“Arithmetic Mean”的概念、定义、习题解题方法及翻译-数学辞典

单词

Arithmetic Mean

释义

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)

and

are Independent Statistics. The ``sample mean,'' which is the mean estimated from a statisticalsample, 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 andBellman 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 populationmean 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 forthe Variance is given by

(21)

where

(22)

(23)

The Square Root of this,

(24)

is called the Standard Error.

(25)

(26)

See also Arithmetic-Geometric Mean, Arithmetic-Harmonic Mean, Carleman'sInequality, Cumulant, Generalized Mean, Geometric Mean, Harmonic Mean, Harmonic-GeometricMean, 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.

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