## The End 2016 Mathematics A To Z: Distribution (statistics)

As I’ve done before I’m using one of my essays to set up for another essay. It makes a later essay easier. What I want to talk about is worth some paragraphs on its own.

## Distribution (statistics)

The 19th Century saw the discovery of some unsettling truths about … well, everything, really. If there is an intellectual theme of the 19th Century it’s that everything has an unsettling side. In the 20th Century craziness broke loose. The 19th Century, though, saw great reasons to doubt that we knew what we knew.

But one of the unsettling truths grew out of mathematical physics. We start out studying physics the way Galileo or Newton might have, with falling balls. Ones that don’t suffer from air resistance. Then we move up to more complicated problems, like balls on a spring. Or two balls bouncing off each other. Maybe one ball, called a “planet”, orbiting another, called a “sun”. Maybe a ball on a lever swinging back and forth. We try a couple simple problems with three balls and find out that’s just too hard. We have to track so much information about the balls, about their positions and momentums, that we can’t solve any problems anymore. Oh, we can do the simplest ones, but we’re helpless against the interesting ones.

And then we discovered something. By “we” I mean people like James Clerk Maxwell and Josiah Willard Gibbs. And that is that we can know important stuff about how millions and billions and even vaster numbers of things move around. Maxwell could work out how the enormously many chunks of rock and ice that make up Saturn’s rings move. Gibbs could work out how the trillions of trillions of trillions of trillions of particles of gas in a room move. We can’t work out how four particles move. How is it we can work out how a godzillion particles move?

We do it by letting go. We stop looking for that precision and exactitude and knowledge down to infinitely many decimal points. Even though we think that’s what mathematicians and physicists should have. What we do instead is consider the things we would like to know. Where something is. What its momentum is. What side of a coin is showing after a toss. What card was taken off the top of the deck. What tile was drawn out of the Scrabble bag.

There are possible results for each of these things we would like to know. Perhaps some of them are quite likely. Perhaps some of them are unlikely. We track how likely each of these outcomes are. This is called the distribution of the values. This can be simple. The distribution for a fairly tossed coin is “heads, 1/2; tails, 1/2”. The distribution for a fairly tossed six-sided die is “1/6 chance of 1; 1/6 chance of 2; 1/6 chance of 3” and so on. It can be more complicated. The distribution for a fairly tossed pair of six-sided die starts out “1/36 chance of 2; 2/36 chance of 3; 3/36 chance of 4” and so on. If we’re measuring something that doesn’t come in nice discrete chunks we have to talk about ranges: the chance that a 30-year-old male weighs between 180 and 185 pounds, or between 185 and 190 pounds. The chance that a particle in the rings of Saturn is moving between 20 and 21 kilometers per second, or between 21 and 22 kilometers per second, and so on.

We may be unable to describe how a system evolves exactly. But often we’re able to describe how the distribution of its possible values evolves. And the laws by which probability work conspire to work for us here. We can get quite precise predictions for how a whole bunch of things behave even without ever knowing what any thing is doing.

That’s unsettling to start with. It’s made worse by one of the 19th Century’s late discoveries, that of chaos. That a system can be perfectly deterministic. That you might know what every part of it is doing as precisely as you care to measure. And you’re still unable to predict its long-term behavior. That’s unshakeable too, although statistical techniques will give you an idea of how likely different behaviors are. You can learn the distribution of what is likely, what is unlikely, and how often the outright impossible will happen.

Distributions follow rules. Of course they do. They’re basically the rules you’d imagine from looking at and thinking about something with a range of values. Something like a chart of how many students got what grades in a class, or how tall the people in a group are, or so on. Each possible outcome turns up some fraction of the time. That fraction’s never less than zero nor greater than 1. Add up all the fractions representing all the times every possible outcome happens and the sum is exactly 1. Something happens, even if we never know just what. But we know how often each outcome will.

There is something amazing to consider here. We can know and track everything there is to know about a physical problem. But we will be unable to do anything with it, except for the most basic and simple problems. We can choose to relax, to accept that the world is unknown and unknowable in detail. And this makes imaginable all sorts of problems that should be beyond our power. Once we’ve given up on this precision we get precise, exact information about what could happen. We can choose to see it as a moral about the benefits and costs and risks of how tightly we control a situation. It’s a surprising lesson to learn from one’s training in mathematics.

## How October 2016 Treated My Mathematics Blog

I do try to get these monthly readership review posts done close to the start of the month. I was busy the 1st of the month, though, and had to fit around the End 2016 Mathematics A To Z. And then I meant to set this to post on Thursday, since I didn’t have anything else going that day, and forgot.

The number of page views declined again in October, part of a trend that’s been steady since June. There were only 907 views, down a slight amount from September’s 922 or more significantly from August’s 1002. I’ll find my way back above a thousand in a month if I can. A To Z months are usually pretty good ones, possibly because of all the fresh posts reminding people I exist.

