Central Limit Theorem (Distribution of Averages)

Assume that there are n independent and identically distributed variables and each of the variables has the same probability distribution as the others and all are mutually independent. The Central Limit Theorem states that the mean of such variables will approach a normal distribution as the number of observations increases.  Examples of normally distributed variables are Intelligent Quotient’s, manufacturing processes, weights to name a few.

Assume that we have a set of variables where each variable has a means µ and standard deviation.  The mean of the value of x is defined as $-\frac{1}{n}\sum_{i}^{n}X_{i}$ .