From 2003 to 2013 — 11 consecutive seasons — at least one NFL team has won its division a year after finishing last in the division. In five of those seasons, including 2013 with Carolina and Philadelphia, multiple teams went from worst to first. In 2006, three teams did it, giving us a total of 17 over the past 11 seasons.

While the factors are numerous, two common denominators seem to be basic but undeniable: improve a lot on defense and/or fix a problem at quarterback.

With free agency looming for the Vikings and 31 other teams, here are the crunched numbers and the blueprint for the Vikings getting this thing back on track:

• Using the most basic metrics of points scored and points allowed, the teams that have gone from worst to first since 2003 have scored, on average, 81 more points and allowed 79 fewer points from year to year in vaulting from the cellar to the top spot in their divisions. It’s an imperfect measurement, but it’s still useful.

• In most cases, the improvement didn’t follow an even split. In the majority of cases, the improvement on one side of the ball was at least 40 points better than it was on the other side of the ball. And in five cases — including each of the past four times a team has gone from worst to first — the improvement has been more than 100 points greater on one side of the ball than the other.

• Those four recent examples are particularly instructive. The 2011 Texans went from worst to first by allowing 149 fewer points than they did in 2010. They actually scored nine fewer points, but their staggering defensive improvement changed their entire dynamic. Same goes for the 2013 Panthers, who improved only nine points on offense but were 122 points better on defense. Washington in 2012, on the other hand, scored 148 more points and actually gave up 21 more than the previous season. The Eagles in 2013 under Chip Kelly improved 162 points on offense and 62 on defense, a gap of 100.

Houston changed its scheme under new defensive coordinator Wade Phillips and saw dramatic results with a combination of new and old talent. The Panthers went from decent to great in pass defense. Washington added dynamic rookie QB Robert Griffin III. The Eagles had an incredible 10-game run from young QB Nick Foles (27 touchdowns, only two interceptions).

• Where does that leave the 2014 Vikings? Well, they brought back Matt Cassel, which is an important step in solidifying the offense. They also figure to draft a QB of the future. But they already ranked 14th in the NFL in points scored in 2013 with 391, so clearly their biggest upside is on defense. They gave up 480 points last year, dead last in the league. They need to invest at least 75 percent of their free agency money on defense — reportedly bringing back Everson Griffen is a nice start — as well as a majority of their draft picks on that side of the ball.

Would that, combined with a new regime led by a defensive-minded coach, lead to a 100-point (or more) improvement and a shot up the standings? It’s not crazy to think that it would.