Some things are not normal.
By that, I mean that if you go out into the world and start measuring things like human height, IQ, or the size of apples on a tree, you will find that most of the data clusters around some average value.
This pattern is so common that we call it the normal distribution.
But not everything in life behaves this way.
courtesy: Varitasium
Pareto’s Discovery
In the late 1800s, Italian engineer Vilfredo Pareto stumbled upon something no one had really seen before.
He suspected there might be a hidden pattern in how much money people make. So he gathered income-tax records from Italy, England, France, and other European countries. For each country, he plotted the distribution of income.
And everywhere he looked, he saw essentially the same pattern—a pattern that still holds in most countries today.
It was not a normal distribution.
Think about a normal distribution such as human height. There is a clearly defined average, and extreme outliers are extraordinarily rare.
You are never going to find someone who is five times the average human height. That would be physically impossible.
But income is different.
Take Pareto’s curve for England. It shows the number of people who earn more than a certain income.
The curve initially declines steeply. Most people earn relatively little. But then it falls away gradually, much more slowly than a normal distribution would.
And it spans several orders of magnitude.
There were people who earned five times, ten times, or even a hundred times more than others.
That kind of spread simply would not occur if income followed a normal distribution.
To compress this enormous range of data, Pareto calculated the logarithms of the values and plotted those instead.
In other words, he used a log-log plot.
When he did that, something remarkable happened.
The broad curve transformed into a straight line.
The gradient was around -1.5.
That means that each time you double the income—for example, from £200 to £400—the number of people earning at least that amount drops by a factor of approximately , or about 2.8.
And this pattern continues for every doubling of income.
So Pareto could describe the distribution of income with a relatively simple equation:
The number of people earning an income greater than or equal to x is proportional to 1/x¹·⁵.
Pareto initially observed this pattern in England, but he performed the same analysis on data from Italy, France, Prussia, and several other countries.
Again and again, the data transformed into a straight line, and the gradients were remarkably similar.
That meant Pareto could describe the income distribution of different countries using essentially the same mathematical relationship:
1 divided by income raised to some power.
This type of relationship is called a power law.
And once you move from the world of normal distributions into the world of power laws, things change dramatically.
Three Games, Three Different Worlds
To understand the difference, imagine a trip to a casino where we play three different games.
Game One: The Normal World
At table number one, you get 100 coin tosses.
Every time the coin lands on heads, you win $1.
How much would you be willing to pay to play?
We first need to calculate the expected value.
The probability of getting heads is 1/2. Multiply that by the $1 payout, and then multiply by 100 tosses.
That gives you an expected payout of $50.
So you should be willing to pay anything less than $50 to play the game.
You may not win exactly $50 every time. But if you play the game hundreds of times, the small variations on either side of the average will tend to cancel out.
Over many trials, the average converges toward the expected value.
One of the first people to study this kind of problem was Abraham de Moivre in the early 1700s.
He showed that if you plot the probability of each outcome, you get a bell-shaped curve, which later became known as the normal distribution.
The traditional explanation for normal distributions is that when many random effects are added together, a normal distribution tends to emerge.
For example, my height depends on many different factors—nutrition, genetics inherited from my parents, environment, and countless other influences.
If these random effects are additive, they tend to produce a normal distribution.
Game Two: The Lognormal World
At table number two, the game is slightly different.
You still get 100 coin tosses, but instead of potentially winning $1 on each flip, your winnings are multiplied by a factor.
You start with $1.
Every time the coin lands on heads, your winnings are multiplied by 1.1.
If it lands on tails, your winnings are multiplied by 0.9.
After 100 tosses, you take home the total.
So how much should you pay to play?
On every flip, your payout can either grow or shrink, and each outcome is equally likely.
The expected factor per turn is therefore:
(1.1 + 0.9) / 2 = 1.
So if you start with $1, the expected payout is also $1.
Does that mean you should be willing to pay anything less than $1?
Not necessarily.
Look at the distribution of payouts.
You could potentially win a huge amount.
