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The Value of a First Down

What is the value of a first down? By that I mean, how many marginal yards is a first down actually worth? Here’s another way to word the question: If 3 first downs and 80 yards are worth X, then 2 first downs and [???] many yards are equal to X?

Calculating the marginal value of a yard isn’t easy. In fact, it’s been bugging me for years, because I’ve never quite been sure how to derive them. Then, a light bulb went off in my head: I needed to reach out to Brian Burke. I had an idea, but not the data or the means to execute.

Burke, of course, runs the fantastic website Advanced Football Analytics (formerly Advanced NFL Stats). I asked him if he would run some queries, and Brian was kind enough to do so. Fortunately, Brian’s not just a guy with access to lots of data, but one of the smartest minds in the industry. I wholeheartedly endorse his methods below, and I’m very thankful for his help. On top of running the numbers, he also provided an excellent writeup on his work. What follows are Brian’s words and analysis.

To estimate the value of achieving a 1st down without counting any of the value of the yardage gained, we can use the Expected Points model. The value of the 1st down itself minus yardage value will be the discontinuity in EPA when a play’s gain crosses the threshold for a 1st down. That discontinuity represents the value of the conversion apart from any yardage gained.

For example, on 2nd and 10, the EPA would increase smoothly for each yard gained up to 9 yards gained, then jump to a much higher EPA crossing the 10-yard mark where the conversion occurs. After that point, the EPA should increase smoothly again with each marginal yard gained above what was needed for the conversion.

Here is an illustration. The Y-axis represents Expected Points Added, the X-axis the amount of yards gained on the play.

EPA 2nd 10

The EPA for a 9-yd gain is 0.57, and the EPA for a 10-yd gain is 1.04. That’s a discontinuity of 0.47 EP, meaning that the 1st down itself is nearly equivalent to the 9-yards gained up to the point of conversion.

But we also need to correct for the yardage value of that 10th yard. One yard of field position is generally worth 0.064 EP. So in this case the discontinuity itself is worth 0.47 – 0.064 = 0.41 EP.

If we wanted to assign a “bonus” of yards to a player who is credited with achieving the conversion over and above the yardage itself, we could use this value’s yardage equivalent. 0.41 EP / 0.064 EP/yd = 6.4 yds. That’s the bonus for 2nd down and 10, but there are many other down and distance situations to consider.

For example, on 3rd and 10, the discontinuity is 1.57 EP, equivalent to nearly 25 yds. First and 10 is very strange because the discontinuity is negative. This makes sense, however, because an offense should prefer a 2nd & 1 to a 1st & 10 anywhere on the field. It would be silly to penalize a player for gaining the extra yard to convert, so my opinion would be to say the EP bonus for a conversion on 1st down is zero.

3rd 10

After examining a smattering of 2nd and 3rd down situations, the 2nd-down bonus EP is about 0.35 and 3rd-down bonus EP is roughly 1.4.

4th down conversions would obviously mean a very large bonus EP. They essentially have the value of a turnover–close to 4 EP or so. Since 4th downs are qualitatively different (and relatively rare) I’m going to set them aside.

In general, 32% of conversions come on 1st down, 38% come on 2nd down, and 30% come on 3rd down. So the weighted value of a conversion alone would roughly be:

[0.32 * 0] + [0.38 * 0.35] + [0.30 * 1.4] = 0.55 EP

The conversion bonus of 0.55 EP can be translated into yards by dividing by 0.064 EP/yd, which ultimately makes the equivalent yardage bonus for a conversion: 8.7 yards.

Figuring out the value of a first down will have many applications for Football Perspective going forward. Please leave your thoughts in the comments, as I’d love to hear what you guys have to say. And thanks again to Brian for his great work.

  • Sunrise089

    Typo Chase – ANS has been renamed to Advanced Football ANALYTICS. To absolve yourself of responsibility you can join me in sticking with the original name though 🙂

    • Chase Stuart

      Doh. Fixed. It’s confusing, because I can’t believe Brian, a Navy guy, wanted to have the acronym AFA, which always make me think of the Air Force Academy.

  • Red

    Awesome work, Brian and Chase! The value of a first down is something I’ve been trying to figure out for a long time; I figured it was somewhere between 5 and 10 yards, but it’s nice to have an exact number. This has applications for player evaluation metrics in several ways: Instead of using completion % for QB’s, first down % would be more meaningful since it would filter out worthless checkdowns and failed 3rd down conversions. The same thing applies to WR’s, by substituting first down % for receptions. And for RB’s, first down % is almost certainly more valuable than YPC.

    I believe the NFL began tracking individual player first downs in 1991, so we have a nice volume of data to play around with.

