Despite finishing 4-12, the Detroit Lions were only outscored by 65 points in 2012. The Philadelphia Eagles and Oakland Raiders, by comparison, also went 4-12, but they were outscored by 164 and 153 points. From a point differential standpoint, the Lions should have had a much better record than 4-12.
According to Grantland's Bill Barnwell, the Lions actually had the make-up of a 6.5-win team based on the Pythagorean Expectation formula, which accounts for points for and points against. The reason the Lions didn't win more often is because of their lousy 3-8 record in close games (decided by one score or less). Between bad luck, costly mistakes at the worst possible time and other factors, the Lions just weren't able to bring home many wins in close games last season.
The bad news here, obviously, is that the Lions underperformed in 2012 despite nearly all of their games being competitive. The good news is that this actually bodes quite well for the 2013 Lions. I'll let Barnwell explain why:
In general, a gap of about two wins between your actual win total and your "expected" win total from your point differential is the spot where you start talking about significantly likely improvements. There have been 42 teams since 1989 that underperformed their Pythagorean win total by two wins or more. The following year, those teams saw their win-loss record improve by an average of 2.6 wins. Last year, two teams fit into that category: the Dolphins, who went from 6-10 to 7-9, and the Vikings, who leaped from 3-13 all the way to 10-6.
This year, the Lions are actually the only underperforming team expected to take a major step forward based on the Pythagorean Expectation formula. There are obviously no guarantees that their luck will change in 2013, but history suggests that they should win at least six games this year. And if they are able to actually catch some breaks and win more close games, getting to eight or more wins is entirely possible.
On the opposite side of the Lions are the Indianapolis Colts and Minnesota Vikings. These two teams are expected to take a step back after having winning seasons in 2012. The Colts finished 11-5 in 2012 despite being outscored by 30 points, whereas the Vikings went 10-6 despite only having a point differential of +31. The Colts' ridiculous 9-1 record in close games helped lead them to the playoffs, just as the Vikings' 5-1 record in close games helped them do the same in the NFC.
Again, these stats don't guarantee anything for the 2013 season. A lot of factors are involved, and what happened last year doesn't mean the Lions are guaranteed to be better this year. However, based on past history, the Lions have a statistical basis for their optimism going into the 2013 season.