Peds96: My average score doesn't change been 68 for weeks although I've played constantly.
Once your average is saturated, it cannot move down. So, after saturation, if all of your scores are above that average, then your average will not move. At Tour Pro, I think you saturate at 25 ranked rounds (at that tier).
By the way, par 3s, par 5s, and custom courses are not included in calculating averages.