How Interesting Is A Low-Scoring Game?

I’m still curious about the information-theory content, the entropy, of sports scores. I haven’t found the statistics I need about baseball or soccer game outcomes that I need. I’d also like hockey score outcomes if I could get them. If anyone knows a reference I’d be glad to know of it.

But there’s still stuff I can talk about without knowing details of every game ever. One of them suggested itself when I looked at the Washington Post‘s graphic. I mean the one giving how many times each score came up in baseball’s history.

I had planned to write about this when one of my Twitter friends wrote —

By “distribution” mathematicians mean almost what you would imagine. Suppose we have something that might hold any of a range of values. This we call a “random variable”. How likely is it to hold any particular value? That’s what the distribution tells us. The higher the distribution, the more likely it is we’ll see that value. In baseball terms, that means we’re reasonably likely to see a game with a team scoring three runs. We’re not likely to see a game with a team scoring twenty runs.

Frequency (in thousands) of various baseball scores. I think I know what kind of distribution this is and I mean to follow up about that.
Philip Bump writes for The Washington Post on the scores of all basketball, football, and baseball games in (United States) major league history. Also I have thoughts about what this looks like.

There are many families of distributions. Feloni Mayhem suggested the baseball scores look like one called the Beta Distribution. I can’t quite agree, on technical grounds. Beta Distributions describe continuously-valued variables. They’re good for stuff like the time it takes to do something, or the height of a person, or the weight of a produced thing. They’re for measurements that can, in principle, go on forever after the decimal point. A baseball score isn’t like that. A team can score zero points, or one, or 46, but it can’t score four and two-thirds points. Baseball scores are “discrete” variables.

But there are good distributions for discrete variables. Almost everything you encounter taking an Intro to Probability class will be about discrete variables. So will most any recreational mathematics puzzle. The distribution of a tossed die’s outcomes is discrete. So is the number of times tails comes up in a set number of coin tosses. So are the birth dates of people in a room, or the number of cars passed on the side of the road during your ride, or the number of runs scored by a baseball team in a full game.

I suspected that, of the simpler distributions, the best model for baseball should be the Poisson distribution. It also seems good for any other low-scoring game, such as soccer or hockey. The Poisson distribution turns up whenever you have a large number of times that some discrete event can happen. But that event can happen only once each chance. And it has a constant chance of happening. That is, happening this chance doesn’t make it more likely or less likely it’ll happen next chance.

I have reasons to think baseball scoring should be well-modelled this way. There are hundreds of pitches in a game. Each of them is in principle a scoring opportunity. (Well, an intentional walk takes three pitches without offering any chance for scoring. And there’s probably some other odd case where a pitched ball can’t even in principle let someone score. But these are minor fallings-away from the ideal.) This is part of the appeal of baseball, at least for some: the chance is always there.

We only need one number to work out the Poisson distribution of something. That number is the mean, the arithmetic mean of all the possible values. Let me call the mean μ, which is the Greek version of m and so a good name for a mean. The probability that you’ll see the thing happen n times is \mu^n e^{-\mu} \div (n!) . Here e is that base of the natural logarithm, that 2.71828 et cetera number. n! is the factorial. That’s n times (n – 1) times (n – 2) times (n – 3) and so on all the way down to times 2 times 1.

And here is the Poisson distribution for getting numbers from 0 through 20, if we take the mean to be 3.4. I can defend using the Poisson distribution much more than I can defend picking 3.4 as the mean. Why not 3.2, or 3.8? Mostly, I tried a couple means around the three-to-four runs range and picked one that looked about right. Given the lack of better data, what else can I do?

The Poisson distribution starts pretty low, with zero, then rises up high at three runs and dwindles down for ever-higher scores.
A simulation of baseball, or other low-scoring games, based on a Poisson distribution with mean of 3.4.

I don’t think it’s a bad fit. The shape looks about right, to me. But the Poisson distribution suggests fewer zero- and one-run games than the actual data offers. And there are more high-scoring games in the real data than in the Poisson distribution. Maybe there’s something that needs tweaking.

And there are several plausible causes for this. A Poisson distribution, for example, supposes that there are a lot of chances for a distinct event. That would be scoring on a pitch. But in an actual baseball game there might be up to four runs scored on one pitch. It’s less likely to score four runs than to score one, sure, but it does happen. This I imagine boosts the number of high-scoring games.

I suspect this could be salvaged by a model that’s kind of a chain of Poisson distributions. That is, have one distribution that represents the chance of scoring on any given pitch. Then use another distribution to say whether the scoring was one, two, three, or four runs.

