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

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 µ for rolling one die is 3.5, and the variance is . If S_n is the number of spots showing when n dice are rolled, then if n is ``large'' the random variable

Should be approximates standard normal, so S_n 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.