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Two kicks are better than one

Posted by Chase Stuart on November 15, 2006

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.