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Yay, another maths lecture!

Click through to see the whole cartoon at XKCD. Really do it. It’s important. Especially if you want the rest of my burblings to make sense.

So. It’s partly funny because it satirises the sensationalism of tabloid news, and the urge to cram as much excitement into a headline as possible only to leave a sober assessment of actual facts to the blogosphere. But it actually addresses a much more common problem with our understanding of probability.

Most people who pay much attention to any kind of sciencey talk are probably familiar with the p-values referenced in the comic. When scientists are testing a hypothesis, they’ll often check whether the p-value (p for probability) of the results from their experiments is less than 5%. The smaller the p-value is, the less likely it is that their results are purely down to chance.

However, the p-value kinda means the exact reverse of what a lot of people assume it means.

When scientists talk about results being “significant at the 5% level”, say, it sounds like this means there’s a 95% chance of a real connection. In this cartoon’s case, it sounds like the scientists are 95% certain of a link between green jelly beans and acne.

Applicants for James Randi’s million dollar challenge are required to meet rather more stringent criteria, but it’s often expressed the same way. For instance, a dowser might have to psychically deduce which of several sealed containers is the one with water in, and repeat it a number of times, so that the p-value becomes very small. They want to be certain there’s really something going on, and it’s not just chance, before the money will be handed over.

But the intuitive idea of what the p-value means in these cases isn’t quite right.

Here’s what you actually need to do. Assume that there is no connection between the things being tested – jelly beans don’t affect acne, and all psychics are just guessing. Then, what are the odds of getting results at least as persuasive as the ones you saw, purely by chance?

That’s your p-value.

So, a p-value of 5% tells us something useful. It means that the results you’ve got are kinda iffy, given what you’d usually expect, if there’s no deeper underlying pattern there. You’d only expect to see results this skewed about 1 time in 20, if you’re relying on randomness. So maybe something’s up.

But if you do a whole bunch of tests, like the jelly bean scientists did, once in a while you will get some iffy results like that just by chance.

Now, clearly one thing this tells us is to be wary of data which has been cherry-picked, like the jelly bean journalists did. There were lots of negative results being ignored, and a single positive outcome highlighted. But the implications for how we assess probabilities more generally are, I think, more interesting.

In particular, it tells us that how likely something is doesn’t just depend on this one set of results. If a 5% p-value means “we’re 95% sure of this”, then this one study has entirely determined your estimate of the likelihood. It fails to take on board any information about how likely or unlikely something seemed before you started – and often this information is really important.

For instance, say you were studying differences between smokers and non-smokers, and the rate at which they get cancer. Any good analysis of data along these lines should easily pass a 5% significance test. It’s a highly plausible link, given what we already know, and 95% sounds like a significant under-estimate of the likelihood of a correlation between smoking and cancer.

But now imagine you’ve done a different test. This time, you just put a bunch of people into two groups, with no information about whether they smoke, or anything else about them, and flipped a coin to decide which group each person would go into. And imagine you get the same, seemingly convincing results as the smoking study.

Are you now 95% convinced that your coin-tossing is either diagnosing or causing cancer in people you’ve never met?

I hope you’re not. I hope you’d check your methodology, look for sources of bias or other things that might have crept in and somehow screwed up your data, and ultimately put it down to a bizarre fluke.

And it makes sense to do that, in this case, even despite the data. The idea that you could accurately sort people by cancer risk simply by flipping a coin is utterly ridiculous. We’d give it virtually zero probability to begin with. The results of your study would nudge that estimate up a little, but not much. Random fluke is still far more likely. If multiple sources kept repeating the experiment and getting the same persuasive results, over and over… then maybe, eventually, the odds would shift so far that your magic coin actually became believable. But they probably won’t.

And this idea of shifting the probability of something, rather than fixing it firmly based on a single outcome, is at the heart of Bayesian probability.

This is something the great Eliezer Yudkowsky is passionate about, and I’m totally with him. That link’s worth a read, though someday I’d like to try and write a similar, even more gently accessible explanation of these ideas for the mathematically un-inclined. He does a great job, but the arithmetic starts to get a bit overwhelming at times.

And if the thrill of counter-intuitive mathematics isn’t enough to convince you that this is fascinating and important stuff, read this. And then this.

Short version: a number of women have been convicted and jailed for murdering their children, then later released when somebody actually did some better statistics.

The expert witness for the prosecution in these trials estimated that the odds of two children in the same family both dying of cot death was 1 in 73,000,000. General population data puts the overall rate of cot deaths at around 1 in 8,500, so multiplying the 8,500s together gives the 1 in 73,000,000 figure for the chance of it happening twice. This was presented as the probability that the children could have died by accident, and thus it was assumed to be overwhelmingly likely that they were in fact deliberately killed.

