## What’s a touchdown worth?

Posted by Chase Stuart on October 1, 2008

On Monday, I looked at the difference between having the ball on first down at the one yard line and scoring a touchdown. Last year, Doug explained why a touchdown was worth ten yards, at least according to The Hidden Game of Football. In this post, I'm going to try and explain what a touchdown really is worth. Later this week, either Doug or I will probably write a postscript to this series.

Every touchdown *except* those that come on a 1st, 2nd, 3rd or 4th and goal situation can be measured by the analysis in yesterday's post. As long as you gain a first down on the play in question, the difference between a touchdown and the ball at the one depends solely on the distribution of results in 1st-and-goal situations. According to the post yesterday, 1st and goal from the one is worth about 0.865 touchdowns.

But what about a touchdown on 4th and goal? That's pretty valuable. Touchdowns on 1st and goal are slightly more valuable than your run of the mill touchdown. Any touchdown on a down-and-goal situation is more valuable than a touchdown in any other situation, since one less yard in the former situation puts you in a worse position than one less yard in the latter position. To use an example, coming up an inch short from the end zone on 3rd and 10 from the 15 gives you 1st and goal from the one; coming up an inch short on 3rd and goal from the one gives you 4th and goal from the one; quite clearly, the former situation is better.

So how many touchdowns come in these X-and-goal situations? In 2007, there were 720 passing touchdowns. Of those, 442 touchdowns came on non-goal to go situations. Of the remaining 278 touchdowns that came in goal-to-go situations, 84 passes were on first down, 95 were on second down, 90 came on third down and nine came on fourth down. This means that 61.4% of passing touchdowns were of the non-goal-to-go variety, 11.7% came on 1st and goal, 13.2% came on 2nd and goal, 12.5% came on 3rd and goal and 1.25% came on 4th and goal. We already know what a passing touchdown is worth in 61.4% of the situations -- what about the rest?

The fourth and goal situation is easy to analyze. If the pass goes for a touchdown, it's worth 6.4 points. If it came up an inch short, your team would now be on defense, with the opponent facing 1st and 10 from their own one yard line. According to Romer, that situation is worth -1.55 points to the offense, or +1.55 points to the defense. This makes a completion on 4th down that crosses the goal line worth 4.85 more points than one to the one inch line.

Grading the value of a third down touchdown pass is a little tricky, because it involves weighing sub-optimal decision making. Your average third-and-goal pass that brings the offense to the one-yard line results in a field goal on the next play -- but that's usually a bad decision. But because we're dealing with averages here, we'll simply concede that a failed play can be made worse by the likelihood of a bad coaching decision. So to get the value of a third down touchdown relative to a third down pass to the one, we need to know the value and likelihood of the alternative options on 4th-and-1: a successful field goal, a missed field goal, a successful fourth and one, and an unsuccessful fourth and one. The odds of a missed field goal from that close are small enough to be ignored, which means we can assume all field goal attempts are successful. The odds of a successful fourth and one, based on data from the past three seasons, is 0.55. To conclude, a successful field goal is worth 2.4, a successful fourth and one is worth 6.4, and an unsuccessful fourth and one is worth about +1.15 (this is because the ball is usually turned over at around the three). In 2007, twenty of forty times the coaches chose to kick the field goal instead of going for the score. Using that ratio, we can find a weighted average value of a fourth and one situation to be +3.22. Therefore, the value of fourth and goal at the one is +3.22, which is 3.18 fewer points than a touchdown.

To get the value of a second down touchdown relative to a play down to the one yard line, we need to know the value and likelihood of the alternative options on 3rd-and-1: a touchdown, a turnover, a play for no gain, or a loss of yards. We know the value of two of those things already - a touchdown is worth 6.4 and the value of a play for no gain is 3.22 -- because that's fourth and one. We can assume that a play that loses yards is simply worth 2.4, because then the team will kick a field goal. In 2007, 59 teams faced a 3rd-and-1 from the 1 and thirty-three of them scored touchdowns. Three more went for lost yardage and ensuing field goals, 21 went for no gain and fourth and one situations, and the final two were interceptions that resulted in touch backs (value of -0.25). The weighted average tells us that the value of 3rd-and-goal from the 1 is +4.84, which is 1.56 fewer points than a touchdown.

