Black Swan Vol: The Idea of Positive Accidental PDF

THE IDEA OF POSITIVE ACCIDENT Recall the empirics, those members of the Greek school of empirical medicine. They considered that you should be open-minded in your medical diagnoses to let luck play a role. By luck, a patient might be cured, say, by eating some food that accidentally turns out...

THE IDEA OF POSITIVE ACCIDENT Recall the empirics, those members of the Greek school of empirical medicine. They considered that you should be open-minded in your medical diagnoses to let luck play a role. By luck, a patient might be cured, say, by eating some food that accidentally turns out to be the cure for his disease, so that the treatment can then be used on subsequent patients. The positive accident (like hypertension medicine producing side bene ts that led to Viagra) was the empirics’ central method of medical discovery. This same point can be generalized to life: maximize the serendipity around you. Sextus Empiricus retold the story of Apelles the Painter, who, while doing a portrait of a horse, was attempting to depict the foam from the horse’s mouth. After trying very hard and making a mess, he gave up and, in irritation, took the sponge he used for cleaning his brush and threw it at the picture. Where the sponge hit, it left a perfect representation of the foam. Trial and error means trying a lot. In The Blind Watchmaker, Richard Dawkins brilliantly illustrates this notion of the world without grand design, moving by small incremental random changes. Note a slight disagreement on my part that does not change the story by much: the world, rather, moves by large incremental random changes. Indeed, we have psychological and intellectual di culties with trial and error, and with accepting that series of small failures are necessary in life. My colleague Mark Spitznagel understood that we humans have a mental hang-up about failures: “You need to love to lose” was his motto. In fact, the reason I felt immediately at home in America is precisely because American culture encourages the process of failure, unlike the cultures of Europe and Asia where failure is met with stigma and embarrassment. America’s specialty is to take these small risks for the rest of the world, which explains this country’s disproportionate share in innovations. Once established, an idea or a product is later “perfected” over there. Volatility and Risk of Black Swan People are often ashamed of losses, so they engage in strategies that produce very little volatility but contain the risk of a large loss—like collecting nickels in front of steamrollers. In Japanese culture, which is ill-adapted to randomness and badly equipped to understand that bad performance can come from bad luck, losses can severely tarnish someone’s reputation. People hate volatility, thus engage in strategies exposed to blowups, leading to occasional suicides after a big loss. Furthermore, this trade-o between volatility and risk can show up in careers that give the appearance of being stable, like jobs at IBM until the 1990s. When laid o , the employee faces a total void: he is no longer t for anything else. The same holds for those in protected industries. On the other hand, consultants can have volatile earnings as their clients’ earnings go up and down, but face a lower risk of starvation, since their skills match demand —fluctuat nec mergitur ( uctuates but doesn’t sink). Likewise, dictatorships that do not appear volatile, like, say, Syria or Saudi Arabia, face a larger risk of chaos than, say, Italy, as the latter has been in a state of continual political turmoil since the second war. I learned about this problem from the nance industry, in which we see “conservative” bankers sitting on a pile of dynamite but fooling themselves because their operations seem dull and lacking in volatility. Barbell Strategy  I am trying here to generalize to real life the notion of the “barbell” strategy I used as a trader, which is as follows. If you know that you are vulnerable to prediction errors, and if you accept that most “risk measures” are awed, because of the Black Swan, then your strategy is to be as hyperconservative and hyperaggressive as you can be instead of being mildly aggressive or conservative. Instead of putting your money in “medium risk” investments (how do you know it is medium risk? by listening to tenure- seeking “experts”?), you need to put a portion, say 85 to 90 percent, in extremely safe instruments, like Treasury bills—as safe a class of instruments as you can manage to nd on this planet. The remaining 10 to 15 percent you put in extremely speculative bets, as leveraged as possible (like options), preferably venture capital–style portfolios.* That way you do not depend on errors of risk management; no Black Swan can hurt you at all, beyond your “ oor,” the nest egg that you have in maximally safe investments. Or, equivalently, you can have a speculative portfolio and insure it (if possible) against losses of more than, say, 15 percent. You are “clipping” your incomputable risk, the one that is harmful to you. Instead of having medium risk, you have high risk on one side and no risk on the other. The average will be medium risk but constitutes a positive exposure to the Black Swan. More technically, this can be called a “convex” combination. Let us see how this can be implemented in all aspects of life. “Nobody Knows Anything” The legendary screenwriter William Goldman was said to have shouted “Nobody knows anything!” in relation to the prediction of movie sales. Now, the reader may wonder how someone as successful as Goldman can gure out what to do without making predictions. The answer stands perceived business logic on its head. He knew that he could not predict individual events, but he was well aware that the unpredictable, namely a movie turning into a blockbuster, would bene t him immensely. So the second lesson is more aggressive: you can actually take advantage of the problem of prediction and epistemic arrogance! As a matter of fact, I suspect that the most successful businesses are precisely those that know how to work around inherent unpredictability and even exploit it. Recall my discussion of the biotech company whose managers understood that the essence of research is in the unknown unknowns. Also, notice how they seized on the “corners,” those free lottery tickets in the world. Here are the (modest) tricks. But note that the more modest they are, the more e ective they will be. 1. First, make a distinction between positive contingencies and negative ones. Learn to distinguish between those human undertakings in which the lack of predictability can be (or has been) extremely bene cial and those where the failure to understand the future caused harm. There are both positive and negative Black Swans. William Goldman was involved in the movies, a positive–Black Swan business. Uncertainty did occasionally pay o there. A negative–Black Swan business is one where the unexpected can hit hard and hurt severely. If you are in the military, in catastrophe insurance, or in homeland security, you face only downside. Likewise, as we saw in Chapter 7, if you are in banking and lending, surprise outcomes are likely to be negative for you. You lend, and in the best of circumstances you get your loan back—but you may lose all of your money if the borrower defaults. In the event that the borrower enjoys great nancial success, he is not likely to o er you an additional dividend. Aside from the movies, examples of positive–Black Swan businesses are: some segments of publishing, scienti c research, and venture capital. In these businesses, you lose small to make big. You have little to lose per book and, for completely unexpected reasons, any given book might take o. The downside is small and easily controlled. The problem with publishers, of course, is that they regularly pay up for books, thus making their upside rather limited and their downside monstrous. (If you pay $10 million for a book, your Black Swan is it not being a bestseller.) Likewise, while technology can carry a great payo , paying for the hyped-up story, as people did with the dot-com bubble, can make any upside limited and any downside huge. It is the venture capitalist who invested in a speculative company and sold his stake to unimaginative investors who is the bene ciary of the Black Swan, not the “me, too” investors. In these businesses you are lucky if you don’t know anything— particularly if others don’t know anything either, but aren’t aware of it. And you fare best if you know where your ignorance lies, if you are the only one looking at the unread books, so to speak. This dovetails into the “barbell” strategy of taking maximum exposure to the positive Black Swans while remaining paranoid about the negative ones. For your exposure to the positive Black Swan, you do not need to have any precise understanding of the structure of uncertainty. I nd it hard to explain that when you have a very limited loss you need to get as aggressive, as speculative, and sometimes as “unreasonable” as you can be. Middlebrow thinkers sometimes make the analogy of such strategy with that of collecting “lottery tickets.” It is plain wrong. First, lottery tickets do not have a scalable payo ; there is a known upper limit to what they can deliver. The ludic fallacy applies here—the scalability of real-life payo s compared to lottery ones makes the payo unlimited or of unknown limit. Secondly, the lottery tickets have known rules and laboratory-style well- presented possibilities; here we do not know the rules and can bene t from this additional uncertainty, since it cannot hurt you and can only bene t you.* 2. Don’t look for the precise and the local. Simply, do not be narrow- minded. The great discoverer Pasteur, who came up with the notion that chance favors the prepared, understood that you do not look for something particular every morning but work hard to let contingency enter your working life. As Yogi Berra, another great thinker, said, “You got to be very careful if you don’t know where you’re going, because you might not get there.” Likewise, do not try to predict precise Black Swans—it tends to make you more vulnerable to the ones you did not predict. My friends Andy Marshall and Andrew Mays at the Department of Defense face the same problem. The impulse on the part of the military is to devote resources to predicting the next problems. These thinkers advocate the opposite: invest in preparedness, not in prediction. Remember that in nite vigilance is just not possible. 3. Seize any opportunity, or anything that looks like opportunity. They are rare, much rarer than you think. Remember that positive Black Swans have a necessary rst step: you need to be exposed to them. Many people do not realize that they are getting a lucky break in life when they get it. If a big publisher (or a big art dealer or a movie executive or a hotshot banker or a big thinker) suggests an appointment, cancel anything you have planned: you may never see such a window open up again. I am sometimes shocked at how little people realize that these opportunities do not grow on trees. Collect as many free nonlottery tickets (those with open-ended payo s) as you can, and, once they start paying o , do not discard them. Work hard, not in grunt work, but in chasing such opportunities and maximizing exposure to them. This makes living in big cities invaluable because you increase the odds of serendipitous encounters—you gain exposure to the envelope of serendipity. The idea of settling in a rural area on grounds that one has good communications “in the age of the Internet” tunnels out of such sources of positive uncertainty. Diplomats understand that very well: casual chance discussions at cocktail parties usually lead to big breakthroughs—not dry correspondence or telephone conversations. Go to parties! If you’re a scientist, you will chance upon a remark that might spark new research. And if you are autistic, send your associates to these events. 