The Central Limit Theorem says that the sample mean of n observations from any population with finite variance gets closer and closer to a normal distribution as n increases. That is, probabilities involving get closer and closer to normal probabilities. The normal distribution has the same mean and standard deviation as the sampling distribution of . If individual observations have mean μ and standard deviation σ, also has mean μ and has standard deviation .

The distribution pictured for n=1 is called an exponential distribution. Its density curve (blue) is far from normal in shape. This distribution has mean 1 and standard deviation 1. For comparison, the red curve is the normal curve with mean 1 and standard deviation 1.

Increase the number of observations n by dragging the slider with the mouse. The blue curve is the density curve of the sample mean of n observations. The red curve is the normal curve with mean 1 and standard deviation . As you increase n, the density curve of looks more and more normal.