## Arithmetic Mean

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

 (1)

where is the Expectation Value. For a Discrete Distribution,
 (2)

The population mean satisfies
 (3)

 (4)

and
 (5)

if and are Independent Statistics. The sample mean,'' which is the mean estimated from a statistical sample, is an Unbiased Estimator for the population mean.

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)

Hoehn and Niven (1985) show that
 (8)

for any Positive constant . The arithmetic mean satisfies
 (9)

where is the Geometric Mean and is the Harmonic Mean (Hardy et al. 1952; Mitrinovic 1970; Beckenbach and Bellman 1983; Bullen et al. 1988; Mitrinovic et al. 1993; Alzer 1996). This can be shown as follows. For ,
 (10)

 (11)

 (12)

 (13)

 (14)

with equality Iff . To show the second part of the inequality,
 (15)

 (16)

 (17)

with equality Iff . Combining (14) and (17) then gives (9).

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 t-Distribution should be used.

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

 (20)

From k-Statistic for a Gaussian Distribution, the Unbiased Estimator for the Variance is given by
 (21)

where
 (22)

so
 (23)

The Square Root of this,
 (24)

is called the Standard Error.
 (25)

so
 (26)

See also Arithmetic-Geometric Mean, Arithmetic-Harmonic Mean, Carleman's Inequality, Cumulant, Generalized Mean, Geometric Mean, Harmonic Mean, Harmonic-Geometric Mean, Kurtosis, Mean, Mean Deviation, Median (Statistics), Mode, Moment, Quadratic Mean, Root-Mean-Square, Sample Variance, Skewness, Standard Deviation, Trimean, Variance

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.