4. Beware of precise plans by governments. As discussed in Chapter 10, let governments predict (it makes o cials feel better about themselves and  justi es their existence) but do not set much store by what they say. Remember that the interest of these civil servants is to survive and self- perpetuate—not to get to the truth. It does not mean that governments are useless, only that you need to keep a vigilant eye on their side e ects. For instance, regulators in the banking business are prone to a severe expert problem and they tend to condone reckless but (hidden) risk taking. Andy Marshall and Andy Mays asked me if the private sector could do better in predicting. Alas, no. Once again, recall the story of banks hiding explosive risks in their portfolios. It is not a good idea to trust corporations with matters such as rare events because the performance of these executives is not observable on a short-term basis, and they will game the system by showing good performance so they can get their yearly bonus. The Achilles’ heel of capitalism is that if you make corporations compete, it is sometimes the one that is most exposed to the negative Black Swan that will appear to be the most t for survival. Also recall from the footnote on Ferguson’s discovery in Chapter 1 that markets are not good predictors of wars. No one in particular is a good predictor of anything. Sorry. 5. “There are some people who, if they don’t already know, you can’t tell ’em,” as the great philosopher of uncertainty Yogi Berra once said. Do not waste your time trying to fight forecasters, stock analysts, economists, and social scientists, except to play pranks on them. They are considerably easy to make fun of, and many get angry quite readily. It is ine ective to moan about unpredictability: people will continue to predict foolishly, especially if they are paid for it, and you cannot put an end to institutionalized frauds. If you ever do have to heed a forecast, keep in mind that its accuracy degrades rapidly as you extend it through time. If you hear a “prominent” economist using the word equilibrium, or normal distribution, do not argue with him; just ignore him, or try to put a rat down his shirt. The Great Asymmetry  All these recommendations have one point in common: asymmetry. Put yourself in situations where favorable consequences are much larger than unfavorable ones. Indeed, the notion of asymmetric outcomes is the central idea of this book: I will never get to know the unknown since, by de nition, it is unknown. However, I can always guess how it might a ect me, and I should base my decisions around that. This idea is often erroneously called Pascal’s wager, after the philosopher and (thinking) mathematician Blaise Pascal. He presented it something like this: I do not know whether God exists, but I know that I have nothing to gain from being an atheist if he does not exist, whereas I have plenty to lose if he does. Hence, this justi es my belief in God. Pascal’s argument is severely awed theologically: one has to be naïve enough to believe that God would not penalize us for false belief. Unless, of course, one is taking the quite restrictive view of a naïve God. (Bertrand Russell was reported to have claimed that God would need to have created fools for Pascal’s argument to work.) But the idea behind Pascal’s wager has fundamental applications outside of theology. It stands the entire notion of knowledge on its head. It eliminates the need for us to understand the probabilities of a rare event (there are fundamental limits to our knowledge of these); rather, we can focus on the payo and bene ts of an event if it takes place. The probabilities of very rare events are not computable; the e ect of an event on us is considerably easier to ascertain (the rarer the event, the fuzzier the odds). We can have a clear idea of the consequences of an event, even if we do not know how likely it is to occur. I don’t know the odds of an earthquake, but I can imagine how San Francisco might be a ected by one. This idea that in order to make a decision you need to focus on the consequences (which you can know) rather than the probability (which you can’t know) is the central idea of uncertainty. Much of my life is based on it. You can build an overall theory of decision making on this idea. All you have to do is mitigate the consequences. As I said, if my portfolio is exposed to a market crash, the odds of which I can’t compute, all I have to do is buy insurance, or get out and invest the amounts I am not willing to ever lose in less risky securities. E ectively, if free markets have been successful, it is precisely because they allow the trial-and-error process I call “stochastic tinkering” on the part of competing individual operators who fall for the narrative fallacy—but are e ectively collectively partaking of a grand project. We are increasingly learning to practice stochastic tinkering without knowing it—thanks to overcon dent entrepreneurs, naïve investors, greedy investment bankers, and aggressive venture capitalists brought together by the free-market system. The next chapter shows why I am optimistic that the academy is losing its power and ability to put knowledge in straitjackets and that more out-of-the-box knowledge will be generated Wiki-style. In the end we are being driven by history, all the while thinking that we are doing the driving. I’ll sum up this long section on prediction by stating that we can easily narrow down the reasons we can’t gure out what’s going on. There are: a) epistemic arrogance and our corresponding future blindness; b) the Platonic notion of categories, or how people are fooled by reductions, particularly if they have an academic degree in an expert-free discipline; and, nally c) awed tools of inference, particularly the Black Swan–free tools from Mediocristan. In the next section we will go deeper, much deeper, into these tools from Mediocristan, into the “plumbing,” so to speak. Some readers may see it as an appendix; others may consider it the heart of the book. * This chapter provides a general conclusion for those who by now say, “Taleb, I get the point, but what should I do?” My answer is that if you got the point, you are pretty much there. But here is a nudge. * Dan Gilbert showed in a famous paper, “How Mental Systems Believe,” that we are not natural skeptics and that not believing required an expenditure of mental e ort. * Make sure that you have plenty of these small bets; avoid being blinded by the vividness of one single Black Swan. Have as many of these small bets as you can conceivably have. Even venture capital rms fall for the narrative fallacy with a few stories that “make sense” to them; they do not have as many bets as they should. If venture capital rms are pro table, it is not because of the stories they have in their heads, but because they are exposed to unplanned rare events. * There is a ner epistemological point. Remember that in a virtuous Black Swan business, what the past did not reveal is almost certainly going to be good for you. When you look at past biotech revenues, you do not see the superblockbuster in them, and owing to the potential for a cure for cancer (or headaches, or baldness, or bad sense of humor, etc.), there is a small probability that the sales in that industry may turn out to be monstrous, far larger than might be expected. On the other hand, consider negative Black Swan businesses. The track record you see is likely to overestimate the properties. Recall the 1982 blowup of banks: they appeared to the naïve observer to be more pro table than they seemed. Insurance companies are of two kinds: the regular diversi able kind that belongs to Mediocristan (say, life insurance) and the more critical and explosive Black Swan–prone risks that are usually sold to reinsurers. According to the data, reinsurers have lost money on underwriting over the past couple of decades, but, unlike bankers, they are introspective enough to know that it actually could have been far worse, because the past twenty years did not have a big catastrophe, and all you need is one of those per century to kiss the business good- bye. Many nance academics doing “valuation” on insurance seem to have missed the point.  It’s time to deal in some depth with four nal items that bear on our Black Swan. Primo, I have said earlier that the world is moving deeper into Extremistan, that it is less and less governed by Mediocristan—in fact, this idea is more subtle than that. I will show how and present the various ideas we have about the formation of inequality. Secondo, I have been describing the Gaussian bell curve as a contagious and severe delusion, and it is time to get into that point in some depth. Terso, I will present what I call Mandelbrotian, or fractal, randomness. Remember that for an event to be a Black Swan, it does not just have to be rare, or just wild; it has to be unexpected, has to lie outside our tunnel of possibilities. You must be a sucker for it. As it happens, many rare events can yield their structure to us: it is not easy to compute their probability, but it is easy to get a general idea about the possibility of their occurrence. We can turn these Black Swans into Gray Swans, so to speak, reducing their surprise e ect. A person aware of the possibility of such events can come to belong to the non-sucker variety. Finally, I will present the ideas of those philosophers who focus on phony uncertainty. I organized this book in such a way that the more technical (though nonessential) sections are here; these can be skipped without any loss to the thoughtful reader, particularly Chapters 15, 17, and the second half of Chapter 16. I will alert the reader with footnotes. The reader less interested in the mechanics of deviations can then directly proceed to Part 4. Chapter Fourteen FROM MEDIOCRISTAN TO EXTREMISTAN, AND BACK I prefer Horowitz—How to fall from favor—The long tail—Get ready for some surprises—It’s not just money Let us see how an increasingly man-made planet can evolve away from mild into wild randomness. First, I describe how we get to Extremistan. Then, I will take a look at its evolution. The World Is Unfair Is the world that unfair? I have spent my entire life studying randomness, practicing randomness, hating randomness. The more that time passes, the worse things seem to me, the more scared I get, the more disgusted I am with Mother Nature. The more I think about my subject, the more I see evidence that the world we have in our minds is di erent from the one playing outside. Every morning the world appears to me more random than it did the day before, and humans seem to be even more fooled by it than they were the previous day. It is becoming unbearable. I nd writing these lines painful; I nd the world revolting. Two “soft” scientists propose intuitive models for the development of this inequity: one is a mainstream economist, the other a sociologist. Both simplify a little too much. I will present their ideas because they are easy to understand, not because of the scienti c quality of their insights or any consequences in their discoveries; then I will show the story as seen from the vantage point of the natural scientists. Let me start with the economist Sherwin Rosen. In the early eighties, he wrote papers about “the economics of superstars.” In one of the papers he conveyed his sense of outrage that a basketball player could earn $1.2 million a year, or a television celebrity could make $2 million. To get an idea of how this concentration is increasing—i.e., of how we are moving away from Mediocristan—consider that television celebrities and sports stars (even in Europe) get contracts today, only two decades later, worth in the hundreds of millions of dollars! The extreme is about (so far) twenty times higher than it was two decades ago! According to Rosen, this inequality comes from a tournament e ect: someone who is marginally “better” can easily win the entire pot, leaving the others with nothing. Using an argument from Chapter 3, people prefer to pay $10.99 for a recording featuring Horowitz to $9.99 for a struggling pianist. Would you rather read Kundera for $13.99 or some unknown author for $1? So it looks like a tournament, where the winner grabs the whole thing—and he does not have to win by much. But the role of luck is missing in Rosen’s beautiful argument. The problem here is the notion of “better,” this focus on skills as leading to success. Random outcomes, or an arbitrary situation, can also explain success, and provide the initial push that leads to a winner-take-all result. A person can get slightly ahead for entirely random reasons; because we like to imitate one another, we will ock to him. The world of contagion is so underestimated! As I am writing these lines I am using a Macintosh, by Apple, after years of using Microsoft-based products. The Apple technology is vastly better, yet the inferior software won the day. How? Luck. The Matthew Effect More than a decade before Rosen, the sociologist of science Robert K. Merton presented his idea of the Matthew e ect, by which people take from the poor to give to the rich.