The number of unique visitors dropped to 536. There had been 576 in September, but then there were only 531 unique visitors in August, if you believe that sort of thing. The number of likes was 115, exactly the same as in September and slightly up from August’s 107. The number of comments rose to 24, up from September’s 20 and August’s 16. That’s certainly been helped by people making requests for the End 2016 Mathematics A To Z. But that counts too.

## Popular Posts:

The most popular post of the month was a surprise to me and dates back to September of 2012, incredibly. I suspect someone on a popular web site linked to it and I never suspected. And the Reading the Comics posts were popular as ever.

I’ve been trying to limit these most-popular posts to just five pieces. But How Mathematical Physics Works was the next piece to make the top ten and I am proud of it, so there.

## Listing Countries:

Where did my readers come from in October? All over, but mostly, from 46 particular countries. Here’s the oddly popular list of them:

United States 466
United Kingdom 78
Philippines 55
India 52
Germany 27
Austria 23
Puerto Rico 19
Australia 14
France 12
Slovenia 10
Spain 9
Brazil 7
Netherlands 7
Italy 6
New Zealand 5
Singapore 5
Denmark 4
Sweden 4
Bulgaria 3
Poland 3
Serbia 3
Argentina 2
European Union 2
Indonesia 2
Norway 2
Bahamas 1
Belgium 1
Czech Republic 1 (**)
Estonia 1 (*)
Finland 1
Greece 1
Ireland 1
Israel 1
Jamaica 1
Japan 1
Mexico 1
Portugal 1 (*)
Russia 1
Saudi Arabia 1
Slovakia 1
South Africa 1
Ukraine 1
United Arab Emirates 1
Uruguay 1
Zambia 1

Estonia and Portugal are on two-month streaks as single-read countries. The Czech Republic’s on a three-month streak so. Nobody’s on a four-month streak, not yet.

## Search Term Non-Poetry:

Once again it wasn’t a truly poetic sort of month. But it was one that taught me what people are looking for, and it’s comics about James Clerk Maxwell. Look at these queries:

• comic strips of the scientist maxwell
• comics trip of james clerk maxwell
• comics about maxwell the scientist
• james clerk maxwell comics trip
• log 10 times 10 to the derivative of 10000
• problems with vinyl lp with too many grooves
• comic strip in advance algebra

I admit I don’t know why someone sees James Clerk Maxwell as a figure for a comics trip. He’s famous for the laws of electromagnetism, of course. Also for great work in thermodynamics and statistical mechanics. Also for color photography. And explaining how the rings of Saturn could work. And for working out the physics of truss bridges, which may sound boring but is important. Great subject for a biography. Just, a comic?

November sees the blog start with 42,250 page views, from 17,747 unique visitors if you can believe that. I’m surprised the mathematics blog still has a higher view count than my humor blog has, just now. That one’s consistently more popular; this one’s just been around longer.

WordPress says I started November with 626 followers, barely up from October’s 624. If you have wanted to follow me, there’s a button on the upper-right corner of the blog for that, at least until I change to a different theme. Also if you know a WordPress theme that would work better for the kind of blog I write let me know. I have a vague itch to change things around and that always precedes trouble. Also you can follow me on Twitter, @Nebusj, or check that out to make sure I’m not one of those people who somehow is hard to Twitter-read.

According to the “Insights” tab my readership’s largest on Sundays, which makes sense. I’ve standardized on Sundays for the Reading the Comics essays. That gets 18 percent of page views, slightly more than one in seven views. The most popular hour is again 6 pm, I assume Universal Time. 14 percent of page views come in that hour. That’s the same percentage as last month and it must reflect when my standard posting hour is.

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• #### Joseph Nebus 5:56 am on Wednesday, 9 November, 2016 Permalink | Reply

Conceivable! Although I suppose I’ve probably hit on a couple of topics that people are perennially if slightly looking for. And I have the advantage of writing in English, which so much of the Internet still depends upon. (I suppose it can’t hurt I’ve been trying to write sentences easier to understand, which is good for all readers as long as I don’t get simpler than the idea I mean to express.)

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• #### davekingsbury 5:03 pm on Wednesday, 9 November, 2016 Permalink | Reply

Popularising maths and the sciences is a valuable art – long may you continue!

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## How September 2016 Treated My Mathematics Blog

I put together another low-key, low-volume month in September. In trade, I got a low readership: my lowest in the past twelve months, according to WordPress, and less than a thousand readers for the first time since May. Well, that’s a lesson to me about something or other.

So there were only 922 page views around here, down from August’s 1,002 and July’s 1,057. The number of distinct readers rose, at least, to 575. There had been only 531 in August. But there were 585 in July, which is the sort of way it goes.

