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. Retrosheet.org 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 ValueOverReplacementGrit.com, 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 Baseball-Fever.com, 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.

Unrelated, but it reminded me of a literature class in High School. The teacher gave multiple-choice quizzes every Friday, and I spotted patterns. By mid-semester I’d compiled a list of likely correct answers for each of the questions (i.e, 1. D; 2. B; 3. A, etc.). The pattern was consistent enough that I sold crib sheets that guaranteed a C for those who hadn’t studied. No one ever asked for a refund, and I never read Ethan Fromme.

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I can believe this. It reminds me of the time in Peanuts when Linus figured he could pass a true-or-false test without knowing anything. The thing students don’t realize about multiple choice questions is they are

hardto write. The instructor has to come up with a reasonable question, and not just the answer but several plausible alternatives, andthenhas to scramble where in the choices the answer comes up.I remember at least once I gave out a five-question multiple choice section where all the answers were ‘B’, but my dim recollection is that I did that on purpose after I noticed I’d made ‘B’ the right answer the first three times. I think I was wondering if students would chicken out of the idea that all five questions had the same answer. But then I failed to check what the results were and if students really did turn away from the right answer just because it was too neat a pattern.

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Sometime professional MCSA test author here. Writing those things can be a bear, especially getting the distractors right.

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My story dates back to the days of mimeograph prints. I never considered the difficulty in generating the tests. In retrospect, we had a very good math department, and some of the teachers would do just what JN said – all answers were “B.” Spooked the hell out of me, and yeah, I punted to the next likely answers.

The bonus questions were always bizarre. You could miss all the questions, but if you got the bonus you got credit for the whole thing. We were still learning how to factor and cross-multiply when we got this:

Given: a = 1, b = 2, c = 3 etc.

[(x-a)(x-b)(x-c) … (x-z)] = ?

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Last one. Got a timed geometry quiz, 10 questions. At the top of the quiz were the directions to read through all of the problems before answering. Each of the problems 1 through 9 were impossible to complete in the time allotted, but Number 10 said, “Disregard problems 1 through 9, sign your name at the top of the page and turn it in.”

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You know, I have a vague memory of getting that sort of quiz myself, back around 1980 or so. It wasn’t in mathematics, although I’m not sure just which class it was. This was elementary school for me so all the classes kind of blended together.

I suspect there was something in the air at the time, since I remember hearing stories about impossible-quizzes like that with a disregard-all-above-problems notes. And I can’t be sure I haven’t conflated a memory of taking one with the stories of disregard-all-above-problems tests being given.

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I only barely make it back to the days of mimeograph machines, as a student, although it’s close.

That bonus question sounds maddening, although its existence makes me suspect there’s a trick I’ll have to poke it with to see.

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I had interviewed once to write mathematics questions for a standardized test corporation. I didn’t get it, though, and I suspect my weakness in coming up with good distractors was the big problem. I suspect I’d do better now.

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