Weird Statistical Result Showing "Optimal" Score
Chicago Odds, Updated, 11 March, 1997
Seattle/Houston Odds, Updated, 15 December, 1996
Michael Jordan has stated a couple times that he is just
hoping to have "a perfect season". He has never explicitly said that he
wanted to go undefeated, but the implication is there. No team has ever
done this before, obviously. College teams that play only 30 games or
so rarely go undefeated. The longest win streak by an NBA team is 33 games,
by the 1971-72 Los Angeles Lakers. It would be one of the greatest
accomplishments in the history of sports if the Chicago Bulls even
So can they do it?
Uhhh, yeah. They can. Seriously. You can calculate the odds
a few different ways, but one way gives them an 35% chance of
doing it. That is not a remote chance, but a realistic one.
Most ways of looking at this issue give the Bulls less than a 1% chance
of achieving a perfect season, but the fact that they are
even within sight is remarkable.
The basic means for calculating the Bulls' chances of
going 82-0 is to use
the team's winning percentage and what is called a binomial distribution.
The Bulls' winning percentage right now is 1.000 (100%) as they
are 12-0. A binomial distribution is a statistical
way to determine what the odds are for a team with a 100% winning
percentage to win 82 games out of 82. [Use the BINOMDIST function
in Microsoft Excel. Enter the following =1-BINOMDIST(81,82,1.000,1)]
At this point, you are either confused or you have a statistics
background and are calling me an idiot. This is because a team
with a 100% chance of winning one game, which is what is implied
by the Bulls' 12-0 record, has a 100% chance of winning all 82
games. That doesn't make sense to anyone, whether you know statistics
or not. What we need to do is to better determine what the Bulls'
chances are of winning any single game.
The best way to make this estimate currently is the
Method, which uses a team's average points scored and allowed,
their consistency in maintaining these averages, and how much
they play up or down to their opponents to estimate a winning
percentage. At this point, the Bulls have outscored their
opponents by an average of 104.1-85.5. They have also been very consistent
in doing this (another thing Jordan has been preaching to his teammates).
The standard deviation of their offensive score
is 7.6 and the standard deviation of the number of points allowed
is 11.1. People who know stats understand these numbers, but for those
who don't, they mean roughly that the Bulls score 104 ppg plus or
minus about 8, allowing 85 ppg plus or minus about 11. Pretty
obviously, if you subtract 8 from 104 and add 11 to 85, you still
get the Bulls tied at 96, implying that they are pretty darn
Method just assigns a probability of winning to a team with this
kind of consistency and it says that their winning percentage
is 0.987 or 98.7% (using a covariance of 55.5).
That number is incredible and unheard of for an NBA team. If we
take that value as correct, we can then ask a binomial distribution
what the chance of winning every game is for this Bulls'
team with a 98.7% chance
of winning each
game. The answer is 35%. [Enter the following into
your Excel spreadsheet: =1-BINOMDIST(81,82,0.987,1) You
may get 34% because of roundoff error.]
Hold on, I hear Las Vegas calling. "No, I won't bet on that!
And, no, Michael Jordan won't either!"
There are a lot of reasons not to bet on that number, but I will
talk specifically about two of them: history and injury.
I once took a class at Duke University from
Bob Winkler (a great
prof and a sports fan), who taught us all sorts of ways to
manipulate probabilities once we had them, but I once asked him
how to obtain the probabilities. He pretty much said, "Aye, that's
the rub." (He didn't say exactly that. He was in the business school
and business profs don't quote Shakespeare.) This is the problem
here. How much do we trust this 98.7% chance obtained from
the Correlated Gaussian Method?
One reason not to trust this number is that the Bulls have not
played a very difficult schedule yet. According to
Jeff Sagarin's ratings of teams, the Bulls have played
a slightly below average schedule and only two legitimately
good teams in Miami and Detroit. That says we should lower the
Bulls' win percentage, but how much? Professor Winkler says he
doesn't know. I have some theoretical ideas, but I won't go into them here other
than to say that Sagarin (and every other web jockey
with a prediction method) has a somewhat ad-hoc way of doing it.
One possible simple way to correct for this is to use more history.
Last year's Bulls were pretty much the same as this year's Bulls
and they had a winning percentage of 88%. If we take that as the
"true" Bulls' chance of winning, then the Bulls have a 0.002% chance (that is 2 in
of winning 82 games, otherwise known as "slim and none". Maybe that
isn't their "true" winning percentage either. Maybe we should add
this year's results on to last years and it should be
(72+12)/(82+12) = 84/94= 89%. Then the Bulls have a 0.010% chance
of winning all 82 games. Again, the odds don't look good. (As a
mathematically-based opinion, also known as a best estimate, the
Bulls now have about a 0.3% chance of winning all their games.
