Statistics – Destroyer of Superstitious Pretension

Statistics-education-research-day1 In Philip Ball’s Critical Mass: How One Thing Leads to Another, he articulates something rather profound: statistics destroys superstition. The idea, once expressed, is simple but does not stem its profundity. Incidents in small numbers sometimes become ‘miraculous’ only because they appear unique, within a context that fuels such thinking. Ball’s own example is Uri Geller: in the 1970’s, the self-proclaimed psychic stated he would stop the watches of several viewers. He, perhaps, twisted his face and furrowed his brow and all over America watches stopped. America, no doubt, turned into an exclamation mark of incredulity. What takes the incident out of the sphere of the miraculous, however, is the consideration of statistics: With so many millions of people watching, what was the likelihood of at least some people’s watches stopping anyway? What about all those watches that did not stop?

Our psychological make-up seeks a chain in disparate events. Our mind is a bridge-builder across chasms of unrelated incidents; a credulity stone-hopper, crouching at each juncture awaiting the next link in a chain of causality. To paraphrase David Hume, we tend to see armies in the clouds, faces in trees, ghosts in shadows, and god in pizza-slices.

Many incidents that people refer to as miraculous, supernatural, and so on, become trivial when placed within their proper context. Consider the implications of this: Nicholas Leblanc, a French chemist, committed suicide in 1806; Ludwig Boltzmann, the physicist who explained the ‘arrow of time’ and gave us the Boltzmann Constant, committed suicide in 1906; his successor, Paul Ehrenfest, also committed suicide, in 1933; the American chemist Wallace Hume Carothers, credited with inventing Nylon, killed himself in 1937. This seems to ‘imply’ a strong link between suicide and science. Of course, as Ball indicates himself, we must look at the contexts: We must ask what the suicide-rating of these different demographics was in general: of Americans, Europeans, males, and any other demographic.

Ball shows that in the 19th- to 20th-century Austria of Boltzmann and Ehrenfest, suicides were quite common: ‘[Suicide in Austria] claimed the lives of three of [the philosopher] Wittgenstein’s brothers, [the composer] Gustav Mahler’s brother, and in 1889, the Crown Prince Rudolf of Austria.’ Seen in the ‘light of the relevant demographic statistics’, says Ball, Ludwig Boltzmann’s death does not indicate something special about suicide and science. Statistics made this incident banal by removing it from isolation; statistics returned these strange facts about the Austrian scientists and their suicides into a context that bridged the chasm where the miraculous or spectacular are birthed. Statistics helps us show the echoes in this Chasm of Credulity harmonise with a larger context, helps us weed out the isolated incidents before they grow into poisoned fruit of proclamations of superstitious awe. Science seeks ways to bridge, if not narrow, this Chasm of Credulity.

Whether the incidents are psychic-telephone calls or astrology charts, nearly all can be minimised, and thus emptied, of their pretensions. Bloated anecdotes of precognitive abilities are drained when we think of their corollary: how many more times have you thought of someone and the telephone hasn’t wrung? What are the chances of several hundred people’s watches stopping in a crowd of a few million? With the millions of combinations of baked dough, tree bark, and mountain cliffs, perhaps it’s more likely for us not to see face in these various phenomena. Statistics can aid us here, bringing us back down to earth, instead of drifting among the clouds of make-believe.

To make sense of this, consider the ‘birthday problem’: what are the chances that, in a small group of people, any two share a birthday? Let us assume a group of 30 people and there are 365 days in a year. Two people must share one of those 365 days. Thus, we first work out the total possible combinations of two people’s birthdays if they asked each other: that would be 365 x 365 x 365 … for the number of people. That means 36530, which is a massive number. This is the denominator. We can now calculate the number of matches that are not birthdays, working our way backwards to figure out the probability.

