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For more from Chase and Jason, check out their work at Football Perspective and The Big Lead.

## Why a touchdown is worth ten yards

Last week Chase posted a through-six-weeks list of quarterbacks ranked by what he called "rearview adjusted yards per attempt." The rearview part means that the numbers had been tweaked to modify the differing schedules that each QB had played against during the first six weeks. The adjusted part means that, following some decades-old research by Pete Palmer, John Thorn, and Bob Carroll, he wasn't just using yards, but was rather giving a 10-yard bonus for each touchdown pass and a 45-yard penalty for each interception thrown.

Over at the footballguys.com message board, Chase decided to use that post as a starting point for his case for David Garrard as the league's through-six-weeks MVP. The resulting thread went about like you'd expect it to go, but a little bit of interesting discussion emerged. One of the main points of contention is that someone who has thrown six TDs in five games can't possibly be as valuable as someone who has thrown 21 in six games. That is, many people thought the 10-yard bonus for a touchdown pass is too low. A bit of discussion about the appropriateness of the 10-yard bonus broke out in the comments to Chase's original post as well. So I figured I'd spend a post talking about why I think it's about right.

First, let's get some disclaimers out of the way. It's clear that there is no static TD-to-yard translation that is going to be right for all situations. Some touchdowns are worth more than others, and some yards are worth more than others. We're trying to hit the average here.

Now, the key to understanding this is to realize that we're not saying that a TD is equivalent to ten yards. We're saying that the marginal value of a TD, compared to a non-TD pass of the same yardage in the same situation, is about the same as the marginal value of ten generic yards. Here is a hypothetical situation, which has been sterilized to remove some of the issues in the disclaimer mentioned above.

It's the first game of the season, the first drive of the game. Your team is on offense. You and your opponent are evenly matched. You have first-and-ten at your own 30 yard line. Which would you rather see on the next play?

- A 69-yard pass that puts you at 1st-and-goal at the one,
- A 70-yard TD pass, followed by a 15-yard excessive celebration penalty (so you'll kick off from the 15 instead of the 30)?

If you have to think about it --- if it's not immediately clear --- then you must think that the marginal value of the TD, compared to a same-yardage pass that is not a TD, is worth approximately 15 yards. Try this experiment next weekend: take every TD pass you see and ask yourself, "if I were coaching the offensive team, and I could trade that TD for the ball at the one and 10 yards of field position on my opponent's next possession, would I do it?" Better yet, write down how many yards of field position you would trade to erase that TD and put the ball at the one instead. Average those numbers.

I don't know if ten is exactly the right answer, but it feels about right.

This entry was posted on Monday, October 22nd, 2007 at 3:26 am and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

Hmm, but if was a trade of a TD vs. 1st and goal at the 5 I ALWAYS take the TD. Only somewhere around the 2.5 yard line is where it even gets worth considering taking the 10 yards or so.

That's making me feel that yards should be weighted based on where they are on the field. Each yard from the 5 to the goaline is worth about 10 yards, from the 10 to the 5 a yard is worth maybe 5 yards, and so on out to the 50 where a yard is worth a yard.

In other words, I'd gladly trade a 15 yard non-scoring pass inside the red zone for a 40 yard bomb that happened starting from my 35 yard line. So it seems this isn't about the marginal value of TDs, but how some yards are far more important than others.

I think if you asked every coach in the league (or Vern), he'd answer, "Take the TD." It's the old "Don't take points off the board" mentality that people like to quote when a team has the chance to cancel their own field goal when there's a penalty on the defense. Statistically, though, he'd probably be off.

I'd assume drives that start 1st and goal from the 1 have a very high chance of scoring a TD. I'll say about 80%, without any real data to back it up. And I'd then guess there's a small (say, 5% chance) of an excessive celebration penalty afterward. So, you're looking at the following options:

100% chance of TD

100% chance of 15-yard penalty on kickoff

vs.

80% chance of TD

5% chance of 15-yard penalty on kickoff

Also, suppose (with more completely made-up numbers) that any drive following a kickoff has a 20% chance of a TD and that a 15-yard penalty on the kicking team increases those chances by 10%. Now your options become:

100% chance of TD for you

30% chance of TD for opponent

vs.

80% chance of TD for you

20.5% chance of TD for opponent

So, by taking the second option, you're cutting your chance of scoring a TD by 20% while decreasing the chances of your opponent scoring on his next drive by about 50%.

