This week’s featured book, The Black Swan, is over ten years old now and as relevant as ever. I think this will always be the case. There are many reasons for its evergreen status but the heart of it all comes from Chapter 15, titled The Bell Curve, That Great Intellectual Fraud.

I hope to write a chapter heading half-as-good someday.

As you can tell, the chapter attacks the broad utility that we prescribe to normal distributions (aka the traditional bell curve, aka Gaussian-curves). It’s really easy to assign a Gaussian curve to virtually everything we do. It’s often wrong or unnecessary, too. This is such an issue that Taleb’s book could have easily been retitled: Fooled By Gaussian Statistics. I don’t think it would have sold as well but, again, it could have worked.

A Brave New (Statistical) World

Today’s world of Big Data and rich computing power gives us unprecedented ability to use the full kit of statistical methods. Yet, those methods are hard to explain to others and harder still to understand. Misinterpretation abounds. Look no further than the notorious misunderstanding of Nate Silver’s predictions for the Trump/Clinton presidential race. The folks at FiveThirtyEight did themselves no favors by mincing words in that whole debacle but the point remains: most people fundamentally misunderstood the nature of a forecast.

We do that a lot. And to be fair, we can’t help it. Data analysis performed at PhD-levels of sophistication must still be presented to audiences that have, at best, an 11th-Grade understanding of the methods.  

Even then, we’ll still misinterpret things.

This is where Taleb comes in. I spent a week reviewing his first book, Fooled By Randomness, to illustrate how Chance plays a tremendous role in our world. We forget this quite regularly and so the book teaches us what to do about that. Today’s article, and the featured book as a whole, goes to another, related issue we suffer from: our strained efforts to fit everything into bell curves.  

I Can Do Anything! Within Two Sigmas.

No one has written a book titled “I Grow Ten Feet Tall And You Can Too”. Because this isn’t something that you just choose to do. It is impossible. The fact that I can write those words with full certainty, based on the broad history of all observations from data, means that a person’s height is something that lives in the world of Mediocristan. You may remember that idea from yesterday’s article. There are such things as sure bets in Mediocristan. I will gladly wager a million dollars that I will never grow to 10 feet in height. No one wants to take me up on that bet. Why? Because of the power of the bell curve.  

The Gaussian-bell curve, when applied to the world of Mediocristan, gives us that one thing we all want in this modern world: certainty. That certainty is built around the Central Limit Theorem and allows us to make fantastic forecasts with nothing more than a well-established mean.

As Taleb puts it:

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 …

We know that the average American man lives about 75ish years. So if I seek a $500,000 life insurance policy at 74 years old, I will be denied by every insurance company in existence. Or, perhaps, they will get crafty and offer me such a policy for the low, low price of … $500,001.

I know that seems obvious but I want this to be useful beyond insurance actuaries. So think about it in Taleb’s terms: as my age gets closer to the mean, my probability for further exceeding the mean diminishes more and more. Past 75, my chances of reaching 76 are lower. My chances of reaching 96 are drastically lower. My chances of reaching 116 are probably 1:1,000,000? I have no idea.

Meanwhile, my chances of a mid-life crisis are greatly increasing with this writing.

This dynamic of the Gaussian curve has tremendous power and fuels our modern world. It creates a very powerful kind of prediction (e.g., linear regression) and fuels certain kinds of controls and production methods (e.g., Six Sigma).

It’s lovely and useful. Until we get to problems that live in Extremistan.

I cut off a portion of what Taleb wrote in the cited passage above. I will now copy/paste it here with the rest of what Taleb has to offer on this idea:

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.

He’s right. That’s most of what we need to know. But what, exactly, does it mean?

What Are Mandelbrotian Variations?

In the aforementioned Chapter 15, Taleb gives a great example of scalable (i.e., Mandelbrotian) distributions through the use of personal income. I won’t recreate this here but suffice to say that a scalable distribution puts the odds of an ever-increasing level of personal wealth do not diminish severely in this type of distribution. This is the stuff of the “fat tail”.

As a small demonstration, take the idea that people with a net worth of > $2M as 1 in 125. In a scalable/Mandelbrotian distribution, we see that people with a net worth of greater than 2x the original amount of $2M are also 2x as unlikely, or 1 in 250.