Low-scoring games I have a harder time accounting for. My suspicion is that each pitch isn’t quite an independent event. Experience shows that pitchers lose control of their game the more they pitch. This results in the modern close watching of pitch counts. We see pitchers replaced at something like a hundred pitches even if they haven’t lost control of the game yet.

If we ignore reasons to doubt this distribution, then, it suggests an entropy of about 2.9 for a single team’s score. That’s lower than the 3.5 bits I estimated last time, using score frequencies. I think that’s because of the multiple-runs problem. Scores are spread out across more values than the Poisson distribution suggests.

If I am right this says we might model games like soccer and hockey, with many chances to score a single run each, with a Poisson distribution. A game like baseball, or basketball, with many chances to score one or more points at once needs a more complicated model.


How Interesting Is A Baseball Score? Some Further Results

While researching for my post about the information content of baseball scores I found some tantalizing links. I had wanted to know how often each score came up. From this I could calculate the entropy, the amount of information in the score. That’s the sum, taken over every outcome, of minus one times the frequency of that score times the base-two logarithm of the frequency of the outcome. And I couldn’t find that.

An article in The Washington Post had a fine lead, though. It offers, per the title, “the score of every basketball, football, and baseball game in league history visualized”. And as promised it gives charts of how often each number of runs has turned up in a game. The most common single-team score in a game is 3, with 4 and 2 almost as common. I’m not sure the date range for these scores. The chart says it includes (and highlights) data from “a century ago”. And as the article was posted in December 2014 it can hardly use data from after that. I can’t imagine that the 2015 season has changed much, though. And whether they start their baseball statistics at either 1871, 1876, 1883, 1891, or 1901 (each a defensible choice) should only change details.

Frequency (in thousands) of various baseball scores. I think I know what kind of distribution this is and I mean to follow up about that.
Philip Bump writes for The Washington Post on the scores of all basketball, football, and baseball games in (United States) major league history. Also I have thoughts about what this looks like.

Which is fine. I can’t get precise frequency data from the chart. The chart offers how many thousands of times a particular score has come up. But there’s not the reference lines to say definitely whether a zero was scored closer to 21,000 or 22,000 times. I will accept a rough estimate, since I can’t do any better.

I made my best guess at the frequency, from the chart. And then made a second-best guess. My best guess gave the information content of a single team’s score as a touch more than 3.5 bits. My second-best guess gave the information content as a touch less than 3.5 bits. So I feel safe in saying a single team’s score is about three and a half bits of information.

So the score of a baseball game, with two teams scoring, is probably somewhere around twice that, or about seven bits of information.

I have to say “around”. This is because the two teams aren’t scoring runs independently of one another. Baseball doesn’t allow for tie games except in rare circumstances. (It would usually be a game interrupted for some reason, and then never finished because the season ended with neither team in a position where winning or losing could affect their standing. I’m not sure that would technically count as a “game” for Major League Baseball statistical purposes. But I could easily see a roster of game scores counting that.) So if one team’s scored three runs in a game, we have the information that the other team almost certainly didn’t score three runs.

This estimate, though, does fit within my range estimate from 3.76 to 9.25 bits. And as I expected, it’s closer to nine bits than to four bits. The entropy seems to be a bit less than (American) football scores — somewhere around 8.7 bits — and college basketball — probably somewhere around 10.8 bits — which is probably fair. There are a lot of numbers that make for plausible college basketball scores. There are slightly fewer pairs of numbers that make for plausible football scores. There are fewer still pairs of scores that make for plausible baseball scores. So there’s less information conveyed in knowing that the game’s score is.

How Interesting Is A Baseball Score? Some Partial Results

Meanwhile I have the slight ongoing quest to work out the information-theory content of sports scores. For college basketball scores I made up some plausible-looking score distributions and used that. For professional (American) football I found a record of all the score outcomes that’ve happened, and how often. I could use experimental results. And I’ve wanted to do other sports. Soccer was asked for. I haven’t been able to find the scoring data I need for that. Baseball, maybe the supreme example of sports as a way to generate statistics … has been frustrating.

The raw data is available. has logs of pretty much every baseball game, going back to the forming of major leagues in the 1870s. What they don’t have, as best I can figure, is a list of all the times each possible baseball score has turned up. That I could probably work out, when I feel up to writing the scripts necessary, but “work”? Ugh.

Some people have done the work, although they haven’t shared all the results. I don’t blame them; the full results make for a boring sort of page. “The Most Popular Scores In Baseball History”, at, reports the top ten most common scores from 1871 through 2010. The essay also mentions that as of then there were 611 unique final scores. And that lets me give some partial results, if we trust that blogger post from people I never heard of before are accurate and true. I will make that assumption over and over here.