* He looked at the performance of scientists and showed how an initial advantage follows someone through life. Consider the following process. Let’s say someone writes an academic paper quoting fty people who have worked on the subject and provided background materials for his study; assume, for the sake of simplicity, that all fty are of equal merit. Another researcher working on the exact same subject will randomly cite three of those fty in his bibliography. Merton showed that many academics cite references without having read the original work; rather, they’ll read a paper and draw their own citations from among its sources. So a third researcher reading the second article selects three of the previously referenced authors for his citations. These three authors will receive cumulatively more and more attention as their names become associated more tightly with the subject at hand. The di erence between the winning three and the other members of the original cohort is mostly luck: they were initially chosen not for their greater skill, but simply for the way their names appeared in the prior bibliography. Thanks to their reputations, these successful academics will go on writing papers and their work will be easily accepted for publication. Academic success is partly (but signi cantly) a lottery.* It is easy to test the e ect of reputation. One way would be to nd papers that were written by famous scientists, had their authors’ identities changed by mistake, and got rejected. You could verify how many of these rejections were subsequently overturned after the true identities of the authors were established. Note that scholars are judged mostly on how many times their work is referenced in other people’s work, and thus cliques of people who quote one another are formed (it’s an “I quote you, you quote me” type of business). Eventually, authors who are not often cited will drop out of the game by, say, going to work for the government (if they are of a gentle nature), or for the Ma a, or for a Wall Street rm (if they have a high level of hormones). Those who got a good push in the beginning of their scholarly careers will keep getting persistent cumulative advantages throughout life. It is easier for the rich to get richer, for the famous to become more famous. In sociology, Matthew e ects bear the less literary name “cumulative advantage.” This theory can easily apply to companies, businessmen, actors, writers, and anyone else who bene ts from past success. If you get published in The New Yorker because the color of your letterhead attracted the attention of the editor, who was daydreaming of daisies, the resultant reward can follow you for life. More signi cantly, it will follow others for life. Failure is also cumulative; losers are likely to also lose in the future, even if we don’t take into account the mechanism of demoralization that might exacerbate it and cause additional failure. Note that art, because of its dependence on word of mouth, is extremely prone to these cumulative-advantage e ects. I mentioned clustering in Chapter 1, and how journalism helps perpetuate these clusters. Our opinions about artistic merit are the result of arbitrary contagion even more than our political ideas are. One person writes a book review; another person reads it and writes a commentary that uses the same arguments. Soon you have several hundred reviews that actually sum up in their contents to no more than two or three because there is so much overlap. For an anecdotal example read Fire the Bastards!, whose author, Jack Green, goes systematically through the reviews of William Gaddis’s novel The Recognitions. Green shows clearly how book reviewers anchor on other reviews and reveals powerful mutual in uence, even in their wording. This phenomenon is reminiscent of the herding of nancial analysts I discussed in Chapter 10. The advent of the modern media has accelerated these cumulative advantages. The sociologist Pierre Bourdieu noted a link between the increased concentration of success and the globalization of culture and economic life. But I am not trying to play sociologist here, only show that unpredictable elements can play a role in social outcomes. Merton’s cumulative-advantage idea has a more general precursor, “preferential attachment,” which, reversing the chronology (though not the logic), I will present next. Merton was interested in the social aspect of knowledge, not in the dynamics of social randomness, so his studies were derived separately from research on the dynamics of randomness in more mathematical sciences. Lingua Franca The theory of preferential attachment is ubiquitous in its applications: it can explain why city size is from Extremistan, why vocabulary is concentrated among a small number of words, or why bacteria populations can vary hugely in size. The scientists J. C. Willis and G. U. Yule published a landmark paper in Nature in 1922 called “Some Statistics of Evolution and Geographical Distribution in Plants and Animals, and Their Signi cance.” Willis and Yule noted the presence in biology of the so-called power laws, atractable versions of the scalable randomness that I discussed in Chapter 3. These power laws (on which more technical information in the following chapters) had been noticed earlier by Vilfredo Pareto, who found that they applied to the distribution of income. Later, Yule presented a simple model showing how power laws can be generated. His point was as follows: Let’s say species split in two at some constant rate, so that new species arise. The richer in species a genus is, the richer it will tend to get, with the same logic as the Mathew e ect. Note the following caveat: in Yule’s model the species never die out. During the 1940s, a Harvard linguist, George Zipf, examined the properties of language and came up with an empirical regularity now known as Zipf’s law, which, of course, is not a law (and if it were, it would not be Zipf’s). It is just another way to think about the process of inequality. The mechanisms he described were as follows: the more you use a word, the less e ortful you will nd it to use that word again, so you borrow words from your private dictionary in proportion to their past use. This explains why out of the sixty thousand main words in English, only a few hundred constitute the bulk of what is used in writings, and even fewer appear regularly in conversation. Likewise, the more people aggregate in a particular city, the more likely a stranger will be to pick that city as his destination. The big get bigger and the small stay small, or get relatively smaller. A great illustration of preferential attachment can be seen in the mushrooming use of English as a lingua franca—though not for its intrinsic qualities, but because people need to use one single language, or stick to one as much as possible, when they are having a conversation. So whatever language appears to have the upper hand will suddenly draw people in droves; its usage will spread like an epidemic, and other languages will be rapidly dislodged. I am often amazed to listen to conversations between people from two neighboring countries, say, between a Turk and an Iranian, or a Lebanese and a Cypriot, communicating in bad English, moving their hands for emphasis, searching for these words that come out of their throats at the cost of great physical e ort. Even members of the Swiss Army use English (not French) as a lingua franca (it would be fun to listen). Consider that a very small minority of Americans of northern European descent is from England; traditionally the preponderant ethnic groups are of German, Irish, Dutch, French, and other northern European extraction. Yet because all these groups now use English as their main tongue, they have to study the roots of their adoptive tongue and develop a cultural association with parts of a particular wet island, along with its history, its traditions, and its customs! Ideas and Contagions The same model can be used for the contagions and concentration of ideas. But there are some restrictions on the nature of epidemics I must discuss here. Ideas do not spread without some form of structure. Recall the discussion in Chapter 4 about how we come prepared to make inferences. Just as we tend to generalize some matters but not others, so there seem to be “basins of attraction” directing us to certain beliefs. Some ideas will prove contagious, but not others; some forms of superstitions will spread, but not others; some types of religious beliefs will dominate, but not others. The anthropologist, cognitive scientist, and philosopher Dan Sperber has proposed the following idea on the epidemiology of representations. What people call “memes,” ideas that spread and that compete with one another using people as carriers, are not truly like genes. Ideas spread because, alas, they have for carriers self-serving agents who are interested in them, and interested in distorting them in the replication process. You do not make a cake for the sake of merely replicating a recipe—you try to make your own cake, using ideas from others to improve it. We humans are not photocopiers. So contagious mental categories must be those in which we are prepared to believe, perhaps even programmed to believe. To be contagious, a mental category must agree with our nature. NOBODY IS SAFE IN EXTREMISTAN There is something extremely naïve about all these models of the dynamics of concentration I’ve presented so far, particularly the socioeconomic ones. For instance, although Merton’s idea includes luck, it misses an additional layer of randomness. In all these models the winner stays a winner. Now, a loser might always remain a loser, but a winner could be unseated by someone new popping up out of nowhere. Nobody is safe. Preferential-attachment theories are intuitively appealing, but they do not account for the possibility of being supplanted by newcomers—what every schoolchild knows as the decline of civilizations. Consider the logic of cities: How did Rome, with a population of 1.2 million in the rst century A.D., end up with a population of twelve thousand in the third? How did Baltimore, once a principal American city, become a relic? And how did Philadelphia come to be overshadowed by New York? A Brooklyn Frenchman When I started trading foreign exchange, I befriended a fellow named Vincent who exactly resembled a Brooklyn trader, down to the mannerisms of Fat Tony, except that he spoke the French version of Brooklynese. Vincent taught me a few tricks. Among his sayings were “Trading may have princes, but nobody stays a king” and “The people you meet on the way up, you will meet again on the way down.” There were theories when I was a child about class warfare and struggles by innocent individuals against powerful monster-corporations capable of swallowing the world. Anyone with intellectual hunger was fed these theories, which were inherited from the Marxist belief that the tools of exploitation were self-feeding, that the powerful would grow more and more powerful, furthering the unfairness of the system. But one had only to look around to see that these large corporate