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

## Two kicks are better than one

In one of the comments to Monday's post, loyal PFR reader Bill M. mentioned that teams that have scored 13 points have won a higher percentage of games than teams that have scored 14 points. This is a bit surprising; all things being equal, you'd expect that teams that score X points would always win more games than teams that have scored X-1 points. Over a long enough time, all other things should be equal. Right?

I looked at all NFL games since 1970, and calculated the winning percentage for each "Points Scored" number. Bill M. was right: teams that have scored 13 points have won 28.5% of their games, while teams that scored 14 points won just 19.9% of their games. Considering over 800 teams have scored 13 points in a game, and over 900 have scored 14 in a single game, there's no way we could quibble about the sample size.

So what's the explanation? You might think Simpson's Paradox comes into play: perhaps in the early years, when scoring was low, teams scored 13 points more often and won more games 13-10 than 14-10; in the more recent years, with scoring up, maybe teams lost more games 17-14 than 17-13. If that was the case, there wouldn't be any real advantage to having scored 13 points, because for each era, scoring 13 points should be less successful.

That's not the case though: In fact, from 1981-2005, teams that scored 13 points had a higher winning percentage in a given season (than teams that scored 14 points) in 23 of those years; in the two years that 14 "won", the difference was worth less than 2.5 percentage points in both years. In other words, this is overwhelming evidence to suggest that scoring 13 points is more highly correlated with winning than scoring 14 points.

I wasn't being sloppy with my language there: I'm describing exactly what we know, nothing more. It's easy to see evidence like this and jump to a causation solution. But I don't think you want to say "I want my team to score 13 points instead of 14, if given the choice" just yet. Remember, we are simply seeing some highly correlated numbers, and we know nothing yet about causation.

So what's the next plan? Look up the winning percentages for *all *points scored totals. PF stands for points for a team in a given game, N is the number of times since 1970 that a team has scored that exact number of points, and the last column shows the team's winning percentage in those games.

PF N WIN%

0 406 0.000

1 0 ----

2 6 0.000

3 466 0.011

4 0 ----

5 15 0.133

6 366 0.071

7 763 0.038

8 22 0.045

9 234 0.218

10 1056 0.140

11 23 0.261

12 153 0.307

13 869 0.285

14 954 0.199

15 126 0.429

16 572 0.495

17 1295 0.390

18 57 0.474

19 306 0.559

20 1076 0.566

21 792 0.461

22 164 0.567

23 623 0.705

24 1030 0.648

25 84 0.667

26 285 0.807

27 793 0.768

28 533 0.700

29 123 0.837

30 393 0.891

31 646 0.824

32 54 0.852

33 155 0.871

34 421 0.884

35 250 0.872

36 62 0.855

37 206 0.942

38 273 0.923

39 20 0.850

40 65 0.969

41 151 0.960

42 115 0.983

43 24 1.000

44 73 0.986

45 95 0.989

46 8 1.000

47 21 0.952

48 35 0.943

49 25 1.000

50 11 1.000

51 15 1.000

52 17 1.000

53 0 ----

54 3 1.000

55 11 1.000

56 6 1.000

57 1 1.000

58 4 1.000

59 3 1.000

60 0 ----

61 3 1.000

62 4 1.000

In general, there's what you would expect: a positive correlation between points scored and winning percentage. The R^2 of 0.861 affirms that. Here's where it gets interesting though.

I restricted the range from teams that scored 5 points to teams that scored 42 points. You might think there's a general linear trend: but a linear trend line has an R^2 of just 0.93. A binomial trend line has an R^2 of 0.96. While these numbers are very close, and show the general trend, there's a reason we won't get to 1.00: the numbers have spikes in them for a reason.

If you look a bit closer at the data, multiples of 7 (+0 and +9) reveal some interesting data.

PF W% PF W%

9 0.218 14 0.199

16 0.495 21 0.461

23 0.705 28 0.700

30 0.891 35 0.872

That chart there explains most of the bumps in the data. Sure 13 has a higher winning percentage than 14 (and higher than 9), 20 higher than 21, and 27 higher than 28, but these ones are particularly interesting. I don't know anyone that would have guessed teams that score 21 points in a game have a worse winning percentage than teams that score sixteen points.

So what's the reason? Once again, I caution anyone from jumping to conclusions here. It's easy to spot the correlation, but not so easy to figure out the causation.

On one level, each pair is an example of Simpson's Paradox. Every time a team scores 20 or 21 points, and lets up fewer than 20, it wins; every time it scores 20 or 21, and allows more than 21 points, it loses. And every time a team scores 21 points and allows 20 or 21, it never loses; every time a team scores 20 points and allows 20 or 21 points, it never wins. The reason for the higher winning percentage is that more often when a team scores 20, its opponents score fewer than 20. The key question, is why.

