Glossary / Normal distribution
A ball falls through rows of pegs and bounces left or right off each one, as if tossing a coin. Nobody can say where this particular ball will land. Yet a thousand balls build the same bell-shaped pile every time. That bell is called the normal distribution, and it turns up wherever many small random pushes add up.
“In plain words” is hidden on all puzzles.
The board of pegs was invented by the English scientist Francis Galton in the 19th century. This one has 12 rows of pegs and 13 bins. Drop one ball and try to guess where it will land. Then drop a thousand.
A single ball is unpredictable, while a thousand balls are wonderfully predictable. A ball in an end bin would have to bounce the same way 12 times in a row, and that happens very rarely. But countless different paths lead to the middle: a little left, a little right, and the bounces cancel each other out. So the middle is always full and the edges are almost empty.
“Rigorous” is hidden on all puzzles.
Any normal distribution is fully set by two numbers. The mean μ says where the top of the bell is. The standard deviation σ says how wide it is: how far values usually stray from the mean. Change μ and the bell moves. Change σ and it becomes wider and lower, or narrower and taller. The shape is always the same.
For any bell, about 68% of values lie within ±1 σ of the mean, 95% within ±2 σ, and 99.7% within ±3 σ. Let's check it on height. The numbers are made up for the example: say adult men somewhere have a mean height of 176 cm and σ = 7 cm. Below are 10,000 random people.
Try 1, 2 and 3. About 3 people in 1,000 fall outside ±3 σ: here that means shorter than 155 cm or taller than 197 cm.
The bell appears when a result is the sum of many small independent pushes, none of which dominates. When these conditions break, the bell lies, most often in the tails: very large values happen far more often than it promises.
“Deeper” is hidden on all puzzles.
P(bin k) = C(12, k) / 2¹² · C(12, 6) = 924, C(12, 0) = 1, 2¹² = 4096
Each ball passes 12 pegs, and each of the 4,096 possible paths is equally likely. The bin depends only on the number of bounces to the right, k. One path leads to an end bin, 924 lead to the middle one. That gives the shape: the middle bin collects 924/4096 ≈ 22.6% of the balls, while each end bin gets just 1/4096, about one ball in four thousand. This is the binomial distribution.
The bars are the shares of the balls you dropped on the board above. The dots are the binomial distribution, the exact path count. The line is the normal curve with μ = 6 and σ = √3 ≈ 1.73 bins. The more balls, the closer the bars get to the dots, and the dots already sit almost on the curve.
balls on the boardbinomialnormal curve
A bounce on the board is a coin. But the coin is not essential. A die is completely flat: each face 1/6, no bell at all. Add up the scores of several dice and the bell appears by itself: two dice give a triangle, three already look like a bell. This is the idea of the central limit theorem: the sum of many independent, identically distributed quantities with any finite spread tends to a normal distribution. Abraham de Moivre noticed it for coins as early as 1733.
20,000 throws for each n. The line is the normal curve with the same mean and σ.
f(x) = 1 / (σ √(2π)) · e−(x − μ)² / (2σ²)
The whole shape lives in e−z²/2, where z = (x − μ) / σ is the distance from the mean measured in σ. At 1 σ the curve is 61% of its peak height, at 2 σ 13.5%, at 3 σ only 1.1%. The square in the exponent makes the tails incredibly thin: the chance of landing more than 10 σ above the mean is less than one in 10²³. The factor 1 / (σ √(2π)) only makes the area under the curve equal 1. These thin tails are the main difference between the bell and incomes or markets, where far-off values happen incomparably more often.
The middle value: line everyone up in order, and half are below it, half above.
A millionaire walks into a bar: the average income shoots up, the median barely moves.
The sum of all values divided by how many there are: what everyone would get if it were shared equally.
Almost everyone has more legs than average: the average is a little under two.
The average result you can expect if you repeat the same thing very many times.
A die's expected value is 3.5, yet no face shows 3.5.
A number from −1 to +1 showing how closely two quantities move together.
In summer both ice cream sales and drownings go up. Ice cream isn't to blame: the heat drives both.
A measure of spread: how far values typically stray from the mean. The standard deviation is its square root, in the same units.
Two towns with the same average temperature: one stays near +15 all year, the other swings from −20 to +50.
The more repetitions, the closer the share of an outcome gets to its true probability.
The share of heads creeps towards 50%, while the gap between heads and tails in coins usually grows.
The chance of an event given that something else is already known.
Almost all NBA players are tall, but almost all tall people are not NBA players.
Events are independent if knowing about one doesn't change the chances of the other.
After five heads in a row tails is no more likely: the coin remembers nothing.
How many times you multiply the base by itself to get a number: log₂ 8 = 3, because 2 · 2 · 2 = 8.
A sheet of paper folded in half 42 times would be thicker than the distance to the Moon.
A sample some people get into more easily than others, so it describes everyone badly.
In 1936 a magazine polled millions of Americans and still picked the wrong election winner.
How often chance alone would produce a gap at least as big as yours if there were really no difference.
When there is no effect at all, p below 0.05 turns up as often as p between 0.50 and 0.55.
A range around a measured number, built by a method that covers the true value in a chosen share of samples, such as 95%.
Of twenty honest 95% intervals, on average one misses the truth, and none of them knows whether it is the one.
The probability that a test notices a real effect of a given size. A power of 80% means the test misses one working effect in five.
A weak A/B test that does find an effect almost always exaggerates it: a real +10% shows up on the dashboard as +24%.
How far an average or a rate computed from a sample usually misses the truth: σ divided by the square root of the sample size.
Halving the error takes four times as much data.