Central Limit Theorem

The Statement

Central Limit Theorem states that if you have population with mean mu and standard deviation sigma, taking sufficiently large random samples from population with replacement would give distribution of sample means to be approximately Normal.


  1. Distribution of population does not matter.
  2. Mean and Variance of X bar can be seen as

3. If X is a random variable then its mean X bar is also a random variable

4. X is a random variable for a pick of an observation, while X bar is random variable over pick of a sample

5. X bar tends to normal when we have sufficiently large number of samples. That means X bar might not be visually normal when few samples are chosen.

This theorem is a blessing to many derivations.

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