monsters dropped like ies. Take a cross section of the dominant corporations at any particular time; many of them will be out of business a few decades later, while rms nobody ever heard of will have popped onto the scene from some garage in California or from some college dorm. Consider the following sobering statistic. Of the ve hundred largest U.S. companies in 1957, only seventy-four were still part of that select group, the Standard and Poor’s 500, forty years later. Only a few had disappeared in mergers; the rest either shrank or went bust. Interestingly, almost all these large corporations were located in the most capitalist country on earth, the United States. The more socialist a country’s orientation, the easier it was for the large corporate monsters to stick around. Why did capitalism (and not socialism) destroy these ogres? In other words, if you leave companies alone, they tend to get eaten up. Those in favor of economic freedom claim that beastly and greedy corporations pose no threat because competition keeps them in check. What I saw at the Wharton School convinced me that the real reason includes a large share of something else: chance. But when people discuss chance (which they rarely do), they usually only look at their own luck. The luck of others counts greatly. Another corporation may luck out thanks to a blockbuster product and displace the current winners. Capitalism is, among other things, the revitalization of the world thanks to the opportunity to be lucky. Luck is the grand equalizer, because almost everyone can bene t from it. The socialist governments protected their monsters and, by doing so, killed potential newcomers in the womb. Everything is transitory. Luck both made and unmade Carthage; it both made and unmade Rome. I said earlier that randomness is bad, but it is not always so. Luck is far more egalitarian than even intelligence. If people were rewarded strictly according to their abilities, things would still be unfair—people don’t choose their abilities. Randomness has the bene cial e ect of reshu ing society’s cards, knocking down the big guy. In the arts, fads do the same job. A newcomer may bene t from a fad, as followers multiply thanks to a preferential attachment–style epidemic. Then, guess what? He too becomes history. It is quite interesting to look at the acclaimed authors of a particular era and see how many have dropped out of consciousness. It even happens in countries such as France where the government supports established reputations, just as it supports ailing large companies. When I visit Beirut, I often spot in relatives’ homes the remnants of a series of distinctively white-leather-bound “Nobel books.” Some hyperactive salesman once managed to populate private libraries with these beautifully made volumes; many people buy books for decorative purposes and want a simple selection criterion. The criterion this series o ered was one book by a Nobel winner in literature every year—a simple way to build the ultimate library. The series was supposed to be updated every year, but I presume the company went out of business in the eighties. I feel a pang every time I look at these volumes: Do you hear much today about Sully Prudhomme (the rst recipient), Pearl Buck (an American woman), Romain Rolland, Anatole France (the last two were the most famous French authors of their generations), St. John Perse, Roger Martin du Gard, or Frédéric Mistral? The Long Tail I have said that nobody is safe in Extremistan. This has a converse: nobody is threatened with complete extinction either. Our current environment allows the little guy to bide his time in the antechamber of success—as long as there is life, there is hope.  This idea was recently revived by Chris Anderson, one of a very few who get the point that the dynamics of fractal concentration has another layer of randomness. He packaged it with his idea of the “long tail,” about which in a moment. Anderson is lucky not to be a professional statistician (people who have had the misfortune of going through conventional statistical training think we live in Mediocristan). He was able to take a fresh look at the dynamics of the world. True, the Web produces acute concentration. A large number of users visit just a few sites, such as Google, which, at the time of this writing, has total market dominance. At no time in history has a company grown so dominant so quickly—Google can service people from Nicaragua to southwestern Mongolia to the American West Coast, without having to worry about phone operators, shipping, delivery, and manufacturing. This is the ultimate winner- take-all case study. People forget, though, that before Google, Alta Vista dominated the search-engine market. I am prepared to revise the Google metaphor by replacing it with a new name for future editions of this book. What Anderson saw is that the Web causes something in addition to concentration. The Web enables the formation of a reservoir of proto- Googles waiting in the background. It also promotes the inverse Google, that is, it allows people with a technical specialty to nd a small, stable audience. Recall the role of the Web in Yevgenia Krasnova’s success. Thanks to the Internet, she was able to bypass conventional publishers. Her publisher with the pink glasses would not even have been in business had it not been for the Web. Let’s assume that Amazon.com does not exist, and that you have written a sophisticated book. Odds are that a very small bookstore that carries only 5,000 volumes will not be interested in letting your “beautifully crafted prose” occupy premium shelf space. And the megabookstore, such as the average American Barnes & Noble, might stock 130,000 volumes, which is still not su cient to accommodate marginal titles. So your work is stillborn. Not so with Web vendors. A Web bookstore can carry a near-in nite number of books since it need not have them physically in inventory. Actually, nobody needs to have them physically in inventory since they can remain in digital form until they are needed in print, an emerging business called print- on-demand. So as the author of this little book, you can sit there, bide your time, be available in search engines, and perhaps bene t from an occasional epidemic. In fact, the quality of readership has improved markedly over the past few years thanks to the availability of these more sophisticated books. This is a fertile environment for diversity.* Plenty of people have called me to discuss the idea of the long tail, which seems to be the exact opposite of the concentration implied by scalability. The long tail implies that the small guys, collectively, should control a large segment of culture and commerce, thanks to the niches and subspecialties that can now survive thanks to the Internet. But, strangely, it can also imply a large measure of inequality: a large base of small guys and a very small number of supergiants, together representing a share of the world’s culture—with some of the small guys, on occasion, rising to knock out the winners. (This is the “double tail”: a large tail of the small guys, a small tail of the big guys.) The role of the long tail is fundamental in changing the dynamics of success, destabilizing the well-seated winner, and bringing about another winner. In a snapshot this will always be Extremistan, always ruled by the concentration of type-2 randomness; but it will be an ever-changing Extremistan. The long tail’s contribution is not yet numerical; it is still con ned to the Web and its small-scale online commerce. But consider how the long tail could a ect the future of culture, information, and political life. It could free us from the dominant political parties, from the academic system, from the clusters of the press—anything that is currently in the hands of ossi ed, conceited, and self-serving authority. The long tail will help foster cognitive diversity. One highlight of the year 2006 was to nd in my mailbox a draft manuscript of a book called Cognitive Diversity: How Our Individual Differences Produce Collective Benefits, by Scott Page. Page examines the e ects of cognitive diversity on problem solving and shows how variability in views and methods acts like an engine for tinkering. It works like evolution. By subverting the big structures we also get rid of the Platoni ed one way of doing things—in the end, the bottom-up theory-free empiricist should prevail. In sum, the long tail is a by-product of Extremistan that makes it somewhat less unfair: the world is made no less unfair for the little guy, but it now becomes extremely unfair for the big man. Nobody is truly established. The little guy is very subversive. Naïve Globalization We are gliding into disorder, but not necessarily bad disorder. This implies that we will see more periods of calm and stability, with most problems concentrated into a small number of Black Swans. Consider the nature of past wars. The twentieth century was not the deadliest (in percentage of the total population), but it brought something new: the beginning of the Extremistan warfare—a small probability of a con ict degenerating into total decimation of the human race, a con ict from which nobody is safe anywhere. A similar e ect is taking place in economic life. I spoke about globalization in Chapter 3; it is here, but it is not all for the good: it creates interlocking fragility, while reducing volatility and giving the appearance of stability. In other words it creates devastating Black Swans. We have never lived before under the threat of a global collapse. Financial institutions have been merging into a smaller number of very large banks. Almost all banks are now interrelated. So the nancial ecology is swelling into gigantic, incestuous, bureaucratic banks (often Gaussianized in their risk measurement)—when one falls, they all fall.* The increased concentration among banks seems to have the e ect of making nancial crisis less likely, but when they happen they are more global in scale and hit us very hard. We have moved from a diversi ed ecology of small banks, with varied lending policies, to a more homogeneous framework of rms that all resemble one another. True, we now have fewer failures, but when they occur … I shiver at the thought. I rephrase here: we will have fewer but more severe crises. The rarer the event, the less we know about its odds. It means that we know less and less about the possibility of a crisis. And we have some idea how such a crisis would happen. A network is an assemblage of elements called nodes that are somehow connected to one another by a link; the world’s airports constitute a network, as does the World Wide Web, as do social connections and electricity grids. There is a branch of research called “network theory” that studies the organization of such networks and the links between their nodes, with such researchers as Duncan Watts, Steven Strogatz, Albert-Laszlo Barabasi, and many more. They all understand Extremistan mathematics and the inadequacy of the Gaussian bell curve. They have uncovered the following property of networks: there is a concentration among a few nodes that serve as central connections. Networks have a natural tendency to organize themselves around an extremely concentrated architecture: a few nodes are extremely connected; others barely so. The distribution of these connections has a scalable structure of the kind we will discuss in Chapters 15 and 16. Concentration of this kind is not limited to the Internet; it appears in social life (a small number of people are connected to others), in electricity grids, in communications networks. This seems to make networks more robust: random insults to most parts of the network will not be consequential since they are likely to hit a poorly connected spot. But it also makes networks more vulnerable to Black Swans. Just consider what would happen if there is a problem with a major node. The electricity blackout experienced in the northeastern United States during August 2003, with its consequential mayhem, is a perfect example of what could take place if one of the big banks