The Tebow Distribution

Tim Tebow wears his faith on his face; but is his "streak" a "miracle"?
Tim Tebow wears his faith on his face; but is his "streak" a "miracle"? The data suggests not; if anything, his current effort is barely keeping the Broncos on statistical track. Photo from Ref. 3.

Author’s Note: in the Dec. 18 game against the Patriots, the Broncos lost spectacularly. Chance giveth, and chance taketh away. I still stick to my punchline: based on the data, the Broncos are statistically behind their own curve…

The first I heard of Tim Tebow, starting quarterback of the Denver Broncos, was last night on NPR’s “All Things Considered.” Tebow has gained some fame in the U.S. of late for, as NPR put it, ” . . . his improbable come-from-behind victories combined with his prominent expressions of faith.” [1]

My interest was piqued further as they went on to explain the reason for “Tebowmania,”

“Earlier this season — his second in the NFL — Tebow orchestrated a stunning comeback in the first game he started. But he played so miserably in the second game that a Denver Post columnist wrote, “Right now, Tebow is the worst quarterback in the NFL.”” [1]

But this young QB went on to lead the team to ” . . . six consecutive victories as a starter . . . ” wherein he ” . . . usually plays awful for a half or more, and then pulls off a comeback.” [1]

I was intrigued. As humans, we are very flawed observers of the natural world. We tend to cherry-pick our observations to confirm our belief systems. We tend to enjoy a good story and ignore statistics. Science is the only mechanism which we have designed and employed which allows us to query the natural world and see if our beliefs are based on fact or whether we are misleading ourselves.

NPR goes on to interview critics who say that this is just a streak of good fortune; it can’t last. Other elements of our society want to believe that Tebow’s religious faith plays a role in this. As NPR reported, “A Denver-area pastor with ties to Tebow’s family says: “It’s not luck. It’s favor. God’s favor.”” [1].

I hate it when people diminish the idea of a wise and powerful superbeing to the role of some fat couch-potato deity that wants the Broncos to win more games. So I decided to use science to see how likely or unlikely Tebow’s streak has been.

First, we have to define numerically what is reportedly so “remarkable” about Tebow’s streak. The NPR story is vague. Requoting them, Tebow ” . . . plays awful for a half or more, and then pulls off a comeback.” [1]

Let’s consider the current Denver Bronco’s season. Here are the dates and final scores (their score is first) of this season’s games, marked as a “win” or “loss” for Denver:

2011-09-12 20 23 [loss]
2011-09-18 24 22 [win]
2011-09-25 14 17 [loss]
2011-10-02 23 49 [loss]
2011-10-09 24 29 [loss]
2011-10-23 15 15 [win]
2011-10-30 10 45 [loss]
2011-11-06 38 24 [win]
2011-11-13 17 10 [win]
2011-11-17 17 13 [win]
2011-11-27 13 13 [win]
2011-12-04 35 32 [win]
2011-12-11 10 10 [win]

(The 4th-quarter ties ended in wins for Denver). In search of a definition of “comeback,” we can look at these last 6 victories and see if the scores in the first half (first pair of scores, Denver first), third quarter (second pair of scores), and fourth quarter (third pair of scores) follow some specific pattern. Here are those scores for those games:

2011-11-06 (7 17) (24 24) (38 24)
2011-11-13 (10 0) (10 7) (17 10)
2011-11-17 (3 3) (10 10) (17 13)
2011-11-27 (7 10) (10 13) (13 13)
2011-12-04 (7 15) (21 22) (35 32)
2011-12-11 (0 0) (0 7) (10 10)

In all but the second game, Denver was behind or tied through the 3rd quarter and then wins in or after the fourth quarter. The second game doesn’t fit the pattern; Denver was winning from the get-go. So, really, there are only FOUR games in a row where Denver begins behind and then pulls off victory.

