# Sampling Distribution of the Sample Mean

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The sample mean is a specific number for a specific sample. The sample mean is a random variable that varies from one random sample to another. Provided the sample size is sufficiently large, the sampling distribution of the sample mean is approximately normal (regardless of the parent population distribution), with mean equal to the mean of the underlying parent population and variance equal to the variance of the underlying parent population divided by the sample size. More formally, if , ,..., are a random sample from an infinite population with mean and variance then and ; and assuming the moment-generating function of the population exists, the limiting distribution of as approaches infinity is the standard normal distribution (Freund 1992).

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Contributed by: Jim R Larkin (March 2011)

Open content licensed under CC BY-NC-SA

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J. E. Freund, *Mathematical Statistics*, 5th ed., Englewood Cliffs, N. J.: Prentice-Hall, 1992.

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