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The Law of Large Numbers

One roll tells you little. The average of thousands settles on the expected value — inside a band that narrows like 1/√n, which is why more data helps, and why it helps slowly.

Step 1 of 10

Share of each value in the first run so far (white ticks: the true probabilities)

Speed

Five people each roll a fair die, again and again, and keep track of their average so far. After one roll, the average is just that roll — anywhere from 1 to 6.

\bar{X}_{1} = \frac{1}{1} \sum_{i=1}^{1} X_i = 5, \qquad \mu = 3.5, \quad |\bar{X}_n - \mu| = 1.5 \\[6pt] \text{band: } \mu \pm \frac{2\sigma}{\sqrt{n}} = 3.5 \pm \frac{2 \times 1.708}{\sqrt{1}} = 3.5 \pm 3.416

n (draws so far)1Total draws2,000Runs5

The Law of Large Numbers

One roll tells you little. The average of thousands settles on the expected value — inside a band that narrows like 1/√n, which is why more data helps, and why it helps slowly.

Step 1 of 10

Share of each value in the first run so far (white ticks: the true probabilities)

Speed

Five people each roll a fair die, again and again, and keep track of their average so far. After one roll, the average is just that roll — anywhere from 1 to 6.

\bar{X}_{1} = \frac{1}{1} \sum_{i=1}^{1} X_i = 5, \qquad \mu = 3.5, \quad |\bar{X}_n - \mu| = 1.5 \\[6pt] \text{band: } \mu \pm \frac{2\sigma}{\sqrt{n}} = 3.5 \pm \frac{2 \times 1.708}{\sqrt{1}} = 3.5 \pm 3.416

n (draws so far)1Total draws2,000Runs5

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The Law of Large Numbers

See also

The Law of Large Numbers

See also