The law of small numbers

A few days ago, I had a perfectly dreadful day.

The events varied from the truly tragic (receiving news that a former student had died) to the bad but mundane (losing a ghostwriting job I'd been asked to do because the person I was working for turned out to be a lunatic, and had decided I was part of a conspiracy against him -- the irony of which has not escaped me) to the "I'll-probably-laugh-about-this-later-but-right-now-I'm-not" (my dog, Guinness, recovering from his recent illness, and feeling chipper enough to swipe and destroy my wife's favorite hat) to the completely banal (my computer demanding an operating system update when I was in the middle of working, tying it up for two and a half hours).

All of this brought to mind the idea of streaks of bad (or good) luck -- something that you find people so completely convinced of that it's nearly impossible to get them to break their conviction that it sometimes happens.  We've all had days when everything seems to go wrong -- when we have what my dad used to call "the reverse Midas touch -- everything you touch turns to shit."  There are also, regrettably fewer, days when we seem to have inordinate good fortune.  My question of the day is: is there something to this?

Of course, regular readers of this blog are already anticipating that I'll answer "no."  There are actually three reasons to discount this phenomenon.  Two have already been the subjects of previous blog posts, so I'll only mention them in brief.

One is the fact that the human brain is wired to detect patterns.  We tend to take whatever we perceive and try to fit it into an understandable whole.  So when several things go wrong in a row -- even when, as with my experiences last week, they are entirely unrelated occurrences -- we try to make them into a pattern.

The second is confirmation bias -- the tendency of humans to use insignificant pieces of evidence to support what we already believe to be true, and to ignore much bigger pieces of evidence to the contrary.  I had four bad things, of varying degrees of unpleasantness, occur one day last week.  By mid-day I had already decided, "this is going to be a bad day."  So any further events -- the computer update, for example -- only reinforced my assessment that "this day is going to suck."  Good things -- like the fact that even though our dog is back to getting into trouble, he is recovering; like the the fact that we've been enjoying the International Ceramics Congress workshops this weekend; like the fact that lovely wife brought me a glass of red wine after dinner -- get submerged under the unshakable conviction that the day was a lost cause.

It's the third one I want to consider more carefully.

I call it the Law of Small Numbers.  Simply put: in any sufficiently small data sample, you will find anomalous, and completely meaningless, patterns.

To take a simple model: let's consider flipping a fair coin.  You would expect that if you flip said coin 1000 times, you will find somewhere near 500 heads and 500 tails. On the other hand, what if you look at any particular run of, say, six flips?

[Image licensed under the Creative Commons ICMA Photos, Coin Toss (3635981474), CC BY-SA 2.0]

In any six-flip run, the statisticians tell us, all possible combinations are equally likely; a pattern of HTTHTH has exactly the same likelihood of showing up as does HHHHHH -- namely, 1/64.  The problem is that the second looks like a pattern, and the first doesn't.  And so if the second sequence is the one that actually emerges, we become progressively more amazed as head after head turns up -- because somehow, it doesn't fit our concept of the way statistics should work.  In reality, if the second pattern amazes us, the first should as well -- when the fifth coin comes up tails, we should be shouting, "omigod, this is so weird" -- but of course, the human mind doesn't work that way, so it's only the second run that seems odd.

Another thing is that in the second case, the six-flip run of all heads, when it come to the seventh flip, what will it be?  It's hard for people to shake the conviction that after six heads, the seventh is bound to be tails, or at least that tails is more likely.  In fact, the seventh flip has exactly the same likelihood of turning up heads as all the others -- 1/2.

All of this brings up how surprisingly hard it is for statisticians to model true randomness.  If a sequence of numbers (for example) is actually random, all possible combinations of two numbers, three numbers, four numbers, and so on should be equally likely.  So, if you have a truly random list of (say) ten million one-digit numbers, there is a possibility that somewhere on that list there are ten zeroes in a row.  It would look like a meaningful pattern -- but it isn't.