If you tossed heads 100 times, you would receive:
1.1¹⁰⁰
which is nearly $14,000.
The probability of that happening is incredibly small—roughly one in .
You would be more likely to win the lottery three times in a row.
On the other hand, the median payout is around $61.
So if you are playing the game only once and want roughly even odds of making a profit, you should probably pay less than $61.
Yet if you play the game hundreds of times, the average payout approaches $1.
Now change the x-axis from a linear scale to a logarithmic scale.
The curve suddenly looks like a normal distribution.
That is why this type of distribution is called a lognormal distribution.
The key difference is that the random effects are not being added together. They are being multiplied.
Imagine that you have a certain amount of wealth. Your wealth increases by a certain percentage next year because of your investments. The following year, it changes by another random percentage.
Each year’s change multiplies the previous year’s wealth.
If you have a large product of random numbers, taking the logarithm turns that product into a sum of logarithms.
And sums of random variables tend to produce normal distributions.
That is what leads to a lognormal distribution.
Lognormal distributions can produce substantial inequality.
You do not simply see a mean. You see a mean with a long tail.
There is a much greater likelihood of extremely large outcomes—in this case, tremendous amounts of wealth—than you would expect from a normal distribution.
The reason the curve is so asymmetric is that the downside is capped at zero.
At most, you can lose the $1 you started with.
But the upside has no equivalent limit. It can grow to nearly $14,000 in our example.
Game Three: The St. Petersburg Paradox
Now let’s move to table number three.
Again, you toss a coin.
You start with $1.
Every time the coin lands on tails, your payout doubles.
You keep tossing until you get heads.
Once you get heads, the game ends.
If you get heads on your first toss, you receive $2.
If you get tails first and heads on the second toss, you receive $4.
If you get two tails followed by heads on the third toss, you receive $8.
And so on.
If it takes n tosses to get heads, your payout is:
2ⁿ.
So how much should you pay to play this game?
Again, we calculate the expected value.
Suppose you get heads on your first toss.
The payout is $2, and the probability is 1/2.
So the expected value is:
$2 × 1/2 = $1.
If it takes two tosses to get heads, the payout is $4 and the probability is 1/4.
Again:
$4 × 1/4 = $1.
If it takes three tosses, you receive $8 and the probability is 1/8.
Again:
$8 × 1/8 = $1.
This continues indefinitely.
Even if you have to toss the coin ten times, or 100 times, before getting heads, the probability becomes extremely small—but the payout becomes so large that the expected value of that outcome is still $1.
So every possible outcome contributes another $1 to the expected value.
Theoretically, the total expected value of the game is therefore infinite.
This is known as the St. Petersburg paradox.
Look at the distribution of payouts.
It is uncapped.
It spans across all orders of magnitude.
You could win $1,000, $100,000, $1 million, or potentially much more.
And while a million-dollar payout is unlikely, it is not impossibly unlikely. In this game, it is roughly a one-in-a-million event.
Now transform both axes to logarithmic scales.
You get a straight line with a gradient of -1.
The payout distribution follows a power law.
Specifically:
P(X) = X⁻¹ = 1/X.
This is fundamentally different from a normal or lognormal distribution.
With a normal distribution, you can measure the width using the standard deviation.
In a normal distribution, about 95% of the data falls within roughly two standard deviations of the mean.
But for a power law like the St. Petersburg distribution, there is no finite, measurable width.
The standard deviation is infinite.
That makes power laws a fundamentally different kind of distribution, with some very strange properties.
When the Average Never Converges
Imagine taking a bunch of random samples, averaging them, and then taking more random samples and averaging those as well.
You might expect the average to settle down.
But with a sufficiently heavy-tailed power-law distribution, it does not.
The average can keep increasing.
The more you measure, the larger the average can become.
This sounds impossible, but it happens because of the heavy tail.
The probability of extremely large events is significant enough that, if you keep measuring, eventually you will encounter an extreme outlier.
And that outlier can completely distort the average.
It is somewhat like standing in a room with Bill Gates or Elon Musk.
