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  • Joseph Nebus 6:00 pm on Thursday, 16 February, 2017 Permalink | Reply
    Tags: , , Crock, , , Mr Lowe, , , statistics,   

    Reading the Comics, February 11, 2017: Trivia Edition 


    And now to wrap up last week’s mathematically-themed comic strips. It’s not a set that let me get into any really deep topics however hard I tried overthinking it. Maybe something will turn up for Sunday.

    Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 7th tries setting arithmetic versus celebrity trivia. It’s for the old joke about what everyone should know versus what everyone does know. One might question whether Kardashian pet eating habits are actually things everyone knows. But the joke needs some hyperbole in it to have any vitality and that’s the only available spot for it. It’s easy also to rate stuff like arithmetic as trivia since, you know, calculators. But it is worth knowing that seven squared is pretty close to 50. It comes up when you do a lot of estimates of calculations in your head. The square root of 10 is pretty near 3. The square root of 50 is near 7. The cube root of 10 is a little more than 2. The cube root of 50 a little more than three and a half. The cube root of 100 is a little more than four and a half. When you see ways to rewrite a calculation in estimates like this, suddenly, a lot of amazing tricks become possible.

    Leigh Rubin’s Rubes for the 7th is a “mathematics in the real world” joke. It could be done with any mythological animals, although I suppose unicorns have the advantage of being relatively easy to draw recognizably. Mermaids would do well too. Dragons would also read well, but they’re more complicated to draw.

    Mark Pett’s Mr Lowe rerun for the 8th has the kid resisting the mathematics book. Quentin’s grounds are that how can he know a dated book is still relevant. There’s truth to Quentin’s excuse. A mathematical truth may be universal. Whether we find it interesting is a matter of culture and even fashion. There are many ways to present any fact, and the question of why we want to know this fact has as many potential answers as it has people pondering the question.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th is a paean to one of the joys of numbers. There is something wonderful in counting, in measuring, in tracking. I suspect it’s nearly universal. We see it reflected in people passing around, say, the number of rivets used in the Chrysler Building or how long a person’s nervous system would reach if stretched out into a line or ever-more-fanciful measures of stuff. Is it properly mathematics? It’s delightful, isn’t that enough?

    Scott Hilburn’s The Argyle Sweater for the 10th is a Fibonacci Sequence joke. That’s a good one for taping to the walls of a mathematics teacher’s office.

    'Did you ever take a date to a drive-in movie in high school?' 'Once, but she went to the concession stand and never came back.' 'Did you wonder why?' 'Yeah, but I kept on doing my math homework.'

    Bill Rechin’s Crock rerun for the 11th of February, 2017. They actually opened a brand-new drive-in theater something like forty minutes away from us a couple years back. We haven’t had the chance to get there. But we did get to one a fair bit farther away where yes, we saw Turbo, that movie about the snail that races in the Indianapolis 500. The movie was everything we hoped for and it’s just a shame Roger Ebert died too young to review it for us.

    Bill Rechin’s Crock rerun for the 11th is a name-drop of mathematics. Really anybody’s homework would be sufficiently boring for the joke. But I suppose mathematics adds the connotation that whatever you’re working on hasn’t got a human story behind it, the way English or History might, and that it hasn’t got the potential to eat, explode, or knock a steel ball into you the way Biology, Chemistry, or Physics have. Fair enough.

     
  • Joseph Nebus 6:00 pm on Friday, 18 November, 2016 Permalink | Reply
    Tags: , , , , , , , statistics, unit vectors,   

    The End 2016 Mathematics A To Z: Hat 


    I was hoping to pick a term that was a quick and easy one to dash off. I learned better.

    Hat.

    This is a simple one. It’s about notation. Notation is never simple. But it’s important. Good symbols organize our thoughts. They tell us what are the common ordinary bits of our problem, and what are the unique bits we need to pay attention to here. We like them to be easy to write. Easy to type is nice, too, but in my experience mathematicians work by hand first. Typing is tidying-up, and we accept that being sluggish. Unique would be nice, so that anyone knows what kind of work we’re doing just by looking at the symbols. I don’t think anything manages that. But at least some notation has alternate uses rare enough we don’t have to worry about it.

    “Hat” has two major uses I know of. And we call it “hat”, although our friends in the languages department would point out this is a caret. The little pointy corner that goes above a letter, like so: \hat{i} . \hat{x} . \hat{e} . It’s not something we see on its own. It’s always above some variable.

    The first use of the hat like this comes up in statistics. It’s a way of marking that something is an estimate. By “estimate” here we mean what anyone might mean by “estimate”. Statistics is full of uses for this sort of thing. For example, we often want to know what the arithmetic mean of some quantity is. The average height of people. The average temperature for the 18th of November. The average weight of a loaf of bread. We have some letter that we use to mean “the value this has for any one example”. By some letter we mean ‘x’, maybe sometimes ‘y’. We can use any and maybe the problem begs for something. But it’s ‘x’, maybe sometimes ‘y’.

    For the arithmetic mean of ‘x’ for the whole population we write the letter with a horizontal bar over it. (The arithmetic mean is the thing everybody in the world except mathematicians calls the average. Also, it’s what mathematicians mean when they say the average. We just get fussy because we know if we don’t say “arithmetic mean” someone will come along and point out there are other averages.) That arithmetic mean is \bar{x} . Maybe \bar{y} if we must. Must be some number. But what is it? If we can’t measure whatever it is for every single example of our group — the whole population — then we have to make an estimate. We do that by taking a sample, ideally one that isn’t biased in some way. (This is so hard to do, or at least be sure you’ve done.) We can find the mean for this sample, though, because that’s how we picked it. The mean of this sample is probably close to the mean of the whole population. It’s an estimate. So we can write \hat{x} and understand. This is not \bar{x} but it does give us a good idea what \hat{x} should be.

    (We don’t always use the caret ^ for this. Sometimes we use a tilde ~ instead. ~ has the advantage that it’s often used for “approximately equal to”. So it will carry that suggestion over to its new context.)

    The other major use of the hat comes in vectors. Mathematics types do a lot of work with vectors. It turns out a lot of mathematical structures work the way that pointing and moving in directions in ordinary space do. That’s why back when I talked about what vectors were I didn’t say “they’re like arrows pointing some length in some direction”. Arrows pointing some length in some direction are vectors, yes, but there are many more things that are vectors. Thinking of moving in particular directions gives us good intuition for how to work with vectors, and for stuff that turns out to be vectors. But they’re not everything.

    If we need to highlight that something is a vector we put a little arrow over its name. \vec{x} . \vec{e} . That sort of thing. (Or if we’re typing, we might put the letter in boldface: x. This was good back before computers let us put in mathematics without giving the typesetters hazard pay.) We don’t always do that. By the time we do a lot of stuff with vectors we don’t always need the reminder. But we will include it if we need a warning. Like if we want to have both \vec{r} telling us where something is and to use a plain old r to tell us how big the vector \vec{r} is. That turns up a lot in physics problems.

    Every vector has some length. Even vectors that don’t seem to have anything to do with distances do. We can make a perfectly good vector out of “polynomials defined for the domain of numbers between -2 and +2”. Those polynomials are vectors, and they have lengths.

    There’s a special class of vectors, ones that we really like in mathematics. They’re the “unit vectors”. Those are vectors with a length of 1. And we are always glad to see them. They’re usually good choices for a basis. Basis vectors are useful things. They give us, in a way, a representative slate of cases to solve. Then we can use that representative slate to give us whatever our specific problem’s solution is. So mathematicians learn to look instinctively to them. We want basis vectors, and we really like them to have a length of 1. Even if we aren’t putting the arrow over our variables we’ll put the caret over the unit vectors.

    There are some unit vectors we use all the time. One is just the directions in space. That’s \hat{e}_1 and \hat{e}_2 and for that matter \hat{e}_3 and I bet you have an idea what the next one in the set might be. You might be right. These are basis vectors for normal, Euclidean space, which is why they’re labelled “e”. We have as many of them as we have dimensions of space. We have as many dimensions of space as we need for whatever problem we’re working on. If we need a basis vector and aren’t sure which one, we summon one of the letters used as indices all the time. \hat{e}_i , say, or \hat{e}_j . If we have an n-dimensional space, then we have unit vectors all the way up to \hat{e}_n .

    We also use the hat a lot if we’re writing quaternions. You remember quaternions, vaguely. They’re complex-valued numbers for people who’re bored with complex-valued numbers and want some thrills again. We build them as a quartet of numbers, each added together. Three of them are multiplied by the mysterious numbers ‘i’, ‘j’, and ‘k’. Each ‘i’, ‘j’, or ‘k’ multiplied by itself is equal to -1. But ‘i’ doesn’t equal ‘j’. Nor does ‘j’ equal ‘k’. Nor does ‘k’ equal ‘i’. And ‘i’ times ‘j’ is ‘k’, while ‘j’ times ‘i’ is minus ‘k’. That sort of thing. Easy to look up. You don’t need to know all the rules just now.

    But we often end up writing a quaternion as a number like 4 + 2\hat{i} - 3\hat{j} + 1 \hat{k} . OK, that’s just the one number. But we will write numbers like a + b\hat{i} + c\hat{j} + d\hat{k} . Here a, b, c, and d are all real numbers. This is kind of sloppy; the pieces of a quaternion aren’t in fact vectors added together. But it is hard not to look at a quaternion and see something pointing in some direction, like the first vectors we ever learn about. And there are some problems in pointing-in-a-direction vectors that quaternions handle so well. (Mostly how to rotate one direction around another axis.) So a bit of vector notation seeps in where it isn’t appropriate.

    I suppose there’s some value in pointing out that the ‘i’ and ‘j’ and ‘k’ in a quaternion are fixed and set numbers. They’re unlike an ‘a’ or an ‘x’ we might see in the expression. I’m not sure anyone was thinking they were, though. Notation is a tricky thing. It’s as hard to get sensible and consistent and clear as it is to make words and grammar sensible. But the hat is a simple one. It’s good to have something like that to rely on.

