Blind lo picked up the cube and counted the sides.
'Come on,' he said wearily. 'Play fair.'
Nature's sample space is often bigger than a conventional statistician would expect. Sample spaces are a human way to model reality: they do not capture all of it. And when it comes to estimating significance, a different choice of sample space can completely change our estimates of probabilities. The reason for this is an extremely important factor, 'selective reporting', which is a type of narra-tivium in action. This factor tends to be ignored in most conventional statistics. That perfect hand at bridge, for instance, is far more likely to make it to the local or even national press than an imperfect one. How often do you see the headline BRIDGE PLAYER GETS ENTIRELY ORDINARY HAND, for instance? The human brain is an irrepressible pattern-seeking device, and it seizes on certain events that it considers significant, whether or not they really are. In so doing, it ignores all the 'neighbouring' events that would help it judge how likely or unlikely the perceived coincidence actually is.
Selective reporting affects the significance of those Formula One times. If it hadn't been them, maybe the tennis scores in the US Open would have contained some unusual pattern, or the football results, or the golf ... Any one of those would have been reported, too, but none of the failed coincidences, the ones that didn't happen, would have hit the headlines. FORMULA ONE DRIVERS RECORD DIFFERENT LAP TIMES ... If we include just ten major sporting events in our list of would-be's that weren't, that one in ten thousand chance comes down to only one in a thousand.
Having understood this, let's go back to the Israeli fighter pilots. Conventional statistics would set up the obvious sample space, assign probabilities to boy and girl children, and calculate the chance of getting 84% girls in a purely random trial. If this were less than one in a hundred, say, then the data would be declared 'significant at the 99% level'. But this analysis ignores selective reporting. Why did we look at the sexes of Israeli fighter pilots' children in the first place? Because our attention had already been drawn to a clump. If instead the clump had been the heights of the children of Israeli aircraft manufacturers, or the musical abilities of the wives of Israeli air traffic controllers, then our clump-seeking brains would again have drawn the fact to our attention. So our computation of the significance level tacitly excludes many other factors that didn 't clump, making it fallacious.
The human brain filters vast quantities of data, seeking things that appear unusual, and only then does it send out a conscious signal: Wow! Look at that! The wider we cast our pattern-seeking net, the more likely it is to catch a clump. For this reason, it's illegitimate to include the data that brought the clump to our attention as part of the evidence that the same clump is unusual. It would be like sorting through a pack of cards until you found the ace of spades, putting it on the table, and then claiming miraculous powers that unerringly accomplish a feat whose probability is one in 52.
Exactly this error was made in early experiments on extra-sensory perception. Thousands of subjects were asked to guess cards from a special pack of five symbols. Anyone whose success rate was above average was invited back, while the others were sent home. After this had gone on for several weeks, the survivors all had an amazing record of success! Then these 'good guessers' were tested some more. Strangely, as time went on, their success rate slowly dropped back towards the average, as if their powers were 'running down'. Actually, that effect wasn't strange at all. It happened because the initial high scores were included in the running total. If they had been omitted, then the scoring rate would have dropped, immediately, to near average.
So it is with the fighter pilots. The curious figures that drew researchers' attention to these particular effects may well have been the result of selective reporting, or selective attention. If so, then we can make a simple prediction: 'From now on, the figures will revert to fifty-fifty.' If this prediction fails, and if the results instead confirm the bias that revealed the clump, then the new data can be considered significant, and a significance level can sensibly be assigned by the usual methods. But the smart money is on a fifty-fifty split.
The alleged decline in the human sperm count may be an example of selective reporting. The story, widely repeated in the press, is that over the past 50 years the human sperm count for 'normal' men has halved. We don't mean selective reporting by the people who published the first evidence, they took pains to avoid all the sources of bias that they could think of. The 'selective reporting' was done by researchers who had contrary evidence but didn't publish it because they thought it must be wrong, by journal referees who accepted papers that confirmed a decline more often than they accepted those that didn't, and by the press, who strung together a whole pile of sex-related defects in various parts of the animal kingdom into a single seamless story, unaware that each individual instance has an entirely reasonable explanation that has nothing to do with falling sperm-counts and often nothing to do with sex.
