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25 Sep 2003

How Many Points is a Turnover Worth?

by Aaron Schatz

How many points is a turnover worth?  If it is returned for a touchdown, clearly the answer is six, right?  But what if it isn't returned for a touchdown?  And even if it is returned for a touchdown, what about taking into account how many points the offense would have scored if they had not turned the ball over?

You would expect that a turnover is worth more the closer you are to the goal line.  After all, give the other team the ball closer to your own goal line, and it is easier for them to score.  Lose the ball closer to the opposing goal line, and you've squandered a chance to score yourself.

In Hidden Game of Football, Pete Palmer and Bob Carroll propose the theory that this seemingly common sense belief is wrong.  According to them, a turnover is always worth -4 points.  How did they figure this out?  Well, they ran a number of seasons worth of data to determine the answer to this question: "If I'm on yard line X, what will be the next score in the game, on average."  It didn't matter whether this score took place on the current drive, or the next drive, or in the next quarter.  They only plays they didn't count were those when halftime or the end of the game came before the next score.

They started with the idea that having the ball on your own 0 yard line is worth -2 points, since you've just given up a safety.  The ball on the opposing yard line is worth 6 points, leaving out the extra point which is pretty much guaranteed anyway.  Graph the results on the rest of the field, and you end up with a graph that looks like this:

The yard lines here are given from 0 to 100, with 51-100 representing the opponent's half of the field, and a negative score reflects that, on average, the defense was more likely to score next than the offense.  Flip the chart around to get the average next score when the opponent has the ball, and you get this table:

Distance from
Your Goal
Team on
Team on
0 -2 -6 -4
25 0 -4 -4
50 2 -2 -4
75 4 0 -4
100 6 2 -4

So a turnover is always worth 4 points -- well, -4 points to the offense and 4 points to the defense -- assuming that you have an average offense and an average defense.  This chart is also important to help explain two of the main precepts of the still-young science of football stat analysis.

First, the idea that field position is fluid.  In fact, if someone asked me, "What is the most important thing you have learned so far that will help people understand why teams win or lose football games," I would answer "Field position is fluid."  While the chances of scoring (or allowing a score) change based on a team's position on the field, it doesn't matter how a team gets to that position.  This is why special teams are so underrated.  When Carolina sends Todd Sauerbrun out there to punt, they are consistently leaving the opposing team about five yards further back, which means that the average value of the next score is, according to Palmer and Carroll, 0.4 points lower.  Punt 100 times in a season -- and when your offense sucks, like Carolina's, you will -- and that adds up.

The idea that field position is fluid also explains why some teams that score a high number of points come out much lower on our Value Over Average ratings for offense.  (What's VOA?  Read here.)  Imagine a team where the offense always gained between 15-20 yards and then punted, but the defense was so good that they always limited the other team to 3-and-out.  If each team had average special teams, our team would gradually move the start of each drive towards the opposing goal line, eventually scoring a field goal or touchdown and starting the whole cycle over again.  Perhaps the opposing defense would score a couple of turnovers, and the game would end 26-14.  But clearly our team's defense is far outplaying the offense, even though the offense scored a lot of points.

So far in 2003, Buffalo is a good example of this.  After three weeks, they are ranked #7 in Defense VOA but only #17 in Offense VOA, despite ranking #6 in the NFL in points scored.   That's because their defense is stopping teams quickly and getting the ball back for the offense in good field position.  Unfortunately, Buffalo's defensive turnovers are balanced by their offensive turnovers.  Against the Patriots in Week 1, all of Buffalo's points, except for an opening drive touchdown, either followed an interception that put the offense in good field position or came from an interception return.  (The ground is still shaking in southern Ontario after that Sam Adams runback.)  Even in their abysmal game against Miami, Buffalo's defense comes out with a -30% VOA.  Miami had the ball 12 times which included three 3-and-outs and three turnovers.  But every time the Buffalo offense took the ball back, they fell flat on their faces with four 3-and-outs, three turnovers, and another three drives with fewer than 20 yards.  Their only points came from the defense on an interception return for a touchdown.

The second idea that comes from this discussion is go for it on fourth down more often.  The conventional wisdom is that if you go for it on fourth down and miss, the other team now has the ball and you've lost your opportunity to score.  The chart above shows that this isn't necessarily so.  Once you are past the opponent's 25-yard line, your team is still more likely to score the next points despite a change of possession.  Less likely to score as many points, sure, but if you go for it on fourth-and-goal and fail, you've backed the other team up in their own end, and most of the time they will need to punt, giving your team the ball back with a loss of 40-60 yards in field position.  Depending on the chances of failing to get that fourth down vs. the advantage of 7 points over 3 points, the odds favor going for the touchdown far more often than current coaches normally try.  No, obviously not with one minute left in a tied game, but in general.  That's the simple version, here's the complicated version

But let's get back to evaluating Carroll and Palmer's original chart.  The ideas seemed great, but a couple of things struck me as not quite right.  First of all, shouldn't a touchdown count as 7 points?  After all, the extra point is part of the reason you're trying to get down to that goal line.  What about the two-point conversion?  It turns out that, in 2002 at least, slightly more than half of the two-point conversions were converted successfully, and and that average slightly higher than one point per conversion balanced out the very small number of mixed extra points.  So a touchdown should be 7 points.