Here's some more data. For teams that score 20 points, the most common points allowed number is 17, which also happens to be the median. For teams that score 21 points, the mode is 24, and the median 23. When teams score 20, 15.8% of the time they have allowed 17 points; when scoring 21, just 7.7% of the time have they allowed 17 points.

This chart shows how many times a team that scored 20 or 21 points, allowed X number of points ** or fewer** in a game.

20 21

0 2.8 2.5

1 2.8 2.5

2 2.8 2.5

3 6.0 3.9

4 6.0 3.9

5 6.0 3.9

6 9.0 5.4

7 13.8 9.7

8 14.0 9.8

9 15.0 10.9

10 21.6 16.3

11 21.6 16.7

12 22.7 17.0

13 28.8 19.9

14 34.1 25.9

15 34.9 26.0

16 37.7 30.2

17 53.5 37.9

18 53.7 38.6

19 55.9 40.7

20 57.2 45.8

21 61.1 46.3

22 62.1 47.6

23 71.3 51.5

24 76.8 61.7

25 77.3 62.2

26 80.1 64.4

27 85.2 70.1

28 86.7 75.0

29 87.1 76.8

30 89.1 78.7

31 91.4 83.8

32 91.4 84.2

33 91.9 85.0

34 93.8 88.5

35 94.7 90.7

36 95.2 90.7

37 96.2 92.6

38 97.4 94.3

39 97.5 94.4

40 97.7 95.2

41 98.6 96.1

42 99.2 97.0

43 99.2 97.0

44 99.3 97.3

45 99.5 98.1

46 99.5 98.1

47 99.5 98.2

48 99.7 98.5

49 99.8 99.0

50 99.8 99.2

51 99.8 99.4

52 99.8 99.6

53 99.8 99.6

54 99.8 99.6

55 99.9 100

56 99.9 100

57 99.9 100

58 100 100

I don't really know what's driving those numbers, but they're certainly thought-provoking.

Now, onto the main topic of the post: Are two kicks (two FGs) better than one (one XP, following a TD)? Teams that score 6 points have higher winning percentages than teams that score 7, and the lower scoring pair has a higher winning percentage at 13/14, 20/21, 27/28 and 34/35. So indeed, it does look like two FGs is better than one TD, plus 0, 1, 2, 3 or 4 TDs.

So what does it mean? Surely if your team is playing and down 17-14, and then scores a TD, you won't root for them to miss the XP so your team hits 20 instead of 21. And I don't quite think you want to root for you team to get 2 FGs instead of a TD just yet, either.

My guess is that there's another factor driving all this. One plausible theory is that it's time of possession. Consider this hypothetical, but realistic, TOP breakdown. Under 20 and 21, I've listed my estimates at the winning percentages.

TOP 20 21

25:00 .400 .420

30:00 .500 .520

35:00 .600 .620

At each of those levels, the team scoring 21 points has a higher winning percentage. But what if the number of games that fits that criteria looks like this:

TOP 20 #20 21 #21

25:00 .400 100 .420 640

30:00 .500 140 .520 310

35:00 .600 760 .620 50

If that was the case, teams that scored 20 points would have won 56.6% of their games, while teams scoring 21 points would have won just 46.1% of their games. But it wouldn't be because scoring 21 is worse than scoring 20; it's just that five minutes of time of possession is worth more than a point, and teams that score 20 points generally hold the ball longer than teams that score 21 points. After all, to score 20 points you usually need four scoring drives, not three, and that *might* mean your offense is on the field less.

This is just one theory, of course. There's got to be something driving the numbers, because the sample size is significant, and the pattern is very clear. I'd hesitate to say I'd want my team to score 9 points this weekend instead of 14, but I wouldn't disagree if you said a team is more likely to win if it scores 9 points instead of two touchdowns. But keep in mind, the reason the team is more likely, is because scoring 9 somehow helps your defense a lot more than scoring fourteen.

The time of possession theory is just one; there may be others. I'm curious to hear the comments today, to see if there are any other explanations for why two kicks are better than one.

This entry was posted on Wednesday, November 15th, 2006 at 10:40 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.

Interesting observation regarding TDs and FGs, Catfish. It does relate to Jason's ATR idea. Taking it one step further, if the number of scores (TDs + FGs) is a better indicator of team strength than points (7*TDs + 3*FGs, ignoring safeties and PATs for the moment), it implies that variations from the mean of a team's TD:FG ratio (normally around 3:2, I believe) is primarily due to chance. In other words, teams have differing abilities to move the ball down the field, but no significant ability to score touchdowns beyond their ability to move the ball. So any illusions that a "big play" team can score TDs but otherwise not move the chains consistently, or that a team can move the ball down the field with the best of them only to choke in the red zone repeatedly, these are just random statistical variations and don't represent persistent characteristics of a team's offense. And the same would apply to defenses. Perhaps differences in FG kicker performances play a factor, but as last week's FO research

showed, even these are very unpredictable and may not represent much differences in skill. I don't know whether any of this is true, but if so, it would seem to have some other important implications as well.