But, as we learned with the cancer stuff earlier, we should consider these substantial odds against our prior assessment of how likely it is that these women would murder their children. This should start off minuscule, because very few women do murder their children. The fact that both their children died should make us adjust our likelihood estimate up a way – someone with two dead children is a more likely candidate for a child murderer than someone whose offspring are alive and well, after all – but it’s still far from conclusive.

Another way of expressing the central point of Bayesian probability is to consider the probability of A given B, for two events A and B. In this case, the odds of two children randomly picked from the population both dying of cot death may well be around 1 in 73,000,000 – but given that the children you’re considering both died in infancy, and were both siblings and so might have genetic or environmental factors in common, the cot death scenario becomes far more likely.

I wanted to expand on that last point some more, and touch on some other interesting things, but I’m hungry and you’re bored.

Ha. I said “briefly”. Classic.

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While I was offline for a month, I kept a note of any links and news stories worth commenting on. Now that I’m back, I’m aiming to post two short items a day here, about stuff that happened during my online absence, until I’ve cleared the backlog. This is one of those.


If you were hoping to be told what card you’re thinking of by a carrion bird, I can only apologise for the disappointment.

Evan Bernstein reported recently about a woman who charges dying hospital patients $125 to spin some yarn about “who’s waiting to greet them on ‘the other side.'”

Talk about the repulsiveness of the image of an ambulance chasing lawyer. Well, this is a psychic chasing the hearse on the way to the hospital to pick up the bodies.

Seems an appropriate analogy to me.

There are people standing up for her, of course, and declaring that anyone who won’t instantly believe in these outlandish claims entirely at face value must be living a life with no hope or meaning. Which is bewildering enough, but okay, let’s work with them a bit. Let’s say she’s real, she genuinely has some power to do what she says she does, and she’s making a really, really good living by providing a legitimate service.

Personally, I find that a real stretch to believe. But let’s run with it. Here’s something that’s absolutely not a stretch to believe:

Somewhere, some unscrupulous con artist would read this article, see this woman making a fortune by telling dying people reassuring things, and think: I have got to get me a piece of this action.

This is easy to imagine. There are undeniably people like this in the world, trying to make a fast buck and not caring who they hurt in the process. Some of them rob banks. So why wouldn’t some of them, somewhere, decide that dispensing a few platitudes to some old suckers desperate for some comfort before they pop their clogs might be an easy gig?

And if it’s obvious that there really could be scammers trying to rip people off with a pale imitation of what the real psychics do, how do you tell the difference?

That’s not at all a rhetorical question. I’m not trying to say that you can’t tell the difference between a real psychic and a con artist, and so you’re a fool for believing that any of this is real. It’s genuinely worth considering how to distinguish the two, and avoid falling for someone’s dishonesty.

It’s something so many believers seem entirely unwilling to consider.

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Just a quick link today to a recent confirmation that Joe Power is still an unpleasant, bitter, cruel, spiteful, nasty little man.

Not that we needed any further reminders. Derren Brown has it covered, in particular. But still. He is quite revolting. Even for a fucking stage psychic who pretends to talk to people’s dead relatives.

Sorry to only bring you negativity today. I’m not feeling creative enough for anything more right now. I wrote over a thousand words of my novel earlier this evening, though, which isn’t too bad as excuses go.

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How easy is it to believe in nonsense, without being notably stupid?

Some years ago – I’d guess I was very approximately 13, so in the mid-to-late ’90s, but certainly long before I had any idea what scientific skepticism was, and even longer before I’d given up believing in God – I was watching some debate show on TV about the paranormal. There was a host/moderator at a desk, with a couple of people on either side of him, each arguing one side of the debate. And there was a fairly large studio audience.

The first person to speak was an alleged psychic, who was over from the US (this was an English show, I’m pretty sure) and had agreed to demonstrate his abilities. He started by doing some readings for people in the crowd, throwing out some fairly specific things straight off, and homing in on some individual people, seeming to know a good deal about each of them. I can’t remember what he said, or whether he claimed to be in direct contact with anybody’s dead relatives, but the audience members he spoke to were definitely very impressed, as were many of the onlookers.

Now, as I recall, I didn’t really have any opinion at the time on things like psychic powers. The subject was interesting enough to me that I was watching a debate that proposed to settle the matter, but I certainly don’t think I was convinced that there was anything to it, though nor was I especially against the idea. I just don’t think I’d given it much thought.

