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The Central Limit Theorem and
Sample Statistics

The Central Limit Theorem

The Central Limit Theorem (CLT) is a powerful and important result of mathematical analysis. In
its standard form it says that if a stochastic variable x has a finite variance then the
distribution of the sums of n samples of x will approach a normal distribution as the sample
size n increases without limit. The CLT provides a basis for some sample statistics having a
normal distribution for large samples. This page provides illustrations of the distribution
of some sample statistics for samples drawn from a population which has a uniform distribution; i.e., the
probability density is constant over a range of values.

Suppose the probability density distribution for z is

p(z) = 1 for -0.5≤z≤+0.5
p(z) = 0 for all other values of z

The following cases shows the distributions for several sample statistics