Sexual abnormalities in fish near sewer outlets, for instance, are probably due to excess nitrites, which all fish-breeders know cause abnormalities of all kinds, and not to oestrogen-like compounds in the water, which would bolster the 'sperm count' story. Current data from fertility clinics, by the way, show no signs of a decline.
Humans add narrativium to their world. They insist in interpreting the universe as if it's telling a story. This leads them to focus on facts that fit the story, while ignoring those that don't. But we mustn't let the coincidence, the clump, choose the sample space -when we do that, we're ignoring the surrounding space of near-coincidences.
Jack and Ian managed to test this theory on a trip to Sweden. On the plane, Jack predicted that a coincidence would happen at Stockholm airport, for reasons of selective reporting. If they looked hard enough, they'd find one. They got to the bus stop outside the terminal, and no coincidences had occurred. But they couldn't find the right bus, so Jack went back to the enquiries desk. As he waited, someone came up next to him, Stefano, a mathematician who normally occupied the office next door to Jack's. Prediction confirmed. But what was really needed was evidence of a near-coincidence, one that hadn't happened, but could have been selectively reported if it had. For instance, if some other acquaintance had shown up at exactly the same time, but on the wrong day, or at the wrong airport, they'd never have noticed. Near coincidences, by definition, are hard to observe ... but not impossible. Ianhappened to mention all of the above to his friend Ted, who was visiting soon after. 'Stockholm?' said Ted: 'When?' Iantold him. 'Which hotel?' Iantold him 'Funny, I was staying there one day later than your Had the trip been one day later, the 'coincidental' encounter with Stefano wouldn't have happened, but the one with Ted would.
What we must not do, then, is to look back at past events and find significance in the inevitable few that look odd. That is the way of the pyramidologists and the tea-leaf readers. Every pattern of raindrops on the pavement is unique. We're not saying that if one such patterns happens to spell your name, this is not to be wondered at, but if your name had been written on the pavement in Beijing during the Ming dynasty, at midnight, nobody would have noticed. We should not look at past history when assessing significance: we should look at all the other things that might have happened instead.
Every event is unique. Until we place that event in a category, we can't work out which background to view it against. Until we choose a background, we can't estimate the event's probability. If we consider the sample space of all possible DNA codes, for instance, then we can calculate the probability of a human being having exactly your DNA code, which is vanishingly small. But it would be silly to conclude that it is impossible for you to exist.
33. STILL BLOODY LIZARDS
'THE FUTURE IS LIZARD,' said Ridcully. 'Obviously.' It was a few days later. The omniscope was focused on a mound of leaves and rotting vegetation a little way from the banks of the river. There was a large depression hanging over the Senior Wrangle
r, and the Dean had a black eye. The war between land and sea had just entered a terminal stage.
'Little portable seas,' said Ponder. 'You know, I never thought of them like that.'
'An egg is an egg, however you look at it,' said Ridcully. 'Look, you two, I don't want to see a scuffle like that again, d'you hear?' The Senior Wrangler dabbed at his bleeding nose. 'He goaded be,' he said. 'Id's still osuns, howeber you look at id.' 'A private ocean full of food,' said Ponder, still entranced. 'Hidden in a heap of... well, compost. Which heats up. That's like having private sunshine.'
The little lizard-like creatures that had hatched from the eggs in the mound slithered and slid down the bank into the water, bright-eyed and hopeful. The first few were instantly snapped up by a large male lying in wait among the weeds.
'However, the mothers still have something to learn about postnatal care,' said Ridcully. 'I wonder if they'll have time to learn? And how did they know how to do this? Who's telling them?'