In fact, I’m sure I must have been open to it and yet unconvinced – because I distinctly remember being very impressed myself by the quality of the readings this guy was giving. I mean, we were assured that he hadn’t met any of these people before, there didn’t seem to be any obvious way he could be finding these things out about them, no way he’d have known who was going to be in the audience tonight so that he could research them beforehand. He was just standing up and somehow providing all this secret information, with startling accuracy.

So I figured maybe there was something to this psychic stuff after all. Maybe the guy was in touch with some sort of magic, or something. It seemed to make sense.

Of course, it didn’t last. It didn’t last more than about thirty seconds, in fact, because as soon as he was done making a very impressive case for the existence of psychic powers, and wowing the crowd, the host told us that this was not in fact Mr So-and-so, the practising psychic as whom he’d been introduced. It was in fact Mr James Randi, experienced skeptical investigator of paranormal claims. I remember a mild uproar from the crowd at this point, which if I had to pin it down to an emotion being expressed I would probably call dismay.

And suddenly my own view of things swung way back in the opposite direction. I hadn’t known it was possible for a psychic to come up with such amazing information – and, I now realised, I hadn’t known it was also possible for some regular guy who knows his shit to put on an equally impressive performance.

And it was really, really cool. It didn’t shake any particular worldviews of mine, and my role as an interested skeptic still didn’t really start for a number of years, but I really liked this. It sounded like some people in the audience objected to being tricked, but I thought it was great. Maybe it says something about the approach to learning that I’d come to adopt in the first few years of my life, or maybe I just hadn’t had time to grow attached to this particular paradigm before it was shattered. But I thought the way I’d been led down one path and then had the rug pulled from under me (if you’ll forgive another mixed metaphor involving carpeted pathways) was really cool.

And it made Randi’s point very well, of course. The fact that I could be tricked like that was quite revealing, and made the case very convincingly that the skeptics were right – or at least that the psychics were going to have to step up their game a great deal if they wanted to be taken seriously. But I didn’t feel upset that I’d been tricked, or like I’d been made to look stupid. And while there may be a lot that I know now, and which seems obvious to me now, which I wasn’t aware of at the time, I don’t think you have to start from a position of total idiocy to make a transition like that.

It’s not for my own ego that I keep reiterating that I wasn’t being stupid, by the way. I know it might seem like I’m just trying to disingenuously maintain my own integrity while continuing to mock anyone of a different opinion to me at present. (“Other people are stupid to believe this stuff. I was just having a momentary but understandable and perfectly human blip in my usually excellent rational and intelligent outlook.”) But I think there’s something important to remember here, and this story is one of my strongest personal reminders.

The most vocal, prominent, or infuriating proponents of woo out there are often infuriatingly stubborn, or committed to irrationality, or strongly mentally blocked to any disconfirming evidence, or otherwise not worth arguing with. But they’re by no means the majority. They’re not the one in three people who believe in telepathy or ghosts. Those people are me a little over a decade ago. They haven’t spent years reading skeptical blogs and listening to science podcasts, learning how to think critically about these things, and about the history and context of all these wacky beliefs, and how people can be fooled into getting things wrong. But they’re not fundamentalists or ideologues. They’ve probably just seen someone on stage, claiming to be a psychic, saying some pretty impressive things.

And if that someone turns out to be James Randi using cold reading (or hot reading, I forget exactly how it went on that particular show), they might find that as cool as I did.

Maybe this is how we should be approaching the skeptical battle, then. Don’t worry about the extremists; there’s no significant hope for some people to ever see any kind of light. But keep talking to the people who are just like you before you knew any of this stuff. Put the emphasis on how much fun it can be to learn you were wrong, to see how you were just fooled by something nifty. Keep some sarcasm in your arsenal, because it’s funny and the sort of thing that can be useful in rallying the troops, but don’t let it become your primary weapon against people who might be willing to learn a thing or two. Always keep it rational. Make sure we’re always the ones explaining why the facts are on our side.

I hope you’re listening to this, writerJames. Seriously, I sometimes think that guy could use this advice more than most.

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Peter Mandelson wants to steal every photograph in the country. Well, that’s not really quite right. I’m being glibly flippant (which is less dirty than it sounds), but I bet it’d be alarmingly easy to make the case for those exact terms. The Digital Economy Bill is, it seems, a terrible terrible idea. I don’t know what to do about that, exactly, but I thought you should know.

Christians: using Jesus as an excuse to be creepy and evil for 2000 years. This slogan was suggested by my own favourite Christian (and let me know if I should credit you), based on an article about some really despicable views on rape and what clothes women choose to wear. Just the fact that those two concepts are even grouped together in a single phrase should tell you that this is going to be painful.