This is part of what makes it hard to create truly randomized multiple-choice tests.  As a former science teacher, I frequently gave my classes multiple-choice quizzes, and I tried to make sure that the correct answers were placed fairly randomly.  But apparently, there's a tendency for test writers to stick the correct answer in the middle of the list -- thus the high school student's rule of thumb, which is, "if you don't know the answer, guess 'c'."

Randomness, it would seem, is harder to detect (and create) than most people think.  And given our tendency to see patterns where there are none, we should be hesitant to decide that the stars are against us on certain days.  In fact, we should expect days where there are strings of bad (or unusually good) occurrences.  It's bound to happen.  It's just that we notice it when several bad things happen on the same day, and don't tend to notice when they're spread out, because that, somehow, "seems more random" -- when, in reality, both distributions are random.

I keep telling myself that.  But it is hard to quell what my mind keeps responding -- "thank heaven it's a new week - it's bound to be better than last week was."

Well, maybe.  I do agree with another thing my dad used to tell me: "I'd rather be an optimist who is wrong than a pessimist who is right."  I'm just hoping that the statisticians don't show up and burst my bubble.

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It's astonishing to see what the universe looks like on scales different from those we're used to.  The images of galaxies and quasars and (more recently) black holes are nothing short of awe-inspiring.  However, the microscopic realm is equally breathtaking -- which you'll find out as soon as you open the new book Micro Life: Miracles of the Microscopic World.

Assembled by a team at DK Publishers and the Smithsonian Institution, Micro Life is a compendium of photographs and artwork depicting the world of the very small, from single-celled organisms to individual fungus spores to nerve cells to the facets of a butterfly's eye.  Leafing through it generates a sense of wonder at the complexity of the microscopic, and its incredible beauty.  If you are a biology enthusiast -- or are looking for a gift for a friend who is -- this lovely book is a sure-fire winner.  You'll never look the same way at dust, pollen, algae, and a myriad of other things from the natural world that you thought you knew.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]

The grand plan

When I'm teaching the unit on evolution in my biology classes, one of the hardest ideas to expunge from my students' brains is that evolution is goal-oriented.

Take, for example, the worn-out example, used in every seventh-grade life science textbook, about giraffes' long necks.  Why do they have these outlandish proportions?

So they can reach food higher up in trees, of course.

The subtle error here is that it implies that a bunch of short-necked giraffes were standing around on the African savanna, looking longingly up at the tempting foliage higher up, and one said, "Dude.  It'd be nice if we could reach higher, don't you think?"  And another said, "Well, we're kinda screwed, because we're short.  But if we had longer-necked kids, that'd be cool, yeah?"

The other giraffes agreed, and lo, they had long-necked kids.

[Image licensed under the Creative Commons Universalwin1222, Lamarckian inheritance- Giraffes, CC BY-SA 4.0]

Okay, I'm oversimplifying, here, but the gist is correct.  The assumption of goal-orientation puts the cart before the horse; that there was some sort of end product evolution had in mind, so organisms headed that way.  Of course, the truth is both simpler and more complex.  What Darwin actually said was that organisms vary from some other cause (being pre-Mendel, he didn't know about genes and mutations), and the environment selects the ones that work the best.  The others die off, taking their deleterious genes with them.

No goal necessary.

The ultimate in goal-orientation, of course, is strict creationism, which posits a designer who created everything as it currently is.  The immediate problem with this -- besides the fact that there's no evidence for it whatsoever -- is that there are a lot of things that seem, well, poorly designed.  The creationists are fond of trotting out some examples of complex structures that work pretty well (such as the eye) and conveniently ignore some examples of seriously poor design (such as the male urinary/reproductive system, which routes the urethra through the prostate gland, making a lot of older guys seriously unhappy).

All of this goal-orientation is known to philosophers as "teleological thinking" -- the attribution of a final cause or goal in natural processes.  And just last week, a paper came out of some research in France that suggests teleological thinking as a commonality between creationism... and conspiracy theories.

