Power Law

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power-law

A Power Law curve.

Take a look at the image to the right. It is very likely the most important chart they never taught you about in school. Failing that, it’s almost certainly the most important chart that you may have seen from time to time and learned a bit about but never realized the significance of.

What is it? Depending upon how you label the axes, it could represent a whole host of things. In fact, it’s kind of shocking just how many things in the real world this curve represents. Depending upon the dataset, you may have to flip the curve.

It goes by several names: Exponential distribution. Logarithmic function. But one of the most popular names, and perhaps the most ominous, is the Power Law curve. From Wikipedia:

In statistics, a power law is a functional relationship between two quantities, where one quantity varies as a power of another.

Or, more simply, it’s what you get when you plot out a function that looks like this:

f(x) = axk

normal

Normal Distribution – the “Bell Curve”

The name “Power Law” comes from the k – the “power” or exponent in the function. If you’ve had any introduction to statistics at all, you’re probably much more familiar with the chart to the left – the famous “Bell Curve,” or, more accurately, the Normal Distribution. Unless you took an actual “full” statistics course from the math department (many science majors these days get away with “statistics for <insert your department> majors” instead of taking the one from the actual math department), you probably didn’t spend enough time studying it. Even if you did take a “real” statistics course, you probably didn’t fully appreciate the significance of it. Don’t feel bad. It’s very likely that your professor didn’t appreciate it, either.

Most people are taught that the Normal Distribution is the most common distribution that you find in nature. Indeed, this is why it’s called the “normal” distribution. It’s very common. Quite a few people, however, mistakenly believe that it’s the only distribution you find in nature – that everything follows a bell curve distribution.

The second variant is patently false. Quite a few things follow other distributions (these are not the only two you will find; there are quite a few others). The first formulation, though, isn’t quite accurate either. Yes, a lot of things follow normal distributions. But quite a lot of things don’t. In fact, the prevalence of the normal distribution is actually what leads to so many cases of the power law distribution.

Because there are some key things about the power law distribution that you were never taught. Here are a few:

Power Law Distributions arise from iteration of Normal Distributions

OK, what does that mean? Let’s consider the following very generic set of circumstances:

  1. A competitive event.
  2. The population of competitors is unequal
  3. The inequality is distributed along something resembling a normal distribution.
  4. Winners from any given round of competition keep their winnings.
  5. The winnings form any round confer an advantage in subsequent rounds.
  6. Competition is iterated over multiple rounds.

Whenever these six conditions are met, after many rounds of competition the results will always form a power law curve. Always. Without exception.

With Power Law Distributions, Averages are Meaningless

Not just averages. The mean, median and mode are all meaningless in a power law distribution. They literally tell us nothing. We are taught as early as middle school to use these numbers to analyze large datasets. Indeed, for many of us they are the only ways we know to get meaning out of those sets. But for anything that follows a power law curve, they literally tell us nothing about the data. Well, OK, not quite nothing:

If the mean (average) and the median are wildly different, that’s a strong hint that the data actually follows a power law curve. In a normal distribution, they will be very close together. In a power law distribution they might be, but they probably won’t be. [The converse is not true: a power law distribution doesn’t necessarily have wildly different medians and means.]

What it all Means

The Power Law curve, combined in some cases with other important information, is the iron law of mathematics defining why:

  • Income and wealth inequality are an inescapable part of the human condition.
  • Top members of some professions (sports, film, tv, music, writing, journalism, and many others) make millions while nobody else can even make enough to live on.
  • Globalism is very bad for everyone except those at the very top.
  • Current levels of immigration in the US are too high.
  • Tenure is bad.
  • Big corporations are terrible.
  • Big government is worse.
  • Most mature industries coalesce around a very small number of very large firms.

And much, much more. But more on those another day.

Side Business – You Should Have One

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I’ve long been a believer in small business. I believe it’s the lifeblood of the economy. Small businesses (those with less than 500 employees) have created 64% of all new jobs in the US since 1995. I also believe it’s the path to true prosperity for most individuals. But for most of my life I’ve believed that you needed your small business to be your full time job for it to be worthwhile.

