Posted by Chase Stuart on October 3, 2008
This week, I've been writing about the value of touchdowns scored on different downs. One of the problems with the data, though, is that the sample size isn't very large for each down. There is some down-to-down variation that exists in the data that doesn't make a lot of sense, and is probably a result of a small sample size. Further, I think the numbers should be consistent -- your odds of getting a 1st and goal from the one should be the same as your odds of getting a 4th and goal at the one (and in fact, the numbers say they pretty much are). Your odds of fumbling should be the same, too.
One of the problems with the results from Wednesday's post is that a first down touchdown is undervalued -- that is, it says the situation of being in 2nd and goal from the one is worth 5.50 points. But according to theory that we're very confident in, 1st and goal from the one is worth 5.55 points.There should be a greater spread than that. So here's what I did.
For every play at the one yard line (designated in the table below as 1st, 2nd, 3rd or 4th and goal), a team has a 55% chance of scoring a touchdown, a 2% chance of a turnover or an interception, a 12% chance of a loss of yards, and a 29% chance of no gain. A touchdown will be worth 6.4 points, a fumble is worth 0.9, and an interception -0.25. A play of "no gain" is worth whatever the next sitaution is worth. So no gain on 2nd and goal at the one is worth 4.746, since that's the value of having 3rd and goal at the one. A loss is worth slightly less than 75% of a play for "no gain". For fourth down, there's considered a 50% chance of a field goal and every other situation is reduced by 50%. The table below sums this up:
1ST + G 2ND + G
TD 0.55 6.4 3.52 0.55 6.4 3.52
FUM 0.02 0.9 0.02 0.02 0.9 0.02
INT 0.02 -0.3 -0.01 0.02 -0.3 -0.01
FG 0.00 2.4 0.00 0.00 2.4 0.00
LOSS 0.12 3.9 0.47 0.12 3.5 0.42
NO GAIN 0.29 5.3 1.55 0.29 4.7 1.38
3RD + G 4TH + G
TD 0.55 6.4 3.52 0.275 6.4 1.76
FUM 0.02 0.9 0.02 0.01 0.9 0.01
INT 0.02 -0.3 -0.01 0.01 -0.3 0.00
FG 0.00 2.4 0.00 0.50 2.4 1.20
LOSS 0.12 2.4 0.28 0.06 1.2 0.07
NO GAIN 0.29 3.2 0.93 0.145 1.2 0.17
That 5.55 number reflects the value of 1st and goal from the one yard line -- which is what our theory predicts. So now a touchdown on a long bomb is worth 0.85 extra points (6.4 - 5.55), a touchdown on 1st and goal is worth 1.07 extra points, on 2nd and goal is worth 1.65 extra points, on 3rd and goal is worth 3.20 points and on 4th and goal is worth, still, 4.85 extra points. Using the numbers from Wednesday's post, this means the average passing touchdown is worth 1.325 extra points. If we convert that number to yards, that would mean each passing touchdown, on average, is worth 18.3 extra yards.
However, it's slightly more complicated than that. Sure we know all passing yards aren't equal -- but leaguewide, passing yards aren't evenly distributed. A pass from the 45 to the 50 isn't as valuable as one from the 5 to the end zone; we know that. But it's also true that the former pass happens very, very often, and the latter is relatively rare. In other words, lots of the passing yards that QBs get are of the less than average value variety. And if that's the case, than the average passing yard isn't as valuable as the yards on the field. And if that's the case, then a passing touchdown is even more valuable than we thought.
Doug looked at every passing play in 2007 that gained at least one yard. Then he individually looked at each yard it covered, computed the value of that yard, and added that value to a giant running total. If you divide that total by the total number of yards and you should get the league wide value of a typical passing yard. Remember 98 yards is worth 7.1 points, meaning 1 yard is worth 0.724 points. Well, according to Doug, the average passing yard is worth .0653 points and that one point is worth 15.3 "average passing yards". So we should be multiplying 15.3 times the 1.325 points the average passing touchdown is worth. In other words, a touchdown is worth 20.3 passing yards.
Except for one more point. As Vince pointed out on Wednesday, we should also be subtracting one yard from our total. Since we're measuring the point value from the 1 into the end zone, we should then subtract out one yard at the very end. So, for the last time for awhile, I'm going with 19.3 yards as the ***official*** value of a passing touchdown. It's still worth remembering, though, that a generic touchdown on a non-X-and-goal play, is still merely worth only 10.7 yards, or 13.0 "average passing yards".