went under today. But banks are in a far worse situation than the Internet. The nancial industry has no signi cant long tail! We would be far better o if there were a di erent ecology, in which nancial institutions went bust on occasion and were rapidly replaced by new ones, thus mirroring the diversity of Internet businesses and the resilience of the Internet economy. Or if there were a long tail of government o cials and civil servants coming to reinvigorate bureaucracies. REVERSALS AWAY FROM EXTREMISTAN There is, inevitably, a mounting tension between our society, full of concentration, and our classical idea of aurea mediocritas, the golden mean, so it is conceivable that e orts may be made to reverse such concentration. We live in a society of one person, one vote, where progressive taxes have been enacted precisely to weaken the winners. Indeed, the rules of society can be easily rewritten by those at the bottom of the pyramid to prevent concentration from hurting them. But it does not require voting to do so— religion could soften the problem. Consider that before Christianity, in many societies the powerful had many wives, thus preventing those at the bottom from accessing wombs, a condition that is not too di erent from the reproductive exclusivity of alpha males in many species. But Christianity reversed this, thanks to the one man–one woman rule. Later, Islam came to limit the number of wives to four. Judaism, which had been polygenic, became monogamous in the Middle Ages. One can say that such a strategy has been successful—the institution of tightly monogamous marriage (with no o cial concubine, as in the Greco-Roman days), even when practiced the “French way,” provides social stability since there is no pool of angry, sexually deprived men at the bottom fomenting a revolution just so they can have the chance to mate. But I nd the emphasis on economic inequality, at the expense of other types of inequality, extremely bothersome. Fairness is not exclusively an economic matter; it becomes less and less so when we are satisfying our basic material needs. It is pecking order that matters! The superstars will always be there. The Soviets may have attened the economic structure, but they encouraged their own brand of übermensch. What is poorly understood, or denied (owing to its unsettling implications), is the absence of a role for the average in intellectual production. The disproportionate share of the very few in intellectual in uence is even more unsettling than the unequal distribution of wealth—unsettling because, unlike the income gap, no social policy can eliminate it. Communism could conceal or compress income discrepancies, but it could not eliminate the superstar system in intellectual life. It has even been shown, by Michael Marmot of the Whitehall Studies, that those at the top of the pecking order live longer, even when adjusting for disease. Marmot’s impressive project shows how social rank alone can a ect longevity. It was calculated that actors who win an Oscar tend to live on average about ve years longer than their peers who don’t. People live longer in societies that have atter social gradients. Winners kill their peers as those in a steep social gradient live shorter lives, regardless of their economic condition. I do not know how to remedy this (except through religious beliefs). Is insurance against your peers’ demoralizing success possible? Should the Nobel Prize be banned? Granted the Nobel medal in economics has not been good for society or knowledge, but even those rewarded for real contributions in medicine and physics too rapidly displace others from our consciousness, and steal longevity away from them. Extremistan is here to stay, so we have to live with it, and nd the tricks that make it more palatable. * These scalable laws were already discussed in the scriptures: “For onto everyone that hath shall be given, and he shall have abundance; but from him that hath not shall be taken away even that which he hath.” Matthew (Matthew 25:29, King James Version). * Much of the perception of the importance of precocity in the career of researchers can be owed to the misunderstanding of the perverse role of this e ect, especially when reinforced by bias. Enough counterexamples, even in elds like mathematics meant to be purely a “young man’s game,” illustrate the age fallacy: simply, it is necessary to be successful early, and even very early at that. * The Web’s bottom-up feature is also making book reviewers more  accountable. While writers were helpless and vulnerable to the arbitrariness of book reviews, which can distort their messages and, thanks to the con rmation bias, expose small irrelevant weak points in their text, they now have a much stronger hand. In place of the moaning letter to the editor, they can simply post their review of a review on the Web. If attacked ad hominem, they can reply ad hominem and go directly after the credibility of the reviewer, making sure that their statement shows rapidly in an Internet search or on Wikipedia, the bottom-up encyclopedia. * As if we did not have enough problems, banks are now more vulnerable to the Black Swan and the ludic fallacy than ever before with “scientists” among their sta taking care of exposures. The giant rm J. P. Morgan put the entire world at risk by introducing in the nineties RiskMetrics, a phony method aiming at managing people’s risks, causing the generalized use of the ludic fallacy, and bringing Dr. Johns into power in place of the skeptical Fat Tonys. (A related method called “Value-at-Risk,” which relies on the quantitative measurement of risk, has been spreading.) Likewise, the government- sponsored institution Fanny Mae, when I look at their risks, seems to be sitting on a barrel of dynamite, vulnerable to the slightest hiccup. But not to worry: their large sta of scientists deemed these events “unlikely.” Chapter Fifteen THE BELL CURVE, THAT GREAT INTELLECTUAL FRAUD* Not worth a pastis—Quételet’s error—The average man is a monster —Let’s deify it—Yes or no—Not so literary an experiment Forget everything you heard in college statistics or probability theory. If you never took such a class, even better. Let us start from the very beginning. THE GAUSSIAN AND THE MANDELBROTIAN I was transiting through the Frankfurt airport in December 2001, on my way from Oslo to Zurich. I had time to kill at the airport and it was a great opportunity for me to buy dark European chocolate, especially since I have managed to successfully convince myself that airport calories don’t count. The cashier handed me, among other things, a ten deutschmark bill, an (illegal) scan of which can be seen on the next page. The deutschmark banknotes were going to be put out of circulation in a matter of days, since Europe was switching to the euro. I kept it as a valedictory. Before the arrival of the euro, Europe had plenty of national currencies, which was good for printers, money changers, and of course currency traders like this (more or less) humble author. As I was eating my dark European chocolate and wistfully looking at the bill, I almost choked. I suddenly noticed, for the rst time, that there was something curious about it. The bill bore the portrait of Carl Friedrich Gauss and a picture of his Gaussian bell curve.  The last ten deutschmark bill, representing Gauss and, to his right, the bell curve of Mediocristan. The striking irony here is that the last possible object that can be linked to the German currency is precisely such a curve: the reichsmark (as the currency was previously called) went from four per dollar to four trillion per dollar in the space of a few years during the 1920s, an outcome that tells you that the bell curve is meaningless as a description of the randomness in currency uctuations. All you need to reject the bell curve is for such a movement to occur once, and only once—just consider the consequences. Yet there was the bell curve, and next to it Herr Professor Doktor Gauss, unprepossessing, a little stern, certainly not someone I’d want to spend time with lounging on a terrace, drinking pastis, and holding a conversation without a subject. Shockingly, the bell curve is used as a risk-measurement tool by those regulators and central bankers who wear dark suits and talk in a boring way about currencies. The Increase in the Decrease  The main point of the Gaussian, as I’ve said, is that most observations hover around the mediocre, the average; the odds of a deviation decline faster and faster (exponentially) as you move away from the average. If you must have only one single piece of information, this is the one: the dramatic increase in the speed of decline in the odds as you move away from the center, or the average. Look at the list below for an illustration of this. I am taking an example of a Gaussian quantity, such as height, and simplifying it a bit to make it more illustrative. Assume that the average height (men and women) is 1.67 meters, or 5 feet 7 inches. Consider what I call a unit of deviation here as 10 centimeters. Let us look at increments above 1.67 meters and consider the odds of someone being that tall.* 10 centimeters taller than the average (i.e., taller than 1.77 m, or 5 feet 10): 1 in 6.3 20 centimeters taller than the average (i.e., taller than 1.87 m, or 6 feet 2): 1 in 44 30 centimeters taller than the average (i.e., taller than 1.97 m, or 6 feet 6): 1 in 740 40 centimeters taller than the average (i.e., taller than 2.07 m, or 6 feet 9): 1 in 32,000 50 centimeters taller than the average (i.e., taller than 2.17 m, or 7 feet 1): 1 in 3,500,000 60 centimeters taller than the average (i.e., taller than 2.27 m, or 7 feet 5): 1 in 1,000,000,000 70 centimeters taller than the average (i.e., taller than 2.37 m,  or 7 feet 9): 1 in 780,000,000,000 80 centimeters taller than the average (i.e., taller than 2.47 m, or 8 feet 1): 1 in 1,600,000,000,000,000 90 centimeters taller than the average (i.e., taller than 2.57 m, or 8 feet 5): 1 in 8,900,000,000,000,000,000 100 centimeters taller than the average (i.e., taller than 2.67 m, or 8 feet 9): 1 in 130,000,000,000,000,000,000,000 … and, 110 centimeters taller than the average (i.e., taller than 2.77 m, or 9 feet 1): 1 in 36,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000, 000,000,000. Note that soon after, I believe, 22 deviations, or 220 centimeters taller than the average, the odds reach a googol, which is 1 with 100 zeroes behind it. The point of this list is to illustrate the acceleration. Look at the di erence in odds between 60 and 70 centimeters taller than average: for a mere increase of four inches, we go from one in 1 billion people to one in 780 billion! As for the jump between 70 and 80 centimeters: an additional 4 inches above the average, we go from one in 780 billion to one in 1.6 million billion!* This precipitous decline in the odds of encountering something is what allows you to ignore outliers. Only one curve can deliver this decline, and it is the bell curve (and its nonscalable siblings). The Mandelbrotian  By comparison, look at the odds of being rich in Europe. Assume that wealth there is scalable, i.e., Mandelbrotian. (This is not an accurate description of wealth in Europe; it is simpli ed to emphasize the logic of scalable distribution.)