     
    • howardat58 7:26 pm on Friday, 18 November, 2016 Permalink | Reply

      Kill the division sign !

      I am posting a problem for you. If you haven’t yet followed me you can do it now!

      Like

      • Joseph Nebus 4:02 am on Sunday, 20 November, 2016 Permalink | Reply

        Minimal Abstract Algebra? Thank you, I’m interested.

        I think — yes, I am following you, and relieved for that as you’ve long been one of my kind readers. I’ve been failing to get to visit other blogs recently and am sorry to be this discourteous. It’s just the demands of each day getting to me.

        Like

  • Joseph Nebus 6:00 pm on Tuesday, 26 July, 2016 Permalink | Reply
    Tags: Plinko, statistics,   

    Something To Read: Galton Boards 


    I do need to take another light week of writing I’m afraid. There’ll be the Theorem Thursday post and all that. But today I’d like to point over to Gaurish4Math’s WordPress Blog, and a discussion of the Galton Board. I’m not familiar with it by that name, but it is a very familiar concept. You see it as Plinko boards on The Price Is Right and as a Boardwalk or amusement-park game. Set an array of pins on a vertical board and drop a ball or a round chip or something that can spin around freely on it. Where will it fall?

    It’s random luck, it seems. At least it is incredibly hard to predict where, underneath all the pins, the ball will come to rest. Some of that is ignorance: we just don’t know the weight distribution of the ball, the exact way it’s dropped, the precise spacing of pins well enough to predict it all. We don’t care enough to do that. But some of it is real randomness. Ideally we make the ball bounce so many times that however well we estimated its drop, the tiny discrepancy between where the ball is and where we predict it is, and where it is going and where we predict it is going, will grow larger than the Plinko board and our prediction will be meaningless.

    (I am not sure that this literally happens. It is possible, though. It seems more likely the more rows of pins there are on the board. But I don’t know how tall a board really needs to be to be a chaotic system, deterministic but unpredictable.)

    But here is the wonder. We cannot predict what any ball will do. But we can predict something about what every ball will do, if we have enormously many of them. Gaurish writes some about the logic of why that is, and the theorems in probability that tell us why that should be so.

     
    • gaurish 6:11 pm on Tuesday, 26 July, 2016 Permalink | Reply

      Thanks for pointing to my blog post. I would like to quote Tim Gowers (A very short introduction to Mathematics, pp. 6) regarding the classical die throwing experiment (“the model”) of probability theory:
      “One might object to this model on the grounds that the dice, when rolled, are obeying Newton’s laws, at least to a very high degree of precision, so the way they land is anything but random…”

      Like

      • Joseph Nebus 6:02 am on Wednesday, 27 July, 2016 Permalink | Reply

        Quite welcome. I’m happy to pass along interesting writing.

        Granted that falling dice, or balls in a Plinko board like this, are moving deterministically. I do wonder if we get to chaotic behavior, in which the toss is nevertheless random. I’m not well-versed enough in the mechanics of this sort of problem to be really sure about my answer. For the balls falling off pins I would imagine that something like twenty rebounds, on either pin or other balls, would be enough to effectively randomize the result.

        (If each rebound doubles the discrepancy between the direction of the ball’s actual velocity and our representation of its direction, then after twenty rebounds the error is about a million times what it started as, and it seems hard to know the direction of a ball’s travel to within a millionth of two-pi radians. But that’s a very rough argument, supposing that randomizing the direction of travel is all we need to have a random ball drop. And maybe two-pi-over-a-million radians is a reasonable precision; maybe we need thirty rebounds, or forty, to be quite sure.)

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Monday, 18 July, 2016 Permalink | Reply
    Tags: , , birds, , statistics,   

    Reading the Comics, July 13, 2016: Catching Up On Vacation Week Edition 


    I confess I spent the last week on vacation, away from home and without the time to write about the comics. And it was another of those curiously busy weeks that happens when it’s inconvenient. I’ll try to get caught up ahead of the weekend. No promises.

    Art and Chip Samson’s The Born Loser for the 10th talks about the statistics of body measurements. Measuring bodies is one of the foundations of modern statistics. Adolphe Quetelet, in the mid-19th century, found a rough relationship between body mass and the square of a person’s height, used today as the base for the body mass index.Francis Galton spent much of the late 19th century developing the tools of statistics and how they might be used to understand human populations with work I will describe as “problematic” because I don’t have the time to get into how much trouble the right mind at the right idea can be.

    No attempt to measure people’s health with a few simple measurements and derived quantities can be fully successful. Health is too complicated a thing for one or two or even ten quantities to describe. Measures like height-to-waist ratios and body mass indices and the like should be understood as filters, the way temperature and blood pressure are. If one or more of these measurements are in dangerous ranges there’s reason to think there’s a health problem worth investigating here. It doesn’t mean there is; it means there’s reason to think it’s worth spending resources on tests that are more expensive in time and money and energy. And similarly just because all the simple numbers are fine doesn’t mean someone is perfectly healthy. But it suggests that the person is more likely all right than not. They’re guides to setting priorities, easy to understand and requiring no training to use. They’re not a replacement for thought; no guides are.

    Jeff Harris’s Shortcuts educational panel for the 10th is about zero. It’s got a mix of facts and trivia and puzzles with a few jokes on the side.

    I don’t have a strong reason to discuss Ashleigh Brilliant’s Pot-Shots rerun for the 11th. It only mentions odds in a way that doesn’t open up to discussing probability. But I do like Brilliant’s “Embrace-the-Doom” tone and I want to share that when I can.

    John Hambrock’s The Brilliant Mind of Edison Lee for the 13th of July riffs on the world’s leading exporter of statistics, baseball. Organized baseball has always been a statistics-keeping game. The Olympic Ball Club of Philadelphia’s 1837 rules set out what statistics to keep. I’m not sure why the game is so statistics-friendly. It must be in part that the game lends itself to representation as a series of identical events — pitcher throws ball at batter, while runners wait on up to three bases — with so many different outcomes.

    'Edison, let's discuss stats while we wait for the opening pitch.' 'Statistics? I have plenty of those. A hot dog has 400 calories and costs five dollars. A 12-ounce root beer has 38 grams of sugar.' 'I mean *player* stats.' 'Oh'. (To his grandfather instead) 'Did you know the average wait time to buy nachos is eight minutes and six seconds?'

    John Hambrock’s The Brilliant Mind of Edison Lee for the 13th of July, 2016. Properly speaking, the waiting time to buy nachos isn’t a player statistic, but I guess Edison Lee did choose to stop talking to his father for it. Which is strange considering his father’s totally natural and human-like word emission ‘Edison, let’s discuss stats while we wait for the opening pitch’.

    Alan Schwarz’s book The Numbers Game: Baseball’s Lifelong Fascination With Statistics describes much of the sport’s statistics and record-keeping history. The things recorded have varied over time, with the list of things mostly growing. The number of statistics kept have also tended to grow. Sometimes they get dropped. Runs Batted In were first calculated in 1880, then dropped as an inherently unfair statistic to keep; leadoff hitters were necessarily cheated of chances to get someone else home. How people’s idea of what is worth measuring changes is interesting. It speaks to how we change the ways we look at the same event.

    Dana Summers’s Bound And Gagged for the 13th uses the old joke about computers being abacuses and the like. I suppose it’s properly true that anything you could do on a real computer could be done on the abacus, just, with a lot ore time and manual labor involved. At some point it’s not worth it, though.

    Nate Fakes’s Break of Day for the 13th uses the whiteboard full of mathematics to denote intelligence. Cute birds, though. But any animal in eyeglasses looks good. Lab coats are almost as good as eyeglasses.

    LERBE ( O O - O - ), GIRDI ( O O O - - ), TACNAV ( O - O - O - ), ULDNOA ( O O O - O - ). When it came to measuring the Earth's circumference, there was a ( - - - - - - - - ) ( - - - - - ).

    David L Hoyt and Jeff Knurek’s Jumble for the 13th of July, 2016. The link will be gone sometime after mid-August I figure. I hadn’t thought of a student being baffled by using the same formula for an orange and a planet’s circumference because of their enormous difference in size. It feels authentic, though.

    David L Hoyt and Jeff Knurek’s Jumble for the 13th is about one of geometry’s great applications, measuring how large the Earth is. It’s something that can be worked out through ingenuity and a bit of luck. Once you have that, some clever argument lets you work out the distance to the Moon, and its size. And that will let you work out the distance to the Sun, and its size. The Ancient Greeks had worked out all of this reasoning. But they had to make observations with the unaided eye, without good timekeeping — time and position are conjoined ideas — and without photographs or other instantly-made permanent records. So their numbers are, to our eyes, lousy. No matter. The reasoning is brilliant and deserves respect.

     
  • Joseph Nebus 3:00 pm on Wednesday, 27 April, 2016 Permalink | Reply
    Tags: , bell curves, , , normal distributions, Six Sigma, statistics, z-score   

    A Leap Day 2016 Mathematics A To Z: Z-score 


    And we come to the last of the Leap Day 2016 Mathematics A To Z series! Z is a richer letter than x or y, but it’s still not so rich as you might expect. This is why I’m using a term that everybody figured I’d use the last time around, when I went with z-transforms instead.

    Z-Score

    You get an exam back. You get an 83. Did you do well?

    Hard to say. It depends on so much. If you expected to barely pass and maybe get as high as a 70, then you’ve done well. If you took the Preliminary SAT, with a composite score that ranges from 60 to 240, an 83 is catastrophic. If the instructor gave an easy test, you maybe scored right in the middle of the pack. If the instructor sees tests as a way to weed out the undeserving, you maybe had the best score in the class. It’s impossible to say whether you did well without context.