  • Red

    Here’s a fun example with David Carr, aka Captain Checkdown. In 2006, Carr led the NFL in completion rate at 68.3%. However, his first down rate (including sacks) was 29.2%, which only ranked 21st. That’s a lot of worthless completions!

    On the flip side, Donovan McNabb ranked 23rd in completion rate at 57.0%, yet placed a solid 8th in first down rate, at 33.5%. He may not have been super accurate, but he still moved the chains better than the average QB (2006 avg was 31.1%).

    Chase, is there any chance you’d let me do a guest post on this very topic?

    • Chase Stuart

      Of course. Guest post submissions are always encouraged.

    • Richie

      I guess it would be interesting to see a larger data sample to see if first down rate is truly more important than completion percentage.

      Let’s compare two game-opening drives:
      David Carr:
      1st down: 5-yard completion
      2nd down: incomplete
      3rd down: 6-yard completion (First down).
      Carr is 2/3 (66%) for 11 yards with a 33% first down rate.

      Donovan McNabb:
      1st down: incomplete
      2nd down: 11-yard completion (First down).
      McNabb is 1/2 (50%) for 11 yards with a 50% first down rate.

      Which is more valuable? (Maybe we need more info.) On one hand, McNabb was more efficient with first downs, and his team didn’t get into a risky 3rd-and-10 situation. But Carr still turned his drive into a first down, and theoretically held possession longer (keeping the ball away from the opponent).

      • Chase Stuart

        Great example, Richie. This gets into the Drive Success Rate idea at Football Outsiders, which I personally love. Here, both teams successfully moved the chains and gained the same number of yards. From a retrodictive standpoint, I’d say they were equal.

      • Red

        From an explanatory standpoint, I think they’re the same. But for predictive purposes, converting first downs on fewer plays will lead to more success. What’s better than converting on third down? Avoiding third down altogether. Passing first down % is highly correlated with NY/A, which is highly correlated with scoring points and winning. The best passers in the league almost always excel at racking up first downs, while the worst passers almost always suck at it.

  • Ben

    This could have interesting applications to Brian’s previous work regarding optimal playcalling that found coaches playcalling to be more conservative than expected as they were focusing on getting a first down, not on getting as many yards as possible on each play. This could possibly help us fins out if they are justified in doing so.

  • Jericho

    Another typo: “0.57 – 0.064 = 0.41 EP.”

    • Chase Stuart

      Good catch. The discontinuity in the first example (2nd and 10) is 0.47, so it should be 0.47 – 0.064 to get to 0.41 EP. I’ve updated the post.

      • Jericho

        Awesome. And I want to let you know how much I appreciate your work. As a big fan of a certain team and somewhat of a stat nerd, it’s sucked that football has lacked a baseball-equivalent smorgasbord of advanced metrics for me (and others) to look at. What you and others are doing…well, just thank you!

    • Quinton

      Should be .506, yes?

      • Quinton

        Whoops, opened this morning and should’ve refreshed before commenting

  • Brian B

    Go Air Force!

  • Chase, will this information have an effect on your formula for ACY and TRY?

  • Tom

    Great stuff Brian and Chase, as always.

    Is the following one way in which this info would be used?

    In 2013, LeGarrette Blount had 153 attempts for 772 total yards, for a Y/A of 5.0.
    He had 35 1st downs (per Sporting Charts) and 7 rushing TD’s (no receiving TD’s).
    I believe TD’s are counted as first down’s, so let’s say he has 7 TD’s and 28 first downs.

    Not counting fumbles (which should be counted, just don’t have the info at my fingertips), and counting a rushing TD as 18 yards (per a previous Chase post), his AY/A would like this:

    (7 TD’s*18)+(28 1st dn’s*8.7)+772 = 1141.2 adjusted yards
    1141.2/153 attempts = 7.5 AY/A

    In other words, would the 8.7 yards per 1st down just be “added” to the total yards…like a bonus in the same way it’s done for passing touchdown’s, etc.

    Thanks again, this is very cool.

  • Richie

    It’s interesting how football has 2 main purposes:

    1) Get a first down
    2) Get a touchdown (or field goal)

    Yet, perhaps the most important goal for an offense is rarely tracked as a stat. I have no idea who achieved the most first downs in 2013. I would assume it was a RB, but can’t be sure. I also have no idea how many first downs that leader might have achieved.

    Does it make sense that this is not a stat that is included in every player’s basic stat line?

    • Red

      Totally agree. First downs should be considered a basic stat right along with yards, attempts, and touchdowns. But remember this is the NFL, the same league where the comically outdated Passer Rating is an official statistic.

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