Central Limit Theorem
The central limit theorem states that under general conditions, the distribution of a variable is well approximated by a normal distribution when the number of observations is large.
(Stock, James H. and Mark. W. Watson. 2007. Introduction to Econometrics, 2nd ed. Boston: Pearson/Addison Wesley, p.52.)
As we consider larger and larger samples, the variance (or standard error) of the estimates becomes smaller and smaller. Put another way, as the sample from the population gets larger, the likelihood that our estimated mean is the actual mean becomes greater.
The central limit theorem holds that the standardized value of the sample mean will be approximately normally distributed when the following two conditions are met:
1.) The sample size is sufficiently large.
2.) The observations are identically and independently distributed.
Phil Oreopoulos, PPG 1004
Paul Grootendorst, PPG2010