The wizards were depressed again. Most days started that way now. Creatures seemed to turn up in the world randomly, and certainly not according to any pictures in a book. If things were changing into other things, and no one had seen that happen yet, why were the original things still the original things? If the land was so great, why were any fish left in the sea?
The air-breathing fishes that Rincewind had seen still seemed to be around, lurking in swamps and muddy beaches. Things changed, but still stayed the same.
And if there was any truth at all in Ponder's tentative theory that things did change into other things, it led to the depressing thought that, well, the world was filling up with quitters, creatures which -instead of staying where they were, and really making a go of life in the ocean or the swamp or wherever, were running away to lurk in some niche and grow legs. The kind offish that'd come out of water was, frankly, a disgrace to the species. It kept coughing all the time, like someone who'd just given up smoking.
And there was no purpose, Ridcully kept saying. Life was on land. According to the book, there should be some big lizards. But nothing seemed to be making much of an effort. The moment anything felt safe, it stopped bothering.
Rincewind, currently relaxing on a rock, rather liked it. There were large animals snuffling around in the greenery near the rock he was sitting on; in general shape and appearance, they looked like a small skinny hippopotamus designed in the dark by a complete amateur. They were hairy. They coughed, too.
Things that were doing sufficiently beetle-like things for him to think of them as beetles ambled across the ground.
Ponder had told him the continents were moving again, so he kept a firm grip on his rock just in case.
Best of all, nothing seemed to be thinking. Rincewind was convinced that no good came of that sort of thing.
The last few weeks of Discworld time had been instructive. The wizards had tentatively identified several dozen embryo civilizations, or at least creatures that seemed to be concerned about more than simply where their next meal was coming from. And where were they now? There was a squid one, HEX said, out in the really deep cold water. Apart from that, ice or fire or both at once came to the thinkers and the stupid alike. There was probably some kind of moral involved.
The air shimmered, and half a dozen ghostly figures appeared in front of him.
There were, in pale shadowy colours, the wizards. Silvery lines flickered across their bodies and, periodically, they flickered.
'Now, remember,' said Ponder Stibbons, and his voice sounded muffled, 'You are in fact still in the High Energy Magic building. If you walk slowly HEX will try to adjust your feet to local ground level. You'll have a limited ability to move things, although HEX will do the actual work...'
'Can we eat?' said the Senior Wrangler.
'No, sir. Your mouth isn't here.'
'Well, then, what am I talking out of?'
'Could be anyone's guess, sir,' said Ponder diplomatically. 'We can hear you because our ears are in the HEM, and you can hear the sounds made here because HEX is presenting you with an analogue of them. Don't worry about it. It'll seem quite natural after a while.'
The ghost of the Dean kicked at the soil. A fraction of a second later, a little heap of earth splashed up.
'Amazing!' he said, happily.
'Excuse me?' said Rincewind.
They turned.
'Oh, Rincewind,' said Ridcully, as one might say 'oh dear, it's raining'. 'It's you.'
'Yes, sir.'
'Mister Stibbons here's found a way of getting HEX to operate more than one virtually-there suit, d'you see? So we thought we'd come down and smell the roses.'
'Not for several hundred million years, sir,' said Ponder.
'Dull, isn't it,' said the Lecturer in Recent Runes, looking around. 'Not a lot going on. Lots of life, but it's just hanging around.'
Ridcully rubbed his hands together.
'Well, we're going to liven it up,' he said. 'We're going to move things forward fast while we're here. A few prods in the right place, that's what these creatures need.'
'The time travelling is not much fun,' said Rincewind. 'You tend to end up under a volcano or at the bottom of the sea.'
'We shall see,' said Ridcully firmly. 'I've had enough of this.
Look at those damn sloppy things over there. 'He cupped his hands and shouted, 'Life in the sea not good enough for you, eh? Skiving off, eh? Got a note from your mother, have you?' He lowered his hands. 'All right, Mister Stibbons ... tell HEX to take us forward, oh, fifty million years, hang on, what was that?'