The study, done by Pascal Wagner-Egger, Sylvain Delouvée, Nicolas Gauvrit, and Sebastian Dieguez, found an interesting set of correspondences:

Although teleological thinking has long been banned from scientific reasoning, it persists in childhood cognition, as well as in adult intuitions and beliefs.  Noting similarities between creationism (the belief that life on Earth was purposefully created by a supernatural agent) and conspiracism, we sought to investigate whether teleological thinking could underlie and associate both types of beliefs. First, we sought to establish whether teleological thinking, classically associated with creationism, was also related to conspiracist beliefs. College students filled a questionnaire including teleological claims and conspiracist statements, as well as measures of analytical thinking, esoteric and magical beliefs, and a randomness perception task.  Promiscuous teleology — the tendency to ascribe function and a final cause to nonintentional natural facts and events — was significantly... correlated with conspiracist beliefs scales.  In addition, teleological thinking was negatively related to analytical thinking, and positively to esoteric beliefs, which in turn were both related to acceptance of conspiracist beliefs.

The results are perhaps not terribly surprising, although I don't know if anyone previously has linked them this way.  Both creationism and conspiracy theories imply a belief in a Grand Plan -- benevolent in the case of creationism, malevolent in the case of conspiracy theories.  Adherents to either tend to be repelled by the idea of chaos, that things just happen because they happen.  (Thus "even when bad things happen, God has a plan" from the former, and the steadfast refusal by the latter to believe that any unpleasant event might just be random bad luck.)

I'd add one more piece to this, however, and that's the determination by both to avoid or explain away facts that contradict their favorite model of how the universe works.  Of course, that unites them with some other groups that aren't necessarily thinking teleologically, such as the anti-vaxxers.

Although the anti-vaxxers tend to believe that there's a huge coverup by "Big Pharma" of the horrific side effects of vaccination, so maybe there's some overlap there, too.

Anyhow, I thought the whole thing was interesting.  And it does bear mention that the students who are the most repelled by evolution for non-religious reasons tend to be the ones who hate the idea that so much of the world could be the result of randomness.  How can the biodiversity on the Earth, with all of its "endless forms most beautiful and most wonderful" (to quote Darwin), be produced by chance mutations?

Of course, the universe is not compelled to be organized in such a way that it makes you comfortable.  The evidence is very much in favor of the idea that mutations plus selection have generated all of the life forms you see around you.  And since selection is a "whatever works" sort of process, it's unsurprising that sometimes it creates designs of dubious logic -- such as the urethra/prostate situation I mentioned above, which a friend of mine calls "routing a sewer pipe through a playground."

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This week's Skeptophilia book recommendation is a classic, and especially for you pet owners: Konrad Lorenz's Man Meets Dog.  In this short book, the famous Austrian behavioral scientist looks at how domestic dogs interact, both with each other and with their human owners.  Some of his conjectures about dog ancestry have been superseded by recent DNA studies, but his behavioral analyses are spot-on -- and will leaving you thinking more than once, "Wow.  I've seen Rex do that, and always wondered why."

[If you purchase the book from Amazon using the image/link below, part of the proceeds goes to supporting Skeptophilia!]

Back to the drawing board

A while back, I was interviewed by Robert Chazz Chute on his online show The Cool People Podcasts, and I was asked an interesting question.

A number of interesting questions, actually, but one specific one stands out.  Chute asked me if my dedication to skepticism and evidence-based argument had ever shown me to be wrong about something I previously believed to be true.

I said, "Sure," but when pressed, the only examples I could think of were fairly low-key, such as when I found out that low-level laser light can stimulate wound healing, something that initially sounded like woo to me.

But that's not really the same as having a prior belief overturned.  So I came up empty-handed, which was a little awkward, although I did maintain that if even my deepest-held beliefs were shown to be false by hard evidence, I would have no choice but to revise my worldview.

I had a more interesting opportunity to walk the talk yesterday, when I came across a new scholarly study of conspiracy theorists,  a topic near and dear to my heart.  I have claimed more than once that I thought that the heart of conspiracy theories was a desire to find meaning in chaos -- that any pattern, even a horrible one, was better than there being no pattern at all.

