Central Limit Theorem? http://en.wikipedia.org/wiki/Central_Limit_Theorem
The sum of die rolls shows a good example of why you get the bell shape: there are more ways (combinations) with which to sum up to the center values than the extremal values.
The sum of die rolls shows a good example of why you get the bell shape: there are more ways (combinations) with which to sum up to the center values than the extremal values.
For the majority of the people I had in the intro statistics courses I TA'ed for, this was enough explanation. In undergrad, we didn't prove Lindeberg-Levy CLT until my third quarter of math stats, and the versions with weaker assumptions we didn't prove until graduate math stats.
When you're struggling to understand the fundamentals of hypothesis testing and simple combinatorics, going through the rigorous process of proving the CLT is daunting, mind-boggling, and largely unnecessary :).
It is slightly easier if you start off assuming Stirling's formula. However there is no need to assume that - you can derive it at the same time. Elsewhere in this thread I gave an outline of how to derive this simple case.