The Bell Curve of Life
& its limiting paradigm of thought
To just assume everyone and everything might fall somewhere within the bell curve of performance would be to dismiss excellence and greatness all together.
If you’ve had any basic academia teachings on statistics then you’re probably familiar with the bell curve. The bell curve or Gaussian distribution, also known as normal distribution. Which is a continuous probability distribution that is symmetrical on both sides of the mean. The area under the normal distribution curve represents probability and the total area under the curve sums to one. Each falling within a fixed standard deviation in relation to the mean, measuring how dispersed the data is.
It’s the paradigm that has shaped most of our understandings of probabilities. Intuitively it satisfies the brain. Beautiful symmetrical graph, where over a long enough period of time and with enough data points, smooths the curve over the mean. No wonder people have been enamored by its occurrence for so long. Quote from the creator:
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the “Law of Frequency of Error”. The law would have been personified by the Greeks and deified, if they had known of it. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
- Sir Francis Galton
With the largest sum of people falling in the center within set parameters, normal, predictable.. boring. Casting a giant umbrella over our thoughts, limiting our minds as to who we could be and where we might fall on the bell curve. As Nassem Taleb so eloquently puts it in his book “Fooled by Randomness”:
“This ubiquity of the Gaussian is not a property of the world, but a problem in our minds, stemming from the way we look at it.”
Maybe probability is not entirely about the odds, but rather the belief in the existence of an alternative outcome, cause, or motive. Before we divulge more into this thought. Let’s take a look at why Normal Distribution is so easily understood and has been taught without much question, for so many years.
The normal distribution is a bell-shaped, symmetrical distribution in which the mean, median and mode are all equal. If the mean, median and mode are unequal, the distribution will be either positively or negatively skewed.
Normal distribution is a continuous probability distribution. It is the distribution that is symmetric with respect to the mean. Showing data near the mean will more frequent in occurrence than data away from the mean. We see it all the time in everyday life, take this weight stack on the gym equipment as a prime example. Most people fall in the middle.
By grading or assessing performance using a bell curve you are forcing groups of people to be categorized as poor, average, or good. For smaller groups, having to categorize a set number of individuals in each category to fit a bell curve will do a disservice to the individuals. Rigid parameters set sometimes misses extreme cases. Although in most cases might in fact fall within the normal bell curve.
To understand the probability factors of a normal distribution, you need to understand the following rules:
- The total area under the curve is equal to 1 (100%)
- About 68% of the area under the curve falls within one standard deviation.
- About 95% of the area under the curve falls within two standard deviations.
- About 99.7% of the area under the curve falls within three standard deviations.
Thanks to the hard work and data provided by Seedata.live
We can look at the results of a coin tossed 50 times. It ended up with this histogram looking like this:
The histogram counts the number of heads after a trial of 50 coin tosses.
After 200 and 10000 trials respectively, the curve approximated a nice Bell curve, centered around a mean of 25 as you would expect. The more data over longer period of time collected the smoother the curves tend to be.
The coin toss might only seem like a specific example, but due to the fact that all kinds of quantities we encounter in the real world are often the sum of many unobserved random events, the CLT helps explain why the Bell curve pops up under so many circumstances. CLT is a statistical premise that, given a sufficiently large sample size from a population with a finite level of variance, the mean of all sampled variables from the same population will be approximately equal to the mean of the whole population.
rolling a pair of dice.
There are 6 × 6 = 36 equally likely outcomes for the pair of dice, but you may know that the median sum of 7 is much likely than the minimum possible sum of 2 or the maximum possible sum of 12.
The probability for each 2-dice sum is proportional to the length of a diagonal on a 6 × 6 grid. The most likely sum of 7 is the main diagonal from top-left to bottom-right.
For 3 dice, we need 3 dimensions to visualize the 6 × 6 × 6 grid of 216 equally likely outcomes…
The outcomes of dice rolls are “discrete” data, because the dice only has whole numbers. The more times we roll the dice, the more possible outcomes exist. The bars on the histogram would become thinner and thinner and the shape would approach a normal distribution giving the bell appearance.
So why are bell curves everywhere?
Because phenomena in the natural world are almost always the result of many contributing factors, we see such examples all over. Such as: baby birth weight, height of males, shoe size, ACT scores, blood pressure, Probability Distribution of the Number of Firing Neurons, Presidential Elections, Mature Buck Antler Scores
Another cool example is grocery shelves. Because stores like to keep popular products together and right in front of your face (the maxim is, eye level + buy level) they tend to stock in a normally-distributed pattern with popular stuff right in the middle. We don’t necessarily see this in action until there is a big sale or a rush in an emergency. When stores can’t restock in time, you can see a kind of bell curve emerge on the empty shelves.
These are all fun and relatively normal occurrences of examples in everyday life. To cast the over arching idea of normal distribution on humans is s stretch and casts us into mediocrity. Because this method of thinking highly underestimates “tail events,” although rare, when they do occur they bring consequential jumps. The obsession with the bell curve deal with not only how people tend to see things, but also want to see the world. Ignoring all black swan events. If You view the world from within a model. You’re missing the point. Take home this trick. 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 restriction.
Taleb invents the metaphors of Mediocristan and Extremistan to explain why Black Swans are important and something we need to be robust against. It goes something like this:
Mediocristan: is where things are scalable, where the law of big numbers applies and you would actually get a Bell curve.
Extremistan: inequalities are such that one single observation can disproportionately impact the aggregate, or the total.