The second problem was the idea that getting closer to the goal line didn't make a turnover any more dangerous.  Is just didn't seem to make sense.  Field goals get easier as you get closer to the goal line, right?  So even if the chances of a touchdown increased in a linear fashion, shouldn't the average next score curve a bit because the chances of getting three are gradually moving upwards?

Faced with these dilemmas, I decided to re-run the numbers using my 2002 database.  Like Palmer and Carroll, I left out plays when there was no next score because of halftime or the game's end.  But this time, if the next score was a touchdown, it was worth 7 points instead of 6 points.  We ended up with the crooked blue line on this next chart (we'll get to those other lines in a second): 

As you see, there's still a pretty clear trend here, although something very strange occurs near a team's own goal line -- the opponent's likelihood of scoring next actually goes down as a team gets backed up to the last 8 yards or so before its own goal.  I'm not sure if this is a one-year glitch, or a general trend.  There are reasons why it would be the case.  Backed up to their own goal lines, teams generally play extremely conservatively, running just to get some room for a successful punt rather than attempting passes.  That cuts down on turnovers, which moves the opposing team's next starting field position back a bit, which means fewer points for them on average.  You also may notice that the average next score stays under 5 points until the last five yards or so.  I guess defenses are better at defending at the goal line than you might expect, meaning lots of fun short field goals.  And we've already established that most coaches will kick that field goal most of the time.

OK, so those lines.  Excel balances out our desire to throttle that annoying talking paper clip by being chock-full of useful features.  One of those features allows us to create a trendline for our little chart.  I started with creating a linear equation just like Palmer and Carroll used.  That's the black line.  You may notice it is a bit different from the line in Hidden Game.  Where the line in Hidden Game runs from -1.96 on your own one-yard line to 5.96 on the opponent's one-yard line, our line from 2002 goes from -1.46 to 5.26.  That means that the value of a turnover based on 2002 numbers is actually a bit lower, 3.8 points.  Somewhere, Ryan Leaf is feeling a little more self-worth.

It's important to note that this gives the value of a turnover without considering the runback by the defense.  For example, this is the value of a fumble at the line of scrimmage where the defense pounces on the ball and goes nowhere.  An interception of a 50-yard pass that only gets run back for a few yards has a lower value, obviously, since the line of scrimmage on the next play has moved down the field.  For the same reason, an interception of a 5-yard pass by a linebacker who gets 20 yards before he's tackled has a higher value.

An added bonus of this trendline is that we can figure out how many yards a turnover is worth based on finding which yard line with the ball has the same average next score as each yard line without the ball.  The -1.46 expected next score with the ball on the one-yard line corresponds to a point about halfway between the 56 and 57-yard lines (a.k.a. the opponent's 43 and 44-yard lines).  That means that a turnover, not counting the length of the runback, is worth on average 55.5 yards.  Which, for Jim who asked, provides part of my answer to this question from the PFRA Forum: How many yards should the interception penalty be in the QB rating?  My answer is 55.5 yards minus whatever is the average distance from the line of scrimmage when the play ends, or 55.5 yards plus the average length of intercepted passes minus the average length of interception returns.  Which I'll get to computing, oh, at some point after I finish fixing my gutter drainers and cooking for Rosh Hashanah.

But I digress, yet again (I do that a lot).  What about our issue of common sense saying that a score should be easier closer to the goal line?  Well, Carroll and Palmer used a nice linear equation for simplicity, but we can get a more complicated trendline.  Excel creates polynomial equation trendlines up to the sixth power!  Unfortunately, all the equations I've gotten from the fourth power and above seem not to work correctly.  I blame the stupid talking paper clip for sabotaging the goal of football science.  He must be a soccer fan.  Nonetheless, our third power equation still provides a more accurate trend that better matches our actual numbers.  For you math geeks, the R-sq has gone up slightly from .9615 to .9634.

Our new trendline is orange on the above chart.  You can see that according to this equation, the average next score is a bit higher closer to the goal lines -- especially near the opponent's goal line, as a touchdown becomes more likely and a field goal becomes child's play for everyone but Todd Peterson -- and a bit lower in the middle of the field.  Since the value of one extra yard of field position now changes depending where we are, the value of a turnover now changes as well.  Which gives us this chart.  Welcome to the Happy Turnover Smile-Time Hour!

