The Law of Large Numbers and Central Limit Theorem both try to approximately tell us the behaviour of the sample mean .

Central Limit Theorem only holds if each is normally distributed (i.e. ).

The Central Limit Theorem gives us the approximate shape of the distribution of .

Central Limit Theorem is not restricted by the underlying distribution of -s.

If the sample size is small (e.g. ), then the Law of Large Numbers on does not hold, but the Central Limit Theorem on is still valid (i.e. if ).

, where is a constant.

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