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Archive for June, 2006

Ten thousand stories

Posted by Doug on June 2, 2006

Following up on a 15-year-old idea of Bill James, I decided to simulate 10,000 NFL seasons and see what would happen. Well I'm going to milk that idea for several posts. So if the idea doesn't intrigue you, you might want to check back in in a week or so. I'll still be here when you get back.

Today I'm just going to post one gnarly table and let you find interesting things in it if you're so inclined. Then, in keeping with Friday tradition, I'm going to get a bit silly.

Here is the table. It shows how often the team whose true quality was ranked Nth in the NFL finished the regular season with each given seed in their conference. Rank is the true quality rank, #1, #2, etc denote how many times the team with the given rank earned that seed, and OOP is the number of times they missed the playoffs:


========= seed ============
Rank #1 #2 #3 #4 #5 #6 OOP
========================================
1 3822 1906 1106 469 1083 567 1047
2 2634 1827 1160 634 1235 753 1757
3 2039 1672 1192 740 1193 865 2299
4 1692 1435 1161 798 1325 879 2710
5 1356 1383 1118 818 1195 978 3152
6 1128 1245 1126 870 1186 952 3493
7 991 1156 1116 880 1087 952 3818
8 881 1008 1050 814 992 953 4302
9 789 957 949 886 954 907 4558
10 672 850 942 896 910 896 4834
11 532 758 882 928 823 901 5176
12 498 651 819 868 802 875 5487
13 447 657 824 797 789 845 5641
14 401 580 769 775 719 809 5947
15 361 540 633 796 665 726 6279
16 282 477 635 757 622 743 6484
17 247 445 572 692 572 683 6789
18 218 386 492 680 517 703 7004
19 174 333 448 695 463 680 7207
20 178 288 410 660 446 573 7445
21 146 263 435 602 412 543 7599
22 102 239 370 592 368 480 7849
23 105 222 298 566 307 461 8041
24 81 164 312 484 298 452 8209
25 58 137 282 459 242 374 8448
26 58 133 214 432 205 320 8638
27 36 88 197 367 189 297 8826
28 27 70 169 332 149 273 8980
29 20 54 106 264 106 225 9225
30 13 38 103 172 84 173 9417
31 9 31 73 165 38 107 9577
32 3 7 37 112 24 55 9762

If you put a decimal point in there you've got percentages. So the best team in football missed the playoffs about 10.47% of the time, roughly once every ten seasons on average. The best team in football got a bye roughly 57% of the time. The worst team in football made the playoffs about 2.4% of the time.

Now we're going to going to play a game called "Am I as much of a freak as Doug is?"

To start with, I am going to pick one of my ten thousand seasons, totally at random, and I'm going to post a summary of it right here. I want you to spend a few minutes looking over it before reading on.


buf 15- 1
nwe 7- 9
mia 6-10
nyj 4-12

bal 9- 7
cle 8- 8
pit 6-10
cin 4-12

jax 12- 4
ten 8- 8
hou 7- 9
ind 7- 9

den 10- 6
sdg 9- 7
kan 8- 8
oak 7- 9

was 10- 6
dal 9- 7
phi 5-11
nyg 1-15

det 11- 5
gnb 10- 6
chi 8- 8
min 8- 8

car 10- 6
tam 9- 7
nor 8- 8
atl 5-11

sfo 13- 3
sea 10- 6
stl 6-10
ari 6-10

AFC Playoffs:
Wildcard: KC over Denver, SD over Baltimore
Divisional: KC over Buffalo, Jacksonville over SD
Championship: KC over Jacksonville

NFC Playoffs:
Wildcard: Washington over Seattle, Carolina over Green Bay
Divisional: San Fran over Carolina, Washington over Detroit
Championship: San Fran over Washington

I know I have a reader who's a Browns' fan, did you notice that your team missed the playoffs on a tiebreaker? Did anyone notice that an 8-8 team beat a 15-1 team in the playoffs, and that that 8-8 team made it to the Super Bowl? Do you have a mental picture of what that 15-1 Buffalo team looked like, what they played like? What about the 8-8 Super Bowl Chiefs? Do you think they were a running-and-defense team or do you think they won those playoff games 45-38? Did you find yourself thinking that the AFC West and NFC North were probably a lot of fun from start to finish that year?

Did you check to see what your favorite team's record was? Did you check to see if it was better than their main rival's record? If I invented an emoto-scope and hooked you up to it, would it have detected some tiny bit of happiness when you saw your team's record was better?

Finally, did you notice that I didn't tell you who won the Super Bowl? Did you find yourself wanting to know who won it?

If you answered yes to any of the above, you might just be as much of a freak as Doug is. The only possible way you could be more of a freak than Doug is, is if you had an urge to gamble on this Super Bowl.

If I were more eloquent, I'd have something really great to say here about the grip that sports has on our minds. You're all smart people, and you were fully aware that the output above was generated by a bunch of random numbers. But because the random numbers were collated in a specific way and attached to some city names, I'm willing to bet that at least a few of you found them interesting.

If I were more familiar with math history and/or the philosophical side of math, I might have something really great to say about the incredible efficiency of numbers in their ability to tell a story. The standings above are essentially a labeled table of numbers containing roughly 200 characters. Is there any way to write 200 characters of prose that would evoke as clear or as many mental images as those standings did?

In my program that plays out these simulated seasons, I built in a flag that alerts me when something odd happens, like a team with a losing record winning the Super Bowl or a team going undefeated. When I go in to check out those flags, I find myself scrolling up to the season above or down to the season below, about which there is nothing special. But somehow those seasons are always just as fascinating as the flagged ones.