What's interesting here is that the bottom of this chart isn't much different from the average value of a turnover that we got from the first equation we used, the straight line!  This graph says that the value of a turnover bottoms out at 3.77 points between the 48-yard lines.  The value hits 3.80 on the 38 and 39-yard lines on each side and goes up gradually.  It hits 4.0 points on the edge of each red zone and 4.10 points between the 9 and 10-yard lines, ending up at 4.25 points on the one-yard line.

The moral of this chart, however, is that Palmer and Carroll, in the effort to simplify, missed that a turnover isn't worth the same amount of points anywhere on the field.  It truly is worse to turn the ball over in the red zone than in the middle of the field.  This curve, however, has the same values on each side of the field, just like the linear equation trendline from Hidden Game.  That should mean that performance in what I call the DEEP zone (your own goal line to 20-yard line) is as important as performance in the red zone on the other side of the field.  Except that it isn't, really, because unless you turn the ball over you are going to either drive out of your own end or punt the ball, giving it back to the opposition not in the red zone but in the middle of the field.  But a turnover in your own DEEP zone is just as deadly as a turnover in the red zone.  In one case, you are losing almost assured points, and in the other, you are handing the other team almost assured points.

There are two more issues here.  First, what about the question of how many yards a turnover is worth?  It turns into a curve, actually.  

This is a rough curve, done with estimates instead of a fancy mathematical equation, but you get the idea.  To make this chart, I had to figure out the value of the opposition having the ball prior to their own goal line, which was sort of silly.  I am willing to entertain comments that this chart shouldn't look like this, peaking at the 20-yard line at around 57.25 yards and dropping on either side.  I think a turnover is worth fewer yards as you go farther down the field because it gets harder for the opposition to score on the ensuing possession.  Why it should go down closer to your own goal line, I'm not sure.

We're not done yet, true believers, because in the comments on the Week 3 team efficiency tables, Vince asked if we knew expected scoring for down and distance.  I had planned to write this article about expected scoring by distance, but it was easy enough to go back and sort by down.  Get out your 3-D glasses for this one, and try not to squint:

Note that we're including punts here.  You can see how the difference between the 2nd down trend and the 3rd down trend is larger than the difference than the 1st down trend and the 2nd down trend, and the difference between the 3rd down and the 4th down trend is even larger.  As an added bonus, the correlation between the trendline and the actual data gets smaller the closer you get to 4th down.  On 1st down, the expected next score becomes positive between the 13 and 14-yard lines; on 4th down, the expected next score isn't positive until just before the 50-yard line because you are almost always turning the ball over for the next down.  One more fun graph, just because I can.  This table shows the value of a turnover in points, only this time it's been split out by down.  4th down isn't here, since that's almost always a "turnover" anyway.  While the point value of field position for the offense changes with the down, the point value of field position for the defense (the team that gets the ball on the turnover) is always the same, since they always start on first down.  That's why the top line is another happy symmetric smile, but the other two lines are not.

OK, so what's the moral of the story?  Having the ball doesn't necessarily mean you are likely the next team to score; losing the ball doesn't necessarily mean you are going to give up the next score.  The likelihood that your team will score -- and the amount of points you are likely to score -- improves gradually as you move towards the goal line, but it only improves slightly with every yard until the last few yards.  Sticking your opponent with the ball near his own goal line is worth about as much as having the ball yourself around your own 40-yard line.  Sometimes a team looks like it has a great offense when what it really has is a great defense that gets the offense the ball in good field position where it is easier to score.

As for the value of a turnover, giving the ball up is worth about 3.8 points in the middle of the field, about 4 points at the 20-yard lines, and 4.25 points at the goal line.  If you want to get technical, it's worth more on first down than second down, and more on second down than third down.  And Excel makes neato graphs.

Posted by: Aaron Schatz on 25 Sep 2003

4 comments, Last at 08 Nov 2011, 2:09pm by bravehoptoad


by The Ninjalectual (not verified) :: Mon, 10/02/2006 - 7:51pm

First comment in nearly 3 years!

I miss graphs like these being posted in articles. Is this research factored into the modern DVOA formula? It may or may not make a difference... Chad Pennington threw his first career red zone interception this week, on fourth down, so does that single play affect his PAR more or less than, say, Matt Hasslebeck, who threw one around midfield? Great research though, and I'm grateful to learn from it.

by V (not verified) :: Mon, 12/03/2007 - 1:14am

Just wanted to let you know that this is my favorite articles on the entire site. Totally changed the way I thought about football.
You guys are amazing.

by PatsFan :: Tue, 10/05/2010 - 9:25pm

Somehow I had missed this article all these years. But a commenter in the 2010 Week 4 DVOA thread pointed to it. Very interesting!

by bravehoptoad :: Tue, 11/08/2011 - 2:09pm

3 comments? Why aren't there 500? This article is great stuff -- good enough to deserve that.