The Central Limit Theorem (CLT) is extremely important in sampling and it shows the power of independent and random sampling in some distribution.

Basically, the CLT states that, if the items are randomly selected from some distribution (the distribution itself does not have to be a normal distribution) then the distribution of the mean values and the standard deviations of those selected items will themselves be normally distributed.

Example

(the History and proof of this theorem (boring and tedious)

http://ccl.northwestern.edu/netlogo/models/run.cgi?CentralLimitTheorem.699.627