Person #1 states his birthday. Person # 2 has 364 days to choose from, Person # 3 has 363, and so on (remember, the birthdays do not match hypothetically). An image useful in considering this is Person # 1 drawing a red cross on yearly-calendar, Person # 2 doing the same in the available spaces, and so on, until thirty people have done it. That is working your way down. So, in trying to work out how many people do not share a birthday, we have to say 365 x 364 x 363 … and so on until you’ve done it 30 times. Thus, we write it as follows:


(365 – n)!

‘N’ equals the amount of participants and ‘!’ indicates a factorial, which works its way down as we indicated above (365 x 364 … 336 x 335). This is the numerator for our example.

Now, we simply combine our figures.


(365 – n)!





(365 – 30)!



We are left with: [365!/335!]/365^30

According to the calculations, we should get: 0.2936. Remember this is the chance of people not sharing a birthday. So, the chance of sharing a birthday is inverted (1 – 0.2936): making it about 70%, between two people, in a group of 30.

Using careful calculations we encounter a counter-intuitive conclusion: in a group of 30 people, the chances that two people share a birthday is above 20-, 50- and even 60-percent. On face value, not many of us would probably think the chances that high. This shows there is actually nothing remarkable or special or spooky about two people sharing a birthday, considering that cold calculation indicates the likelihood being more than a coin-toss.

How does this reflect in superstition? Using the horizon this little but wonderful example provides, we can eclipse all manner of abysmal superstitious exclamations: What were the chances that we would meet again? What was the likelihood that I should win the lottery/win at Blackjack after I wore my lucky-jacket, prayed to my god, etc.? What were the chances of recovery from my cancer, after I went to a homeopath, a crystal-healer, a witch-doctor? All these are important questions, but are asked in a rhetorical flourish meant to indicate that the chances ‘were slim’ or ‘highly unlikely’, thus it must be the magic-man that heals, or your hidden psychic connection that provoked meeting your friend.

Consider the danger of ignoring proper calculations in medicine. People often tell us they go to a homeopath after going to a doctor; the doctor who is merely a puppet to ‘big-pharma’, who treats ‘me like a machine’ and so on. The medicines ‘Western’ doctors supply ‘do not work’, so people attend something more catchy, comforting and casual: the homeopath, the angel-healer, the witch-doctor. Strangely, one thing doctors can learn from these hucksters is the attention given to patients: the care, the pampering and the dignity conveyed. These all appear to play a factor, though people, like the great Barbara Ehrenreich, destroyer of all positive thinking, remain sceptical of how much attitude really affects health. If for no other reason than to keep patients, doctors could learn from these practitioners (they may be ‘practitioners’ but they are not medical ‘practitioners’). However, in the most important engagements of medicine, there is no time for pampering or it is simply inappropriate in an environment where, for example, the most important thing is to immunise a child.

Back to the patient: Firstly, what were the chances of you getting cured of your ailment anyway? Secondly, are we talking about a cold or a cancer? Is it absolutely impossible for cancers to suddenly go into remission without medical foresight? Of course not; oncologists will relate many stories where this has happened suddenly. The irony of course is that people imagine medical treatment as a coin-toss; you flip a coin once, the chances of getting tails are fifty-fifty. If you flip it again, the chances of getting tails remain fifty-fifty. The chances are ‘reset’ each time (this is different to asking how many tails I can get in a row, for example). But medical treatment does not ‘reset’ (similar to Ian Hacking’s Inverse Gambler’s Fallacy). Medical effects carry over.

People forget that medicine takes time to have an effect. When the effect happens to coincide with you drinking glorified water or smelling pretty aromas, many will point to homeopathy or aromatherapy as being the curing solution. But you might as well point to closing your car door or scratching your chin, since these might also have coincided with your body’s defence recognising the aid you had taken months or weeks ago. This false attribution to alternate stuffs gives them undeserved recognition and detracts from the things that actually cured you: even if it was not the medicine, we can safely say it was just your body! Health, though incredibly advanced, is still swathed in mystery but it does not mean we resort to made-up answers or whatever is convenient.