The number of likes rose to 115, which is technically up from August’s 107. It’s well down from July’s 177. There were 20 comments in September, up from August’s 16 yet down from July’s 33. I think this mostly reflects how many fewer posts I’ve been publishing. There were just eleven original posts in August and September, compared to, for example, July’s boom of 17. I am thinking about doing a new A To Z round to close out the year, which if past performance is any indication would bring me all sorts of readers as well as make me spend every day writing, writing, writing and hoping for any kind of mathematics word that starts with ‘y’.

## Popular Posts:

I’m not surprised that my most popular post for September was a Reading the Comics post. With hindsight I realize it’s almost perfectly engineered for reliable readership. It’s about something light but lets me, at least in principle, bring up the whole spectrum of mathematics. That said I am surprised two of the most popular posts were stepped deep into Freshman Calculus, threatening to be inaccessible to casual readers. But then both of those posts started out when online friends needed help with their calculus work, so maybe it better matches stuff people need to know. The most-read articles around here in September were:

## Listing Countries:

United States 808
India 53
United Kingdom 34
New Zealand 24
Australia 23
Germany 18
Philippines 17
France 9
Argentina 8
Spain 7
Singapore 6
Brazil 6
Kenya 5
Switzerland 5
Austria 3
Denmark 3
Indonesia 3
Italy 3
Netherlands 3
South Africa 3
Uruguay 3
Bulgaria 2
Croatia 2
Cyprus 2
Greece 2
Israel 2
Japan 2
Malaysia 2
Mexico 2
Norway 2
Puerto Rico 2
Sweden 2
Turkey 2
Costa Rica 1
Czech Republic 1 (*)
Estonia 1
European Union 1
Hong Kong SAR China 1
Hungary 1
Mauritius 1
Poland 1
Portugal 1
Romania 1
Taiwan 1

Czech Republic was the only single-reader country last month, and no country’s on a two- or more-month single-reader streak. European Union dropped from three page views so I don’t know what they’re looking for that they aren’t finding here.

## Search Term Non-Poetry:

Nothing all that trilling among the search terms, although someone’s on a James Clerk Maxwell kick. Among things that brought people here in September were:

• how many grooves on a record
• james clerk maxwell comics strip
• james clerk maxwell comics
• james clerk maxwell comics stript about scientiest
• james clerk maxwell comics streip photos
• james clerk maxwell comics script scientist
• record groove width in micrometers
• example of comics strip of maxwell

Definitely have to commission someone to draw a bunch of James Clerk Maxwell comics.

October starts with the mathematics blog at 41,318 page views from 17,189 recorded distinct visitors. (The first year or so of the blog WordPress didn’t keep track of distinct visitors, though, or at least didn’t tell us about them.)

WordPress’s “Insights” tab tells me the most popular day for reading stuff here is Sunday, with 18 percent of page views coming then. Since that’s the designated day for Reading the Comics posts that doesn’t surprise me. The most popular hour is 6 pm, which gets 14 percent of readers in. That must be because I’ve set 6 pm Universal Time as the standard moment when new posts should be published.

WordPress says I start October with 624 total followers, up modestly from September’s 614 base. There’s a button on the upper-right corner to follow this blog on WordPress. Below that is a button to follow this blog by e-mail. And if you’d like you can follow me on Twitter too, where I try to do more than just point out I’ve posted stuff here. But also to not post so often that you wonder if or when I rest.

• #### davekingsbury 9:52 pm on Wednesday, 5 October, 2016 Permalink | Reply

I wonder if readership is down generally. My own seems to have slumped a bit …

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• #### Joseph Nebus 12:13 am on Tuesday, 11 October, 2016 Permalink | Reply

I wonder. I ought to poke around other people’s readership reports and see what their figures are like, and whether there’s any correlations. But that’s also a lot of work, by which I mean any work at all. I’m not sure about going to the trouble of actually doing it.

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## Some More Mathematics Stuff To Read

And some more reasy reading, because, why not? First up is a new Twitter account from Chris Lusto (Lustomatical), a high school teacher with interest in Mathematical Twitter. He’s constructed the Math-Twitter-Blog-o-Sphere Bot, which retweets postings of mathematics blogs. They’re drawn from his blogroll, and a set of posts comes up a couple of times per day. (I believe he’s running the bot manually, in case it starts malfunctioning, for now.) It could be a useful way to find something interesting to read, or if you’ve got your own mathematics blog, a way to let other folks know you want to be found interesting.

