I just took 15 mins to google any study that shows that replacing a head coach leads to better results. You can guess what I found -- just the opposite is true.

Most of the statistical research that came up in my short search was on NFL coaches, but I did find one paper published specifically on college football coaches from 1997-2010. Here's an excerpt:

"Results. We find that for particularly poorly performing teams, coach replacements have little effect on team performance as measured against comparable teams that did not replace their coach. However, for teams with middling records—that is, teams where entry conditions for a new coach appear to be more favorable—replacing the head coach appears to result in worse performance over subsequent years than comparable teams who retained their coach."

(Social Science Quarterly 2013 -- http://onlinelibrary.wiley.com/doi/10.1111/j.1540-6237.2012.00929.x/full)

Any actual mathematical evidence you can point me to that shows that changing up head coaches leads to better results? (even if for a small subset, like coaches on a decline or something?)