Thunder rolled around the horizon.
'Probably just another snowball landing,' said Rincewind morosely. 'There's generally one around just when things are settled. It was in the sea, I expect. Stand by for the tidal wave.' He nodded at the browsing creatures, who had glanced up briefly.
'The Dean thinks all this hammering from rocks is making the life on this world very resilient,' said Ridcully.
'Well, that's certainly a point of view,' said Rincewind. 'But in a little while a wave the size of the University is going to wash this beach on to the top of those mountains over there. Then I expect the local volcanoes will all let go ... again ... so stand by for a country-sized sea of lava coming the other way. After that there'll probably be outbreaks of rain that you could use to etch copper, followed by a bit of a cold spell for a few years and some fog you could cut up in lumps.' He sniffed. 'That which does not kill you can give you a really bad headache.'
He glanced at the sky. Strange lightning was flickering between the clouds, and now there was a glow on the horizon.
'Damn,' he said, in the same tone of voice. 'This is going to be one of the times when the atmosphere catches fire. I hate it when that happens.'
Ridcully gave him a long blank look, and then said, 'Mister Stibbons?'
'Archchancellor?'
'Make that seventy thousand years, will you? And, er ... right now, if you would be so good.'
The wizard vanished.
All the insects stopped buzzing in the bushes.
The hairy lizards carried on placidly eating the leaves. Then, something made them look up…
The sun jerked across the sky, became very briefly a reddish-yellow band across a twilight hemisphere, and then the world was simply a grey mist. Below Rincewind's feet it was quite dark, and above him it was almost white. Around him, the greyness flickered.
'Is this what it always looks like?' said the Dean.
'Something has to stand still for a couple of thousand years before you see it at all,' said Rincewind.
'I thought it would be more exciting...'
The light flickered, and sun exploded into the sky, the wizards saw waves around them for a moment, and then there was darkness.
'I told you,' said Rincewind. 'We're under water.'
'The land sank under all the volcanoes?' said Ridcully.
'Probably just moved away,' said Rincewind. 'There's a lot of that sor
t of thing down here.'
They rose above the surface as HEX adjusted to the new conditions. A landmass was smeared on the horizon, under a bank of cloud.
'See?' said Rincewind. 'It's a pain. Time travel always means you end up walking.'
'hex, move us to the nearest land, please. Inland about ten miles,' said Ponder.
'You mean I could have just asked?' said Rincewind. 'All this time, I needn't have been walking?'
'Oh, yes.'
The landscape blurred for a second.
'You could have said,' said Rincewind accusingly, as they were rushed past, and sometimes through, a forest of giant ferns.
The view stabilized. The wizard had been through to the edge of the forest. Low-growing shrubs stretched away towards more ferns.
'Not much about,' said Ridcully, leaning against a trunk. 'Can I smoke my pipe here, Stibbons?'
'Since technically you'll be smoking in the High Energy Building, yes, sir.'
Rincewind apparently struck a match on the tree trunk. 'Amazing,' he said.
'That's odd, sir,' said Ponder. 'I didn't think there would be any proper trees yet.'
'Well, here they are,' said Ridcully. 'And I can see at least another three more ...'
Rincewind had already started to run. The fact that nothing can harm you is no reason for not being scared. An expert can always find a reason for being scared.
The fact that the nearest trunk had toenails was a good one.
From among the ferns above, a large head appeared on the end of too much neck.
'Ah,' said Ridcully calmly. 'Still bloody lizards, I see.'
Ponder was working the Rules again. Now they read:
THE RULES
1 Things fall apart, but centres hold
2 Everything moves in curves
3 You get balls
4 Big balls tell space to bend
5 There are no turtles anywhere
The Science of Discworld I tsod-1 Page 26