Humans have evolved to live in Mediocristan. In this familiar and safe land, no single observation significantly impacts the average. For example, pick a group of 1,000 random humans and measure their weight or height (which are examples of attributes belonging to Mediocristan). The average weight or height would not be significantly impacted if you were to add the single heaviest or tallest human to the group. The bell curve, or normal distribution, adequately represents the statistics of most phenomena in Mediocristan. It is possible to study past data and make predictions for the future, without much risk of loss.
Lets look at an example where extrimistan is definitely the case in “The Strange Country of Extremistan,” where Taleb writes:
“Consider by comparison the net worth of the thousand people you lined up in the stadium. Add to them the wealthiest person to be found on the planet — say Bill Gates, the founder of Microsoft. Assume his net worth to be close to $80 billion — with the total capital of the others around a few million. How much of the total wealth would he represent? 99.9 percent?”
So while weight, height, and calorie consumption are from Mediocristan, wealth is not. Almost all social matters are from Extremistan. Another way to say it is that social quantities are informational, not physical: you cannot touch them.
Almost everything in social life is produced by rare, but consequential shocks and jumps; all the while almost everything studied about social life focuses on the “normal,” particularly with “bell curve” methods of inference that tell you close to nothing. Why? Because the bell curve ignores large deviations, cannot handle them, yet makes us confident that we have tamed uncertainty.
1. The disproportionate role of high-profile, hard-to-predict, and rare events that are beyond the realm of normal expectations in history, science, finance, and technology.
2. The non-computability of the probability of the consequential rare events using scientific methods (owing to the very nature of small probabilities).
3. The psychological biases that blind people, both individually and collectively, to uncertainty and to a rare event’s massive role in historical affairs.
In other words: A Black Swan is a highly impactful event which was completely unexpected and could not be predicted, but which (due to human biases) is often rationalized after the fact as if it could indeed have been expected at the time.
“Perhaps the wise one is the one who knows that he cannot see things far away.”
Attempting to predict Black Swans is much more complicated than trying to make yourself prepared for them. Since the world is assymetric.
Remember that the Wednesday before Thanksgiving might be a Black Swan for the turkey, but it is not for the butcher. The point is how to avoid being the turkey. In Taleb’s words, first and foremost you must accept the fact that randomness is a large factor in life, both personal and on a societal scale. To blanket your thoughts with normal distribution outcomes, would be to both limit your abilities and put you at higher risk in case of of extreme events.
A “Power Law” distribution is also known as a “long tail.” It indicates that people are not “normally distributed.” In this statistical model there are a small number of people who are hyper high performers, a broad swath of people who are good performers and a smaller number of people who are low performers. It essentially accounts for a much wider variation in performance among the sample and is nowhere close to normal distributed.
Consider the Research conducted in 2011 and 2012 by Ernest O’Boyle Jr. and Herman Aguinis (633,263 researchers, entertainers, politicians, and athletes in a total of 198 samples). found that performance in 94 percent of these groups did not follow a normal distribution. Rather these groups fall into what is called a “Power Law” distribution.
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 eighty percent of the land in Italy was owned by twenty percent of the people. Some use it to imply that eighty percent of the work is done by twenty percent of the people. Or eighty percent worth of effort contributes to only twenty percent of results, and vice versa. As far as slogans go, this one wasn’t phrased to impress you the most. It can easily be called the 50/01 rule, that is, fifty percent of the work comes from one percent of the workers. This formulation makes the world look even more unfair –yet the two formulas are exactly the same.
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 perhaps there is a better different way to view things, which takes the exceptional as a starting point and treats the ordinary as subordinate.
You cannot use one single measure for randomness called standard deviation (and call it “risk”); you cannot expect a simple answer to characterize uncertainty. To go the extra step required courage, commitment, ability to connect the dots, a desire to understand randomness fully.
Let me show how the Gaussian bell curve sucks randomness out of life –which is why it is popular. Why? Because it allows certainties! How? Through averaging, as I will discuss next.
Casino operators understand this well –which is why they never (if they do it right) lose money. They just 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 and they cap 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 size of the total gambling activity. You will not see anyone leaving the casino with a billion dollars — not in the lifetime of this universe. This is an application of the supreme law of Mediocristan.
Maybe the bell curve makes sense in minor occurrences being observed. But it’s important to not limit your thinking to such rigid set rules for probabilities in a world fueled by randomness. The only constant in life is change, good luck predicting future change in such a complex world with by staying only in your set parameters.
“Society was not a “social pyramid” with the proportion of rich to poor sloping gently from one class to the next. Instead, it was more of a “social arrow”- very fat at the bottom where the mass of men live, and very thing at the top where sit the wealthy elite. Nor was this effect by chance; the data did not remotely fit a bell curve, as one would expect if wealth were distributed randomly. It is a social law, he wrote: something “in the nature of man. That something, though expressed in a neat equation, is harsh and Darwinian, in Pareto’s view. At the very bottom of the wealth curve, he wrote, men and women starve and children die young. In the broad middle of the curve all is turmoil and motion: people rising and falling, climbing by talent or luck and falling by alcoholism, tuberculosis, or other forms of unfitness. At the very narrow top sit the elite of the elite, who control wealth and power for a time-until they are unseated through revolution or upheaval by a new aristocratic class. There is not progress in human history. Democracy is a fraud. Human nature is primitive, emotional, unyielding. The smarter, abler, stronger, and shrewder take the lion’s share. — Benoît B. Mandelbrot, The (Mis)Behavior of Markets
