For a Continuous Distribution function, the arithmetic mean of the population, denoted , ,
, or
, is given by

(1) |

(2) |

(3) |

(4) |

(5) |

For small samples, the mean is more efficient than the Median and approximately
less (Kenney and Keeping 1962, p. 211). A general expression which often holds approximately is

(6) |

Given a set of samples , the arithmetic mean is

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

Given independent random Gaussian Distributed variates , each with population
mean and Variance
,

(18) |

(19) |

so the sample mean is an Unbiased Estimator of population mean. However, the distribution of depends on the sample size. For large samples, is approximately Normal. For small samples, Student's

The Variance of the sample mean is independent of the distribution.

(20) |

From

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 10, 1972.

Alzer, H. ``A Proof of the Arithmetic Mean-Geometric Mean Inequality.'' *Amer. Math. Monthly* **103**, 585, 1996.

Beckenbach, E. F. and Bellman, R. *Inequalities.* New York: Springer-Verlag, 1983.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, p. 471, 1987.

Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. *Means & Their Inequalities.* Dordrecht, Netherlands: Reidel, 1988.

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. *Inequalities.* Cambridge, England: Cambridge University Press, 1952.

Hoehn, L. and Niven, I. ``Averages on the Move.'' *Math. Mag.* **58**, 151-156, 1985.

Kenney, J. F. and Keeping, E. S. *Mathematics of Statistics, Pt. 1, 3rd ed.* Princeton, NJ: Van Nostrand, 1962.

Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. *Classical and New Inequalities in Analysis.*
Dordrecht, Netherlands: Kluwer, 1993.

Vasic, P. M. and Mitrinovic, D. S. *Analytic Inequalities.* New York: Springer-Verlag, 1970.

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1999-05-25