Central Limit Theorem

where

then, as n tends to infinity, the limiting distribution of Z_n is standard normal.

Notes:

• The above result essentially states that the standardized sample mean Z_n has a standard normal distribution when n is large enough.
• In practical terms, the Central Limit Theorem (CLT) says that you can use a normal distribution to approximate the distribution of the sample mean when the sample size n is sufficiently large, i.e.,
  
  
  (the tilde with an a above it reads "approximately distributed as").
• It is worthwhile noting that CLT does not assume anything about the distribution of the X_i's used to define the sample mean! It only assumes the existence of the common mean and variance of these random variables. This makes the CLT a useful tool for solving problems which involve the sample mean when the underlying probability model for the X_i's is not known. Indeed, even in situations where you know the distribution of the X_i's, you may find it expedient to use CLT to solve such problems.