The average wealth of everyone in that room could be enormous because one extraordinarily wealthy person dominates the calculation.
Why Does the St. Petersburg Game Produce a Power Law?
Look again at the third coin game.
The payout X grows exponentially with each toss:
X = 2ⁿ.
But the probability of needing that many tosses to get heads shrinks exponentially.
The probability of reaching n tosses is:
1/2ⁿ.
But we are not really interested in the number of tosses.
We are interested in the payout.
Since:
X = 2ⁿ,
we can replace in the probability equation with X.
We then get:
P(X) = 1/X.
Or:
P(X) = X⁻¹.
That is a power law.
So two exponentials—the exponential growth of the payout and the exponential decline in probability—combine to produce a power law.
And this is a surprisingly common pattern in nature.
Many times when we see a power law, we find that two underlying exponential processes are interacting to produce it.
Earthquakes: Two Exponentials Become a Power Law
Earthquakes provide another example.
If you look at earthquake data, you find that small earthquakes are very common.
As earthquake magnitude increases, earthquakes of that magnitude become exponentially rarer.
But the destruction caused by an earthquake is not proportional simply to its magnitude.
It is related to the energy released.
And as earthquake magnitude increases, the energy released increases exponentially.
So we have two processes:
- The frequency of earthquakes decreases exponentially with magnitude.
- The energy released increases exponentially with magnitude.
When you combine those two exponentials and eliminate magnitude, you can end up with a power-law relationship.
Power Laws Reveal the Structure of Systems
Power laws do more than describe data.
They can reveal something fundamental about the underlying structure of a system.
Go back to the St. Petersburg game.
You can draw all the possible outcomes as a tree diagram, where the length of each branch corresponds to its probability.
You start with a single line of length one.
Then it splits into two branches, each with length 1/2.
Those branches split again, producing four branches, each with length 1/4.
And so on.
Now zoom in.
You keep seeing the same structure repeating at smaller and smaller scales.
It is self-similar, like a fractal.
And that is not a coincidence.
We see similar fractal-like patterns in the veins of leaves, river networks, blood vessels in our lungs, and even lightning.
In many of these systems, the underlying patterns can be described using power laws.
Power laws and fractals are therefore deeply connected.
That is because power laws reveal something fundamental about the structure of a system.
Netflix, YouTube and the Power of the Few
We see the same pattern on streaming platforms.
On Netflix, the top 6% of shows account for more than half of all viewing hours on the platform.
On YouTube, fewer than 4% of videos ever reach 10,000 views, yet those videos account for more than 93% of all views.
All of these examples follow the same broad principle that Pareto identified more than a century ago:
A disproportionate share of wealth, attention, or success goes to a very small number of winners.
The entire system can be dominated by rare runaway hits.
But not every industry operates this way.
Imagine running a restaurant.
You need to fill tables night after night.
You cannot have one exceptionally busy summer evening that brings in millions of customers and use it to compensate for months of quiet nights.
Over the course of a year, busy nights and quiet nights tend to balance out.
The restaurant’s performance is therefore largely determined by its average performance.
Airlines are similar.
An airline needs to fill seats on each flight.
You cannot squeeze a million passengers onto one airplane and make up for hundreds of empty flights.
The average number of passengers carried over time determines the airline’s success.
Know What Kind of Game You Are Playing
We are used to living in a world of normal distributions.
We make decisions as if averages are stable, extreme outcomes are rare, and past performance gives us a reasonable idea of what to expect in the future.
But as soon as you enter a world governed by a power law, you need to start thinking and acting very differently.
In a normal world, the average is often a useful summary of reality.
In a power-law world, the average can be misleading.
A few extraordinary events can dominate the entire system.
A single company can become worth more than an entire industry.
A small number of videos can account for almost all the views.
A handful of people can hold a huge share of total wealth.
A single earthquake can release vastly more energy than thousands of smaller earthquakes combined.
The lesson is simple:
It really pays to know what kind of world you are in—and what kind of game you are playing.
Because the rules are different.
And once the rules change, the strategies that work in one world may fail completely in another.