It's fine, I do this for a living and I get much worse criticism on that front than I've gotten here!
Since a few people asked, I performed the same analysis for programs in the others receiving votes of this week's AP top 25. As before, I excluded programs that are considered elite (USC, Penn State). I also had to exclude Pitt, Wisconsin and Boise State (*edit and Tennessee) since their coaches haven't been on the job 3 years yet. Although Narduzzi and Chryst seem to be off to solid starts, while what Boise State accomplishes while cycling through coaches is nothing sort of remarkable.
That leaves 6 more coaches to analyze:
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*edit* messed up the order of the data for Butch Jones. He hasn't completed his 3rd year so I'll exclude him for now. Change in analysis reflected below...
Five of these six had more wins in year #3 than the program did the year before they took over (just 2 had to exceed winning records though). 4 of the 6 had more wins in year 3 than they did in year 1, while one matched his year 1 total, and one fell short (Cutliffe).
David Cutliffe is the guy most often referenced around here as the case study of needing to be patient, and the data paints an interesting picture. Duke was absolutely horrendous before Cutliffe arrived. I mean worse then Gergian. Much worse. That program had 0 wins (zero) THREE times in the 8 years before he took over. They also had one win twice, and two wins twice. Duke was 13-90 in the eight seasons before Cutliffe started. Look at that again... THIRTEEN and NINETY. Cutliffe had Duke at 5 wins in his second season, which was the best record they had in fifteen years! He backslid in years 3 and 4, but that program is such an outlier compared to almost everyone else in college football I don't think he's a useful benchmark.
Honestly Cutliffe winning 5 games in his second season, including 3 ACC wins, might be the most impressive coaching job of this century. I'm not kidding. I don't think he's really the poster child for patience, it was clear he was good very early on.
This additional data supports the prior top 25 data. I'm sticking to my finding from earlier, generally speaking a coach who is in the top 25 today, or others receiving votes, was showing tangible improvement in his program's record by year #3.