You may have been given this leaflet because of the way you are dressed,

is a phrase included on a tract handed to a 19-year-old girl working in a drive-through. My own suggestion was some counter-literature to hand back. “You may have been given this leaflet because of the offensive anti-feminist victim-blaming propaganda bullshit you were pushing on people.”

Adrian Pengelly cannot cure cancer. But he says he can, and charges for it, and is a fucking twat who deserves to go down for way longer than three months, which it sounds like might be all he’s up for. I can’t stand the picture of him in that article, either. Would I want to punch him in the smfugly (smug + fugly) face quite so many times if I didn’t already dislike him for other perfectly good reasons? I suppose we’ll never know.

– But more importantly than all that, the first series of Chance In A Million is out on DVD! This was a sitcom that my dad wrote about 25 years ago, and I can say with perfect objectivity that it’s brilliant. As I write this, it has six reviews on Amazon, all giving it five stars (out of five). And only one of those was written by my mum.

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Yes, I’m still moaning about the lack of time I find to get anything done. Couple of quick links.

James Randi on Uri Geller, summing things up nicely. The guy’s made a decades-long career out of convincing people that he does real, genuine, no-foolin’ magic using psychic alien powers and no trickery whatever. If he’s now claiming to be an illusionist relying on natural means, he’s a disingenuous twat. Randi put it more eloquently.

And when I go home for Christmas, I’m going to have to remember to try out at least some of Richard Wiseman’s quirky science tricks.

And that’s everything that happened today.

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I used to believe in some wacky stuff.

It didn’t seem all that wacky at the time, of course. When I first started taking an interest in the stuff I was reading online, about people’s religious experiences and psychics and mind readers and dowsing and so on, it sounded fascinating, and wasn’t obviously bullshit at all. I guess I tend to think about things a bit differently now, or maybe there are just more things that I’ve learnt aren’t real in the intervening years.

Anyway, there was a lot of stuff about dowsing that caught my eye, and made it seem like an accessible skill. There was reams of advice and personal experiences people wanted to share, and it sounded like you didn’t need to be whisked away from your cupboard under the stairs to a wizards’ school by a hairy giant in order to be a part of it. It sounded like anyone could join in, and learn to access some spiritual dimension which could provide insight and knowledge from beyond this world.

So I bought a crystal pendulum from a new age shop.

It feels so weird typing that sentence now.

It was cheap, but kinda pretty, and looked a lot like this quartz one. The idea, as described on that page, is to clear your mind and mentally ask a series of yes/no questions, while letting the pendulum hang loosely from your fingers. There are various ways the pendulum might swing – circular motions, clockwise or anticlockwise, back and forth, diagonally – and you can calibrate it with some control questions.

I don’t remember exactly how it went when I tried it, but it would have been something like: “Is my name James?” – and I saw it swing forward and back, so I knew that meant yes. “Is today Wednesday?” – another yes, with the same swinging motion. “Is there a dragon in my room?” – and it swung side to side, meaning no.

This was really exciting.

So I decided to test it out properly, and see if I could find out something that I didn’t know, and prove that I was really tapping into some amazing psychic source of power.

I think this is the point where my strategy departed from that of a lot of new age fans.

I got a deck of playing cards and placed one face down in front of me. I didn’t know what card it was, but I held the pendulum over it, and asked yes/no questions to narrow it down. “Is it black?” – no. “Is it red?” – yes. “Is it a picture card?” – no. And so on.

Eventually I narrowed it down to “Is it the five of diamonds?” and got a yes. It had given me a definite answer to everything I asked. It had never contradicted itself. I’d started with absolutely no knowledge or assumptions or preconceptions about the card in front of me, and my pendulum had honed directly in on its identity as the five of diamonds.

I still remember the fluttering in my chest – half excitement and half genuine fear – in the second or two before I turned over the king of clubs.

Aw, crap.

It turns out that there’s a bunch of reasons why people believe in this kind of thing, and post articles to the internet about their powerfully moving personal experiences with it. And these reasons don’t require magic to actually be real.

When I first started looking into it, it didn’t require any particular daftness on my part to take it seriously – it just seemed to be a part of the world. A somewhat secretive, not generally known, exclusive part, but that just made it all the more fun. At the depth at which I explored it at the time, I didn’t find any good reason to suppose that it was all completely fictitious. People were taking it for granted, writing detailed accounts of their achievements, and beginners’ guides to the basic techniques.

But once you start thinking about it more critically, you realise that magic powers aren’t the only explanation. They’re not the best explanation. In fact, they’re not even a very good explanation.

