Does the bell curve accurately describe the world? When does the bell curve work, and when does it fail? How can we make better predictions and more accurately describe the phenomena of real life?<\/p>\n\n\n\n
We’ll cover the situations in which the normal bell curve distribution is a good predictor of the real world, the situations where it’s not, and better ways to represent randomness in an uncertain world.<\/p>\n\n\n\n\n\n\n\n
Before we look at how the bell curve describes the world, let’s look at Nassim Nicholas Taleb’s Extremistan<\/a> and Mediocristan<\/a>. To explain how and why Black Swans<\/a> occur, Taleb<\/a> coins two categories to describe the measurable facets of existence: Extremistan and Mediocristan. In Mediocristan, randomness is highly constrained, and deviations from the average are minor. <\/strong>Physical characteristics such as height and weight are from Mediocristan: They have upper and lower bounds, their distribution is the bell curve, and even the tallest or lightest human being isn\u2019t much taller or lighter than the average. In Mediocristan, prediction is possible.<\/em> In Extremistan, however, randomness is wild, and deviations from the average can be, well, extreme.<\/strong> Most social, man-made aspects of human society\u2014the economy, the stock market, politics\u2014hail from Extremistan: They have no known upper or lower bounds, their behavior can\u2019t be graphed on the bell curve, and individual events or phenomena\u2014i.e., Black Swans\u2014can have exponential impacts on averages. Imagine you put ten people in a room. Even if one of those people is Shaquille O\u2019Neal, the average height in the room is likely to be pretty close to the human average (Mediocristan). If one of those people is Jeff Bezos, however, suddenly the wealth average changes drastically (Extremistan).<\/p>\n\n\n\n When discussing uncertainty, experts will frequently invoke physicist Werner Heisenberg\u2019s uncertainty principle. In its specific application, the uncertainty principle holds that the more precisely one measures the position of a given particle, the less precisely one can measure its momentum, and vice versa. The problem with this analogy is that subatomic uncertainty actually hails from Mediocristan\u2014particles\u2019 deviations from their expected positions or momentums can be graphed using the normal bell curve, and the sheer number and tininess of particles in existence mean those mild deviations average out. Real uncertainty arises in the real world<\/strong>\u2014like, for example, when Taleb can\u2019t reach his ancestral village in Lebanon from New York because Israel is battling Hezbollah.<\/p>\n\n\n\n The classic normal bell curve\u2014which is also called the \u201cGaussian distribution\u201d after German mathematician Carl Friedrich Gauss\u2014is an accurate description of Mediocristan phenomena, but it is dangerously misleading when it comes to Extremistan. Consider human height, an eminently Mediocristan phenomenon. With every increase or decrease in height relative to the average, the odds of a person being that tall or short decline. For example, the odds that a person (man or woman) is three inches taller than the average is 1 in 6.3; 7 inches taller, 1 in 44; 11 inches taller, 1 in 740; 14 inches taller, 1 in 32,000. It\u2019s important to note that the odds not only decline as the height number gets further and further away from the average, but they decline at an accelerating rate. For example, the odds of someone being 7\u20191\u201d are 1 in 3.5 million, but the odds of someone being just four inches taller are 1 in 1 billion, and the odds of someone being four inches taller than that <\/em>are 1 in 780 billion! Human height can be accurately graphed on the bell curve. Now consider an Extremistan phenomenon like wealth. In Europe, the probability that someone has a net worth higher than 1 million euros is 1 in 62.5; higher than 2 million euros, 1 in 250; higher than 4 million, 1 in 1,000; higher than 8 million, 1 in 4,000; and higher than 16 million, 1 in 16,000. The odds decrease at a constant, rather than accelerating, rate<\/strong>, indicating that their distribution doesn\u2019t conform to the normal bell curve.<\/strong> (Note: The European wealth statistics cited above aren\u2019t precise, but they illustrate the central point: that wealth is scalable<\/a> and does not look like a bell curve distribution.) The upshot is that in Extremistan, as the name suggests, extreme events have much better odds of occurring than in Mediocristan. Simply put, <\/strong>the bell curve does not apply<\/em><\/strong>.