MathsSeeing how numbers work9 / 10▾

  1. Square roots

    01What is a square root?
  2. 02Perfect squares
  3. 03Heron's method
  4. 04Roots that never end

  5. Probability

    05What is probability?
  6. 06The law of large numbers
  7. 07Two dice
  8. 08Bayes' theorem
  9. 09The bell curve
  10. 10Three meanings of probability

09 / 10 · Probability

The bell curve

Order out of many random bounces

Drop a ball onto a board of pegs. At every peg it bounces left or right, 50/50. Where does it end up?

To land at the far left it must bounce left every single time, which is very unlikely. To land in the middle it needs about as many lefts as rights, and there are a huge number of ways to do that. So most balls pile up in the middle.

ways to land in bin k after n pegs = C(n, k)

many small random pushes added up  →  bell curve

Try it · Drop balls and watch the bins fill. The dashed curve is the ideal bell. Tilt the board to make bounces favour one side.

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Drop hundreds of balls and the piles form a smooth hill: the normal distribution, or bell curve. Francis Galton built this machine in 1873 to show it.

The bell curve appears whenever many small random effects add up: heights of people, measurement errors, test scores. It is the two-dice triangle from the last section, with many more dice.

Key ideas

Words worth remembering

Normal distribution
The symmetric, bell-shaped curve that sums of many random effects follow.
Mean
The centre of the distribution, where it balances.
Central limit theorem
Adding up many independent random things tends to give a bell curve.

Think about it

What happens to the shape if every bounce goes right 80% of the time?