    The z-score is a way to provide that context. It draws that context by comparing a single score to all the other values. And underlying that comparison is the assumption that whatever it is we’re measuring fits a pattern. Usually it does. The pattern we suppose stuff we measure will fit is the Normal Distribution. Sometimes it’s called the Standard Distribution. Sometimes it’s called the Standard Normal Distribution, so that you know we mean business. Sometimes it’s called the Gaussian Distribution. I wouldn’t rule out someone writing the Gaussian Normal Distribution. It’s also called the bell curve distribution. As the names suggest by throwing around “normal” and “standard” so much, it shows up everywhere.

    A normal distribution means that whatever it is we’re measuring follows some rules. One is that there’s a well-defined arithmetic mean of all the possible results. And that arithmetic mean is the most common value to turn up. That’s called the mode. Also, this arithmetic mean, and mode, is also the median value. There’s as many data points less than it as there are greater than it. Most of the data values are pretty close to the mean/mode/median value. There’s some more as you get farther from this mean. But the number of data values far away from it are pretty tiny. You can, in principle, get a value that’s way far away from the mean, but it’s unlikely.

    We call this standard because it might as well be. Measure anything that varies at all. Draw a chart with the horizontal axis all the values you could measure. The vertical axis is how many times each of those values comes up. It’ll be a standard distribution uncannily often. The standard distribution appears when the thing we measure satisfies some quite common conditions. Almost everything satisfies them, or nearly satisfies them. So we see bell curves so often when we plot how frequently data points come up. It’s easy to forget that not everything is a bell curve.

    The normal distribution has a mean, and median, and mode, of 0. It’s tidy that way. And it has a standard deviation of exactly 1. The standard deviation is a way of measuring how spread out the bell curve is. About 95 percent of all observed results are less than two standard deviations away from the mean. About 99 percent of all observed results are less than three standard deviations away. 99.9997 percent of all observed results are less than six standard deviations away. That last might sound familiar to those who’ve worked in manufacturing. At least it des once you know that the Greek letter sigma is the common shorthand for a standard deviation. “Six Sigma” is a quality-control approach. It’s meant to make sure one understands all the factors that influence a product and controls them. This is so the product falls outside the design specifications only 0.0003 percent of the time.

    This is the normal distribution. It has a standard deviation of 1 and a mean of 0, by definition. And then people using statistics go and muddle the definition. It is always so, with the stuff people actually use. Forgive them. It doesn’t really change the shape of the curve if we scale it, so that the standard deviation is, say, two, or ten, or π, or any positive number. It just changes where the tick marks are on the x-axis of our plot. And it doesn’t really change the shape of the curve if we translate it, adding (or subtracting) some number to it. That makes the mean, oh, 80. Or -15. Or eπ. Or some other number. That just changes what value we write underneath the tick marks on the plot’s x-axis. We can find a scaling and translation of the normal distribution that fits whatever data we’re observing.

    When we find the z-score for a particular data point we’re undoing this translation and scaling. We figure out what number on the standard distribution maps onto the original data set’s value. About two-thirds of all data points are going to have z-scores between -1 and 1. About nineteen out of twenty will have z-scores between -2 and 2. About 99 out of 100 will have z-scores between -3 and 3. If we don’t see this, and we have a lot of data points, then that’s suggests our data isn’t normally distributed.

    I don’t know why the letter ‘z’ is used for this instead of, say, ‘y’ or ‘w’ or something else. ‘x’ is out, I imagine, because we use that for the original data. And ‘y’ is a natural pick for a second measured variable. z’, I expect, is just far enough from ‘x’ it isn’t needed for some more urgent duty, while being close enough to ‘x’ to suggest it’s some measured thing.

    The z-score gives us a way to compare how interesting or unusual scores are. If the exam on which we got an 83 has a mean of, say, 74, and a standard deviation of 5, then we can say this 83 is a pretty solid score. If it has a mean of 78 and a standard deviation of 10, then the score is better-than-average but not exceptional. If the exam has a mean of 70 and a standard deviation of 4, then the score is fantastic. We get to meaningfully compare scores from the measurements of different things. And so it’s one of the tools with which statisticians build their work.

     
  • Joseph Nebus 3:00 pm on Tuesday, 15 March, 2016 Permalink | Reply
    Tags: , , , , , statistics, , , ,   

    Reading the Comics, March 14, 2016: Pi Day Comics Event 


    Comic Strip Master Command had the regular pace of mathematically-themed comic strips the last few days. But it remembered what the 14th would be. You’ll see that when we get there.

    Ray Billingsley’s Curtis for the 11th of March is a student-resists-the-word-problem joke. But it’s a more interesting word problem than usual. It’s your classic problem of two trains meeting, but rather than ask when they’ll meet it asks where. It’s just an extra little step once the time of meeting is made, but that’s all right by me. Anything to freshen the scenario up.

    'Please answer this math question, Mr Wilkins. John is traveling east from San Francisco on a train at a speed of 80 miles per hour. Tom is going to that same meeting from New York, headed west, on a train traveling 100 miles per hour. In what state will they meet?' 'Couldn't they just Skype?'

    Ray Billingsley’s Curtis for the 11th of March, 2016. I am curious what the path of the rail line is.

    Tony Carrillo’s F Minus for the 11th was apparently our Venn Diagram joke for the week. I’m amused.

    Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 12th of March name-drops statisticians. Statisticians are almost expected to produce interesting pictures of their results. It is the field that gave us bar charts, pie charts, scatter plots, and many more. Statistics is, in part, about understanding a complicated set of data with a few numbers. It’s also about turning those numbers into recognizable pictures, all in the hope of finding meaning in a confusing world (ours).

    Brian Anderson’s Dog Eat Doug for the 13th of March uses walls full of mathematical scrawl as signifier for “stuff thought deeply about’. I don’t recognize any of the symbols specifically, although some of them look plausibly like calculus. I would not be surprised if Anderson had copied equations from a book on string theory. I’d do it to tell this joke.

    And then came the 14th of March. That gave us a bounty of Pi Day comics. Among them:

    'Happy Pi Day.' 'Mmm. I love apple pie.' 'Pi day, not Pie Day. Pi ... you know ... 3.14 ... March 14th. Get it?' 'Today is a pie-eating holiday?' 'Sort of. They do celebrate it with pie, but it's mostly about pi.' 'I don't understand what that kid says half the time.'

    John Hambrock’s The Brilliant Mind of Edison Lee for the 14th of March, 2016. The strip is like this a lot.

    John Hambrock’s The Brilliant Mind of Edison Lee trusts that the name of the day is wordplay enough.

    Scott Hilburn’s The Argyle Sweater is also a wordplay joke, although it’s a bit more advanced.

    Tim Rickard’s Brewster Rockit fuses the pun with one of its running, or at least rolling, gags.

    Bill Whitehead’s Free Range makes an urban legend out of the obsessive calculation of digits of π.

    And Missy Meyer’s informational panel cartoon Holiday Doodles mentions that besides “National” Pi Day it was also “National” Potato Chip Day, “National” Children’s Craft Day, and “International” Ask A Question Day. My question: for the first three days, which nation?

    Edited To Add: And I forgot to mention, after noting to myself that I ought to mention it. The Price Is Right (the United States edition) hopped onto the Pi Day fuss. It used the day as a thematic link for its Showcase prize packages, noting how you could work out π from the circumference of your new bicycles, or how π was a letter from your vacation destination of Greece, and if you think there weren’t brand-new cars in both Showcases you don’t know the game show well. Did anyone learn anything mathematical from this? I am skeptical. Do people come away thinking mathematics is more fun after this? … Conceivably. At least it was a day fairly free of people declaring they Hate Math and Can Never Do It.

     
  • Joseph Nebus 10:00 pm on Saturday, 9 January, 2016 Permalink | Reply
    Tags: statistics, ,   

    How 2015 Treated My Mathematics Blog 


    Oh yeah, I also got one of these. WordPress put together a review of what all went on around here last year. The most startling thing to me is that I had 188 posts over the course of the year. A lot of that is thanks to the A To Z project, which gave me something to post each day for 31 days in a row. If I’d been thinking just a tiny bit harder I’d have come up with two more posts and made a clean sweep of June.

    The unit of comparison for my readership this year was the Sydney Opera House. That’s a great comparison because everybody thinks they know how big an opera house is. It reminds me of a bit in Carl Sagan and and Ann Druyan’s Comet in which they compare the speed of an Oort cloud comet puttering around the sun to the speed of a biplane. We may have only a foggy idea how fast that is (I guess maybe a hundred miles per hour?) but it sounds nice and homey.

     
    • Matthew Wright 1:00 am on Sunday, 10 January, 2016 Permalink | Reply

      I got one of those emails and was wondering if they first had to assume humans were spherical. But maybe that joke is too corny to repeat. In my case the number should have been ‘bloggers standing outside the Sydney Opera House’ – which is where I was when my wife took the pic I still use on WordPress.

      Like

      • Joseph Nebus 3:49 am on Saturday, 16 January, 2016 Permalink | Reply

        Hey, you’re right. Bloggers standing outside the Sydney Opera House would be a good measure to use for a count of things. Unless there were a fire drill inside requiring everybody to leave, since that would artificially inflate the number of bloggers outside.

        Liked by 1 person

    • Problems With Infinity 5:40 pm on Wednesday, 13 January, 2016 Permalink | Reply

      woah 188 posts that’s crazy! I wish I could get that many out, nice work!

      Like

      • Joseph Nebus 3:51 am on Saturday, 16 January, 2016 Permalink | Reply

        Well, thank you. It’s not as much work as you might think, partly because I do a fair bit of reblogging, and partly because things like the monthly statistics review or the Reading the Comics posts don’t take so much effort.

        Like

  • Joseph Nebus 3:00 pm on Thursday, 31 December, 2015 Permalink | Reply
    Tags: , , , , statistics   

    Reading the Comics, December 30, 2015: Seeing Out The Year Edition 


    There’s just enough comic strips with mathematical themes that I feel comfortable doing a last Reading the Comics post for 2015. And as maybe fits that slow week between Christmas and New Year’s, there’s not a lot of deep stuff to write about. But there is a Jumble puzzle.

    Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips gives us someone so wrapped up in measuring data as to not notice the obvious. The obvious, though, isn’t always right. This is why statistics is a deep and useful field. It’s why measurement is a powerful tool. Careful measurement and statistical tools give us ways to not fool ourselves. But it takes a lot of sampling, a lot of study, to give those tools power. It can be easy to get lost in the problems of gathering data. Plus numbers have this hypnotic power over human minds. I understand Lard’s problem.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of December messes with a kid’s head about the way we know 1 + 1 equals 2. The classic Principia Mathematica construction builds it out of pure logic. We come up with an idea that we call “one”, and another that we call “plus one”, and an idea we call “two”. If we don’t do anything weird with “equals”, then it follows that “one plus one equals two” must be true. But does the logic mean anything to the real world? Or might we be setting up a game with no relation to anything observable? The punchy way I learned this question was “one cup of popcorn added to one cup of water doesn’t give you two cups of soggy popcorn”. So why should the logical rules that say “one plus one equals two” tell us anything we might want to know about how many apples one has?

    Words: LIHWE, CAQUK, COYKJE, TALAFO. Unscramble to - - - O O, O O - - -, - - - - - O, - - O - O -, and solve the puzzle: 'The math teacher liked teaching addition and subtraction - - - - - - -'.

    David L Hoyt and Jeff Knurek’s Jumble for the 28th of December, 2015. The link will probably expire in late January 2016.

    David L Hoyt and Jeff Knurek’s Jumble for the 28th of December features a mathematics teacher. That’s enough to include here. (You might have an easier time getting the third and fourth words if you reason what the surprise-answer word must be. You can use that to reverse-engineer what letters have to be in the circles.)

    Richard Thompson’s Richard’s Poor Almanac for the 28th of December repeats the Platonic Fir Christmas Tree joke. It’s in color this time. Does the color add to the perfection of the tree, or take away from it? I don’t know how to judge.

    A butterfly tells another 'you *should* feel guilty --- the flutter of your wings ended up causing a hurricane that claimed thousands of lives!'

    Rina Piccolo filling in for Hilary Price on Rhymes With Orange for the 29th of December, 2015. It’s a small thing but I always like the dog looking up in the title panel for Cartoonist Showcase weeks.

    Hilary Price’s Rhymes With Orange for the 29th of December gives its panel over to Rina Piccolo. Price often has guest-cartoonist weeks, which is a generous use of her space. Piccolo already has one and a sixth strips — she’s one of the Six Chix cartoonists, and also draws the charming Tina’s Groove — but what the heck. Anyway, this is a comic strip about the butterfly effect. That’s the strangeness by which a deterministic system can still be unpredictable. This counter-intuitive conclusion dates back to the 1890s, when Henri Poincaré was trying to solve the big planetary mechanics question. That question is: is the solar system stable? Is the Earth going to remain in about its present orbit indefinitely far into the future? Or might the accumulated perturbations from Jupiter and the lesser planets someday pitch it out of the solar system? Or, less likely, into the Sun? And the sad truth is, the best we can say is we can’t tell.

    In Brian Anderson’s Dog Eat Doug for the 30th of December, Sophie ponders some deep questions. Most of them are purely philosophical questions and outside my competence. “What are numbers?” is also a philosophical question, but it feels like something a mathematician ought to have a position on. I’m not sure I can offer a good one, though. Numbers seem to be to be these things which we imagine. They have some properties and that obey certain rules when we combine them with other numbers. The most familiar of these numbers and properties correspond with some intuition many animals have about discrete objects. Many times over we’ve expanded the idea of what kinds of things might be numbers without losing the sense of how numbers can interact, somehow. And those expansions have generally been useful. They strangely match things we would like to know about the real world. And we can discover truths about these numbers and these relations that don’t seem to be obviously built into the definitions. It’s almost as if the numbers were real objects with the capacity to surprise and to hold secrets.

    Why should that be? The lazy answer is that if we came up with a construct that didn’t tell us anything interesting about the real world, we wouldn’t bother studying it. A truly irrelevant concept would be a couple forgotten papers tucked away in an unread journal. But that is missing the point. It’s like answering “why is there something rather than nothing” with “because if there were nothing we wouldn’t be here to ask the question”. That doesn’t satisfy. Why should it be possible to take some ideas about quantity that ravens, raccoons, and chimpanzees have, then abstract some concepts like “counting” and “addition” and “multiplication” from that, and then modify those concepts, and finally have the modification be anything we can see reflected in the real world? There is a mystery here. I can’t fault Sophie for not having an answer.

     
  • Joseph Nebus 3:00 pm on Friday, 24 July, 2015 Permalink | Reply
    Tags: , , , , statistics,   

    A Summer 2015 Mathematics A to Z Roundup 


    Since I’ve run out of letters there’s little dignified to do except end the Summer 2015 Mathematics A to Z. I’m still organizing my thoughts about the experience. I’m quite glad to have done it, though.

    For the sake of good organization, here’s the set of pages that this project’s seen created:

     
  • Joseph Nebus 2:24 pm on Sunday, 12 July, 2015 Permalink | Reply
    Tags: , , , , , , statistics, survival   

    Reading the Comics, July 12, 2015: Chuckling At Hart Edition 


    I haven’t had the chance to read the Gocomics.com comics yet today, but I’d had enough strips to bring up anyway. And I might need something to talk about on Tuesday. Two of today’s strips are from the legacy of Johnny Hart. Hart’s last decades at especially B.C., when he most often wrote about his fundamentalist religious views, hurt his reputation and obscured the fact that his comics were really, really funny when they start. His heirs and successors have been doing fairly well at reviving the deliberately anachronistic and lightly satirical edge that made the strips funny to begin with, and one of them’s a perennial around here. The other, Wizard of Id Classics, is literally reprints from the earliest days of the comic strip’s run. That shows the strip when it was earning its place on every comics page everywhere, and made a good case for it.

    Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (July 8) shows how a compass, without straightedge, can be used to ensure one’s survival. I suppose it’s really only loosely mathematical but I giggled quite a bit.

    Ken Cursoe’s Tiny Sepuku (July 9) talks about luck as being just the result of probability. That’s fair enough. Random chance will produce strings of particularly good, or bad, results. Those strings of results can look so long or impressive that we suppose they have to represent something real. Look to any sport and the argument about whether there are “hot hands” or “clutch performers”. And Maneki-Neko is right that a probability manipulator would help. You can get a string of ten tails in a row on a fair coin, but you’ll get many more if the coin has an eighty percent chance of coming up tails.

    Brant Parker and Johnny Hart’s Wizard of Id Classics (July 9, rerun from July 12, 1965) is a fun bit of volume-guessing and logic. So, yes, I giggled pretty solidly at both B.C. and The Wizard of Id this week.

    Mell Lazarus’s Momma (July 11) identifies “long division” as the first thing a person has to master to be an engineer. I don’t know that this is literally true. It’s certainly true that liking doing arithmetic helps one in a career that depends on calculation, though. But you can be a skilled songwriter without being any good at writing sheet music. I wouldn’t be surprised if there are skilled engineers who are helpless at dividing fourteen into 588.

    In the panel of interest, Loretta says the numbers (presumably the bills) don't add up, but they subtract down fine.

    Bunny Hoest and John Reiner’s Lockhorns for the 12th of July, 2015.

    Bunny Hoest and John Reiner’s Lockhorns (July 12) includes an example of using “adding up” to mean “make sense”. It’s a slight thing. But the same idiom was used last week, in Eric Teitelbaum and Bill Teitelbaum’s Bottomliners. I don’t think Comic Strip Master Command is ordering this punch line yet, but you never know.

    And finally, I do want to try something a tiny bit new, and explicitly invite you-the-readers to say what strip most amused you. Please feel free to comment about your choices, r warn me that I set up the poll wrong. I haven’t tried this before.

     
  • Joseph Nebus 2:49 pm on Wednesday, 1 July, 2015 Permalink | Reply
    Tags: , , , median, , quintiles, statistics, word counts   

    A Summer 2015 Mathematics A To Z: quintile 


    Quintile.

    Why is there statistics?

    There are many reasons statistics got organized as a field of study mostly in the late 19th and early 20th century. Mostly they reflect wanting to be able to say something about big collections of data. People can only keep track of so much information at once. Even if we could keep track of more information, we’re usually interested in relationships between pieces of data. When there’s enough data there are so many possible relationships that we can’t see what’s interesting.

    One of the things statistics gives us is a way of representing lots of data with fewer numbers. We trust there’ll be few enough numbers we can understand them all simultaneously, and so understand something about the whole data.

    Quintiles are one of the tools we have. They’re a lesser tool, I admit, but that makes them sound more exotic. They’re descriptions of how the values of a set of data are distributed. Distributions are interesting. They tell us what kinds of values are likely and which are rare. They tell us also how variable the data is, or how reliably we are measuring data. These are things we often want to know: what is normal for the thing we’re measuring, and what’s a normal range?

    We get quintiles from imagining the data set placed in ascending order. There’s some value that one-fifth of the data points are smaller than, and four-fifths are greater than. That’s your first quintile. Suppose we had the values 269, 444, 525, 745, and 1284 as our data set. The first quintile would be the arithmetic mean of the 269 and 444, that is, 356.5.

    The second quintile is some value that two-fifths of your data points are smaller than, and that three-fifths are greater than. With that data set we started with that would be the mean of 444 and 525, or 484.5.

    The third quintile is a value that three-fifths of the data set is less than, and two-fifths greater than; in this case, that’s 635.

    And the fourth quintile is a value that four-fifths of the data set is less than, and one-fifth greater than. That’s the mean of 745 and 1284, or 1014.5.

