The mathematically-themed comic strips of the past week tended to touch on some classic topics and classic motifs. That’s enough for me to declare a title for these comics. Enjoy, won’t you please?
John McPherson’s Close To Home for the 9th uses the classic board full of mathematics to express deep thinking. And it’s deep thinking about sports. Nerds like to dismiss sports as trivial and so we get the punch line out of this. But models of sports have been one of the biggest growth fields in mathematics the past two decades. And they’ve shattered many longstanding traditional understandings of strategy. It’s not proper mathematics on the board, but that’s all right. It’s not proper sabermetrics either.
Vic Lee’s Pardon My Planet for the 10th is your classic joke about putting mathematics in marketable terms. There is an idea that a mathematical idea to be really good must be beautiful. And it’s hard to say exactly what beauty is, but “short” and “simple” seem to be parts of it. That’s a fine idea, as long as you don’t forget how context-laden these are. Whether an idea is short depends on what ideas and what concepts you have as background. Whether it’s simple depends on how much you’ve seen similar ideas before. π looks simple. “The smallest positive root of the solution to the differential equation y”(x) = -y(x) where y(0) = 0 and y'(0) = 1” looks hard, but however much mathematics you know, rhetoric alone tells you those are the same thing.
Scott Hilburn’s The Argyle Sweater for the 10th is your classic anthropomorphic-numerals joke. Well, anthropomorphic-symbols in this case. But it’s the same genre of joke.
Randy Glasbergen’s Glasbergen Cartoons rerun for the 10th is your classic sudoku-and-arithmetic-as-hard-work joke. And it’s neat to see “programming a VCR” used as an example of the difficult-to-impossible task for a comic strip drawn late enough that it’s into the era of flat-screen, flat-bodied desktop computers.
Bill Holbrook’s On The Fastrack for 11th is your classic grumbling-about-how-mathematics-is-understood joke. Well, statistics, but most people consider that part of mathematics. (One could mount a strong argument that statistics is as independent of mathematics as physics or chemistry are.) Statistics offers many chances for intellectual mischief, whether deliberately or just from not thinking matters through. That may be inevitable. Sampling, as in political surveys, must talk about distributions, about ranges of possible results. It’s hard to be flawless about that.
That said I’m not sure I can agree with Fi in her example here. Take her example, a political poll with three-point margin of error. If the poll says one candidate’s ahead by three points, Fi asserts, they’ll say it’s tied when it’s as likely the lead is six. I don’t see that’s quite true, though. When we sample something we estimate the value of something in a population based on what it is in the sample. Obviously we’ll be very lucky if the population and the sample have exactly the same value. But the margin of error gives us a range of how far from the sample value it’s plausible the whole population’s value is, or would be if we could measure it. Usually “plausible” means 95 percent; that is, 95 percent of the time the actual value will be within the margin of error of the sample’s value.
So here’s where I disagree with Fi. Let’s suppose that the first candidate, Kirk, polls at 43 percent. The second candidate, Picard, polls at 40 percent. (Undecided or third-party candidates make up the rest.) I agree that Kirk’s true, whole-population, support is equally likely to be 40 percent or 46 percent. But Picard’s true, whole-population, support is equally likely to be 37 percent or 43 percent. Kirk’s lead is actually six points if his support was under-represented in the sample and Picard’s was over-represented, by the same measures. But suppose Kirk was just as over-represented and Picard just as under-represented as they were in the previous case. This puts Kirk at 40 percent and Picard at 43 percent, a Kirk-lead of minus three percentage points.
So what’s the actual chance these two candidates are tied? Well, you have to say what a tie is. It’s vanishingly impossible they have precisely the same true support and we can’t really calculate that. Don’t blame statisticians. You tell me an election in which one candidate gets three more votes than the other isn’t really tied, if there are more than seven votes cast. We can work on “what’s the chance their support is less than some margin?” And then you’d have all the possible chances where Kirk gets a lower-than-surveyed return while Picard gets a higher-than-surveyed return. I can’t say what that is offhand. We haven’t said what this margin-of-tying is, for one thing.
But it is certainly higher than the chance the lead is actually six; that only happens if the actual vote is different from the poll in one particular way. A tie can happen if the actual vote is different from the poll in many different ways.
Doing a quick and dirty little numerical simulation suggests to me that, if we assume the sampling respects the standard normal distribution, then in this situation Kirk probably is ahead of Picard. Given a three-point lead and a three-point margin for error Kirk would be expected to beat Picard about 92 percent of the time, while Picard would win about 8 percent of the time.
Here I have been making the assumption that Kirk’s and Picard’s support are to an extent independent. That is, a vote might be for Kirk or for Picard or for neither. There’s this bank of voting-for-neither-candidate that either could draw on. If there are no undecided candidates, every voter is either Kirk or Picard, then all of this breaks down: Kirk can be up by six only if Picard is down by six. But I don’t know of surveys that work like that.
Not to keep attacking this particular strip, which doesn’t deserve harsh treatment, but it gives me so much to think about. Assuming by “they” Fi means news anchors — and from what we get on panel, it’s not actually clear she does — I’m not sure they actually do “say the poll is tied”. What I more often remember hearing is that the difference is equal to, or less than, the survey’s margin of error. That might get abbreviated to “a statistical tie”, a usage that I think is fair. But Fi might mean the candidates or their representatives in saying “they”. I can’t fault the campaigns for interpreting data in ways useful for their purposes. The underdog needs to argue that the election can yet be won. The leading candidate needs to argue against complacency. In either case a tie is a viable selling point and a reasonable interpretation of the data.
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 12th is a classic use of Einstein and general relativity to explain human behavior. Everyone’s tempted by this. Usually it’s thermodynamics that inspires thoughts that society could be explained mathematically. There’s good reason for this. Thermodynamics builds great and powerful models of complicated systems by supposing that we never know, or need to know, what any specific particle of gas or fluid is doing. We care only about aggregate data. That statistics shows we can understand much about humanity without knowing fine details reinforces this idea. The Wingartens and Clark probably shifted from thermodynamics to general relativity because Einstein is recognizable to normal people. And we’ve all at least heard of mass warping space and can follow the metaphor to money warping law.
In vintage comics, Dan Barry’s Flash Gordon for the 14th (originally run the 28th of November, 1961) uses the classic idea that sufficient mathematics talent will outwit games of chance. Many believe it. I remember my grandmother’s disappointment that she couldn’t bring the underaged me into the casinos in Atlantic City. This did save her the disappointment of learning I haven’t got any gambling skill besides occasionally buying two lottery tickets if the jackpot is high enough. I admit that an irrational move on my part, but I can spare two dollars for foolishness once or twice a year. The idea of beating a roulette wheel, at least a fair wheel, isn’t absurd. In principle if you knew enough about how the wheel was set up and how the ball was weighted and how it was launched into the spin you could predict where it would land. In practice, good luck. I wouldn’t be surprised if a good roulette wheel weren’t chaotic, or close to it. If it’s chaotic then while the outcome could be predicted if the wheel’s spin and the ball’s initial speed were known well enough, they can’t be measured well enough for a prediction to be meaningful. The comic also uses the classic word balloon full of mathematical symbols to suggest deep reasoning. I spotted Einstein’s famous quote there.
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