Some people are very keen to find evidence that supports the idea that their dangling crystal can tell them things – so confirmation bias plays a big part in explaining why it’s so widely believed, as well as a host of other logical fallacies. But the ideomotor effect is one of the most persuasive aspects if you don’t know what it is. And it’s the one I’m supposed to be talking about here.

When I was asking myself those questions, I really was trying to hold the pendulum as still as possible. I know I wasn’t deliberately swinging it around to make myself seem like an amazing wizard (“Look, it knows my name!!”), but it’s worth asking: how good am I at holding my hand perfectly still? When I look closely at my outstretched digits as I try to remain motionless, I seem surprisingly wobbly. If I’m going to hold something on a thin and flexible cord or chain, it seems likely that my natural shakiness is going to have some effect.

And it turns out that the pendulum picks up more than just a general jiggle from my unsteady muscles. Let’s say I know a forward-swing means yes, because of my first test question. If I then ask something else which I know, or expect, has the answer yes, then on some level of consciousness I’m going to be imagining getting a forward-swing answer from the pendulum. My hand will then actually twitch, without my being aware of it, to make the pendulum swing forward.

The mental processes to do this can really happen inside your head, without the part where you’re conscious of it. It “bypasses volition”, to be a bit technical (volition being your capacity to do something by your own will).

You can try it easily yourself with any weight on some sort of dangling cord. I’m trying it now with one of the earphones from my mp3 player on its lead, and it’s still quite odd to see. I concentrate on a clockwise spinning motion, and it starts spinning clockwise, even though I’m still trying to hold it as steady as I can.

If you’re thinking that this might be evidence that I was secretly psychic all along, you’re still leaping to a more complicated explanation than is necessary. If I’m not directly touching the cord, or holding it in such a way that my hand movements won’t affect its swing, then it doesn’t respond in the same way. It only moves like this when I have the capacity to be swinging it around unconsciously. The best explanation is that I’m simply moving my hand.

There’s also a common hypnotic trick, where you’re asked to close your eyes and stick your arms out, then vividly imagine a heavy weight in one hand pulling it down, and a balloon tied to the other pulling it up. You focus on the respective feelings of pressure and lightness for a while, and if you’re anything like me, after a couple of minutes you open your eyes and find that you’ve lifted and lowered your hands accordingly by several inches, without being aware of doing it.

The point is, your mind’s good at doing stuff like this without telling you about it.

Now, this doesn’t mean that nobody can dowse anything, or that we’ve proved that Ouija boards are universally a load of crap (yes, the people are just pushing the glass around even if they don’t realise it). But it reminds us the importance of asking the question “Is there a simpler, less Harry Potter explanation?” when we see something we think might be magic.

If I was doing actual magic over my playing card that time, then my skills make Neville Longbottom look like Gandalf. I must really suck at magic. I didn’t even get close to getting the card right. Magic just isn’t a good enough explanation for what happened there. But the idea that my hand wasn’t perfectly still, and made the pendulum swing a little by entirely natural means? Yep, that fits.

But what if I had got it right? What if I had no way of knowing what card I was staring at the back of, and wasn’t being provided the information by any means except the pendulum, and I actually got it right? And it kept happening, consistently?

Well, the ideomotor effect wouldn’t cover that. And I’d be a millionaire.

But it does cover, y’know, every case that’s ever been examined of any kind of dowsing ever. Except the ones that are outright fraud, where there’s conscious deception taking place. But there really doesn’t need to be any malice or dishonesty for people to make magical claims that aren’t based in reality. If you don’t know what the ideomotor effect is, and maybe don’t test out your new idea all that rigorously, and kinda let slide the few occasions where it doesn’t work… then I can imagine this being pretty convincing.

People who do things like dowsing aren’t being stupid or evil. But they are claiming that they can do magic, and it’s a big ask that we should take that at face value without daring to question it any further, even if we don’t doubt their sincerity. It’s the kind of massive claim that we should probably, y’know, check.

And, unfortunately for any aspiring Weasleys out there, natural phenomena like the ideomotor effect provide a better explanation for every instance of “magic” that’s yet been observed. They account perfectly for what’s going on, but the magical explanation fails to explain why the effect always vanishes when studied closely. It just doesn’t work. The five of diamonds was not my card.

Sorry, Hermione. Muggles win.

A more academic and less chatty approach to this topic can be found at The Skeptic’s Dictionary, RationalWiki, SkepticWiki, and all over the place really. Barrett Dorko and Ray Hyman, among others, have written rather more scientifically rigorous documents about the ideomotor effect in action, with examples of experiments in which it’s been seen.

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