1.16 Law of large numbers and the CLT
You know why averages behave and single samples do not.
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The law of large numbers says averages settle down; the central limit theorem says how they fluctuate on the way — together they explain why sample means are trustworthy and why the normal distribution appears everywhere. It sits here because estimation and testing both stand on these two results. The trap is applying the CLT where its conditions fail: heavy tails and dependence break it, and knowing where it quietly fails is worth as much as knowing the statement.
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Weak and strong law of large numbers
Averages of many independent observations settle down toward the true value, and there are two versions of the statement with different strengths. This is why sampling works at all.
Central limit theorem and its conditions
The sum of many independent contributions tends toward a normal shape whatever the individual pieces look like, provided some conditions hold. Those conditions are the part people forget.
Sampling distributions of the mean
The distribution of a sample average is itself a distribution, and knowing its spread is what makes error bars possible. This is the bridge from probability to statistics.
Where the CLT quietly fails
Heavy tails, strong dependence and small samples are the situations where the usual reassurance does not apply. Knowing when the theorem fails is more useful than reciting it.
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