The Central Limit Theorem
The Central Limit Theorem by itself makes the NormalDistribution the most important probability distribution. It statesthat any distribution can be approximated to some extent by thenormal distribution. Specifically, if a sample of size n is takenfrom a population with mean
andvariance
thenthe sample mean has the approximate distribution 
The mean and variance of some common distributions of given in thetable below.
Distribution | Parameters and Meaning of Parameters | Mean | Variance | Approximate Distribution of Sample Mean for Sample of Size n |
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Uniform, |
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Binomial, |
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Geometric, |
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Poisson, |
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–standard deviation
and
are the minimum and maximum possible values of the random variable
is the number of trials,
is the probability of success
is the average number of events per time period
Example: For a sample of size 10 taken from a uniform distribution between the limits
and
what is the probability that the average is less than 12?
This corresponds to a probability of 0.791.