Re: points against (#16, #18), it would be interesting to compile a table of the average number of points against next to each distinct value of points for. In theory, they should all hover around 20, but I wonder if we'd see some similar blips around the seven-point multiples.

That's interesting, I didn't know punts and kickoffs average the same field position. The difference is that there is a lot more variation in field position after a punt.

Jim A, I would guess that the points allowed trends downward as points scored increases. A better offense gives their defense better field position and therefore allows fewer points. It would also work the other way, where a good defense would give the offense good field position as well.

What would be the steps to gaining a definitive understanding of all this?

Going through old boxscores, Bill M.. I can't promise anything, but it's on my "to do list" to go through about 100 boxscores over winter break to work on this. Obviously, I'll keep you guys updated.

What can I do to help?

Catfish, that was my initial thought also about points allowed, but there might be field surface/weather conditions or pace (total number of possessions) effects that would pull points allowed closer to points scored.

I guess the real question is still why scoring 13 points is so much better for your points allowed than scoring 14 points.

Jim A, here is the average opponents' score for each offensive point total. This is all games since 1970.

Given the other information in this post, it's about what you would expect. Interesting nonetheless.

I find those last set of numbers to be very interesting indeed. First of all, there doesnt seem to be a downward trend as you score more (as catfish and I expected). This might be becuase as teams score more they loosen up their defense.

Secondly, all the "normal" scores seem to have peaks. 3,7,10,14,17,21... This is somewhat consistent with the rest of this discussion but includes scores with 1 FG (3,10,17). Any thoughts?

Lastly, it seems that before 22 you are usually losing the game but after 22 you win. I would have thought that number would be slightly higher (24?).

I looked at weeks 1-5 of this season to see what the scoring margins were at the time teams were kicking field goals. Here is some support for what Doug said in #2, Alex said in #6, and I said in #17.

50 field goals were made by teams that had already made at least 2 field goals in a game at the time of the kick. Only 6 of those 50 were made by a team that was trailing by 4 or more at the time of the kick. (In other words, by a team that would still be trailing even after the made field goal). 14 were made by a team trailing by 3 or less, or tied, at the time of the kick. 15 were made by a team already leading by a touchdown or less, and 15 were made by a team already ahead by more than 7.

This ties into the average points against list in #58. Setting aside the small sample size numbers at "5 points scored" and at 40 points for and above, the smallest average points against happen to be at the following scores: 12, 9, 16, 26, 15, and 19. These are all scores that correlate strongly with 3+ field goal games for the team in question. My guess is that the teams that are getting to 9 and 16 points for example, are in close games at the time they are kicking that 3rd field goal. If they are down big, such as 28-6, they are probably not kicking a 3rd field goal.

This thought leads me on another tangent. What if the teams scoring 9, 13, 16, etc., which show a higher winning percentage than the teams scoring 1 more point, are actually leading in an even higher percentage of games? The 2 FG's are better than a TD may actually be a mirage, and may be just reflecting scores that are more highly related to being achieved in close games, but those teams may also be losing leads late due to conservative play (after all, if you are sitting on 13, 20 is certainly better than 16, even if "13" is better than "14").

To follow up on my last thought, teams that scored 13 points are 6-14 this year, teams that score 14 are 4-12. However, 5 of the "13 point scorers" were actually leading 13 to "X" at some point in the 4th quarter and lost. (Ten-Ind, Sd-Bal, Car-Min, Cin-TB, Oak-Den). Only 1 of the "14 point scorers" blew a late lead for the loss. (Cle-Bal). Most of the teams that scored 14 and lost were already trailing, sometimes by large margins. On the other hand, over half of the 13 point teams had the lead at some point late in the game.

"ATR is the inverse of net scores (net scoring drives)"

That only works if you count a safety as two turnovers. I know, I already went through this with you at Football Outsiders, and I know I said I'd drop it, but if you only count a safety as one turnover, then it's not really the inverse of net scoring drives. But, since safeties are rare, I'm sure it won't make enough of a difference to matter.

How many of you followed the scores just a wee bit more intently this week, in light of this ongoing discussion? I definitely did.

Another thing I just realized that probably contributes to this effect is that overtime games (about 6% of total) rarely end with the winning team scoring 7, 14, 21, 28, etc. for the simple reason that the extra point is not attempted in overtime.