Which conclusion was completely unsupported by Sebastian Dieguez, Pascal Wagner-Egger, and Nicole Gauvrit of the Department of Psychology at the University of Fribourg.  Their paper, entitled "Nothing Happens by Accident, or Does It?  A Low Prior for Randomness Does Not Explain Belief in Conspiracy Theories," found no correlation between belief in conspiracies and a belief that things can't happen at random.

Here's how Dieguez et al. explain their findings:

Belief in conspiracy theories has often been associated with a biased perception of randomness, akin to a nothing-happens-by-accident heuristic.  Indeed, a low prior for randomness (i.e., believing that randomness is a priori unlikely) could plausibly explain the tendency to believe that a planned deception lies behind many events, as well as the tendency to perceive meaningful information in scattered and irrelevant details; both of these tendencies are traits diagnostic of conspiracist ideation. In three studies, we investigated this hypothesis and failed to find the predicted association between low prior for randomness and conspiracist ideation, even when randomness was explicitly opposed to malevolent human intervention.  Conspiracy believers’ and nonbelievers’ perceptions of randomness were not only indistinguishable from each other but also accurate compared with the normative view arising from the algorithmic information framework.  Thus, the motto “nothing happens by accident,” taken at face value, does not explain belief in conspiracy theories.

I was pretty surprised by this, largely because I was so certain that I was on to the root cause of conspiracy theories.  But apparently, the Truth-Is-Out-There Cadre are no more likely to see meaning in noise than the rest of us.

So what, then, does unite the True Believers?  Because they have some pretty wacko ideas, and those have to come from somewhere, you know?  Just in the last few days, we have had:

• Michael Savage claiming that President Obama has staged the Syrian refugee crisis in order to overthrow the US government
• Not only the Moon landing, but every space mission launched by any country is a hoax
• The Rothschild family deliberately sank the Titanic in order to set up the Federal Reserve
• Anti-HPV vaccine Gardasil is a "bioweapon"
• Archaeologists are staging a systematic coverup of evidence that there have been intelligent life-forms on the Earth for tens of millions of years, including a 5.3 million-year-old site that has vehicle tire tracks

This last one generated the greatest number of wackos coming out of the woodwork, and resulted in comments like the following:

What people fail to take into account is the molecular destabilization and rapid metamorphosis that occurred when the the pyramids power source failed.  This likely occurred in the time of the flood of Noah.  So the Noah’s pickup truck hypothesis is not that far fetched as it would seam [sic].

Look this up: Limestone, Concrete and Granite are the same material only in different metamorphic states.  I think when the pyramids went haywire things got very molecularly unstable for a period of time.  This theory explains all the crazy imprints in what should have been solid rock found all around the world. Particularly in granite.

In fact perhaps that is what actually weakened the crust enough to open “the fountains of the deep” (reference: hydroplate theory). 

For anyone who has been slacking the pyramids where [sic] something like a Tesla coil energy system and at one time likely housed a power source known as the Tetragrammaton.

So yeah.  What would make someone believe that, if not a desire to make sense of a world that is mostly composed of chaos?  I mean, that's honestly why science appeals to me; it puts at least some sense of order to the randomness, gives us deep explanations of the perplexing, provides a heuristic for winnowing out fact from fiction.

And science has a pretty good track record for being right.  Unlike crazy talk about Tesla coils powering pyramids to cause "molecular destabilization."

[image courtesy of the Wikimedia Commons]

But if Dieguez et al.'s research bears up under scrutiny, the appeal of conspiracy theories must lie elsewhere.  Are they generated from fear, from the same primitive drive that makes us imagine monsters when we hear noises at night?  Is it a misfire of our application of the scientific method, where we try to apply the rules, but make mistakes in judging evidence or constructing arguments, and come to the wrong conclusions?

Or is it something else entirely?

That's another thing about science; you can't engage in scientific thought without being willing to say the dreadful words, "I don't know."  Once your hypothesis is shown to be unsupported, it's back to the drawing board you go.  But there's nothing so very bad about that, honestly.  As Neil deGrasse Tyson said, "Scientists are always at the drawing board.  If you're not at the drawing board, you're not doing science.  You're doing something else."