There’s, in principle, no limit to how many scores are possible. Baseball contains many implied infinities, and it’s not impossible that a game could end, say, 580 to 578. But it seems likely that after 139 seasons of play there can’t be all that many more scores practically achievable.

Suppose then there are 611 possible baseball score outcomes, and that each of them is equally likely. Then the information-theory content of a score’s outcome is negative one times the logarithm, base two, of 1/611. That’s a number a little bit over nine and a quarter. You could deduce the score for a given game by asking usually nine, sometimes ten, yes-or-no questions from a source that knew the outcome. That’s a little higher than the 8.7 I worked out for football. And it’s a bit less than the 10.8 I estimate for college basketball.

And there’s obvious rubbish there. In no way are all 611 possible outcomes equally likely. “The Most Popular Scores In Baseball History” says that right there in the essay title. The most common outcome was a score of 3-2, with 4-3 barely less popular. Meanwhile it seems only once, on the 28th of June, 1871, has a baseball game ended with a score of 49-33. Some scores are so rare we can ignore them as possibilities.

(You may wonder how incompetent baseball players of the 1870s were that a game could get to 49-33. Not so bad as you imagine. But the equipment and conditions they were playing with were unspeakably bad by modern standards. Notably, the playing field couldn’t be counted on to be flat and level and well-mowed. There would be unexpected divots or irregularities. This makes even simple ground balls hard to field. The baseball, instead of being replaced with every batter, would stay in the game. It would get beaten until it was a little smashed shell of unpredictable dynamics and barely any structural integrity. People were playing without gloves. If a game ran long enough, they would play at dusk, without lights, with a muddy ball on a dusty field. And sometimes you just have four innings that get out of control.)

What’s needed is a guide to what are the common scores and what are the rare scores. And I haven’t found that, nor worked up the energy to make the list myself. But I found some promising partial results. In a September 2008 post on, user weskelton listed the 24 most common scores and their frequency. This was for games from 1993 to 2008. One might gripe that the list only covers fifteen years. True enough, but if the years are representative that’s fine. And the top scores for the fifteen-year survey look to be pretty much the same as the 139-year tally. The 24 most common scores add up to just over sixty percent of all baseball games, which leaves a lot of scores unaccounted for. I am amazed that about three in five games will have a score that’s one of these 24 choices though.

But that’s something. We can calculate the information content for the 25 outcomes, one each of the 24 particular scores and one for “other”. This will under-estimate the information content. That’s because “other” is any of 587 possible outcomes that we’re not distinguishing. But if we have a lower bound and an upper bound, then we’ve learned something about what the number we want can actually be. The upper bound is that 9.25, above.

The information content, the entropy, we calculate from the probability of each outcome. We don’t know what that is. Not really. But we can suppose that the frequency of each outcome is close to its probability. If there’ve been a lot of games played, then the frequency of a score and the probability of a score should be close. At least they’ll be close if games are independent, if the score of one game doesn’t affect another’s. I think that’s close to true. (Some games at the end of pennant races might affect each other: why try so hard to score if you’re already out for the year? But there’s few of them.)

The entropy then we find by calculating, for each outcome, a product. It’s minus one times the probability of that outcome times the base-two logarithm of the probability of that outcome. Then add up all those products. There’s good reasons for doing it this way and in the college-basketball link above I give some rough explanations of what the reasons are. Or you can just trust that I’m not lying or getting things wrong on purpose.

So let’s suppose I have calculated this right, using the 24 distinct outcomes and the one “other” outcome. That makes out the information content of a baseball score’s outcome to be a little over 3.76 bits.

As said, that’s a low estimate. Lumping about two-fifths of all games into the single category “other” drags the entropy down.

But that gives me a range, at least. A baseball game’s score seems to be somewhere between about 3.76 and 9.25 bits of information. I expect that it’s closer to nine bits than it is to four bits, but will have to do a little more work to make the case for it.

How Interesting Is A Football Score?

Last month, Sarcastic Goat asked me how interesting a soccer game was. This is “interesting” in the information theory sense. I describe what that is in a series of posts, linked to from above. That had been inspired by the NCAA “March Madness” basketball tournament. I’d been wondering about the information-theory content of knowing the outcome of the tournament, and of each game.

This measure, called the entropy, we can work out from knowing how likely all the possible outcomes of something — anything — are. If there’s two possible outcomes and they’re equally likely, the entropy is 1. If there’s two possible outcomes and one is a sure thing while the other can’t happen, the entropy is 0. If there’s four possible outcomes and they’re all equally likely, the entropy is 2. If there’s eight possible outcomes, all equally likely, the entropy is 3. If there’s eight possible outcomes and some are likely while some are long shots, the entropy is … smaller than 3, but bigger than 0. The entropy grows with the number of possible outcomes and shrinks with the number of unlikely outcomes.

