I got a request on Facebook to take a look at how coaches do in their first year with a new team. This is a decidedly diferent question than "How do first-time coaches do with their team?" An experienced coach is somewhat handicapped -- he doesn't have the same element of surprise as some new coaches might have, and he has a reputation to live up to (or to disprove).
So, how do coaches do in their first year with a team, regardless of whether they are a new coach or seasoned vet like Parcells or Gibbs? I'm so glad you asked. First of all, they tend to do a bit better than their successor. On average, a team with a new coach will add 1.34 wins the following season. Since 2000, teams with new coaches (TWNC) have kept the same win percentage only twice (2000 Bengals under Dick LeBeau and 2001 Redskins under Marty Schottenheimer). Check the chart for more detail:
Teams that have gone 4-12 in previous years have generally added 2.7 wins to their season. That would lead one to expect the Seahawks to go 7-9 last year, based only on that fact. That, of course, would make no sense, but hey, whatevah. When Jim L. Mora took over the helm for the Falcons in 2004, he added 6 wins from the prior year. In 2003, the Falcons had a lot of injuries including missing their quarterback for more than half the season. That sounds oddly familiar...
Still, facts are facts, and the greatest single season turnaround since 2000 is 11 games (Tony Sparano in 2008 with the Dolphins), which didn't make it ot the chart above because its kind of an outlier and I screwed up, haha. The next highest is +7 games, which has been done three times (Mike Smith with the Falcons last year, Sean Payton with the Saints in 2006, and Jim Haslett with the Saints in 2000). Most often, a new coach adds 2 wins to their teams totals, with crappy teams (6 - 10 or worse) fairing better than average. This has plenty of other correlative factors -- higher draft picks, a full "regime change" that allows the new coach to bring in some of his people to make his system work better, and just playing the odds. (Last year's 4-12 record for the Hawks was a combination of playing terribly and massive injury troubles -- odds are, the injuries won't come back this year.)
Teams that go 6-10 or worse in one year, add, on average, 3.0 wins the following year. That doesn't sound like a ton, but remember: that's equivalent to an MLB team winning an extra 30 games. Three games is huge in football, and with a couple of extra breaks, it can be gigantic (see: 2008 Falcons).
How will Mora do? Who knows! But if history is any precedent, it seems very likely that it will be better than 4-12. Only three (9%) "bad teams" had 1st Year Coaches who had more losses than the year prior -- Cam Cameron (Dolphins 2007, 5 fewer wins), Rod Marinelli (Lions 2006, 2 fewer wins), and Jack Del Rio (Jaguars 2003, 1 fewer win).