To get the value of a first down touchdown relative to a play that ends at the one, we need to know the value and likelihood of the alternative options on 2nd-and-1: a touchdown, a turnover, a play for no gain, or a loss of yards. Once again, we know the value of a touchdown and of the potential 3rd-and-1 situation. In 2007, there were 97 2nd-and-goal situations from the one, and 55 went for touchdowns (+6.4) and 28 went for no gain (and thus third and one, +4.84). Of the remaining fourteen plays, there was one fumble (+0.9), one interception (-0.25) and twelve plays that averaged into a third and two situation. For simplicity’s sake, I’m going to approximate a third and two as around +3.8. Add up all the possibilities, and this comes to a 5.50 weighted average, the value of 2nd-and-goal at the one.

Here's a final table of the value of each touchdown pass:

To the 1 TD Diff % of TD passes Non-goal to go 5.55 6.4 0.85 61.39 1st and goal 5.50 6.4 0.90 11.67 2nd and goal 4.84 6.4 1.56 13.19 3rd and goal 3.22 6.4 3.18 12.50 4th and goal 1.55 6.4 4.85 1.25

Simple multiplication tells us, then, that the *average* touchdown pass is worth 1.29 points. The majority of touchdowns are worth only 0.85 points, but enough touchdowns are worth over 3 points to bring that average up to 1.29. Now depending on who you listen to, a yard is worth about 10-15 yards. According to Romer, the 98 yards on the field (from the 1 to the 1) span from a value of -1.55 to +5.55; in other words, 98 yards are worth 7.1 points. Therefore, one point is worth about 13.8 yards, and 1.29 points are worth about 17.8 yards. This is a significant increase from the 10 yards given in The Hidden Game of Football, and depending on blog reaction (and the next post), going to be my new standard for valuing touchdown passes.

Note: Rushing touchdowns are very easy to value here, too. Why's that? There were 386 rushing touchdowns in 2007. Of those, 130 occurred in non-goal to go situations. 117 occurred on 1st and goal, 94 on 2nd and goal, 40 on 3rd and goal and five on 4th and goal. Because the value of the situation that follows being tackled at the one is the same whether the previous play was a run or a pass, we can use the same numbers as above.

Non-goal to go 5.55 6.4 0.85 33.68 1st and goal 5.50 6.4 0.90 30.31 2nd and goal 4.84 6.4 1.56 24.35 3rd and goal 3.22 6.4 3.18 10.36 4th and goal 1.55 6.4 4.85 1.30

While the average passing touchdown was worth 1.29 points, the average rushing touchdown was worth 1.33 points. And of course, 1.33 points is equal to 18.4 yards.

It also follows from this analysis that return touchdowns by Josh Cribbs or anyone else are worth "only" 11.7 yards. That number isn't too far from the number used by The Hidden Game of Football.

So this would affect your recent QB rankings, yes? It'd add 350-400 yards to the crazy Marino, Manning, and Brady years, probably pushing up their NAY/A by over 3/4 of a yard...

If it's possible to get a reasonable value number given LOS, yards to first, and down, then one could essentially assign points gained per play. 🙂

MattieShoes---What difference does it really make--those 3 Regular Season performances will remain as 3 of the very best Regular Season performances ever, and one other thing won't change either--all 3 will remain as 3 of the most disappointing Post-seasons ever FOR THEIR FANS.

Shouldn't you subtract a yard from those values (17.8, 18.4, and 11.7) to account for the fact that you're comparing a TD to life at the 1? The difference between goal to go at the 1 and a TD is a touchdown and one yard. For instance, a 14 yard run on 3rd & 10 from the 15 leaves you with 5.55 pts of value (1st & goal at the 1), and a 15 yard TD run from there leaves you with 6.40 pts of value, so that TD plus the additional yard is worth .85 pts (which converts to 11.7 yards). The player with the 15 yard run is already getting credit for 1 more yard than the player with the 14 yard run, so to reflect his value you need to credit him for an additional 10.7 yards of value for the TD. That means that the value of an average TD should be 10.7 for return TDs, 16.8 for passing TDs, and 17.4 for rushing TDs.

Given that yards approaching the end zone are more difficult to gain than your average yard, perhaps one could argue that even 17.8 yards underestimates the value of a passing touchdown. Maybe that's how Doug is going to argue that the bonus should be 20 yards.