All these are factors made apparent when we put it into a proper context, asking for calculations and chances. Statistics is also wonderful since numbers do not discriminate, though obviously people may use them to do so.

The only thing remarkable about the strange world of ‘alternative medicine’ is the extent to which we allow ourselves to be duped, paying billions of our currencies into industries that consistently prove the power of the placebo. We are watching the pretensions of assertions squander our money. These fraudsters are using the Chasm of Credulity, the gap of isolated incidents, where the echoes of events removed from their context reside, leading to the fruition of bad thinking and anecdotal justifications. This same chasm across which people take leaps of faith and jump to conclusions.

The main reason scientists do not automatically trust anecdotal evidence is primarily because we need to put it into a context, test it, prod it, poke it. Anecdotes were and could be the first stirrings of something magnificent. But if the scientific eye turns toward the phenomena and it shrivels up and dies under scrutiny, it probably was not worth pursuing anyway. Someone’s clouds of hot air dissipate when cold reason enters the room.

For example: simply saying I felt better after being ‘touched’ by a magic man tells us nothing. Even if millions of people testify to the abilities of holy men and women, as they do in India with certain gurus, we need to obtain a context, the likelihood of their abilities occurring naturally (for example, did he really cure someone or was the patient’s disease likely to disappear anyway? What are the chances the storm clouds had been gathering for days and not summoned by a rain-dance?) Anecdotes are by definition after the fact, often not repeatable, and often, and most important, divorced of their context. Remember: what makes an event miraculous or supernatural is, more often than not, ignorance about the statistics of its occurrence within a specific context; as we saw with science and suicides, and sharing a birthday between random people.

To give you a further idea of this, consider this seemingly incredible find: Ben Goldacre relates a story from England in which ‘drinking the Queen’s Royal Deeside spring water improved arthritis symptoms in two-thirds of patients.’ Sounds remarkable until we put it into context, as Goldacre does: ‘It was a study of 34 patients over three months and there was no control group.’ To truly engage us, it should have much more patients, over a longer time and have a control group: i.e. a group that serves as a foil to the original and has similar characteristics as the experimental group but are given a placebo. To create a context for this remarkable find, we must offer a control to see whether it was truly the Queen’s Royal Deeside spring water or something else (if the control group gets similar results it does not mean the control was the cure, but that, more ldrinking the Queen’s Royal Deeside spring water improved arthritis symptoms in two-thirds of patientsikely, it was neither the experimental cure nor the control). Goldacre quips: ‘It’s hard to imagine an experiment where it would have been easier to come up with a convincing placebo [for a control group]. Water.’ Remember the birthday experiment: it sounds remarkable until we actually use statistics. Similarly, things become remarkable when we are unaware of the likelihood of, for example, arthritis being improved anyway due to the body’s own resistance.

Michael Shermer, in Scientific American, wrote: ‘thinking anecdotally comes naturally, whereas thinking scientifically does not.’ Because thinking scientifically is, most often, counter-intuitive to our ape minds: we are not computers or calculators. Would anyone think that there was above 50% chance that two people, in a random group of 30, share a birthday? Would anyone automatically think tiny things called bacteria and germs and viruses can cause untold misery and death, sometimes able to destroy entire civilisations?

No wonder for this latter we invoked gods since it seemed there was no other explanation: the irony being that both explain the death of crops but: which has been more useful, helped with preventative measures and so on?

We could say (1) sacrificing a virgin and letting her blood drain into the soil satisfied the gods resulting in our crops being restored or (2) we could point out that specific bacteria are infecting our plants and getting rid of these leads to restored crops. We face enormous problems if we use the first considering, for example, that not all virgins seem to pacify the gods. At the least, for simply practical, testable reasons – not to mention that crops have been restored despite no sacrifices over the years – the latter is more helpful and indeed more people realise as such for this simple, pragmatic reason. Yet we can’t escape the fact that both explain the same phenomenon. To explain is not to justify or even to reasonably justify. It is simply a story we tell to narrate our target events. Gods or bacteria, both result in the same thing. The duty of statistics and indeed of science can help disconnect the two showing that, whilst it is true both are explanations, only one survives objective testing so that even outsiders can ‘cure the appetites of the gods’.

