I of course am speaking only from a practical reading of games that have happened in the past. I'm sure having field goals be worth almost half of a touchdown makes them feel relevant while watching a game. I'm sure it's exciting to imagine catching up to a touchdown incrementally through field goals and conversions.
But it doesn't happen.
Without a good way of accessing game history data I was limited by my patience in looking up final game scores and, more importantly, the scoring patterns that make them up. I ended up looking up the past twenty Super Bowls on Wikipedia. Each one supports my hypothesis, though some more obviously than others.
First, a quick overview of the types of scoring we see.
| Touchdowns (7 points) -- These make up by far the bulk of points scored in Super Bowls. A touchdown is achieved by running the ball to the end zone or throwing it to a receiver there. Strictly speaking a touchdown is worth only six points, the seventh being a kick through the goalpost immediately following the touchdown. |
 One million points to Gryffindor! |
Two-Point Conversions (±1 point) -- In some cases a team will choose to attempt a two-point conversion instead of kicking the extra point for a touchdown. This essentially amounts to gambling a point since the kick is almost never missed. In most games no team will attempt a two-point conversion, though in some games they will be attempted several times. According to Wikipedia, these succeed about half the time.
Field Goal (3 points) -- If a drive is unsuccessful at reaching the end zone for a touch down, the team can choose to kick a field goal instead. The kick takes place from the current field position, so the rate of success of the field goal depends on the penetration of the drive into opposition territory.
Safety (2 points) -- Sometimes the team in possession of the ball will be driven backwards and tackled in their own end zone. In this case the defense is awarded two points. This almost never happens.
It was my hypothesis that seven points, being more than double three points, might as well be infinitely more. In order to make up a deficit of a single touchdown, a team has to attain a three field goal surplus, or two field goals plus a successful two-point conversion. Since possession changes hands after each scoring event (meaning that the team that's behind must also prevent the other team from scoring on multiple drives) this seems unlikely to happen.
 Don't bother, kid. | It turns out that this is true; touchdowns seem to always determine the winner of the game, with all other scoring methods being only important as tiebreakers. It doesn't even really matter how exactly we rate conversions, field goals, and safeties with regard to one another. From the past twenty Super Bowls, we see the following: |
In thirteen of the past twenty Super Bowls, one team has scored more touchdowns. All thirteen of those teams have won.
In six of the remaining seven, the teams have earned the same number of touchdowns, but one team has had more field goals. The team with more field goals has won all five of these. There was a safety and a two-point conversion attempt here and there, but none changed the outcome of the game.
The final game was the 2004 Super Bowl, in which the Panthers and the Patriots each scored four touchdowns and a single field goal. The Panthers attempted a pair of two-point conversions, failing both; the Patriots attempted only one and got it.
The same winners would result in each case if, for example, touchdowns were worth ten points, safeties and field goals were worth a point each, and after each touchdown a team had the option of wagering a single point on a conversion.
This would perhaps be less of a tease on the viewer, who can no longer reasonably imagine that you can make up a touchdown deficit, but my (admittedly small) sample does not show that the outcomes of games would change. I do not intend to continue researching this topic, but would be interested to hear of a game that defies this rule.
EDIT: Alex found an easily-processed play-by-play list of every football game played in the NFL since 2002 on http://www.advancednflstats.com/. More thorough analysis will follow but our preliminary estimate is that field goals matter in about 5% of games (one in twenty). This holds up pretty well for each of the past ten years. We are not able to comment on years before 2002.