† Scalable Wealth Distribution People with a net worth higher than €1 million: 1 in 62.5 Higher than €2 million: 1 in 250 Higher than €4 million: 1 in 1,000 Higher than €8 million: 1 in 4,000 Higher than €16 million: 1 in 16,000 Higher than €32 million: 1 in 64,000 Higher than €320 million: 1 in 6,400,000 The speed of the decrease here remains constant (or does not decline)! When you double the amount of money you cut the incidence by a factor of four, no matter the level, whether you are at €8 million or €16 million. This, in a nutshell, illustrates the di erence between Mediocristan and Extremistan. Recall the comparison between the scalable and the nonscalable in Chapter 3. Scalability means that there is no headwind to slow you down. Of course, Mandelbrotian Extremistan can take many shapes. Consider wealth in an extremely concentrated version of Extremistan; there, if you double the wealth, you halve the incidence. The result is quantitatively di erent from the above example, but it obeys the same logic. Fractal Wealth Distribution with Large Inequalities People with a net worth higher than €1 million: 1 in 63 Higher than €2 million: 1 in 125 Higher than €4 million: 1 in 250 Higher than €8 million: 1 in 500 Higher than €16 million: 1 in 1,000 Higher than €32 million: 1 in 2,000 Higher than €320 million: 1 in 20,000 Higher than €640 million: 1 in 40,000 If wealth were Gaussian, we would observe the following divergence away from €1 million. Wealth Distribution Assuming a Gaussian Law  People with a net worth higher than €1 million: 1 in 63 Higher than €2 million: 1 in 127,000 Higher than €3 million: 1 in 14,000,000,000 Higher than €4 million: 1 in 886,000,000,000,000,000 Higher than €8 million: 1 in 16,000,000,000,000,000,000,000,000,000,000,000 Higher than €16 million: 1 in … none of my computers is capable of handling the computation. What I want to show with these lists is the qualitative di erence in the paradigms. As I have said, the second paradigm is scalable; it has no headwind. Note that another term for the scalable is power laws. Just knowing that we are in a power-law environment does not tell us much. Why? Because we have to measure the coe cients in real life, which is much harder than with a Gaussian framework. Only the Gaussian yields its properties rather rapidly. The method I propose is a general way of viewing the world rather than a precise solution. What to Remember Remember this: the Gaussian–bell curve variations face a headwind that makes probabilities drop at a faster and faster rate as you move away from the mean, while “scalables,” or Mandelbrotian variations, do not have such a restriction. That’s pretty much most of what you need to know.* Inequality Let us look more closely at the nature of inequality. In the Gaussian framework, inequality decreases as the deviations get larger—caused by the increase in the rate of decrease. Not so with the scalable: inequality stays the same throughout. The inequality among the superrich is the same as the inequality among the simply rich—it does not slow down.† Consider this e ect. Take a random sample of any two people from the U.S. population who jointly earn $1 million per annum. What is the most likely breakdown of their respective incomes? In Mediocristan, the most likely combination is half a million each. In Extremistan, it would be $50,000 and $950,000. The situation is even more lopsided with book sales. If I told you that two authors sold a total of a million copies of their books, the most likely combination is 993,000 copies sold for one and 7,000 for the other. This is far more likely than that the books each sold 500,000 copies. For any large total, the breakdown will be more and more asymmetric. Why is this so? The height problem provides a comparison. If I told you that the total height of two people is fourteen feet, you would identify the most likely breakdown as seven feet each, not two feet and twelve feet; not even eight feet and six feet! Persons taller than eight feet are so rare that such a combination would be impossible. Extremistan and the 80/20 Rule Have you ever heard of the 80/20 rule? It is the common signature of a power law—actually it is how it all started, when Vilfredo Pareto made the observation that 80 percent of the land in Italy was owned by 20 percent of the people. Some use the rule to imply that 80 percent of the work is done by 20 percent of the people. Or that 80 percent worth of e ort contributes to only 20 percent of results, and vice versa. As far as axioms go, this one wasn’t phrased to impress you the most: it could easily be called the 50/01 rule, that is, 50 percent of the work comes from 1 percent of the workers. This formulation makes the world look even more unfair, yet the two formulae are exactly the same. How? Well, if there is inequality, then those who constitute the 20 percent in the 80/20 rule also contribute unequally—only a few of them deliver the lion’s share of the results. This trickles down to about one in a hundred contributing a little more than half the total. The 80/20 rule is only metaphorical; it is not a rule, even less a rigid law. In the U.S. book business, the proportions are more like 97/20 (i.e., 97 percent of book sales are made by 20 percent of the authors); it’s even worse if you focus on literary non ction (twenty books of close to eight thousand represent half the sales). Note here that it is not all uncertainty. In some situations you may have a concentration, of the 80/20 type, with very predictable and tractable properties, which enables clear decision making, because you can identify beforehand where the meaningful 20 percent are. These situations are very easy to control. For instance, Malcolm Gladwell wrote in an article in The New Yorker that most abuse of prisoners is attributable to a very small number of vicious guards. Filter those guards out and your rate of prisoner abuse drops dramatically. (In publishing, on the other hand, you do not know beforehand which book will bring home the bacon. The same with wars, as you do not know beforehand which con ict will kill a portion of the planet’s residents.) Grass and Trees  I’ll summarize here and repeat the arguments previously made throughout the book. Measures of uncertainty that are based on the bell curve simply disregard the possibility, and the impact, of sharp jumps or discontinuities and are, therefore, inapplicable in Extremistan. Using them is like focusing on the grass and missing out on the (gigantic) trees. Although unpredictable large deviations are rare, they cannot be dismissed as outliers because, cumulatively, their impact is so dramatic. The traditional Gaussian way of looking at the world begins by focusing on the ordinary, and then deals with exceptions or so-called outliers as ancillaries. But there is a second way, which takes the exceptional as a starting point and treats the ordinary as subordinate. I have emphasized that there are two varieties of randomness, qualitatively di erent, like air and water. One does not care about extremes; the other is severely impacted by them. One does not generate Black Swans; the other does. We cannot use the same techniques to discuss a gas as we would use with a liquid. And if we could, we wouldn’t call the approach “an approximation.” A gas does not “approximate” a liquid. We can make good use of the Gaussian approach in variables for which there is a rational reason for the largest not to be too far away from the average. If there is gravity pulling numbers down, or if there are physical limitations preventing very large observations, we end up in Mediocristan. If there are strong forces of equilibrium bringing things back rather rapidly after conditions diverge from equilibrium, then again you can use the Gaussian approach. Otherwise, fuhgedaboudit. This is why much of economics is based on the notion of equilibrium: among other bene ts, it allows you to treat economic phenomena as Gaussian. Note that I am not telling you that the Mediocristan type of randomness does not allow for some extremes. But it tells you that they are so rare that they do not play a signi cant role in the total. The e ect of such extremes is pitifully small and decreases as your population gets larger. To be a little bit more technical here, if you have an assortment of giants and dwarfs, that is, observations several orders of magnitude apart, you could still be in Mediocristan. How? Assume you have a sample of one thousand people, with a large spectrum running from the dwarf to the giant. You are likely to see many giants in your sample, not a rare occasional one. Your average will not be impacted by the occasional additional giant because some of these giants are expected to be part of your sample, and your average is likely to be high. In other words, the largest observation cannot be too far away from the average. The average will always contain both kinds, giants and dwarves, so that neither should be too rare—unless you get a megagiant or a microdwarf on very rare occasion. This would be Mediocristan with a large unit of deviation. Note once again the following principle: the rarer the event, the higher the error in our estimation of its probability—even when using the Gaussian. Let me show you how the Gaussian bell curve sucks randomness out of life —which is why it is popular. We like it because it allows certainties! How? Through averaging, as I will discuss next. How Coffee Drinking Can Be Safe Recall from the Mediocristan discussion in Chapter 3 that no single observation will impact your total. This property will be more and more signi cant as your population increases in size. The averages will become more and more stable, to the point where all samples will look alike. I’ve had plenty of cups of co ee in my life (it’s my principal addiction). I have never seen a cup jump two feet from my desk, nor has co ee spilled spontaneously on this manuscript without intervention (even in Russia). Indeed, it will take more than a mild co ee addiction to witness such an event; it would require more lifetimes than is perhaps conceivable—the odds are so small, one in so many zeroes, that it would be impossible for me to write them down in my free time. Yet physical reality makes it possible for my co ee cup to jump—very unlikely, but possible. Particles jump around all the time. How come the co ee cup, itself composed of jumping particles, does not? The reason is, simply, that for the cup to jump would require that all of the particles jump in the same direction, and do so in lockstep several times in a row (with a compensating move of the table in the opposite direction). All several trillion particles in my co ee cup are not going to jump in the same direction; this is not going to happen in the lifetime of this universe. So I can safely put the co ee cup on the edge of my writing table and worry about more serious sources of uncertainty. FIGURE 7: How the Law of Large Numbers Works In Mediocristan, as your sample size increases, the observed average will present itself with less and less dispersion—as you can see, the distribution will be narrower and narrower. This, in a nutshell, is how everything in statistical theory works (or is supposed to work). Uncertainty in Mediocristan vanishes under averaging. This illustrates the hackneyed “law of large numbers.”  