Doesn’t The Other Team Count? How Much?

I’d worked out an estimate of how much information content there is in a basketball score, by which I was careful to say the score that one team manages in a game. I wasn’t able to find out what the actual distribution of real-world scores was like, unfortunately, so I made up a plausible-sounding guess: that college basketball scores would be distributed among the imaginable numbers (whole numbers from zero through … well, infinitely large numbers, though in practice probably not more than 150) according to a very common distribution called the “Gaussian” or “normal” distribution, that the arithmetic mean score would be about 65, and that the standard deviation, a measure of how spread out the distribution of scores is, would be about 10.

If those assumptions are true, or are at least close enough to true, then there are something like 5.4 bits of information in a single team’s score. Put another way, if you were trying to divine the score by asking someone who knew it a series of carefully-chosen questions, like, “is the score less than 65?” or “is the score more than 39?”, with at each stage each question equally likely to be answered yes or no, you could expect to hit the exact score with usually five, sometimes six, such questions.

But How Interesting Is A Basketball Score?

When I worked out how interesting, in an information-theory sense, a basketball game — and from that, a tournament — might be, I supposed there was only one thing that might be interesting about the game: who won? Or to be exact, “did (this team) win”? But that isn’t everything we might want to know about a game. For example, we might want to know what a team scored. People often do. So how to measure this?

The answer was given, in embryo, in my first piece about how interesting a game might be. If you can list all the possible outcomes of something that has multiple outcomes, and how probable each of those outcomes is, then you can describe how much information there is in knowing the result. It’s the sum, for all of the possible results, of the quantity negative one times the probability of the result times the logarithm-base-two of the probability of the result. When we were interested in only whether a team won or lost there were just the two outcomes possible, which made for some fairly simple calculations, and indicates that the information content of a game can be as high as 1 — if the team is equally likely to win or to lose — or as low as 0 — if the team is sure to win, or sure to lose. And the units of this measure are bits, the same kind of thing we use to measure (in groups of bits called bytes) how big a computer file is.

Gaussian distribution of NBA scores

The Prior Probability blog points out an interesting graph, showing the most common scores in basketball teams, based on the final scores of every NBA game. It’s actually got three sets of data there, one for all basketball games, one for games this decade, and one for basketball games of the 1950s. Unsurprisingly there’s many more results for this decade — the seasons are longer, and there are thirty teams in the league today, as opposed to eight or nine in 1954. (The Baltimore Bullets played fourteen games before folding, and the games were expunged from the record. The league dropped from eleven teams in 1950 to eight for 1954-1959.)

I’m fascinated by this just as a depiction of probability distributions: any team can, in principle, reach most any non-negative score in a game, but it’s most likely to be around 102. Surely there’s a maximum possible score, based on the fact a team has to get the ball and get into position before it can score; I’m a little curious what that would be.

Prior Probability itself links to another blog which reviews the distribution of scores for other major sports, and the interesting result of what the most common basketball score has been, per decade. It’s increased from the 1940s and 1950s, but it’s considerably down from the 1960s.

prior probability

You can see the most common scores in such sports as basketball, football, and baseball in Philip Bump’s fun Wonkblog post here. Mr Bump writes: “Each sport follows a rough bell curve … Teams that regularly fall on the left side of that curve do poorly. Teams that land on the right side do well.” Read more about Gaussian distributions here.

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