These make-believe seasons were intended to simulate reality in the present day NFL. If you're like me, and you find a random made-up season interesting, then I think the lesson here is that the present day NFL is incapable of producing an uninteresting season. There may be a 24% chance that the best team will win and a 10% chance they'll miss the playoffs, but there is apparently a 100% chance that I will enjoy the NFL in 2006.

Unless Dallas wins.

17 Comments | Posted in Statgeekery

Ten thousand seasons

Posted by Doug on June 1, 2006

You'd better read yesterday's post if you haven't yet.

So the plan is to simulate an NFL season a bazillion times and observe what kind of wacky stuff happens. Here are the particulars.

For each simulated season, I will assign each team a true strength which is a random number from a normal distribution with mean 0 and standard deviation 6. This means that the teams' true strengths are mostly somewhat close to zero. In particular, roughly two-thirds of all teams will have true strengths between -6 and +6, about 95% of all teams will have true strengths between -12 and +12. As you probably guessed, these numbers were rigged so that they generally agree with the values that the simple rating system produces for real NFL seasons in this decade.

You'll note that, even though it will be true for a real NFL season, I am not requiring that the teams' strengths in a given year average zero. Even though we can't observe it (at least not easily), there must surely be years when the league is stronger and years when it's weaker. And in any case, since we are primarily interested in questions like "how often does the best team in football (for that year) win the Super Bowl," it doesn't matter much.

Each simulated season had the same league structure and schedule as the 2005 NFL. That is, there were 32 teams divided into eight divisions of four teams each, and the schedule is just like that of the 2005 NFL.

There is one potential complication here, but I think it's minor. In the simulated world, each season is independent of the previous one, so the two intra-conference games in each team's schedule that are determined by last season's finish are instead essentially against random teams. In the real NFL, the seasons are not independent and good teams probably end up playing very slightly stronger schedules in general than bad teams do. Fortunately, this effect isn't nearly as dramatic now as it was in the 80s and 90s.

Also, I was too lazy to program the tiebreakers. All ties were broken by coin flip. I don't think this will affect anything, but let me know if you think I'm wrong about that.

Finally, the individual games are played by using the same formula we used in this post:


Home team prob. of winning =~ 1 / (1 + e^(-.438 - .0826*diff))

where diff is the home team's true strength minus the visiting team's true strength.

OK, that's that. Let's get to the question of the day, which is: how often does the best team in the NFL win the Super Bowl?

The answer is roughly 24% of the time.

I simulated 10,000 seasons. The table below shows that the best team won the Super Bowl 2,399 times, the second-best team won it 1,448 times, and so.


Tm# SBwins
==========
1 2399
2 1448
3 1060
4 846
5 670
6 584
7 464
8 388
9 327
10 285
11 231
12 189
13 188
14 151
15 141
16 122
17 113
18 72
19 70
20 55
21 42
22 35
23 36
24 22
25 22
26 15
27 12
28 4
29 4
30 3
31 1
32 1

[NOTE: if you thought this table looked slightly different earlier, you're not seeing things. I accidentally inlcuded the wrong table at first, so I updated it about an hour later.]

Very nearly 50% of the time, the Super Bowl champion was one of the best three teams in football. And let me reiterate that when I say "the best team," I am not necessarily talking about the team with the best record. I am talking about the best team. Remember, we're omniscient here. We know which team really was the best.

I'm sure what caught your eye was that the 32nd-best (i.e. the worst) team in the NFL won the title once. Let me tell you about that season.

It was simulated season #6605. The Seattle Seahawks were truly a great team (true strength +15.1) and they played up to their potential, posting a 15-1 regular season record. The Chicago Bears were the worst team in football, but with a true strength of -9.0, they really weren't that bad, at least by worst-team-in-football standards. The NFC North was relatively weak, and Chicago took the division with an 8-8 record.

The Bears' first round playoff opponent was the Carolina Panthers, who were not great (+2.8) but had posted a 10-6 record to finish second in the NFC South. The game was in Chicago, of course, and it was therefore only a mild upset when Chicago won it. Chicago then beat the Saints in New Orleans and the Seahawks in Seattle to reach the Super Bowl.

The AFC was weak in 6605. The best they had to offer was the Jets (+7.2) who had gone 12-4 in the regular season and had beaten the Colts on the road to reach the Super Bowl. The Bears beat the Jets to win the title.

As James points out in his article, there is no single event here that is too hard to believe. It's not unlikely that there wouldn't be any truly terrible teams in the NFL in a given year. It's not unlikely that an entire division would be weak, and it's not unlikely that the worst team in such a division could win the title with an 8-8 record. In their four playoff games, their probabilites of victory were 37%, 10%, 8%, and 21%. That they'd win those four games is certainly unlikely, but no more unlikely than, say, an NL team getting four straight hits at the bottom of their batting order, and I'll bet you've seen that.

No one of those things is terribly bizarre. Yet they all come together to create an almost-unbelievable occurrence. Almost unbelievable. Ten thousand years is a long time. Most of you have probably been watching NFL football for 20 or 30 years, and think of all the crazy stuff you've seen in that time. If you lived another 500 lifetimes, you'd see some even crazier stuff.

Do you think you'd ever see a team like the 2005 Jets win a Super Bowl? And I'm not talking about the Jets if Pennington and Curtis Martin had stayed healthy. I'm talking about the Brooks Bollinger Cedric Houston 2005 New York Jets. If you gave that team 10,000 tries, would they win a Super Bowl? Before you say no, think about all the times you've seen a really bad team rattle off three or four unexpected victories; think of the Craig Krenzel-led Bears during that stretch in 2004, for example. Such runs are unlikely, but you've seen lots of them. Don't you think that, in 10,000 years, some team could string a couple of those runs together, get some breaks from the schedule, and then fluke out in the playoffs?

It could happen.

25 Comments | Posted in Statgeekery

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