In the last five years, I’ve learned better. Most people should be operating a small side business. My wife and I currently operate two side businesses (a third business we operate is technically and legally part of one of the others) – and they’ve been really great for us.

Side businesses have several huge advantages over businesses that require you to work full time.

  • You don’t have to quit your day job, so you don’t lose that income.
  • You don’t have to find the cash to pay yourself a full time salary that replaces your day job, so it’s much easier to make the business profitable. In fact, with some kinds of businesses they can be profitable – or nearly so – from day 1 (Ever After Videography was profitable after Morgon’s first paying job).
  • If you pick the right business, the major investment can be your time rather than your money.
  • You don’t have to make a full time living at it, so you can scale the business up or down to fit your needs. If the workload is too high, scale it back.
  • Keeping it part time can make it a great fit for a “stay at home” spouse.
  • You get (almost) all the same tax benefits as if you had a full time business.

A quick example that combines several points: my wife is a stay-at-home-mother. Her side business does videography. She does all kinds of video work, but the main “bread and butter” work is weddings. This is a wonderful fit. Most of her workload occurs on Friday evenings or Saturdays – times when I’m already home. So we very seldom have to pay extra for childcare for her to work (we do, very occasionally – 4 to 5 times a year – have to make use of a drop-in daycare facility near us; the kids go so seldom that they view it as a treat).

The business has been profitable since the first paying job we landed her. Now… it’s not making us enough money for me to quit my day job. It’s not even making her what a typical “professional” job would make. But for about a dozen weekends a year worth of work, she makes more money than she would if she worked full time at a fair amount above minimum wage. That alone isn’t bad.

But on top of that, there are quite a few tax benefits to be had for having income in the stay-at-home-spouse’s name. You can now contribute to retirement accounts for the second spouse, not just the first. Dependent care expenses become tax deductible, if they help the second spouse work (or go to school). A lot of things you’re already paying for can become legitimate tax deductions (be careful with this, though – you do need to understand the rules well; or better yet, get an accountant who does).

I just finished doing our taxes for the year, and the business deductions alone from my wife’s business just impacted our personal tax bill to the point where they effectively increased her net profit by 25% (anybody need 20% off on TurboTax?). Now… if you grow your business past a certain point, it will definitely increase your tax bill – but then, you’ll be making a tidy profit at that point, too.

Think carefully before you jump in – not every business model will work as a side business. And you need a plan, not just an idea. But there can be a lot of benefits. What side business are you thinking of?

Hillary’s EMail Server Might Actually Have Been Legal

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In typical Clintonian fashion, today’s press conference leaves a lot of questions unanswered. However, it does provide a small clue around which the whole thing might hinge:

She also said her server, set up for President Bill Clinton’s office, contained “numerous safeguards,” was protected by the Secret Service and experienced “no security breaches.”

The question is, set up by whom? As a former US President, Bill Clinton is entitled to an office and a staff paid for on the taxpayer dime. If the server was paid for via government funds and operated by government staff, then technically Hillary never violated the law – her “private” email was a government email all along.

If this is the case, though, then I wonder why she didn’t just come back with that defense right away. My guess? That despite their hinting at it, this isn’t actually the case.

Update 3/12/15: Nevermind, it wasn’t legal after all:

A Clinton spokesman also emphasized that no taxpayer money was used to purchase or maintain a private email server operated for the former president’s staff starting in 2007, which Hillary Clinton started using in 2009.

Ghost of the Frost Giant King FLASH SALE!

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Not only is the Ghost of the Frost Giant King PDF version finalized and ready for download, but we messed up and it got released early. Now we COULD roll it back and turn it off for another week… but why do that when we could have a FLASH SALE instead! Purchase the PDF version this weekend only for only $2.99! To get this special price, you must use this special link!

For those who would rather have a physical product, the print edition will be available very soon!

Ghost of the Frost Giant King

Click this link to get the special FLASH SALE price!

 

Transhuman and Subhuman

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The second title from Larry Correia’s book bomb that I’ll be reviewing today is “Transhuman and Subhuman: Essays on Science Fiction and Awful Truth” by Mr. John C. Wright. Yes, I just reviewed another of Mr. Wright’s works. And yes, Mr. Correia just book bombed that one as well.