I'd also say that if the Bulls win their first 17 games of this season,
then they breach the 1% chance of winning them all. If they win their
first 27, then they're at about 5%.)
The second reason the 35% chance makes little sense is that
this Bulls team relies upon Michael Jordan. If he gets injured
or suspended for betting that the Bulls will go undefeated,
the Bulls are an ordinary playoff team. The odds that Jordan
will get injured for just one game are high enough to affect
this percentage to an important degree. The odds that Jordan
will get suspended for betting are none -- right, Michael?
Despite all this, there is no dispute that the Bulls are
again one of the greatest teams in the history of sports. Though
they probably won't be as "perfect" as Mike would like, I think
he just might re-retire if they win 75 games. At that point,
there really is nothing left for him to prove.
There are perhaps more appropriate ways of
calculating some of the probabilities above, including
accounting for the fact that the Bulls only have
to win 70 more (which I do account for in my "best estimate").
But my point here was not necessarily to
be completely correct and completely incoherent; rather, I'd
like to be first-order correct and mostly coherent. So, if
you are a stats nut, feel free to extend the analysis for
a class or your research. Then send it on to me and I'll
pass it on here with your name. But don't tell me I did it
wrong unless you can back it up and explain it
Weirdness.... That said, I'd like to point out that the Bulls don't necessarily make
themselves look any better by running up the score. In fact,
they can actually make themselves look worse. This is primarily
because the Bulls are so incredibly good that it doesn't make sense for them
to run up the score. As an example of this, if the Bulls had beaten
their tenth opponent, Atlanta, by the score 102-69 rather
than 97-69, their expected winning percentage went down
from 98.4% to 98.1%. Mathematically, this is because the additional
gain in their scoring average is outweighed by the decrease
in their covariance. Physically (or, dare I say, psychologically),
Chicago is proving nothing in terms of their ability to win by running
up the score. A typical winner lets the other team play without
letting them be a threat to the outcome of the game. In my opinion,
the following result is a little weird: the score that would have
improved the Bulls' theoretical winning percentage the most was 91-69.
You gotta love science....
Now that the Bulls have (predictably) lost, it's time to look at more
reasonable goals they might set for themselves. Unfortunately, probably
the only goal that will last for the Bulls is that of winning the
Championship again. When things get tough, breaking the regular
season win record probably won't provide sufficient motivation....
But the Bulls may be so much better as a team than anyone else in the
league that motivation isn't
necessary. Until the Jazz loss, the Bulls had been challenged for a victory
only once or twice and neither time seriously. Numerically then, my best
estimate of how good they are comes by averaging last year's numbers
with this year's. Specifically, I use the
method to estimate their winning percentage this season, then weight
that against last season's record. The Correlated Gaussian
method yields an estimate Bulls' winning percentage of 0.842
through 61 games (Avg Pts Scored-Allowed: 103.1-91.1. Std Dev
Pts Scored-Allowed: 12.4-12.1. Covariance of Pts Scored and Allowed: 78.1).
Combining this with the Bulls' 72-10 (0.878) record of last year,
I estimate the "true" Bulls' winning percentage as
61*(0.842) + 20*(0.878)
Win % = ------------------------- = 0.851
(61 + 20)
You can vary the weights any way you personally feel, but clearly
the more games the Bulls' play this season, the more we should believe
their numbers this season vs. last season. However, last season's numbers
do represent the Bulls over the course of a full season with
the injuries and the varied schedule that have not yet been introduced
into this season. So if you want to vary the weights, they should
honor these concepts.
With this winning percentage, the chance that the Bulls will
win 72 or more games stands at about 38%. You can calculate this
by plugging the following formula in to your Excel spreadsheet:
=1-BINOMDIST(71-53,82-61,0.851,1). Notice that I subract off
53 from 71 and 61 from 82 -- this is because the Bulls only have to win 19
more (as opposed to 72) of their remaining 21 in order to reach 72 wins.
Updated 12/3: Seattle now has
only about a 3% chance of winning 64 or more games, their win total of
last season, using the
Houston with their 20-2 record has about a 0.017% chance
of winning 75 and a 47% chance of winning 64. These somewhat small percentages
are for two primary reasons. First, last season
indicates that Houston struggles over
the course of a long season, even though I only weighted
last season's record with 10 games (rather than 20) because of the
addition of Barkley. Given that Houston's stars are old and
injury-prone, it would be remarkable if they don't suffer a breakdown
at some point in the season. The second reason is that this year's Rockets
have been squeaking by and, as a result, do not look as good as
their record. An unusual aspect of Houston's
season so far is that they have played to the level of their
competition to a remarkable degree, winning games against Utah,
Denver, Indiana, Portland, and Minnesota by less than three points
and additional games against Boston, Washington, and the Lakers by
five points or less. That is
usually called LUCK.