Also possibly of interest is Gregory Taylor’s Any ~Qs comic strip blog. Taylor is a high school teacher and an amateur cartoonist. He’s chosen the difficult task of drawing a comic about “math equations as people”. It’s always hard to do a narrowly focused web comic. You can see Taylor working out the challenges of writing and drawing so that both story and teaching purposes are clear. I would imagine, for example, people to giggle at least at “tangent pants” even if they’re not sure what a domain restriction would have to do with anything, or even necessarily mean. But it is neat to see someone trying to go beyond anthropomorphized numerals in a web comic. And, after all, Math With Bad Drawings has got the hang of it.

Finally, an article published in Notices of the American Mathematical Society, and which I found by some reference now lost to me. The essay, “Knots in the Nursery:(Cats) Cradle Song of James Clerk Maxwell”, is by Professor Daniel S Silver. It’s about the origins of knot theory, and particularly of a poem composed by James Clerk Maxwell. Knot theory was pioneered in the late 19th century by Peter Guthrie Tait. Maxwell is the fellow behind Maxwell’s Equations, the description of how electricity and magnetism propagate and affect one another. Maxwell’s also renowned in statistical mechanics circles for explaining, among other things, how the rings of Saturn could work. And it turns out he could write nice bits of doggerel, with references Silver usefully decodes. It’s worth reading for the mathematical-history content.

• #### elkement (Elke Stangl) 1:55 pm on Friday, 22 January, 2016 Permalink | Reply

Your blog is really an awesome resource for all things math, no doubt!!

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• #### Joseph Nebus 5:02 am on Sunday, 24 January, 2016 Permalink | Reply

That’s awfully kind of you to say. I’ve really just been grabbing the occasional thing that comes across my desk and passing that along, though, part of the great chain of vaguely sourced references.

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• #### elkement (Elke Stangl) 8:48 am on Sunday, 24 January, 2016 Permalink | Reply

But ‘curating’ as they say today is an art, too, and after all you manage to make things accessible, e.g. by summarizing posts you reblog so neatly…. and manage to do so without much images!!

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• #### Joseph Nebus 10:12 pm on Tuesday, 26 January, 2016 Permalink | Reply

Well, thank you again. I do feel like if I’m pointing to or reblogging someone else’s work I should provide a bit of context and original writing. It’s too easy to just pass around a link and say “here’s a good link”, which I wouldn’t blame anyone for doubting.

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## The Geometry of Thermodynamics (Part 2)

I should mention — I should have mentioned earlier, but it has been a busy week — that CarnotCycle has published the second part of “The Geometry of Thermodynamics”. This is a bit of a tougher read than the first part, admittedly, but it’s still worth reading. The essay reviews how James Clerk Maxwell — yes, that Maxwell — developed the thermodynamic relationships that would have made him famous in physics if it weren’t for his work in electromagnetism that ultimately overthrew the Newtonian paradigm of space and time.

The ingenious thing is that the best part of this work is done on geometric grounds, on thinking of the spatial relationships between quantities that describe how a system moves heat around. “Spatial” may seem a strange word to describe this since we’re talking about things that don’t have any direct physical presence, like “temperature” and “entropy”. But if you draw pictures of how these quantities relate to one another, you have curves and parallelograms and figures that follow the same rules of how things fit together that you’re used to from ordinary everyday objects.

A wonderful side point is a touch of human fallibility from a great mind: in working out his relations, Maxwell misunderstood just what was meant by “entropy”, and needed correction by the at-least-as-great Josiah Willard Gibbs. Many people don’t quite know what to make of entropy even today, and Maxwell was working when the word was barely a generation away from being coined, so it’s quite reasonable he might not understand a term that was relatively new and still getting its precise definition. It’s surprising nevertheless to see.

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James Clerk Maxwell and the geometrical figure with which he proved his famous thermodynamic relations

Historical background

Every student of thermodynamics sooner or later encounters the Maxwell relations – an extremely useful set of statements of equality among partial derivatives, principally involving the state variables P, V, T and S. They are general thermodynamic relations valid for all systems.

The four relations originally stated by Maxwell are easily derived from the (exact) differential relations of the thermodynamic potentials:

dU = TdS – PdV   ⇒   (∂T/∂V)S = –(∂P/∂S)V
dH = TdS + VdP   ⇒   (∂T/∂P)S = (∂V/∂S)P
dG = –SdT + VdP   ⇒   –(∂S/∂P)T = (∂V/∂T)P
dA = –SdT – PdV   ⇒   (∂S/∂V)T = (∂P/∂T)V

This is how we obtain these Maxwell relations today, but it disguises the history of their discovery. The thermodynamic state functions H, G and A were yet to…

View original post 1,262 more words

• #### elkement 11:24 am on Tuesday, 23 September, 2014 Permalink | Reply

carnotcycle has to be turned into a book :-)

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• #### Joseph Nebus 11:37 pm on Wednesday, 24 September, 2014 Permalink | Reply

It should become one! I wouldn’t be surprised if that’s in mind, actually, particularly given the deliberate pace with which the articles are being written.

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