The safety of my co ee cup illustrates how the randomness of the Gaussian is tamable by averaging. If my cup were one large particle, or acted as one, then its jumping would be a problem. But my cup is the sum of trillions of very small particles. Casino operators understand this well, which is why they never (if they do things right) lose money. They simply do not let one gambler make a massive bet, instead preferring to have plenty of gamblers make series of bets of limited size. Gamblers may bet a total of $20 million, but you needn’t worry about the casino’s health: the bets run, say, $20 on average; the casino caps the bets at a maximum that will allow the casino owners to sleep at night. So the variations in the casino’s returns are going to be ridiculously small, no matter the total gambling activity. You will not see anyone leaving the casino with $1 billion—in the lifetime of this universe. The above is an application of the supreme law of Mediocristan: when you have plenty of gamblers, no single gambler will impact the total more than minutely. The consequence of this is that variations around the average of the Gaussian, also called “errors,” are not truly worrisome. They are small and they wash out. They are domesticated uctuations around the mean. Love of Certainties If you ever took a (dull) statistics class in college, did not understand much of what the professor was excited about, and wondered what “standard deviation” meant, there is nothing to worry about. The notion of standard deviation is meaningless outside of Mediocristan. Clearly it would have been more bene cial, and certainly more entertaining, to have taken classes in the neurobiology of aesthetics or postcolonial African dance, and this is easy to see empirically. Standard deviations do not exist outside the Gaussian, or if they do exist they do not matter and do not explain much. But it gets worse. The Gaussian family (which includes various friends and relatives, such as the Poisson law) are the only class of distributions that the standard deviation (and the average) is su cient to describe. You need nothing else. The bell curve satis es the reductionism of the deluded. There are other notions that have little or no signi cance outside of the Gaussian: correlation and, worse, regression. Yet they are deeply ingrained in our methods; it is hard to have a business conversation without hearing the word correlation. To see how meaningless correlation can be outside of Mediocristan, take a historical series involving two variables that are patently from Extremistan, such as the bond and the stock markets, or two securities prices, or two variables like, say, changes in book sales of children’s books in the United States, and fertilizer production in China; or real-estate prices in New York City and returns of the Mongolian stock market. Measure correlation between the pairs of variables in di erent subperiods, say, for 1994, 1995, 1996, etc. The correlation measure will be likely to exhibit severe instability; it will depend on the period for which it was computed. Yet people talk about correlation as if it were something real, making it tangible, investing it with a physical property, reifying it. The same illusion of concreteness a ects what we call “standard” deviations. Take any series of historical prices or values. Break it up into subsegments and measure its “standard” deviation. Surprised? Every sample will yield a di erent “standard” deviation. Then why do people talk about standard deviations? Go gure. Note here that, as with the narrative fallacy, when you look at past data and compute one single correlation or standard deviation, you do not notice such instability. How to Cause Catastrophes If you use the term statistically significant, beware of the illusions of certainties. Odds are that someone has looked at his observation errors and assumed that they were Gaussian, which necessitates a Gaussian context, namely, Mediocristan, for it to be acceptable. To show how endemic the problem of misusing the Gaussian is, and how dangerous it can be, consider a (dull) book called Catastrophe by Judge Richard Posner, a proli c writer. Posner bemoans civil servants’ misunderstandings of randomness and recommends, among other things, that government policy makers learn statistics … from economists. Judge Posner appears to be trying to foment catastrophes. Yet, in spite of being one of those people who should spend more time reading and less time writing, he can be an insightful, deep, and original thinker; like many people, he just isn’t aware of the distinction between Mediocristan and Extremistan, and he believes that statistics is a “science,” never a fraud. If you run into him, please make him aware of these things. QUÉTELET’S AVERAGE MONSTER This monstrosity called the Gaussian bell curve is not Gauss’s doing. Although he worked on it, he was a mathematician dealing with a theoretical point, not making claims about the structure of reality like statistical-minded scientists. G. H. Hardy wrote in “A Mathematician’s Apology”: The “real” mathematics of the “real” mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly “useless” (and this is as true of “applied” as of “pure” mathematics). As I mentioned earlier, the bell curve was mainly the concoction of a gambler, Abraham de Moivre (1667–1754), a French Calvinist refugee who spent much of his life in London, though speaking heavily accented English. But it is Quételet, not Gauss, who counts as one of the most destructive fellows in the history of thought, as we will see next. Adolphe Quételet (1796–1874) came up with the notion of a physically average human, l’homme moyen. There was nothing moyen about Quételet, “a man of great creative passions, a creative man full of energy.” He wrote poetry and even coauthored an opera. The basic problem with Quételet was that he was a mathematician, not an empirical scientist, but he did not know it. He found harmony in the bell curve. The problem exists at two levels. Primo, Quételet had a normative idea, to make the world t his average, in the sense that the average, to him, was the “normal.” It would be wonderful to be able to ignore the contribution of the unusual, the “nonnormal,” the Black Swan, to the total. But let us leave that dream for utopia. Secondo, there was a serious associated empirical problem. Quételet saw bell curves everywhere. He was blinded by bell curves and, I have learned, again, once you get a bell curve in your head it is hard to get it out. Later, Frank Ysidro Edgeworth would refer to Quételesmus as the grave mistake of seeing bell curves everywhere. Golden Mediocrity Quételet provided a much needed product for the ideological appetites of his day. As he lived between 1796 and 1874, so consider the roster of his contemporaries: Saint-Simon (1760–1825), Pierre-Joseph Proudhon (1809– 1865), and Karl Marx (1818–1883), each the source of a di erent version of socialism. Everyone in this post-Enlightenment moment was longing for the aurea mediocritas, the golden mean: in wealth, height, weight, and so on. This longing contains some element of wishful thinking mixed with a great deal of harmony and … Platonicity. I always remember my father’s injunction that in medio stat virtus, “virtue lies in moderation.” Well, for a long time that was the ideal; mediocrity, in that sense, was even deemed golden. All-embracing mediocrity. But Quételet took the idea to a di erent level. Collecting statistics, he started creating standards of “means.” Chest size, height, the weight of babies at birth, very little escaped his standards. Deviations from the norm, he found, became exponentially more rare as the magnitude of the deviation increased. Then, having conceived of this idea of the physical characteristics of l’homme moyen, Monsieur Quételet switched to social matters. L’homme moyen had his habits, his consumption, his methods. Through his construct of l’homme moyen physique and l’homme moyen moral, the physically and morally average man, Quételet created a range of deviance from the average that positions all people either to the left or right of center and, truly, punishes those who nd themselves occupying the extreme left or right of the statistical bell curve. They became abnormal. How this inspired Marx, who cites Quételet regarding this concept of an average or normal man, is obvious: “Societal deviations in terms of the distribution of wealth for example, must be minimized,” he wrote in Das Kapital. One has to give some credit to the scienti c establishment of Quételet’s day. They did not buy his arguments at once. The philosopher/ mathematician/economist Augustin Cournot, for starters, did not believe that one could establish a standard human on purely quantitative grounds. Such a standard would be dependent on the attribute under consideration. A measurement in one province may di er from that in another province. Which one should be the standard? L’homme moyen would be a monster, said Cournot. I will explain his point as follows. Assuming there is something desirable in being an average man, he must have an unspeci ed specialty in which he would be more gifted than other people—he cannot be average in everything. A pianist would be better on average at playing the piano, but worse than the norm at, say, horseback riding. A draftsman would have better drafting skills, and so on. The notion of a man deemed average is different from that of a man who is average in everything he does. In fact, an exactly average human would have to be half male and half female. Quételet completely missed that point. God’s Error A much more worrisome aspect of the discussion is that in Quételet’s day, the name of the Gaussian distribution was la loi des erreurs, the law of errors, since one of its earliest applications was the distribution of errors in astronomic measurements. Are you as worried as I am? Divergence from the mean (here the median as well) was treated precisely as an error! No wonder Marx fell for Quételet’s ideas. This concept took o very quickly. The ought was confused with the is, and this with the imprimatur of science. The notion of the average man is steeped in the culture attending the birth of the European middle class, the nascent post-Napoleonic shopkeeper’s culture, chary of excessive wealth and intellectual brilliance. In fact, the dream of a society with compressed outcomes is assumed to correspond to the aspirations of a rational human being facing a genetic lottery. If you had to pick a society to be born into for your next life, but could not know which outcome awaited you, it is assumed you would probably take no gamble; you would like to belong to a society without divergent outcomes. One entertaining e ect of the glori cation of mediocrity was the creation of a political party in France called Poujadism, composed initially of a grocery- store movement. It was the warm huddling together of the semi-favored hoping to see the rest of the universe compress itself into their rank—a case of non-proletarian revolution. It had a grocery-store-owner mentality, down to the employment of the mathematical tools. Did Gauss provide the mathematics for the shopkeepers? Poincaré to the Rescue Poincaré himself was quite suspicious of the Gaussian. I suspect that he felt queasy when it and similar approaches to modeling uncertainty were presented to him. Just consider that the Gaussian was initially meant to measure astronomic errors, and that Poincaré’s ideas of modeling celestial mechanics were fraught with a sense of deeper uncertainty. Poincaré wrote that one of his friends, an unnamed “eminent physicist,” complained to him that physicists tended to use the Gaussian curve because they thought mathematicians believed it a mathematical necessity; mathematicians used it because they believed that physicists found it to be an empirical fact. Eliminating Unfair Influence Let me state here that, except for the grocery-store mentality, I truly believe in the value of middleness and mediocrity—what humanist does not want to minimize the discrepancy between humans? Nothing is more repugnant than the inconsiderate ideal of the Übermensch! My true problem is epistemological. Reality is not Mediocristan, so we should learn to live with it. “The Greeks Would Have Deified It” The list of people walking around with the bell curve stuck in their heads, thanks to its Platonic purity, is incredibly long. Sir Francis Galton, Charles Darwin’s rst cousin and Erasmus Darwin’s grandson, was perhaps, along with his cousin, one of the last independent gentlemen scientists—a category that also included Lord Cavendish, Lord Kelvin, Ludwig Wittgenstein (in his own way), and to some extent, our überphilosopher Bertrand Russell. Although John Maynard Keynes was not quite in that category, his thinking epitomizes it. Galton lived in the Victorian era when heirs and persons of leisure could, among other choices, such as horseback riding or hunting, become thinkers, scientists, or (for those less gifted) politicians. There is much to be wistful about in that era: the authenticity of someone doing science for science’s sake, without direct career motivations. Unfortunately, doing science for the love of knowledge does not necessarily mean you will head in the right direction. Upon encountering and absorbing the “normal” distribution, Galton fell in love with it. He was said to have exclaimed that if the Greeks had known about it, they would have dei ed it. His enthusiasm may have contributed to the prevalence of the use of the Gaussian. Galton was blessed with no mathematical baggage, but he had a rare obsession with measurement. He did not know about the law of large numbers, but