Again, I'm completely guessing on these numbers, but it seems like a decent choice to me.

I think TD passes are overvalued. In fact, I think individual TDs are overvalued for all players. Many times, scoring a TD is just the result of a coach choosing to run a play that gets you a TD.

How many times do you see a player make a long play and take the ball inside the 5, only to have some other player score on a short TD run (where he's untouched because the line blocked perfectly)?

Shaun Alexander's 2005 season is a great example.

“if I were coaching the offensive team, and I could trade that TD for the ball at the one and 10 yards of field position on my opponent’s next possession, would I do it?”

If I got the ball at the one with a first and goal, then I would take the ball at the one and the extra field position in virtually all situations, barring having a historically inept offense or facing Chicago 1985.

Jason W, I think your 80% estimate is conservative. I would put it above 90% of scoring a touchdown with 1st and goal from the 1. Last year, the 4th and goal from 1 conversion rate was 15/24. If we assume 60% chance of scoring on any run from the 1, the chances of scoring are in excess of 90%, even if we discount for the occasional fumble or large loss of yards.

I also just had a random thought about touchdowns, which we can file under "never would happen." Football is derived from rugby, and my limited understanding of rugby is that when a player scores and touches the ball down, the try attempt (similar to our extra point) is attempted on the same line as where the ball was touched down. Thus, there is value to getting the "touchdown" in the middle of the field to improve the chances on the try.

How much more strategy would there be if the extra point had to be kicked from the same line as the touchdown was scored, and all touchdowns scored along the sideline or outside the numbers were only brought to the numbers for the try? Then, going down at the one would have even greater value in allowing the team to try to score in the middle of the field on the next play.

Why just 69 yards and first and goal at the one? Why not 60 yards and first and goal at the ten ... versus a TD and 25 yards of penalties? I'd take the TD.

And interceptions are overvalued at 45 yards... it's closer to 35 yards according the current regressions.

I don't see where lost fumbles enter into this QB ranking. Heck... they're probably even worse than interceptions due to field position.

And what about lost yardage due to sacks? Is this in the formula?

What's the difference between coughing the ball up to the other team... or simply throwing it to them? They are both QB error, aren't they?

The Wages of Wins site includes these, I believe...

Here's an idea about sacks. I agree with T.G. that yardage lost due to sacks ought to figure into any credible passer rating system.

Consider an incomplete pass. It affects the passer rating by increasing the number of attempts, thereby reducing completion percentage, touchdown percentage, and yards per attempt. (It improves INT percentage by reducing it-- that makes sense, because an INC is more helpful than an INT.) The rating generally goes down when an INC is thrown.

What is an INC? It is a wasted down in which no yardage is gained. What is a sack? It is a wasted down in which yardage is lost.

It begs the question: Why should an INC, which is better than a sack, hurt the rating, but a sack doesn't?

Those who know me know I rarely just present a problem without posing a solution Count a sack as a pass attempt (which it really is) and deduct the yardage lost from passing yardage (which it really does). In this way, the passer rating would more accurately reflect the effects of sacks.

--

For extra credit (or extra "cred-eet", if you prefer), you could figure out the passer rating with sacks and subtract that from the passer rating the way it's traditionally figured (i.e., without regard to sacks), and you will have a solid number as to how sacks hurt a particular passer.

Let's have an example!

Let's take a look at D. Culpepper's numbers against the Chiefs this week...

Pass attempts: 29

Completions: 18

Pass yards: 228

TD Passes: 1

INTs: 1

Was sacked: 2

Yards lost on sacks: 15

--

His passer rating was a respectable 83.7

Now if you factor in the sacks, his attempts become 31 and his yardage goes down to 213. The new passer rating would be 76.4

--

This could be interpreted as 7.3 rating units that were lost to sacks. The offensive line could take credit for 7.3 pass protection. The defense of the Chiefs could take credit for a 7.3 pass rush. The possibilities are endless!

In my fantasy league, the qb gets 1 point per 50 passing yards and 6 points for a TD. So we basically give qbs a 300-yard bonus for each TD. Compared to the other values you mention, this is off the charts.

I think I'd rather kick from the 1 yard line and have the touchdown than have the ball on the one yard line and not have the touchdown.

