Homological stability is now well established as an organizing principle and computational tool in algebraic topology and other areas. In many cases it is of interest to obtain homological stability with twisted coefficients, and the standard choice of such coefficients are the *polynomial coefficient systems*. All known approaches to homological stability with polynomial coefficients produce a stable range depending on the degree of polynomiality. I will explain a method of obtaining uniform stable ranges for some classes of groups and coefficients of natural interest. This has important consequences in arithmetic statistics, discussed in the number theory seminar on Nov 2. (Joint with Jeremy Miller, Peter Patzt, Oscar Randal-Williams.)