    From looking at the quintiles we can say … well, not much, because this is a silly made-up problem that demonstrates how quintiles are calculated rather instead of why we’d want to do anything with them. At least the numbers come from real data. They’re the word counts of my first five A-to-Z definitions. But the existence of the quintiles at 365.5, 484.5, 635, and 1014.5, along with the minimum and maximum data points at 269 and 1284, tells us something. Mostly that numbers are bunched up in the three and four hundreds, but there could be some weird high numbers. If we had a bigger data set the results would be less obvious.

    If the calculating of quintiles sounds much like the way we work out the median, that’s because it is. The median is the value that half the data is less than, and half the data is greater than. There are other ways of breaking down distributions. The first quartile is the value one-quarter of the data is less than. The second quartile a value two-quarters of the data is less than (so, yes, that’s the median all over again). The third quartile is a value three-quarters of the data is less than.

    Percentiles are another variation on this. The (say) 26th percentile is a value that 26 percent — 26 hundredths — of the data is less than. The 72nd percentile a value greater than 72 percent of the data.

    Are quintiles useful? Well, that’s a loaded question. They are used less than quartiles are. And I’m not sure knowing them is better than looking at a spreadsheet’s plot of the data. A plot of the data with the quintiles, or quartiles if you prefer, drawn in is better than either separately. But these are among the tools we have to tell what data values are likely, and how tightly bunched-up they are.

     
  • Joseph Nebus 8:00 am on Tuesday, 2 June, 2015 Permalink | Reply
    Tags: , , , , , Romania, statistics, Vietnam, ,   

    How May 2015 Treated My Mathematics Blog 


    For May 2015 I tried a new WordPress theme — P2 Classic — and I find I rather like it. Unfortunately it seems to be rubbish on mobile devices and I’m not WordPress Theme-equipped-enough to figure out how to fix that. I’m sorry, mobile readers. I’m honestly curious whether the theme change affected my readership, which was down appreciably over May.

    According to WordPress, the number of pages viewed here dropped to 936 in May, down just over ten percent from April’s 1047 and also below March’s 1022. Perhaps the less-mobile-friendly theme was shooing people away. Maybe not, though: in March and April I’d posted 14 articles each, while in May there were a mere twelve. The number of views per post increased steadily, from 73 in March to just under 75 in April to 78 in May. I’m curious if this signifies anything. I may get some better idea next month. June should have at least 13 posts from the Mathematics A To Z gimmick, plus this statistics post, and there’ll surely be at least two Reading The Comics posts, or at least sixteen posts. And who knows what else I’ll feel like throwing in? It’ll be an interesting experiment at least.

    Anyway, the number of unique visitors rose to 415 in May, up from April’s 389 but still below March’s 468. The number of views per visitor dropped to 2.26, far below April’s 2.68, but closer in line with March’s 2.18. And 2.26 is close to the normal count for this sort of thing.

    The number of likes on posts dropped to 259. In April it was 296 likes and in March 265. That may just reflect the lower number of posts, though. Divide the number of likes by the number of posts and March saw an average of 18.9, April 21.14, and May 21.58. That’s all at least consistent, although there’s not much reason to suppose that only things from the current month were liked.

    The number of comments recovered also. May saw 83 comments, up from April’s 64, but not quite back to March’s 93. That comes to, for May, 6.9 comments for each post, but that’s got to be counting links to other posts, including pingbacks and maybe the occasional reblogging. I’ve been getting chattier with folks around here, but not seven comments per post chatty.

    June starts at 24,820 views, and 485 people following specifically through WordPress.

    I’ve got a healthy number of popular posts the past month; all of these got at least 37 page views each. I cut off at 37 because that’s where the Trapezoids one came in and we already know that’s popular. More popular than that were:

    I have the suspicion that comics fans are catching on, quietly, to all this stuff.

    Now the countries report. The nations sending me at least twenty page views were the United States (476), the United Kingdom (85), Canada (65), Italy (53), and Austria (20).

    Sending just a single reader were Belgium, Bulgaria, Colombia, Nigeria, Norway, Pakistan, Romania, and Vietnam. Romania is on a three-month single-reader streak; Vietnam, two. India sent me a mere two readers, down from six last month. The European Union sent me three.

    And among the interesting search terms this past month were:

    • origin is the gateway to your entire gaming universe.
    • how to do a cube box (the cube is easy enough, it’s getting the boxing gloves on that’s hard)
    • popeye “computer king” (Remember that comic?)
    • google can you show me in 1 trapezoid how many cat how many can you make of 2 (?, although I like the way Google is named at the start of the query, like someone on Next Generation summoning the computer)
    • plato “divided line” “arthur cayley” (I believe that mathematics comes in on the lower side of the upper half of Plato’s divided line)
    • where did negative numbers originate from

    Someday I must work out that “origin is the gateway” thing.

     
    • Ken Dowell 11:25 am on Tuesday, 2 June, 2015 Permalink | Reply

      It would make a nice addition to WordPress stats if it broke down views by device or at minimum separated out mobile views.

      Like

      • Joseph Nebus 10:34 pm on Friday, 5 June, 2015 Permalink | Reply

        I’d quite like if it did that. I am growing a little fonder of the breakdown by week, at least. I’m not sure this new “Insight” panel is any good, though. Especially since its display about what days I post most often are just plain wrong, at least for my humor blog. (That one gets exactly one post a day, and their Insight panel lists some days with two and some with none.)

        Like

    • abyssbrain 12:42 pm on Tuesday, 2 June, 2015 Permalink | Reply

      It’s quite strange that the person who searched for “origin is the gateway…” ended up on your blog since Origin is the DRM platform used by the video game company Electronic Arts.

      Like

      • Joseph Nebus 10:34 pm on Friday, 5 June, 2015 Permalink | Reply

        Oh, is that what it is? Thank you. I had no idea.

        Of course, with that explained now I’ve got no more idea why the search would bring people here.

        Liked by 1 person

    • balauru 5:43 am on Wednesday, 3 June, 2015 Permalink | Reply

      Reblogged this on Sharing Maniak.

      Like

  • Joseph Nebus 3:50 pm on Friday, 1 May, 2015 Permalink | Reply
    Tags: , , , Origin, , statistics, ,   

    How April 2015 Treated My Mathematics Blog 


    (I apologize if the formatting is messed up. For some reason preview is not working, and I will not be trying the new page for entering posts if I can at all help it. I will fix when I can, if it needs fixing.)

    As it’s the start of the month I want to try understanding the readership of my blogs, as WordPress gives me statistics. It’s been a more confusing month than usual, though. One thing is easy to say: the number of pages read was 1,047, an all-time high around these parts for a single month. It’s up from 1,022 in March, and 859 in February. And it’s the second month in a row there’ve been more than a thousand readers. That part’s easy.

    The number of visitors has dropped. It was down to 389 in April, from a record 468 in March and still-higher 407 in April. This is, if WordPress doesn’t lead me awry, my fifth-highest number of viewers. This does mean the number of views per visitor was my highest since June of 2013. The blog had 2.69 views per visitor, compared to 2.18 in March and 2.11 in February. It’s one of my highest views-per-visitor on record anyway. Perhaps people quite like what they see and are archive-binging. I approve of this. I’m curious why the number of readers dropped so, though, particularly when I look at my humor blog statistics (to be posted later).

    I’m confident the readers are there, though. The number of likes on my mathematics blog was 297, up from March’s 265 and February’s 179. It’s the highest on record far as WordPress will tell me. So readers are more engaged, or else they’re clicking like from the WordPress Reader or an RSS feed. Neither gets counted as a page view or a visitor. That’s another easy part. The number of comments is down to 64, from March’s record 93, but March seems to have been an exceptional month. February had 56 comments so I’m not particularly baffled by April’s drop.

    May starts out with 23,884 total views, and 472 people following specifically through WordPress.

    It’s a truism that my most popular posts are the trapezoids one and the Reading The Comics posts, but for April that was incredibly true. Most popular the past thirty days were:

    1. How Many Trapezoids I Can Draw.
    2. Reading The Comics, April 10, 2015: Getting Into The Story Problem Edition.
    3. Reading The Comics, April 15, 2015: Tax Day Edition.
    4. Reading The Comics, April 20, 2015: History Of Mathematics Edition.
    5. Reading The Comics, March 31, 2015: Closing Out March Edition.

    I am relieved that I started giving all these Comics posts their own individual “Edition” titles. Otherwise there’d be no way to tell them apart.

    The nations sending me the most readers were, as ever, the United States (662), Canada (82), and the United Kingdom (47), with Slovenia once again strikingly high (36). Hong Kong came in with 24 readers, Italy 23, and Austria a mere 18. Elke Stangl’s had a busy month, I know.

    This month’s single-reader countries were Czech Republic, Morocco, the Netherlands, Puerto Rico, Romania, Taiwan, and Vietnam. Romania’s the only one that sent me a single reader last month. India bounced back from five readers to six.

    Among the search terms bringing people to me were no poems. Among the interesting phrases were:

    • what point is driving the area difference between two triangles (A good question!)
    • how do you say 1,898,600,000,000,000,000,000,000,000 (I almost never do.)
    • is julie larson still drawing the dinette set (Yes, to the best of my knowledge.)
    • jpe fast is earth spinning? (About once per day, although the answer can be surprisingly difficult to say! But also figure about 465 times the cosine of your latitude meters per second, roughly.)
    • origin is the gateway to your entire gaming universe. (Again, I don’t know what this means, and I’m a little scared to find out.)
    • i hate maths 2015 photos (Well, that just hurts.)
    • getting old teacher jokes (Again, that hurts, even if it’s not near my birthday.)
    • two trapezoids make a (This could be a poem, actually.)
    • how to draw 2 trapezoids (I’d never thought about that one. Shall have to consider writing it.)

    I don’t know quite what it all means, other than that I need to write about comic strips and trapezoids more somehow.

