite group\, and also consider the case of rings. Specifically, we demonstrate the existence of word maps (and polynomial maps) whose set of image ratios is dense in $[0,1]$ for both groups (and rings).