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09 · Large numbers and the CLT

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Probability, lesson 9

Large numbers and the CLT

Why averages settle down, and why the bell curve keeps appearing.

The last lesson used the bell curve as a shortcut for counts. This lesson shows why the shortcut works: averages of many independent draws settle on the mean, and their wobble around it takes the shape of a bell. You need the square root rule and z-scores.

By the end, you can:

  1. Work out how far an average of n draws typically lands from the mean.
  2. Explain why sums and averages of many draws look like a bell.
  3. Work out how many draws it takes to see a small edge.

Part 1. Averages settle down

Work out how far an average of n draws typically lands from the mean.

Take a lopsided die with faces 1, 1, 1, 2, 3 and 10. Half its rolls are 1, and now and then it throws a 10. Its mean is 18 / 6 = 3, but one roll is a poor guess at that 3. The average of many rolls is a much better one. The question is how much better.

Example 1.1

Averages of 30 rolls

Roll the lopsided die 30 times and take the average. How far from 3 does that average typically land?

  1. Spread of one rollσ = √(116/6 − 3²) ≈ 3.21

    The average square is (1 + 1 + 1 + 4 + 9 + 100) / 6 = 116/6, so the variance is 31/3, about 10.33.

  2. Spread of the total√30 × 3.21 ≈ 17.6

    The square root rule from two lessons back.

  3. Spread of the average17.6 / 30 ≈ 0.59

    Dividing the total by 30 divides its spread by 30 too.

AnswerAbout 0.59, against 3.21 for a single roll.

Put the last two steps together: √30 × σ / 30 = σ / √30. The total's spread grows like √n, but dividing by n shrinks it by n, so the average's spread falls like 1 / √n. Every extra roll pulls the average closer to 3. That's the law of large numbers, with a size attached.

Law of large numbers and standard error

The law of large numbers says the average of many independent draws settles on the expected value. The typical distance of an average of n draws from that value is its standard error, σ / √n, read "sigma over root n".

SE = σ / √n

One run of 300 rolls of the lopsided die 0 3 6 0 100 200 300 rolls so far running average 3 ± 2 standard errors
One run of 300 rolls of the lopsided die 0 3 6 0 100 200 300 rolls so far running average 3 ± 2 standard errors
Figure 1One simulated run of the lopsided die. The running average jumps around at first, then settles towards 3, inside a band 2 standard errors either side that narrows like 1 / √n.

Settling isn't correcting. A fair coin that starts with 8 heads in 10 flips is still 50:50 on every later flip. After 1,000 more, you expect about 508 heads in 1,010, or 50.3%. The early lead never goes away. It gets diluted by everything after it.

Your turn

You average 100 rolls of the lopsided die instead of 25. The standard error of the average:

The standard error is σ / √n. Four times the rolls makes √n twice as big, so the spread halves: from 3.21 / 5 ≈ 0.64 to 3.21 / 10 ≈ 0.32. Halving the error costs four times the data.

Part 2. Why the bell keeps appearing

Explain why sums and averages of many draws look like a bell.

Keep going with Pro

You've done the first part. Pro opens the rest of this lesson, and every other one.

Still to come
  1. Why the bell keeps appearing
  2. How much data you need
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Next lessonBetting gamesFair games, edges, and how much to bet when the odds favour you. Start lesson 10

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