I guess I am missing the point of this, but if you'd ask me, I'd rather have my QB throw for 200 yards and ten touchdowns than 300 yars and zero touchdowns. I might be wrong, but I think the former gives your team a slightly better chance to win.

I have a question/research request for you. The colts haven't played a team with a losing record yet, and may not until they face the falcons in week 12! Has that ever happened before? What's the hardest schedule ever faced? Weakest? By record and SRS?

monkeytime: I think the point that these systems try to get across is that passing touchdowns are not necessarily the result of the quarterback doing anything "special," that he couldn't normally do at any other point on the field. He just happens to be doing it near the opponent's goal line.

For instance, is a 10-yard completion from your opponent's 10 really any different from a 10-yard completion from your own 20? For that matter, is a 5-yard run from your opponent's 5-yard line different from a 5-yard from from the 50?

Put another way, which seems more "skillful" by your QB and receivers? A 10-yard pass from the opponent's 10 or a 60-yard pass from your own 20? Yes the 10-yard TD gives you 6 points, but, in theory, just about any QB-receiver tandem can make that play, whereas it takes something more along the lines of Brady-Moss or Manning-Harrison/Wayne to come up with the 60-yard play. If Mike Karney catches 8 2-yard touchdown passes for the Saints this year, is he a better receiver than Marques Colston, who catches only 6 touchdowns? Probably not.

I thought Mike Karney was a running back.

In determining the "value" of a TD, first we need to decide whether you're talking about performance or ability (a variant of the retrodictive vs. predictive rating systems discussion). Clearly a TD pass from 1st and goal at the 3 yard line is more valuable than a 3-yard pass on 1st and 10 from your own 20, even though both look the same to yards/attempt. I'd bet that the marginal value of a TD (beyond the yards gained) is greater the closer you get to the end zone, so the 70-yard example is probably on the extreme low end of the TD-worth scale. As you get closer to the end zone, yards become more difficult to gain and completion rates decrease.

Moreover, is it possible that some QBs have extra repeatable ability to throw TD passes, beyond what can be measured by yards? If there are runners and receivers who seemingly have a nose for the end zone (LT, Cris Carter), shouldn't a QB who can place the ball in tight spaces have some value? That value probably varies from team to team. We'll probably never know how many TD passes Troy Aikman would have had without Emmitt Smith and the Dallas power running game because he didn't have as many opportunities. On the other hand, is it the fault of Marino and Favre that their teams never had a great short-yardage RB that would have been useful near the goal line, thus compelling their coaches to fully utilize their passing skills inside the red zone? I'm not sure any existing rating system has addressed these issues satisfactorily.

I agree with almost all the above. I'm actually relieved that most people here are not suckered by the TD pass as a good measure of QB skill.

The 10 yd bonus makes sense to me, but there is another consideration. Inside the red zone, the field shortens and the secondary has less real estate to cover. It would be expected to make passing somewhat more difficult than in other parts of the field.

Fantasy football and the idiotic NFL passer rating have really warped most fans' perceptions of how to measure a QB's effectiveness.

Interceptions are a negative against the QB in this site's rating system. But lost fumbles aren't.

So if the QB coughs up the ball three or four times, that's OK in this rating system. "No harm done", the author of this site is saying.

And that is WHAT he is saying or he would include them in his formula.

But whenever I played QB and lost the ball... my teammates were pissed.

t.g., perhaps the "author of the site" is saying that he doesn't have complete data on fumbles.

This system (and any statistical system, for that matter) is being used to measure a QB's ability to pass the ball. It doesn't take into account rushing yardage, sacks, or fumbles, all of which are certainly important but don't figure into the formula. Nor does it say that the QB with the best "score" by this method is the best QB overall, just that he has the best numbers as a passer.

I think the very first post (Vern) has it right. The question of "marginal value of TD" is equivalent to "value of last yard on the field" - and that one yard isn't the only yard on the field with a different value. In fact, if you look at historical data of the "likely next score" from possessing the ball at each yard line on the field, you can draw a curve that will give you a slightly different marginal value for every yard on the field. As you might expect, yards near each endzone have the highest value - but not just the very last yard. Football Outsiders has actually done the research, and I believe the full chart appears in Pro Football Prospectus 2006.

All that said, if you're working from yardage/TD totals rather than play-by-play data, the best you can do is a very rough approximation of this curve by giving a yardage bonus for TDs.