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!




Faith-based dodge

Today, the Supreme Court ruled on a number of cases that have piled up as its current term comes to an end. One of the issues the court ruled on today was whether funding President Bush’s “faith-based programs initiative” is a constitutional use of taxpayer money. The court didn’t rule on the case, but instead ruled that taxpayers have no standing in the court to bring such a case. From a technical perspective, this is a useful ruling. If every taxpayer or taxpayer organization had standing, we could tie up the court with questions about every law on the books.

The larger question – whether organizations with religious affiliations should have a special program that gives them money to do their work – is left open. Let’s investigate this question by looking, if possible, at the rules of the program. We begin with the website for the President’s Office of Faith-Based and Community Initiatives (OFBCI) [1]. The President’s vision of the program is summarized as follows: “Yet, all too often, the Federal government has put in place complicated
rules and regulations preventing FBCOs from competing for funds on an
equal footing with other organizations. President Bush believes that
besides being inherently unfair, such an approach can waste tax-payer
dollars and cut off the poor from successful programs” [2].  There are several principles on which the office is founded, but one of interest is the following: “The underlying premise of the President’s Initiative is that a more
open and competitive Federal grant-making process will increase the
delivery of effective social services to those whose needs are

What does the U.S. Constitution have to say about religion? A lot of people throw around the idea of “separation”; however, this word doesn’t appear explicitly in the Constitution. Instead, we must look to the language in Article 6 (Debts, Supremacy, Oaths).

The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States.

Of course, there is the most famous language in the first amendment – Freedom of Religion, Press, Expression:

Congress shall make no law respecting an establishment of religion, or
prohibiting the free exercise thereof; or abridging the freedom of speech, or of
the press; or the right of the people peaceably to assemble, and to petition the
Government for a redress of grievances.

This is where things get gray. Is the OFBCI an instance of “establishment”, or it is, as the President states, a case of removing prohibition by “leveling the playing field”? This is, perhaps, the question that the Court could have addressed had the taxpayers had standing.

From my perspective, having such an office in the President’s own house, so close to the Executive, does smack of establishment. I don’t have a problem with religious organizations offering aid, even if I have noted in my own experience with such organizations that there are spiritual strings attached to that aid. If the President thought the playing field wasn’t level, why not just have an Office of Community Initiatives, dropping “Faith” from the name? If the idea is to just level the playing field, why not make it fair (at least, in name) for all kinds of organizations to do community work, holding all organizations to the same standards and offering no religious test of any kind which determines funding?

One question I had was, “Is a religious test required to apply for funding from OFBCI?” I learned a few things. First, the White House has an OFBCI, but so does Health and Human Services. As of March 2006, the Department of Homeland Security ALSO has an OFBCI. Why three are needed to “level the playing field” is really beyond me. This looks more like a pattern of infection than an concerted centralized effort to level a playing field.

It turns out be be quite hard to answer my question. I can find lots of short paragraphs guiding those unfamiliar with grant programs generally about how to proceed. To find the criteria applied to determine grant award winners, you have to read each grant’s rules. I tried going to some of the grant websites to get info. For instance, I tried to go to the Community Block Grants for HHS,, but it was unavailable. That was the first one I tried. I then tried a State Abstinence Education Program website,  It was also unavailable. I then tried the Temporary Assistance to Needy Families program website,
.  This page also didn’t exist or wasn’t available. I then tried a Child Care and Development fund page,

At that point, I gave up trying. It seems that an honest citizen trying to learn the rules for getting a grant, as recommended by a given OFBCI, is blocked by a complete lack of actual information about the program. This gives me no faith in the Constitutionality of the faith-based programs. I want to give them a fair shake, but it seems like failing to supply information about the rules in a reliable way is a first small step for mis- for dis-information. Is OFBCI leveling the playing field, or creating a special opportunity for those only with ties to religious organizations to get federal money to do work? Is the grant application process a religious test, or not? Will the Supreme Court ever be given a chance to rule on this issue by a party with standing? I have faith, at least, that some of these might be answered.


Bush Calls U.S. a “Nation of Prayer”

News of the weird, my friends. Our “politics-and-religion-mixing President has declared, on this the national day of prayer, that the U.S. is a nation of prayer”:;_ylt=Auh2YG3m0lrz_xYEcS4QLD6s0NUE;_ylu=X3oDMTA3OXIzMDMzBHNlYwM3MDM-. I wonder if, on the national day of reason, he’ll declare the U.S. a nation of reason? Oh, right, we don’t have that national day. Yikes.

Thinking back to that recent study on prayer [TAOMPH245], it is now clear to me why the U.S. is split down the middle politically. If you have a 50% chance of suffering complications after heart bypass surgery whether people pray for you or not, I guess a nation of prayer has a 50% chance of being Republican or Democrat. Double yikes.

.. [TAOMPH245]


Thoughts on Science and Spirit from a man of the Spirit

My good pal Mandeep sent me a lovely blog article from “Jim Burklo, minister at the Sausalito Presbyterian Church”: Jim talks about how intelligent design is a discredit to both science and religion. Well, Jim puts it best so let me just quote him:

But [intelligent design] isn’t a “theory” at all. “Intelligent design” posits that the structure of life is so complex and delicate that it is unimaginable that it could have come into existence without having been designed by some intelligent force. Therefore such an intelligence must be responsible for it. But this is a conclusion that can be reached only by assuming that it is true in the first place – a classic tautology, or example of circular reasoning, which has no place in science. It is not a theoretical alternative to evolution, because it suggests no other credible means by which this outside intelligence created the complexity of life. There is nothing in the theory of evolution, the only theory that holds any water in explaining the origin of the species, that proves or disproves the existence of such an intelligent “designer”. Even if one thinks of God as a separate, distinct being that manipulates the universe, “intelligent design” offers no intelligent reason that God didn’t employ evolution as his or her means for creating life on earth.

Circular reasoning doesn’t belong in science education. “Intelligent design” is a thinly-veiled and inappropriate attempt to inject religious indoctrination into public schools. If it gets into school science textbooks, it would damage both science and religion in this country.

The emphasis was added by me to highlight the key conclusions he draws. Hear, hear, Jim. Let us both hope that together science can continue to map out the order of the natural world and religion can map out the order of the spirit, and through both we can gain a deeper insight into this wonderous universe.