     
    • scifihammy 4:34 pm on Friday, 1 May, 2015 Permalink | Reply

      While I think it is great fun looking at all the stats – and who doesn’t? – I still think the main point is to enjoy blogging. If you enjoy it, chances are your readers will :)
      As for trying to make sense if the numbers, if you ever do – let me know!! :)

      Like

      • Joseph Nebus 6:08 am on Tuesday, 5 May, 2015 Permalink | Reply

        Oh, certainly, the important thing is enjoying the blogging. And mostly I do enjoy the writing. Sometimes I end up afraid of the comments, but I try to overcome that.

        Liked by 1 person

    • Angie Mc 7:15 pm on Friday, 1 May, 2015 Permalink | Reply

      OK, Joseph, I’m inching ever closer to doing something with my stats! I’m at least looking at them now :D One thing I like about them is the personal challenge of trying to work hard to engage with others. Looking at the bar graph go up and down helps to keep me motivated in its own way. Congrats on your all time high page reads!

      Like

      • Joseph Nebus 6:09 am on Tuesday, 5 May, 2015 Permalink | Reply

        I love looking at the statistics that say how much more a particular essay is read one week compared to the week before. I’m not so fond of the statistics that say how much less a particular essay is read one week compared to the week before. But I don’t know how to get the first without the second. Yet.

        Liked by 1 person

    • Ken Dowell 9:01 pm on Friday, 1 May, 2015 Permalink | Reply

      The WordPress stats are all very fascinating but I sometimes think they raise more questions than they answer.

      Like

      • Joseph Nebus 6:12 am on Tuesday, 5 May, 2015 Permalink | Reply

        Oh, do they ever. Mostly “counting trapezoids? That’s all people want of me?” Sometimes I fear I know what a one-hit wonder’s life is like.

        Like

    • elkement 7:31 pm on Saturday, 2 May, 2015 Permalink | Reply

      Only 18?? ;-) I would have guessed something more like 30-40? I wonder if one of my AdBlocker thingies on one of my computers prevents WP Stats from collecting my clicks?

      Like

      • Joseph Nebus 6:27 am on Tuesday, 5 May, 2015 Permalink | Reply

        Only 18, so it says. But I have wondered if something is blocking reads from showing up in WordPress statistics. I had a weird blip in my humor blog’s readership in April and I’m still not sure what (if anything) accounts for it.

        Liked by 1 person

  • Joseph Nebus 6:39 pm on Thursday, 26 February, 2015 Permalink | Reply
    Tags: animal research, , , , , , mice, , , statistics   

    How Not To Count Fish 


    I’d discussed a probability/sampling-based method to estimate the number of fish that might be in our pond out back, and then some of the errors that have to be handled if you want to have a reliable result. Now, I want to get into why the method doesn’t work, at least not without much greater insight into goldfish behavior than simply catching a couple and releasing them will do.

    Catching a sample, re-releasing it, and counting how many of that sample we re-catch later on is a logically valid method, provided certain assumptions the method requires are accurately — or at least accurately enough — close to the way the actual thing works. Here are some of the ways goldfish fall short of the ideal.

    First faulty assumption: Goldfish are perfectly identical. In this goldfish-trapped we make the assumption that there is some, fixed, constant probability of a goldfish being caught in the net. We have to assume that this is the same number for every goldfish, and that it doesn’t change as goldfish go through the experience of getting caught and then released. But goldfish have personality, as you learn if you have a bunch in a nice setting and do things like try feeding them koi treats or introduce something new like a wire-mesh trap to their environment. Some are adventurous and will explore the unfamiliar thing; some are shy and will let everyone else go first and then maybe not bother going at all. I empathize with both positions.

    If there are enough goldfish, the variation between personalities is probably not going to matter much. There’ll be some that are easy to catch, and they’ll probably be roughly as common as the ones who can’t be coaxed into the trap at all. It won’t be exactly balanced unless we’re very lucky, but this would probably only throw off our calculations a little bit.

    Whether the goldfish learn, and become more, or less, likely to be trapped in time is harder. Goldfish do learn, certainly, although it’s not obvious to me that the trapping and releasing experience would be one they draw much of a lesson from. It’s only a little inconvenience, really, and not at all harmful; what should they learn? Other than that there’s maybe an easy bit of food to be had here so why not go in? So this might change their behavior and it’s hard to predict how.

    (I note that animal capture studies get quite frustrated when the animals start working out how to game the folks studying them. Bil Gilbert’s early-70s study of coatis — Latin American raccoons, written up in the lovely popularization Chulo: A Year Among The Coatimundis — was plagued by some coatis who figured out going into the trap was an easy, safe meal they’d be released from without harm, and wouldn’t go back about their business and leave room for other specimens.)

    Second faulty assumption: Goldfish are not perfectly identical. This is the biggest challenge to counting goldfish population by re-catching a sample of them. How do you know if you caught a goldfish before? When they grow to adulthood, it’s not so bad, since they grow fairly distinctive patterns of orange and white and black and such, and they’ll usually settle into different sizes. (That said, we do have two adult fish who were very distinct when we first got them, but who’ve grown into near-twins.)

    But baby goldfish? They’re basically all tiny black things, meant to hide into the mud at the bottom of ponds and rivers — their preferred habitat — and pretty near indistinguishable. As they get larger they get distinguishable, a bit, and start to grow patterns, but for the vast number of baby fish there’s just no telling one from another.

    When we were trying to work out whether some mice we found in the house were ones we had previously caught and put out in the garage, we were able to mark them by squiring some food dye at their heads as they were released. The mice would rub the food dye from their heads onto their whole bodies and it would take a while before the dye would completely fade out. (We didn’t re-catch any mice, although it’s hard to dye a wild mouse efficiently because they will take off like bullets. Also one time when we thought we’d captured one there were actually three in the humane trap and you try squiring the food dye bottle at two more mice than you thought were there, fleeing.) But you can see how the food dye wouldn’t work here. Animal researchers with a budget might go on to attach collars or somehow otherwise mark animals, but if there’s a way to mark and track goldfish with ordinary household items I can’t think of it.

    (No, we will not be taking the bits of americium in our smoke detectors out and injecting them into trapped goldfish; among the objections, I don’t have a radioactivity detector.)

    Third faulty assumption: Goldfish are independent entities. The first two faulty assumptions are ones that could be kind of worked around. If there’s enough goldfish then the distribution of how likely any one is to get caught will probably be near enough normal that we can pretend there’s an identical chance of catching each, and if we really thought about it we could probably find some way of marking goldfish to tell if we re-caught any. Independence, though; this is the point on which so many probability-based schemes fall.

    Independence, in the language of probability, is the principle that one thing’s happening does not affect the likelihood of another thing happening. For our problem, it’s the assumption that one goldfish being caught does not make it any more or less likely that another goldfish will be caught. We like independence, in studying probability. It makes so many problems easier to study, or even possible to study, and it often seems like a reasonable supposition.

    A good number of interesting scientific discoveries amount to finding evidence that two things are not actually independent, and that one thing happening makes it more (or less) likely the other will. Sometimes these turn out to be vapor — there was a 19th-century notion suggesting a link between sunspot activity and economic depressions (because sunspots correlate to solar activity, which could affect agriculture, and up to 1893 the economy and agriculture were pretty much the same thing) — but when there is a link the results can be profound, as see the smoking-and-cancer link, or for something promising but still (to my understanding) under debate, the link between leaded gasoline and crime rates.

    How this applies to the goldfish population problem, though, is that goldfish are social creatures. They school, loosely, forming and re-forming groups, and would much rather be around another goldfish than not. Even as babies they form these adorable tiny little schools; that may be in the hopes that someone else will get eaten by a bigger fish, but they keep hanging around other fish their own size through their whole lives. If there’s a goldfish inside the trap, it is hard to believe that other goldfish are not going to follow it just to be with the company.

    Indeed, the first day we set out the trap for the winter, we pulled in all but one of the adult fish, all of whom apparently followed the others into the enclosure. I’m sorry I couldn’t photograph that because it was both adorable and funny to see so many fish just station-keeping beside one another — they were even all looking in the same direction — and waiting for whatever might happen next. Throughout the months we were able to spend bringing in fish, the best bait we could find was to have one fish already in the trap, and a couple days we did leave one fish in a few more hours or another night so that it would be joined by several companions the next time we checked.

    So that’s something which foils the catch and re-catch scheme: goldfish are not independent entities. They’re happy to follow one another into trap. I would think the catch and re-catch scheme should be salvageable, if it were adapted to the way goldfish actually behave. But that requires a mathematician admitting that he can’t just blunder into a field with an obvious, simple scheme to solve a problem, and instead requires the specialized knowledge and experience of people who are experts in the field, and that of course can’t be done. (For example, I don’t actually know that goldfish behavior is sufficiently non-independent as to make an important difference in a population estimate of this kind. But someone who knew goldfish or carp well could tell me, or tell me how to find out.)

    Several dozen goldfish, most of them babies, within a 150-gallon rubber stock tank, their wintering home.

    Goldfish brought indoors, to a stock tank, for the winter.

    For those curious how the goldfish worked out, though, we were able to spend about two and a half months catching fish before the pond froze over for the winter, though the number we caught each week dropped off as the temperature dropped. We have them floating about in a stock tank in the basement, waiting for the coming of spring and the time the pond will be warm enough for them to re-occupy it. We also know that at least some of the goldfish we didn’t catch made it to, well, about a month ago. I’d seen one of the five orange baby fish who refused to go into the trap through a hole in the ice then. It was holding close to the bottom but seemed to be in good shape.

    This coming year should be an exciting one for our fish population.

     
    • AR 3:06 am on Friday, 27 February, 2015 Permalink | Reply

      I’m interested in this idea of statistical independence. By this measure, people are not very independent either!

