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Sample Mean

The sample mean of a set {x_1,...,x_n} of n observations from a given distribution is defined by

m=1/nsum_(k=1)^nx_k.

It is an unbiased estimator for the population mean mu. The notation mu^^_n is therefore sometimes used, with the hat indicating that this quantity is an estimator for mu.

The sample mean of a list of data is implemented directly as Mean[list].

An interesting empirical relationship between the sample mean, statistical median, and mode which appears to hold for unimodal curves of moderate asymmetry is given by

mean-mode approx 3(mean-median)

(Kenney and Keeping 1962, p. 53), which is the basis for the definition of the Pearson mode skewness.

See also

Arithmetic Mean, Mean, Population Mean, Sample, Sample Central Moment, Sample Raw Moment, Sample Size, Sample Variance

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References

Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, p. 16, 2000.Kenney, J. F. and Keeping, E. S. "Averages," "Relation Between Mean, Median, and Mode," and "Relative Merits of Mean, Median, and Mode." §3.1 and §4.8-4.9 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 32 and 52-54, 1962.

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Sample Mean

Cite this as:

Weisstein, Eric W. "Sample Mean." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SampleMean.html

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