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First Principles of PhysicsLimits & Uncertainty · Rule 31

Limits & Uncertainty

Rule 31Chapter V

Randomness is structure seen at scale

Why this rule exists

Individually unpredictable events can produce sharply predictable aggregates because independent fluctuations tend to cancel. The law of large numbers guarantees averages converge, and the central limit theorem explains why sums of many small independent effects fall into the same bell-shaped distribution. Randomness at the level of one event becomes structure at the level of many.

First Principles of Physics

In practice

When single outcomes are noisy, work with distributions and averages instead. Estimate the mean, then the spread: statistical fluctuations shrink like 1/sqrt(N), so more samples buy tighter results. Use this to justify treating a gas by its pressure and temperature, or to size the sample you need for a target precision.

When it doesn't apply

Convergence assumes many independent events with finite variance. Correlated events, heavy-tailed distributions, or small N break it — rare extremes then dominate and averages mislead. At small numbers, fluctuations are the story, not the noise.

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