San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

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

The Distributions of Sample Statistics

Sample Mean Sample VarianceSample Standard
Sample MedianSample MaximumSample Minimum
Sample RangeSample Skewness Sample Kurtosis
Sample QuartileSample Percentile Sample Regression

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