They’re both worthy of it.

But first, a story: A friend of mine joined the Roman Catholic Church last Easter. My wife and I are also converts to the church, and even though we weren’t able to sponsor him or follow him through RCIA due to other commitments on the nights that it meets, we were happy to see him join and wanted to welcome him. So we bought him a book. No, it wasn’t this one. We bought him G. K. Chesterton’s Christian Writings – a solid choice for anyone interested in Catholic thought.

A few weeks later we were having lunch with our friend and somehow it came up that we had both read another of Mr. Wright’s works, Awake in the Night Land. Conversation progressed a bit until eventually it emerged that my friend had not read any Chesterton before we bought him his gift. After that revelation, conversation went something like this:

Me: Yeah, I really feel like John C. Wright is Chesterton come again for our modern age.

My friend: It’s funny you say that, because I was just about to say that Chesterton reminded me of John C. Wright.

Nowhere is that more apparent than in this work, a collection of philosophical ruminations on modern culture. From his thorough and complete explanation of why the recent Hobbit movies are so completely terrible to an Aristotelian explanation of why Snow White has animal helpers to ruminations on the value of science fiction itself… these essays are truly amazing in every way. He even managed to explain why I didn’t really like The Golden Compass, which was a book that never satisfied but I couldn’t quite put my finger on why – until now.

I have exactly one complaint about this collection, and Mr. Wright will probably hate me for saying this as it might impact his sales: almost all of these essays are also available for FREE (albeit in less polished form) on his blog. If you feel like wading through all the other stuff (and you should actually go read all of that, too), you can save yourself a whopping $4.99. Or you could just pay a little bit, give the man what he’s due for such amazing writing, and get the nicely collected, well-edited version. Yeah, do that one.

The Hot Equations

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Riding the Red Horse

Riding the Red Horse

The indefatigable Larry Correia is running another book bomb today. Again, as it happens, I’ve already purchased and read two of the works on the slate. So today I will contribute by reviewing those two works.

First on today’s list is “The Hot Equations: Thermodynamics and Military SF” by Ken Burnside. This essay is part of a wider collection of military science fiction stories and essays about military science fiction entitled Riding the Red Horse.

There’s no other way around it: this essay is essential reading for aspiring writers of space based “hard” science fiction. The main gist of the essay is this: the laws of thermodynamics impose limitations on space travel. Space opera and science fantasy mostly just ignore these problems, but even most “hard” science fiction barely acknowledges them, much less actually account for them. If you’re going to write “hard” science fiction, you need to deal with these issues.

The essay covers the relevant material in a thorough but concise manner. Importantly for most laymen – especially of the aspiring writer variety – he also touches on the concepts with minimal amounts of math. You will understand this essay. And if you’re planning to write about space travel, you should read and understand this essay.

Non-writers who are simply interested in the topic will also find this essay to be of interest… unless you’re an actual rocket scientist, in which case you’ll probably find it boring.

Update 6/3/15:

This essay was actually the direct inspiration for my story “The Fourth Fleet” which appears in the anthology Make Death Proud to Take Us. I leave it to the readers to decide if the story honors the science of “The Hot Equations” – or if it’s any good.

Violence Solves Everything

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Violence Solves Everything

Violence Solves Everything

Whoever said that violence never solved anything had no connection to reality. While it is not quite so that violence solves everything, it is a readily observable truth that violence does, in fact, solve rather a lot of problems. What we should be telling ourselves instead is that it’s a bad solution.

But sometimes – just sometimes – it is actually the least bad solution.

A 15-30 minute read!

Little Johnny isn't Afraid of the Dark

He's afraid of the things that come out in the dark.

Enter yBig boys aren't afraid of the dark - and at five, Little Johnny is a big boy. No, he isn't afraid of the dark. He's afraid of the things that come out in the dark, and what they might do to his baby sister. But his parents don't believe him - so it's up to Johnny to keep her safe, armed only with his toy sword and shield.our text here...

The first of the

Tales of Peter Bishop

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