Random Bounces Pile Up into a Bell Curve

The Board

Steel balls drop from the hopper on top through twelve rows of brass pins. At every pin a ball jumps left or right to the next row, and at the bottom it falls into a bin.

Where one ball will land can’t be told beforehand. After a thousand, the bins hold a curve that is high in the middle and low at the sides, and nearly the same curve every time. The gold line on the glass is that curve: twelve rows of pins, a thousand balls.

Each row has one pin more than the row above, set half a pitch across, like the five spots on a die. That arrangement is called a quincunx.

One Ball’s Path

At a pin the ball is as likely to go left as right. Twelve rows of pins are twelve coin tosses in a row.

Which bin it ends in depends only on how many times it went right: never, and it falls into bin 0 on the far left; all twelve times, and it falls into bin 12 on the far right. Right then left and left then right arrive at the same place.

In slow motion the red line is the path the ball took.

Counting Paths

A path to bin k is a choice of which k of the twelve jumps go right: C(12, k) paths in all. Each path has a chance of 1/2¹² = 1/4096. So a ball lands in bin k with probability C(12, k)/4096.

The middle bin, bin 6, has 924 paths, about 22.6%; each end bin has only one, 1/4096. The middle is high only because the most paths lead there.

This is the binomial distribution. With n rows the average bin is n/2 and the variance is n/4: for twelve rows the variance is 3, a standard deviation of about 1.73 bins. The red line on the glass is the exact distribution for the number of balls on the board now; the more balls, the closer the bins come to it.

Into a Bell

The more rows of pins, the closer the binomial distribution comes to a smooth curve, the normal curve: centred on n/2, its width growing as √n/2.

The approximation was published in 1733, and later extended to chances of going right other than one half:

C(n, k)/2ⁿ ≈ √(2/(πn)) · e^(−2(k − n/2)²/n)

The model changes to 1, 2, 4, 8 and 14 rows in turn. The more rows, the lower, wider and smoother the curve.

Swap in a Die

A coin is only the simplest nudge. Make each level’s nudge the throw of a die, moving the ball its score less 3½ bins; or move it any distance between −3 and 3 bins, all equally likely.

With one nudge the bins show the nudge’s own shape: six equal bins for the die, a flat plateau for the uniform nudge. Two nudges add up to a triangle; three are already close to a bell.

That is the central limit theorem: add up many independent nudges, each with a finite variance and none outweighing the rest, and the total is close to normal, whatever the shape of a single nudge. The figure rescales the total to one width and sets it against the normal curve; the more nudges, the smaller the gap.

Where It Shows Up

An error of measurement comes from many small, independent disturbances: temperature, vibration, the slip of a reading. Added up, such errors are often close to normal. In 1809 the normal curve was already used as the law of errors in astronomical observation.

Adult heights within one population and one sex are also close to normal, because height depends on many genes and circumstances, none of them large on its own.

Not everything is normal. Incomes, city sizes, the energy of earthquakes and daily stock-market moves have heavy tails: extremes come far more often than the normal curve says. One factor may be very large, the factors may not be independent, or they may multiply instead of add. Some distributions have no mean at all, such as the Cauchy distribution: averaging more values never settles down.

A Real Board

A real board only comes close to the ideal. Pins don’t split exactly half and half: a pin a fraction of a millimetre out of place, or a board slightly tilted, and every pin favours one side. Balls also jump more than one row at once, hit each other, and carry speed from one bounce into the next.

So a board has to stand level; the spirit level on top is there for that. Studies of the board’s dynamics find that, depending on how elastic the impacts are, the bins can stray clearly from a bell.

If every pin sends the ball right with probability p, the result is still binomial, now centred on bin np with variance np(1−p). The model shows p = 0.5, 0.65 and 0.35 in turn: the whole bell moves across and gets a little narrower.