So let’s define our algorithm for this process: Denver must begin a game behind or tied, and then end either ahead or tied. By “begin,” I mean in either the second or third quarter Denver must be tied or behind. They must repeat this pattern at least 4 times.

I looked at ALL data from ALL NFL games in which Denver participated from 1960 onward [2]. If I apply the above algorithm to these most recent games, indeed it tells me that Denver achieved the goal outlined in the algorithm.

Let’s then look at the following frequencies: how often does Denver achieve the pattern one game in a row, two games in a row, three games in a row, and four games in a row?

  1. One game in a row:  173/817
  2. Two games in a row: 42/817
  3. Three games in a row: 11/817
  4. Four games in a row: 1/817

Let’s look at the progression. Out of 817 games, 173 involve the Broncos winning a single game by the above pattern (regardless of whether the next game is won or not by the same pattern). 42 involve winning at least two games in a row by this pattern.  That’s about 1/4 of the games involving at least 1 win in a row by the above pattern. 11 involve at least 3 wins in a row by the above pattern – this is already a remarkable fact: that 11 times the Broncos have delivered a sequence of three games in which each is won by starting behind or tied and ending the fourth quarter at least in a tie (definitely a “comeback”).

Interestingly, 11/42 is also about 1/4. This suggests a pattern in the data: the Broncos, based on historical information, can deliver a sequence of N wins by this pattern, W(N), which follows the equation 4*W(N+1) = W(N). The trend is almost exponential, following a function with a form close to W(N) = 817*exp(-1.52*N). Plugging in N=0, we get 817; N=1 yields 179; N=2 yields 39; N=3 yields 9; N=4 yields 2.

This would then suggest that the number of times the Broncos should have won 4 games in a row by this pattern is about 3 times; they’ve done it once. If anything, based on the progression in the data, the Broncos are due for something like this and have done it less often than, statistically, they should have.

At first glance, it seems that the current pattern of wins is not a remarkable feat; it’s potentially a statistical by-product of the distribution of N consecutive wins by this pattern.  If anything, the current “streak” should have happened more than once in 817 games.

Let’s look at another team and see if this pattern holds. I chose the Packers, widely considered a historically remarkable team. Let’s see how their record holds up under the same analysis. The Packers have data going back to 1940, so there is more data on them. This will help us see if the above “pattern” is really some kind of pattern or just a fluke feature of the Denver data.

  1. One game in a row:  187/1284
  2. Two games in a row: 34/1284
  3. Three games in a row: 5/1284
  4. Four games in a row: 0/1284

Interesting! So out of 1284 games, the Packers have only started behind/tied and then won/tied 187 times. 34 times, they have done this in 2 consecutive games, or about 1/5 of the time they did it 1 consecutive times. 5 times, they did it 3 consecutive times, or about 1/6 of the times they did it 2 consecutive times. They’ve never done it 4 times in a row, although that’s consistent with a nearly exponential decay pattern from the Packer data. Based on the data, we would have expected the Packers to have done 4 in a row about once in their history.

This is a small study, albeit with a lot of individual game data from two separate teams. That said, the data suggest that nothing remarkable or improbable is happening with the Broncos right now. In fact, based on the trend in their historical data one would have expected AT LEAST 1 such “streak” by now.

Getting the data from the referenced website is not simple, so I’ve only done two teams. It would be interesting to repeat this for 4-5 teams and see what happens. But, looking at two teams – the Broncos and the Packers – there is nothing remarkable in the current trend in Tebow’s tenure as QB. His team is basically “on track” statistically.

If anything, the Packers are due for a little drama like this. C’mon Packers! Stop being such a good team and act mediocre in the first half, then go for the kill in the second half. Stop being so consistently good!

[1] http://www.npr.org/2011/12/16/143859116/tebow-mania-why-the-quarterback-is-so-popular

[2] http://www.pro-football-reference.com/

[3] http://www.flickr.com/photos/opensports/3339588859/sizes/m/in/photostream/

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