<\/strong> Adolphe Qu\u00e9telet<\/a> (1796\u20131874) was a French mathematician who developed the idea of the \u201caverage human\u201d (l\u2019homme moyen<\/em>) through the use of \u201cmeans\u201d\u2014golden averages that represented the ideal human form. At first, his inquiries were limited to human beings\u2019 physical characteristics\u2014height, weight, newborn weight, chest size\u2014but before long, he began to seek averages in the social realm by studying human habits and morals. Qu\u00e9telet developed bell curve distributions to describe humans\u2019 deviance\u2014literally\u2014from both physical and moral norms. Although Qu\u00e9telet\u2019s ideas were influential in his historical moment, there were skeptics. Augustin Cournot, a fellow mathematician and philosopher, pointed out that social averages would necessarily depend on exactly whom <\/em>and what attribute <\/em>was being studied. An obvious example is human diet: If you were trying to determine the amount of seafood an average human consumes, you might get vastly different measurements if you were surveying inland populations versus coastal ones, or populations whose cultural traditions included a seafood-heavy diet. In our contemporary moment, the most egregious misuse of normal bell curves can be found among economists and financial-industry analysts. Most economists, including many Nobel Prize\u2013winners, treat financial markets as though they hail from Mediocristan when, in fact, they\u2019re from Extremistan. A stunning example of economists\u2019 misinterpretation of the market came in 1998, with the collapse of Long-Term Capital Management (LCTM), a hedge fund that included among its founding partners two Nobel Prize\u2013winners, Myron Scholes and Robert C. Merton (Robert K. Merton\u2019s son). Scholes and Merton won their Nobel Prize for a theory of stock option pricing; the theory, however, was based on a Gaussian model of the market that excluded the possibility of Black Swans. When markets were disrupted by the Asian financial crisis of 1997 and the Russian financial crisis of 1998\u2014a one-two punch of Black Swans\u2014Scholes and Merton\u2019s theory was revealed for the fantasy it was. The twin crises resulted in a crushing $4.6 billion loss for LCTM in fewer than four months. The firm had to be bailed out by a consortium of private banks, at the behest of the Federal Reserve. The problem is that economists\u2019 models are thoroughly Platonified: self-consistent, but bearing almost no resemblance to reality. Economists are like philosopher John Locke\u2019s madmen: they reason \u201ccorrectly from erroneous premises.\u201d<\/strong> The bell shaped curve isn’t always the best tool.<\/p>\n\n\n\n We’ve seen that the bell curve distribution doesn’t always accurately represent the world. So can the phenomena of Extremistan be modeled\u2014and thereby predicted\u2014at all? Not precisely; but they can <\/em>be approximated\u2014by the use of fractals<\/a><\/em>. The reason fractals are helpful in representing Black Swans is that their internal ratios stay constant across scales. Unlike bell curves, in which ratios decline at accelerating rates the further one gets from the average, fractals exhibit no (or mild) acceleration. They obey power laws<\/a><\/em>, which describe a functional relationship between two quantities in which the relationship remains constant no matter what the initial size of the quantities.<\/p>\n","protected":false},"excerpt":{"rendered":" Does the bell curve accurately describe the world? When does the bell curve work, and when does it fail? How can we make better predictions and more accurately describe the phenomena of real life? We’ll cover the situations in which the normal bell curve distribution is a good predictor of the real world, the situations where it’s not, and better ways to represent randomness in an uncertain world.<\/p>\n","protected":false},"author":4,"featured_media":4525,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[24,25],"tags":[60],"class_list":["post-4507","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-society","category-statistics","tag-black-swan","","tg-column-two"],"yoast_head":"\n
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<\/p>\n\n\n\nForget Heisenberg<\/strong><\/h3>\n\n\n\n
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<\/p>\n\n\n\nThe Limits of the Bell Curve<\/strong>
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<\/p>\n\n\n\nThe Mistakes of Adolphe Qu\u00e9telet<\/strong> and the Misuse of the Bell Curve
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<\/p>\n\n\n\nRepresenting Randomness: Fractals and Power Laws<\/strong><\/h2>\n\n\n\n