But it’s easy to calculate. List all the possible outcomes. Find the probability of each of those possible outcomes happening. Then, calculate minus one times the probability of each outcome times the logarithm, base two, of that outcome. For each outcome, so yes, this might take a while. Then add up all those products.

I’d estimated the outcome of the 63-game basketball tournament was somewhere around 48 bits of information. There’s a fair number of foregone, or almost foregone, conclusions in the game, after all. And I guessed, based on a toy model of what kinds of scores often turn up in college basketball games, that the game’s score had an information content of a little under 11 bits of information.

Sarcastic Goat, as I say, asked about soccer scores. I don’t feel confident that I could make up a plausible model of soccer score distributions. So I went looking for historical data. Surely, a history of actual professional soccer scores over a couple decades would show all the possible, plausible, outcomes and how likely each was to turn out.

I didn’t find one. My search for soccer scores kept getting contaminated with (American) football scores. But that turned up something interesting anyway. Sports Reference LLC has a table which purports to list the results of all professional football games played from 1920 through the start of 2016. There’ve been, apparently, some 1,026 different score outcomes, from 0-0 through to 73-0.

As you’d figure, there are a lot of freakish scores; only once in professional football history has the game ended 62-28. (Although it’s ended 62-14 twice, somehow.) There hasn’t been a 2-0 game since the second week of the 1938 season. Some scores turn up a lot; 248 games (as of this writing) have ended 20-17. That’s the most common score, in its records. 27-24 and 17-14 are the next most common scores. If I’m not making a dumb mistake, 7-0 is the 21st most common score. 93 games have ended with that tally. But it hasn’t actually been a game’s final score since the 14th week of the 1983 season, somehow. 98 games have ended 21-17; only ten have ended 21-18. Weird.

Anyway, there’s 1,026 recorded outcomes. That’s surely as close to “all the possible outcomes” as we can expect to get, at least until the Jets manage to lose 74-0 in their home opener. But if all 1,026 outcomes were equally likely then the information content of the game’s score would be a touch over 10 bits. But these outcomes aren’t all equally likely. It’s vastly more likely that a game ended 16-13 than it is likely it ended 16-8.

Let’s suppose I didn’t make any stupid mistakes in working out the frequency of all the possible outcomes. Then the information content of a football game’s outcome is a little over 8.72 bits.

Don’t be too hypnotized by the digits past the decimal. It’s approximate. But it suggests that if you were asking a source that would only answer ‘yes’ or ‘no’, then you could expect to get the score for any particular football game with about nine well-chosen questions.

I’m not surprised this is less than my estimated information content of a basketball game’s score. I think basketball games see a wider range of likely scores than football games do.

If someone has a reference for the outcomes of soccer games — or other sports — over a reasonably long time please let me know. I can run the same sort of calculation. We might even manage the completely pointless business of ranking all major sports by the information content of their scores.

What Do I Need To Pass This Class? (December 2014 Edition)

I don’t mean to repeat myself too much, but it is finals season for United States colleges on a semesterly schedule so, here. Good luck, people who’re minutes away from their final exams.


It’s finals season, at least for colleges that run on a semesterly schedule, and a couple of my posts are turning up in search query results again. So I thought it worth drawing a little more attention to them and hopefully getting people what they need sooner.

The answer: you need to study a steady but not excessive bit every night from now to before the exam; you need to get a full night of sleep before the exam; and you really needed to pay attention in class and do the fiddly little assignments all semester, so, sorry it’s too late for that. Also you need to not pointlessly antagonize your professor; even if you don’t like this class, you could have taken others to meet your academic requirement, so don’t act like you were dragged into Topics in Civilization: Death against your will even if it does satisfy three

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What Do I Need If The Final Is Worth 40 Percent?

I suspected that my little pair of articles about what scores you need on the final to pass a class (or get an A, or such) would prove useful, which is almost as good as being informative. I noticed in the search queries bringing people to my pages a question about what a person needed for a course where the final was 40 percent of the class score. I hadn’t put that in my original set of tables, and if the searcher followed my first article — about how to work out what you need for any weighting of the final exam — then she or he got what was needed. I just didn’t think of finals being quite that heavily weighted. But, what the heck, if people want to see the tables worth 40 percent, it’s easy enough to generate them, and I added that to the collection of scores-needed tables. Good luck next term.