From: Stan Hilliard
Date: 5/12/2005
Time: 2:46:17 PM
The centra limit theorem says that as you increase the sample size, n, the distribution of sample averages and of sample sums both approache a normal distribution.
And the standard deviation of the averages, the standard error is SE=SD/sqrt(n)
The following website simulates a square distribution of dice sums going toward normal as you go from one die to five. http://www.stat.sc.edu/~west/javahtml/CLT.html
The following website gives a mathematical proof that the SD of sample averages is SD/sqrt(n), where n is the number of data points going into each average. http://mathworld.wolfram.com/CentralLimitTheorem.html
An engineering approach to explaining it is here. Near the bottom of the page you can see an animated distributiom change as n changes. http://www.statisticalengineering.com/central_limit_theorem.htm
This principle is not an estimate but is a property of nature. If it is not consistent with your experience then you probably misinterpret what it means.