The Central Limit Theorem

The central limit theorem explains why many distributions tend to be close to the normal distribution. The key ingredient is that the random variable being observed should be the sum or mean of many independent identically distributed random variables. One version of the theorem is

 

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In this applet, we look at rolling dice again. Let X be the number of spots showing when one die is rolled. The mean value tex2html_wrap_inline35 for rolling one die is 3.5, and the variance is tex2html_wrap_inline37 . If Sn is the number of spots showing when n dice are rolled, then if n is ``large'' the random variable

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Should be approximates standard normal, so Sn itself should be approximately normal with mean 3.5*n and variance 35n/12. The red curve is the graph of the density function with these parameters.