I am reminded of Wittgenstein’s pertinent question: ‘Why did people think the sun went around the Earth?’ A reply given was: ‘Well, it just looks that way!’ Wittgenstein looked up at the sky and said: ‘But what does it look like when the Earth revolves around the Sun?’ Both heliocentrism and geocentrism arise from the same platform: looking up at the ‘movement’ of the Sun. We now know which is true (it’s heliocentrism in case you’re wondering).

Today, our societies face even worse submission before the altar of intuition, upon which bleeds all the evidence to the contrary.

The recent horror of anti-vaccination foolishness is a direct point that could be sharpened with an awareness of statistics. Shermer relates that there were a number of ‘parents who noticed that shortly after having their children vaccinated autistic symptoms began to appear.’ It was the beginning of the furore that would claim children’s lives, all because people believed anecdotal dogma above scientific reasoning. This was then compounded by the fraudulent blathering of Andrew Wakefield. Indeed, Wakefield is an excellent case-study for showing the power of statistics to empower us against charlatans like him.

Wakefield published an article in the prestigious Lancet journal, in 1998. It was more a speculative piece not warranting the media’s salacious transformation. In it, Wakefield reported 12 cases about his topic, the first stirrings of the link between autism and vaccination. Depending on the criteria, this is either remarkable or statistically negligible: 12 people sprouting wings or extra limbs from touching a wall warrants attention. Goldacre says: ‘For things as common as MMR and autism, finding 12 people with both is entirely unspectacular.’ This plugs the case back into context, stripping it of anything remarkable. Johann Hari agrees that the pool of test-subjects was too small: ‘It was based on a tiny pool of infants, most of whom were in the study because their parents believed in the link [between vaccines and autism] and wanted to sue for compensation.’

Wakefield then went on a media-rampage, publishing wherever he could to poke and prod the current medical procedures involved (specifically saying that the vaccinations should be separated by perhaps a year). Hari and Goldacre correctly condemn the media as being the main culprit in this saga of salaciousness, this epic of idiocy; giving an equal platform for health professionals and grieving parents, as if both had a basis for the scientific justifications. Here’s a clue: tears aren’t evidence. No doubt there is nothing worse than to lose a child; but when the danger of your own anger and hatred will lead to the death and suffering of other children, you deserve no compassion. I am looking at you, Jenny McCarthy.

Parents were given a tangible culprit in the form of the ‘Western medical establishment’ to blame for their impaired child instead of facing the facts of an indifferent universe, with no cosmic balance or care for us. Using no scientific facts, except sometimes Wakefield’s now completely discredited authority, mothers could invoke their own intuition to decide whether ‘stabbing their child three times’ was a good thing; they were encouraged to consult something as unfounded as psychics: their gut-feelings.

If we want a possible half-definition of science, perhaps it is this: whatever is counter-intuitive, perhaps most upsetting to our axiomatic assumptions, that thoroughly and clearly and elegantly explains the phenomena we are encountering. It’s not a perfect definition, but then, it’s not meant to be. Consulting your gut-feelings is precisely how not to do science using this sense of it: we are not so ‘made’ that consultation with our internal organs will lead to a proper explanation of the world; indeed, we will get the same results by consulting the innards of other animals, like cows or chickens. The point being, time and time again, science has shown the world to be other than we expected it to be. (Of course, there is the opposite, too, but we are not relating that for now.) Mothers being encouraged to consult an animal’s innards, whether their own person or a bovine’s, were being encouraged to be antithetical to evidence-based medicine, a history long and hard fought to combat a disease, and our greatest achievement as a species which continues to save millions of lives.