This factor of 2 (i.e., the coefficient) continues in a nice, linear pattern …

Higher than $2 million: 1 in 125

Higher than $4 million: 1 in 250

Higher than $8 million: 1 in 500

Meanwhile, consider a Gaussian distribution that follows the traditional bell curve. You’ll see that the odds decrease drastically. There is no nice, easy, linear application of a fixed factor. This gets to Taleb’s expression that variations in a Gaussian bell curve “face a headwind that makes probabilities drop at a faster and faster rate”. I like that expression. It implies the natural forces at work in Mediocristan.

Higher than $2 million: 1 in 127,000

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

Here’s an illustration that can help. It comes from a good article on the same topic at

The point of all this is two-fold: first, within the world of casino games, short-term weather patterns, and human lifespans, we can develop a really good sense of the odds thanks to mountains of data and Gaussian distributions. The data shows trends and patterns that tell us, with fantastic confidence, that it will not snow in Guatemala in July. The odds are the stuff of a Six Sigma event, I’m sure.

Such confidence is the stuff of certainty. Such certainty is the pinnacle of what we crave in the modern world. It’s great!

Second, we are romanced by symmetry. Because normal distribution is so easily understood, we think it should apply to everything. Including income. I try to stay away from political issues but I think it’s still important to recognize that we talk about income inequality as if income, as a by-product of certain activities, is somehow naturally led to a normal distribution and that “crony capitalists” have upset that normalcy to create a massive hoarding of wealth to the 1%.

That concentration is a feature, not a bug. To say it ought to be different makes a whole lot of sense to 99% of the population. But we’re dealing with something that inhabits Extremistan. It does not naturally comport to a normal distribution. That means we have to rethink the concept of “average”. It doesn’t apply here. And without the typical use of “average” we have a hard time finding what is “fair”. Fairness, in Extremistan, doesn’t mean that we all huddle to the mean. Fairness in Extremistan means we identify and manage power laws.

Power Laws in Extremistan

Consider Pareto’s 80/20 rule. As a power law, it suggests that 80% of a given whole is driven by 20% of a given player. So 80% of the work is done by 20% of the people. Or 80% of the money is recouped by 20% of the people.

The numerical distribution can, and should, change. As Taleb puts it,

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). This is funny but probably true.

So who among us budding writers will be the next to join that 20% club? Can we predict who will be the Next Big Author? No. We cannot.


Because there is no reliable average here. If you tried to find one, you’d see that the average author earns something like $150 a year. You can’t take the entire population of authors and say what each one’s odds are of making it big. We’re in Extremistan.

By comparison, we can take the entire population of male humans and define the odds of each person growing to a height of 5 feet, 11 inches.

Back to our authors: you can’t predict the next member of that 20% club from the whole dataset. But if you narrow the dataset, you get closer to the power law and closer to a prediction. Screen out self-published authors. Then you might find an average income, as an expression of book sales, that reaches $15,000. And then, if you screened out all the small publishers, you’d probably climb to $50,000. And then, if you only selected those authors working for large publishers and with elite literary agents, you probably get to $150,000.

Even so, it’s not very predictable. As examined in yesterday’s article, there are still plenty of flops in Extremistan. You can have the best book publisher, best agent, best book and still not reach that 20%. All it takes is one bad review in the New York Times. Or some ancillary negative story about the author. Or bad timing.

These aren’t “Black Swans”. This is just Murphy’s Law. Which is itself a power law, I’m sure.

This leads me to two final points in this article.

Magnitudes Must Shift Attitudes

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 profits.

Taleb offers this sage advice and I like it. There is an expression that you “need money to make money”. That’s because magnitude matters in investing. Exponential growth from compounding interest is a miracle. But it only seems miraculous when it happens at scale.

But to get to that scale requires a certain amount of luck. As Taleb explores in Fooled By Randomness. Because this is Extremistan. This is about power laws. And anything that grows rapidly can also disappear rapidly when the dynamics in those power laws change. We call those events “Black Swans” but the kinda aren’t. We’ll explore that more tomorrow.

The beauty of Extremistan is that it can take some people on a rocket ship ride to fantastic highs. And fantastic lows.

The beauty of Mediocristan is that, where it exists, it creates a reliable foundation for all that you don’t want to lose. This is the stuff of the “acceptable minimum” I wrote about before with this article. So with the things you don’t want to lose, operate in Mediocristan. This is where the bell curve, and all its lovely certainty, offers its greatest strength. As Taleb explains,

My technique, instead of studying the possible models generating non-bell curve randomness, hence making the same errors of blind theorizing, is to do the opposite: to know the bell curve as intimately as I can and identify where it can and cannot hold.

I think this is really the best way to play with risk, ventures, and speculations. Everything from writing a book to starting a business. I’ll explore it further tomorrow.