A Thought Experiment on Where the Bell Curve Comes From
Repeated independent left-or-right steps produce a binomial distribution: middle outcomes can occur in more ways than extreme ones. With many steps and suitable scaling, its shape approaches a Gaussian bell curve.
THE IDEA
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 flips 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 first flip, you will either win or lose.
At the second flip, 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 each extreme—making or losing $2—and a 50 percent chance of breaking even. The finite walk is binomial; a Gaussian approximation becomes useful as the number of independent steps grows.
When to use it
Modeling the distribution of test scores where many independent factors contribute.
Describing measurement errors in scientific experiments due to numerous small perturbations.
Assessing whether a Gaussian approximation is appropriate for a sum of independent, finite-variance contributions, rather than assuming that a financial-return distribution must have that shape.
Understanding the behavior of particles undergoing diffusion or Brownian motion.
FROM THEORY TO EVERYDAY LIFE
In practice
Pinball Machine Analogy
A well-balanced pinball board with equal chances of a ball going left or right at each pin. After launching many balls, the central columns receive the most balls, forming a bell-shaped distribution.
Coin Flip Random Walk
Flipping a fair coin to decide whether to gain or lose $1 each round. After two rounds, there are four equally likely outcomes, but winning and losing once (in either order) occurs twice as often as winning twice or losing twice, illustrating why middle outcomes are more likely. With many independent steps, the binomial distribution can be approximated by a Gaussian.
A MOMENT TO REFLECT
Put it to work.
How does increasing the number of steps in a random walk affect the shape of the resulting distribution?
In what real-world situations might the bell curve emerge from simple binary decisions?
Why does the middle outcome have the highest probability in a symmetric random walk?
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)