Here is how statistics killed Wakefield’s reputation. His findings in the Lancet from his tiny, biased pool of patients, were overshadowed by a later investigation into 1.8 million, randomlychosen children, in Finland. They found nothing unstable when the children encountered the MMR-vaccine. Hari tells us:

Even more startlingly, it was found that when MMR was suspended in Japan due to production problems, autism rates held steady – but 90 extra children died of measles. This evidence was waved away by much of the press as difficult and indigestible; they preferred to focus instead on brain-dead trivia… (italics added)

The statistics tell us then that there is no fruitful or engaging relation between MMR and autism. The size and measurement completely undermines Wakefield’s biased nonsense. Of course, these weren’t the only tests but the sheer size indicates why, just from them, we can at the least be highly suspicious and, at best, completely dismissive of Mr Wakefield.

Wakefield was, however, not main the problem. It was the media’s coverage, their dismissal of important statistics it took me merely seconds to find. If you want the actual culprits, all you need to do is investigate. My point is this: through merely putting Wakefield’s findings into a proper context, we can see whether he is worth taking seriously or is biased and mistaken, if not lying. Like the assertion that drinking from the Queen’s pond heals arthritis, we can just increase the size, look wider and farther, investigate other explanations or ponder if there is an explanation at all worth pursuing. For example, the MMR and autism link was not worth pursuing and was better off not pursuing: considering that even one child died from not being immunised. That is one child too many. However, as one powerful website has indicated, we can (at time of writing) attribute 612 preventable deaths, in the US, to the furore and madness that was targeted at vaccines. And another number is close to it: 66, 515 preventable illnesses.

If we need any more reasons, I can provide them. A problem close to home, in the metaphorical and literal sense, came about through the public poli(dio)cy of Thabo Mbeki, in his ‘denial’ of a link between HIV and AIDS. There is much speculation whether he really thought so, but he very fervently thought of it as a colonial problem, instead of a medical one. So much so that anti-retrovirals were affected in their distribution because Mbeki denied the ‘Western’ science’s diagnosis of HIV/AIDS. Here is the great Raymond Tallis, quoted in full, from his brilliant Hippocratic Oaths:

Of the 70,000 children born annually to HIV-positive mothers in South Africa, about half could have been protected from becoming HIV-positive themselves, and suffering a painful, protracted death, with a single dose of a cheap anti-retroviral drug. Mbeki did what he could to stop this happening. Many of the 800,000 non-infant deaths a year from Aids could also be prevented by making antiretroviral drugs available, but Mbeki’s ideological views did not permit it. According to a recent study (suppressed by the South African Government, which maintains that anti-HIV drugs are toxic and will primarily benefit pharmaceutical companies) immediate provision of such drugs could save up to 1.7 million people by 2010. As one of his former supporters, the Anglican Archbishop of Cape Town, the Most Rev Njongonkulu Ndungane, has said, Mbeki's Aids policies are as serious a crime as apartheid — and have already killed many more people.

Mbeki was and indeed is aware of the statistics, which highlights another problem: statistics can be ignored. But then, so can the preventable deaths of infants who die as a result of your bigoted delusions. Ignorance, like a flood, does not discriminate in what it sweeps away.

Empowering ourselves with numbers might seem strange, until we recall how statistics can destroy the pretentions of charlatans or miraculous happenings. Indifferent in itself, statistics displays information anyone is welcome to assess. You would be hard-pressed in defending Wakefield’s tiny Lancet study, with twelve children, over the thorough Finnish one, with 1.8 million. However, numbers are not the end: control-groups, double-blind mechanisms and sensitivity to scientific reasoning that comes with studying statistics are also necessary. In many instances of outrage, like the anti-vaccination uproar or Mbeki’s idiocy, we can reasonably assume that their assertions have no backing with regard to control-groups, alternative hypotheses, and so on. It is invariably the ape-man bursting out the lab coat, to pound his chest, beating out the rhythm of his own bias and delusion.

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