      Like

      • Joseph Nebus 3:26 am on Friday, 27 February, 2015 Permalink | Reply

        Well, it depends on the frame, really. In some ways people can be treated as statistically independent entities: all the people in a mall food court, for example, could be modeled as equally likely as every other person to go to the McDonald’s, the sandwich shop, the Sbarro’s, or the mediterranean food grill, at least for the purpose of figuring out pedestrian traffic flows, or how tables would be best arranged, or matters like that.

        Like

        • AR 11:11 am on Friday, 27 February, 2015 Permalink | Reply

          Is that because there are so many people going through a mall, making group decision-making and following statistically unimportant?

          I think I see dependence sometimes on facebook or Amazon reviews… whoever makes the first comment on a post or leaves the first review on a movie usually sets the tone for the majority of the other reviews and comments. It seems like social media has revealed people’s overall lack of individuality and their tendency to follow. (Blogging is, of course, the shining exception – the online bastion of originality.)

          So for instance, I wonder what the outcome would be if one of those restaurants got models to walk in and out of their doors all day, and sit at outside tables laughing and eating and raving about the food.

          Like

          • Joseph Nebus 8:10 pm on Saturday, 28 February, 2015 Permalink | Reply

            Well, independence in this context just means that the probability that (say) this party is going to that food stall doesn’t depend on what any other parties are doing. That is, it doesn’t matter if the last four parties entering all went to the hot dog place; that doesn’t make the next party to enter more — or less — likely to go to the pizza place.

            (There are limits on this, of course. If there’s a line of 200 people at the hot dog place and nobody anywhere else, I would skip my hot dog plans. Or suppose that the hot dogs have got to be too good to pass up, I guess.)

            The sort of dependence you’re describing seems to be more of an anchoring effect, one of the strange and creepy aspects of human decision-making. If asked to give their opinion on something, people will tend to give an answer that’s close to whatever the last thing they heard was, even if it had nothing to do with whatever they’re supposed to decide. The tone-setting effect of the first comments is probably a reflection of that effect in less-quantifiable matters.

            Liked by 1 person

    • Aquileana 5:25 am on Friday, 27 February, 2015 Permalink | Reply

      This is so interesting Joseph… I agree with the commenter above as I found that the excerpt called faulty assumption: Goldfish are independent entities, were outstanding.
      Thanks for sharing this information and problems on statistics. Best wishes :star: Aquileana :D

      Like

    • Angie Mc 5:56 pm on Friday, 6 March, 2015 Permalink | Reply

      I’m going to read together with my 9 year old son who loves fish and real life math. Thanks, Joseph :D

      Like

      • Joseph Nebus 1:05 am on Sunday, 8 March, 2015 Permalink | Reply

        Aw, good. Hope you enjoy.

        We’re hoping the weather will be warm enough in the next month that we can transfer the fish back into the pond. I can think of a mathematical problem that results from this, but it’s a less obviously appealing one than simply “estimate the pond’s fish population”.

        Liked by 1 person

        • Angie Mc 4:16 am on Monday, 9 March, 2015 Permalink | Reply

          Have I mentioned my son-in-law who is a grad student in physics? Once he gets through prelims this spring (he’s hyper studying), I will send him your blog. I think he’ll really like it.

          Like

          • Joseph Nebus 12:01 am on Tuesday, 10 March, 2015 Permalink | Reply

            I don’t remember that you had. That’s neat and I do hope he enjoys, although I also remember keenly how much work went into prelims. Analysis particularly was a hard one for me to get through.

            Liked by 1 person

            • Angie Mc 1:13 am on Tuesday, 10 March, 2015 Permalink | Reply

              I love my son-in-law has a son. I met him when he was 19 and contemplating going to college for math. He’s first generation college and I couldn’t be more proud of him. We are close and I’m feeling his prelim pain!

              Like

              • Joseph Nebus 7:54 pm on Thursday, 12 March, 2015 Permalink | Reply

                Oh, that’s great for him. I hope the prelims go well. Grad school was wonderful at least for me, and my love, although it is a lot of very particular challenges, many of them more of endurance and tolerance for the grad school lifestyle than anything else. I was able to come out the far end successfully, but it was a close-run thing at points, with prelims one of those points.

                Like

  • Joseph Nebus 7:46 pm on Saturday, 14 February, 2015 Permalink | Reply
    Tags: advertising, art collectives, Dewey decimal system, , , existential dread, , lying, statistics,   

    Reading the Comics, February 14, 2015: Valentine’s Eve Edition 


    I haven’t had the chance to read today’s comics, what with it having snowed just enough last night that we have to deal with it instead of waiting for the sun to melt it, so, let me go with what I have. There’s a sad lack of strips I feel justified including the images of, since they’re all Gocomics.com representatives and I’m used to those being reasonably stable links. Too bad.

    Eric the Circle has a pair of strips by Griffinetsabine, the first on the 7th of February, and the next on February 13, both returning to “the Shape Single’s Bar” and both working on “complementary angles” for a pun. That all may help folks remember the difference between complementary angles — those add up to a right angle — and supplementary angles — those add up to two right angles, a straight line — although what it makes me wonder is the organization behind the Eric the Circle art collective. It hasn’t got any nominal author, after all, and there’s what appear to be different people writing and often drawing it, so, who does the scheduling so that the same joke doesn’t get repeated too frequently? I suppose there’s some way of finding that out for myself, but this is the Internet, so it’s easier to admit my ignorance and let the answer come up to me.

    Mark Anderson’s Andertoons (February 10) surprised me with a joke about the Dewey decimal system that I hadn’t encountered before. I don’t know how that happened; it just did. This is, obviously, a use of decimal that’s distinct from the number system, but it’s so relatively rare to think of decimals as apart from representations of numbers that pointing it out has the power to surprise me at least.

    (More …)

     
    • ivasallay 8:40 am on Sunday, 15 February, 2015 Permalink | Reply

      My favorites this time were Andertoons, The Daily Drawing, and One Big Happy.

      Like

    • ioanaiuliana 9:49 pm on Wednesday, 18 February, 2015 Permalink | Reply

      I think One Big Happy is great… From my tutoring experience so far, children just stop thinking about the problem if you change the variables (in our case change the apples with potatoes). If they are used with finding x, and you want them to find e – they just block. Or if they get used with calling functions f, and I called it t, they could not do a thing. So, I find it really bad, that children are used with some specific notations in school math, that they get stressed when that notation is changed.

      Like

      • Joseph Nebus 8:45 pm on Friday, 20 February, 2015 Permalink | Reply

        I’m a bit surprised by this, but shouldn’t be. It is an extra step of abstraction to think of not just doing arithmetic in which you don’t know (or don’t care) what value “x” has, but also to realize that you don’t even care whether it’s called “x” or some other name. It does take time to feel as comfortable with an abstraction as with a specific concrete example — probably why it often helps work out a difficult general problem by working out a simple test case — and I really ought to be better at sympathizing with people learning these abstractions.

        Like

  • Joseph Nebus 11:47 pm on Saturday, 24 January, 2015 Permalink | Reply
    Tags: , airlines, , , , , polls, , , statistics   

    Reading the Comics, January 24, 2015: Many, But Not Complicated Edition 


    I’m sorry to have fallen behind on my mathematics-comics posts, but I’ve been very busy wielding a cudgel at Microsoft IIS all week in the service of my day job. And since I telecommute it’s quite hard to convincingly threaten the server, however much it deserves it. Sorry. Comic Strip Master Command decided to send me three hundred billion gazillion strips, too, so this is going to be a bit of a long post.

    Jenny Campbell’s Flo and Friends (January 19) is almost a perfect example of the use of calculus as a signifier of “something really intelligent people think of”. Which is flattening to mathematicians, certainly, although I worry that attitude does make people freeze up in panic when they hear that they have to take calculus.

    The Amazing Yet Tautological feature of Ruben Bolling’s Super-Fun-Pak Comix (January 19) lives up to its title, at least provided we are all in agreement about what “average” means. From context this seems to be the arithmetic mean — that’s usually what people, mathematicians included mean by “average” if they don’t specify otherwise — although you can produce logical mischief by slipping in an alternate average, such as the “median” — the amount that half the results are less than and half are greater than — or the “mode” — the most common result. There are other averages too, but they’re not so often useful. On the 21st Super-Fun-Pak Comix returned with another installation of Chaos Butterfly, by the way.

    (More …)

     
    • The Chaos Realm 11:43 pm on Sunday, 25 January, 2015 Permalink | Reply

      …somewhat off the mark, but do you know how many times I catch the spelling of “discrete” substituted for “discreet” as in “She tried to be discreet in her actions…” LOL Maybe they are meant to be mathematicians, not writers :-)

      Like

      • Joseph Nebus 10:01 pm on Tuesday, 27 January, 2015 Permalink | Reply

        Oh, I worry excessively about the difference between ‘discrete’ and ‘discreet’ and every time I have to use one of the words I over-think whether I’ve picked out the right one. Sometimes I rework the whole sentence so I can avoid the question.

        Still, there was that time back in grad school when the homework papers for Discrete and for Vector Calculus were put into separate piles labelled “Discrete” and “Flamboyant”.

        Like

    • ivasallay 8:02 am on Monday, 26 January, 2015 Permalink | Reply

      I think the funniest thing was that Bob Newhart was an accountant. That previously unknown-to-me fact fired up my imagination the most!

      Like

      • Joseph Nebus 10:02 pm on Tuesday, 27 January, 2015 Permalink | Reply

        And it’s even true! He mentions it in passing in the start of his “Retirement Party” routine, and in his biography mentions how this led him to working in the Unemployment Office, until he found that he did better per hour without a job, collecting unemployment rather than working for what they would pay him.

        Like

    • Aquileana 11:15 pm on Saturday, 31 January, 2015 Permalink | Reply

      Stunning deliver… I much enjoyed your post Nebus!… All my best wishes, Aquileana :D

      Like

  • Joseph Nebus 5:03 pm on Monday, 5 January, 2015 Permalink | Reply
    Tags: annual report, , statistics,   

    WordPress’s 2014 in review, Mathematics Blog Edition 


    It’s a little more formal than my usual monthly roundups, but WordPress makes this nice little animated report and everything for the year as a whole, and I’d like to share it now while I work on the first mathematics-comics roundup for 2015.