rediscovered it from the data itself. He built the quincunx, a pinball machine that shows the development of the bell curve—on which, more in a few paragraphs. True, Galton applied the bell curve to areas like genetics and heredity, in which its use was justi ed. But his enthusiasm helped thrust nascent statistical methods into social issues. “Yes/No” Only Please Let me discuss here the extent of the damage. If you’re dealing with qualitative inference, such as in psychology or medicine, looking for yes/no answers to which magnitudes don’t apply, then you can assume you’re in Mediocristan without serious problems. The impact of the improbable cannot be too large. You have cancer or you don’t, you are pregnant or you are not, et cetera. Degrees of deadness or pregnancy are not relevant (unless you are dealing with epidemics). But if you are dealing with aggregates, where magnitudes do matter, such as income, your wealth, return on a portfolio, or book sales, then you will have a problem and get the wrong distribution if you use the Gaussian, as it does not belong there. One single number can disrupt all your averages; one single loss can eradicate a century of pro ts. You can no longer say “this is an exception.” The statement “Well, I can lose money” is not informational unless you can attach a quantity to that loss. You can lose all your net worth or you can lose a fraction of your daily income; there is a di erence. This explains why empirical psychology and its insights on human nature, which I presented in the earlier parts of this book, are robust to the mistake of using the bell curve; they are also lucky, since most of their variables allow for the application of conventional Gaussian statistics. When measuring how many people in a sample have a bias, or make a mistake, these studies generally elicit a yes/no type of result. No single observation, by itself, can disrupt their overall ndings. I will next proceed to a sui generis presentation of the bell-curve idea from the ground up. A (LITERARY) THOUGHT EXPERIMENT ON WHERE THE BELL CURVE COMES FROM Consider a pinball machine like the one shown in Figure 8. Launch 32 balls, assuming a well-balanced board so that the ball has equal odds of falling right or left at any juncture when hitting a pin. Your expected outcome is that many balls will land in the center columns and that the number of balls will decrease as you move to the columns away from the center. Next, consider a gedanken, a thought experiment. A man ips a coin and after each toss he takes a step to the left or a step to the right, depending on whether the coin came up heads or tails. This is called the random walk, but it does not necessarily concern itself with walking. You could identically say that instead of taking a step to the left or to the right, you would win or lose $1 at every turn, and you will keep track of the cumulative amount that you have in your pocket. Assume that I set you up in a (legal) wager where the odds are neither in your favor nor against you. Flip a coin. Heads, you make $1, tails, you lose $1. At the rst ip, you will either win or lose. At the second ip, the number of possible outcomes doubles. Case one: win, win. Case two: win, lose. Case three: lose, win. Case four: lose, lose. Each of these cases has equivalent odds, the combination of a single win and a single loss has an incidence twice as high because cases two and three, win-lose and lose-win, amount to the same outcome. And that is the key for the Gaussian. So much in the middle washes out—and we will see that there is a lot in the middle. So, if you are playing for $1 a round, after two rounds you have a 25 percent chance of making or losing $2, but a 50 percent chance of breaking even. FIGURE 8: THE QUINCUNX (SIMPLIFIED)—A PINBALL MACHINE Drop balls that, at every pin, randomly fall right or left. Above Is the most probable scenario, which greatly resembles the bell curve (a.k.a. Gaussian disribution). Courtesy of Alexander Taleb. Let us do another round. The third ip again doubles the number of cases, so we face eight possible outcomes. Case 1 (it was win, win in the second ip) branches out into win, win, win and win, win, lose. We add a win or lose to the end of each of the previous results. Case 2 branches out into win, lose, win and win, lose, lose. Case 3 branches out into lose, win, win and lose, win, lose. Case 4 branches out into lose, lose, win and lose, lose, lose. We now have eight cases, all equally likely. Note that again you can group the middling outcomes where a win cancels out a loss. (In Galton’s quincunx, situations where the ball falls left and then falls right, or vice versa, dominate so you end up with plenty in the middle.) The net, or cumulative, is the following: 1) three wins; 2) two wins, one loss, net one win; 3) two wins, one loss, net one win; 4) one win, two losses, net one loss; 5) two wins, one loss, net one win; 6) two losses, one win, net one loss; 7) two losses, one win, net one loss; and, nally, 8) three losses. Out of the eight cases, the case of three wins occurs once. The case of three losses occurs once. The case of one net loss (one win, two losses) occurs three times. The case of one net win (one loss, two wins) occurs three times. Play one more round, the fourth. There will be sixteen equally likely outcomes. You will have one case of four wins, one case of four losses, four cases of two wins, four cases of two losses, and six break-even cases. The quincunx (its name is derived from the Latin for ve) in the pinball example shows the fth round, with thirty-two possibilities, easy to track. Such was the concept behind the quincunx used by Francis Galton. Galton was both insu ciently lazy and a bit too innocent of mathematics; instead of building the contraption, he could have worked with simpler algebra, or perhaps undertaken a thought experiment like this one. Let’s keep playing. Continue until you have forty ips. You can perform them in minutes, but we will need a calculator to work out the number of outcomes, which are taxing to our simple thought method. You will have about 1,099,511,627,776 possible combinations—more than one thousand billion. Don’t bother doing the calculation manually, it is two multiplied by itself forty times, since each branch doubles at every juncture. (Recall that we added a win and a lose at the end of the alternatives of the third round to go to the fourth round, thus doubling the number of alternatives.) Of these combinations, only one will be up forty, and only one will be down forty. The rest will hover around the middle, here zero. We can already see that in this type of randomness extremes are exceedingly rare. One in 1,099,511,627,776 is up forty out of forty tosses. If you perform the exercise of forty ips once per hour, the odds of getting 40 ups in a row are so small that it would take quite a bit of forty- ip trials to see it. Assuming you take a few breaks to eat, argue with your friends and roommates, have a beer, and sleep, you can expect to wait close to four million lifetimes to get a 40-up outcome (or a 40-down outcome) just once. And consider the following. Assume you play one additional round, for a total of 41; to get 41 straight heads would take eight million lifetimes! Going from 40 to 41 halves the odds. This is a key attribute of the nonscalable framework to analyzing randomness: extreme deviations decrease at an increasing rate. You can expect to toss 50 heads in a row once in four billion lifetimes! FIGURE 9: NUMBERS OF WINS TOSSED Result of forty tosses. We see the proto-bell curve emerging. We are not yet fully in a Gaussian bell curve, but we are getting dangerously close. This is still proto-Gaussian, but you can see the gist. (Actually, you will never encounter a Gaussian in its purity since it is a Platonic form—you just get closer but cannot attain it.) However, as you can see in Figure 9, the familiar bell shape is starting to emerge. How do we get even closer to the perfect Gaussian bell curve? By re ning the ipping process. We can either ip 40 times for $1 a ip or 4,000 times for ten cents a ip, and add up the results. Your expected risk is about the same in both situations—and that is a trick. The equivalence in the two sets of ips has a little nonintuitive hitch. We multiplied the number of bets by 100, but divided the bet size by 10—don’t look for a reason now, just assume that they are “equivalent.” The overall risk is equivalent, but now we have opened up the possibility of winning or losing 400 times in a row. The odds are about one in 1 with 120 zeroes after it, that is, one in 1,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000 times. Continue the process for a while. We go from 40 tosses for $1 each to 4,000 tosses for 10 cents, to 400,000 tosses for 1 cent, getting close and closer to a Gaussian. Figure 10 shows results spread between −40 and 40, namely eighty plot points. The next one would bring that up to 8,000 points. FIGURE 10: A MORE ABSTRACT VERSION: PLATO’S CURVE An in nite number of tosses. Let’s keep going. We can ip 4,000 times staking a tenth of a penny. How about 400,000 times at 1/1000 of a penny? As a Platonic form, the pure Gaussian curve is principally what happens when he have an in nity of tosses per round, with each bet in nitesimally small. Do not bother trying to visualize the results, or even make sense out of them. We can no longer talk about an “in nitesimal” bet size (since we have an in nity of these, and we are in what mathematicians call a continuous framework). The good news is that there is a substitute. We have moved from a simple bet to something completely abstract. We have moved from observations into the realm of mathematics. In mathematics things have a purity to them. Now, something completely abstract is not supposed to exist, so please do not even make an attempt to understand Figure 10. Just be aware of its use. Think of it as a thermometer: you are not supposed to understand what the temperature means in order to talk about it. You just need to know the correspondence between temperature and comfort (or some other empirical consideration). Sixty degrees corresponds to pleasant weather; ten below is not something to look forward to. You don’t necessarily care about the actual speed of the collisions among particles that more technically explains temperature. Degrees are, in a way, a means for your mind to translate some external phenomena into a number. Likewise, the Gaussian bell curve is set so that 68.2 percent of the observations fall between minus one and plus one standard deviations away from the average. I repeat: do not even try to understand whether standard deviation is average deviation—it is not, and a large (too large) number of people using the word standard deviation do not understand this point. Standard deviation is just a number that you scale things to, a matter of mere correspondence if phenomena were Gaussian. These standard deviations are often nicknamed “sigma.” People also talk about “variance” (same thing: variance is the square of the sigma, i.e., of the standard deviation). Note the symmetry in the curve. You get the same results whether the sigma is positive or negative. The odds of falling below −4 sigmas are the same as those of exceeding 4 sigmas, here 1 in 32,000 times. As the reader can see, the main point of the Gaussian bell curve is, as I have been saying, that most observations hover around the mediocre, the mean, while the odds of a deviation decline faster and faster (exponentially) as you move away from the mean. If you need to retain one single piece of information, just remember this dramatic speed of decrease in the odds as you move away from the average. Outliers are increasingly unlikely. You can safely ignore them. This property also generates the supreme law of Mediocristan: given the paucity of large deviations, their contribution to the total will be vanishingly small. In the height example earlier in this chapter, I used units of deviations of ten centimeters, showing how the incidence declined as the height increased. These were one sigma d