    A static scene of fireworks to tease people into the whole (visual) report.

    Here's an excerpt:

    A New York City subway train holds 1,200 people. This blog was viewed about 7,000 times in 2014. If it were a NYC subway train, it would take about 6 trips to carry that many people.

    Here’s the complete report.

     
    • Carrie Rubin 5:07 am on Sunday, 11 January, 2015 Permalink | Reply

      I’m always amazed by the number of countries that visit our blogs. I suppose many of these visits are by accident–someone stumbles upon our blog in search of something else–but still, it’s pretty cool.

      Like

      • Joseph Nebus 9:24 pm on Monday, 12 January, 2015 Permalink | Reply

        I’m regularly amazed by readership from non-Anglosphere countries, myself. I understand someone looking for trapezoid information going to a search engine and finding me, and while I know intellectually that there are plenty of English-reading people in any country, to think that I might have something of use to somebody in Slovenia or Cambodia is pretty heady.

        Liked by 1 person

    • ioanaiuliana 2:36 pm on Monday, 12 January, 2015 Permalink | Reply

      You finally had the courage to look at it :) You did a great job :) Hope 2015 will be a googol times better :)

      Like

  • Joseph Nebus 3:00 pm on Sunday, 28 December, 2014 Permalink | Reply
    Tags: BMI, , , , , G.I.Joe, , , , statistics   

    Reading the Comics, December 27, 2014: Last of the Year Edition? 


    I’m curious whether this is going to be the final bunch of mathematics-themed comics for the year 2014. Given the feast-or-famine nature of the strips it’s plausible we might not have anything good through to mid-January, but, who knows? Of the comics in this set I think the first Peanuts the most interesting to me, since it’s funny and gets at something big and important, although the Ollie and Quentin is a better laugh.

    Mark Leiknes’s Cow and Boy (December 23, rerun) talks about chaos theory, the notion that incredibly small differences in a state can produce enormous differences in a system’s behavior. Chaos theory became a pop-cultural thing in the 1980s, when Edward Lorentz’s work (of twenty years earlier) broke out into public consciousness. In chaos theory the chaos isn’t that the system is unpredictable — if you have perfect knowledge of the system, and the rules by which it interacts, you could make perfect predictions of its future. What matters is that, in non-chaotic systems, a small error will grow only slightly: if you predict the path of a thrown ball, and you have the ball’s mass slightly wrong, you’ll make a proportionately small error on what the path is like. If you predict the orbit of a satellite around a planet, and have the satellite’s starting speed a little wrong, your prediction is proportionately wrong. But in a chaotic system there are at least some starting points where tiny errors in your understanding of the system produce huge differences between your prediction and the actual outcome. Weather looks like it’s such a system, and that’s why it’s plausible that all of us change the weather just by existing, although of course we don’t know whether we’ve made it better or worse, or for whom.

    Charles Schulz’s Peanuts (December 23, rerun from December 26, 1967) features Sally trying to divide 25 by 50 and Charlie Brown insisting she can’t do it. Sally’s practical response: “You can if you push it!” I am a bit curious why Sally, who’s normally around six years old, is doing division in school (and over Christmas break), but then the kids are always being assigned Thomas Hardy’s Tess of the d’Urbervilles for a book report and that is hilariously wrong for kids their age to read, so, let’s give that a pass.

    (More …)

     
    • gpcox 12:52 pm on Tuesday, 30 December, 2014 Permalink | Reply

      Lovin’ the funny papers! I stumbled on a copy of The Smithsonian Collection of Newspaper Comics, it includes the Toonerville Trolley and all!!

      Like

      • Joseph Nebus 8:49 pm on Tuesday, 30 December, 2014 Permalink | Reply

        Thank you; I’m glad you like them.

        I used to hover around a used book store which had a collection of the Toonerville Trolley strips in it, but I was just poor enough (grad school, you know) that I couldn’t quite justify buying it. I’m a bit surprised nobody’s brought that back online; surely a wide swath of its panels are in the public domain by now, I’d imagine, but it’s only in the past couple years that Percy Crosby’s Skippy and Winsor McKay’s Little Nemo in Slumberland have gotten onto gocomics.com. King Features has a broader selection of vintage comic strips online, though they don’t let you see them much before buying a subscription.

        Like

  • Joseph Nebus 10:50 pm on Thursday, 18 December, 2014 Permalink | Reply
    Tags: , , , , , statistics   

    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.

    Like

    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.

    View original post

     
  • Joseph Nebus 8:19 pm on Tuesday, 2 December, 2014 Permalink | Reply
    Tags: , , statistics,   

    Advanced November 2014 Statistics 


    So that little bit I added in my last statistics post, tracking how many days went between the first and the last reading of an article according to WordPress’s figures? I was curious, and went through my posts from mid-October through mid-November to see how long the readership lifespan of an average post was. I figured stuff after mid-November may not have quite had long enough for people to gradually be done with it.

    I’d expected the typical post to have what’s called a Poisson distribution, in number of page views per day, with a major peak in the first couple days after it’s published and then, maybe, a long stretch of exceedingly minor popularity. I think that’s what’s happening, although the problem of small numbers means it’s a pretty spotty pattern. Also confounding things is that a post can sometimes get a flurry of publicity long after its main lifespan has passed. So I decided to count both how long each post had between its first and last-viewed days, and also the “first span”, how many days it was until the first day without page views, to use as proxy for separating out late revivals.

    Post Days Read First Span
    How To Numerically Integrate Like A Mathematician 45 8
    Reading the Comics, October 14, 2014: Not Talking About Fourier Transforms Edition 25 7
    How Richard Feynman Got From The Square Root of 2 to e 41 4
    Reading The Comics, October 20, 2014: No Images This Edition 5 5
    Calculus without limits 5: log and exp 25 3
    Reading the Comics, October 25, 2014: No Images Again Edition 28 2
    How To Hear Drums 14 6
    My Math Blog Statistics, October 2014 30 4
    Reading The Comics, November 4, 2014: Will Pictures Ever Reappear Edition 9 6
    Echoing “Fourier Echoes Euler” 12 5
    Some Stuff About Edmond Halley 11 2
    Reading The Comics, November 9, 2014: Finally, A Picture Edition 11 4
    About An Inscribed Circle 13 5
    Reading The Comics, November 14, 2014: Rectangular States Edition 15 1
    Radius of the inscribed circle of a right angled triangle 12 5

    For what it’s worth, the mean lifespan of a post is 19.7 days, with standard deviation of 12.0 days. The mean lifespan of the first flush of popularity is 4.5 days, with a standard deviation of 1.9 days.

    I suspect the thing that brings out these late rushes of popularity are things like the monthly roundup posts, which send people back to articles whose lifespans had expired weeks before; or when there’s a running thread as in the circle-inscribed-in-a-triangle theme that encourages people to go back again and again. And I’m curious how long articles would last without this sort of threading between them.

     
    • elkement 8:50 am on Thursday, 4 December, 2014 Permalink | Reply

      I noticed there are two different kinds of posts (of mine): Those that ‘decay’ as you describe – and about 5-10 that slowly get more and more popular over time.
      Or I see the combination: An initial decay in the first months, then after 6 months the posts gets more and more hits. This can be due to a backlink from a popular site (my most popular post is my review of that QFT book as the author linked back from his site) or because I obviously have included the “right” search terms. Due to whatever reason people search for “sniffing router” or the like and my post on network sniffing is gradually becoming my top post – months after the initial decay.
      And then there is seasonal popularity: Every year in October or November my post about mice getting in microwaves through the rear-side vent has a peak. In summer people search for tomatoes and espalier. So I conclude most of my readers are from the Northern hemisphere :-)

      Like

      • Joseph Nebus 6:24 am on Saturday, 6 December, 2014 Permalink | Reply

        I really only have the one perennial now, my guide to all the different kinds of trapezoids that I could think of. (I’m happy nobody has pointed out an obvious other one that I failed to list, so far.) Now and then one of the Price Is Right posts makes a comeback, and I suspect that the pair about what grade you need on the final to pass the class should be seasonally successful.

        I have spotted a couple of people searching for my Arthur Christmas posts too. I should make a page that collects all those into a reasonably seamless whole for better fundability.

        Liked by 1 person

    • samkhan13 9:18 pm on Thursday, 11 December, 2014 Permalink | Reply

      hmmm… i read a whole bunch of articles by merely scrolling down your home page. wordpress would note the fact that i reached your home page and this particular one but it wont tell you what else i happened to linger on ;)

      in any case, articles that address a very peculiar technical issue seem to get really high number of hits and articles that talk about sex, religion, politics and drugs are also top sellers. unadulterated mathematics or physics, as i see, does not interest the masses to a very great extent. but the bunch who do seem to care about such topics are rather dedicated ^_^

      Like

      • Joseph Nebus 7:06 am on Friday, 12 December, 2014 Permalink | Reply

        This is true. Properly speaking the most popular post I ever have is my home page, but since that’s always there it seems trivial to include. And there’s no way for me to guess what people are lingering over. I’d imagine it to be the most recent couple of posts, but that’s just a simplifying assumption.

        I suspect that my best chance for really well-read posts is to come up with a bunch of slightly mathematical questions — “What is an ansatz?” “What are conjugate variables?” “What is a symplectic integrator?” — and go through each of those. I don’t suppose it’s a coincidence my most popular post is “How many trapezoids are there?”.

        And, alas, my subject matter doesn’t really get me much excuse to talk about politics or religion or drugs or sex that get so thrilling. (Well, they can, but they end up being the religious controversies of mid-17th century England, which rouse the passions relatively slowly these days.)

        Like

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