The Arithmetic Mean Standard Deviation Distribution: A Geometrical Framework

Author(s)
R. Caimmi

ABSTRACT

The current attempt is aimed to
outline the geometrical framework of a well known statistical problem,
concerning the explicit expression of the arithmetic mean standard deviation
distribution. To this respect, after a short exposition, three steps are
performed as 1)
formulation of the arithmetic mean standard deviation, , as a function of the errors, , which, by themselves, are
statistically independent; 2)
formulation of the arithmetic mean standard deviation distribution, , as a function of the errors, ; 3) formulation of the arithmetic mean
standard deviation distribution, , as a function of the
arithmetic mean standard deviation, , and the arithmetic mean rms
error, . The integration domain can
be expressed in canonical form after a change of reference frame in the *n*-space, which is recognized as an
infinitely thin *n*-cylindrical corona
where the symmetry axis coincides with a coordinate axis. Finally, the solution
is presented and a number of (well known) related parameters are inferred for
sake of completeness.

Cite this paper

R. Caimmi, "The Arithmetic Mean Standard Deviation Distribution: A Geometrical Framework,"*Applied Mathematics*, Vol. 4 No. 11, 2013, pp. 1-10. doi: 10.4236/am.2013.411A4001.

R. Caimmi, "The Arithmetic Mean Standard Deviation Distribution: A Geometrical Framework,"

References

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[1] C. W. Misner, J. A. Wheeler and K. S. Thorne, “Gravitation,” W.H. Freeman & Company, 1973.

[2] B. Greene, “The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory,” W.W. Norton, 1999.

[3] J. Douthett and R. Krantz, “Energy Extremes and Spin Configurations for the One-Dimensional Antiferromagnetic Ising Model with Arbitrary-Range Interactions,” Journal of Mathematical Physics, Vol. 37, No. 7, 1996, pp. 3334-3353. http://dx.doi.org/10.1063/1.531568

[4] A. M. Kosevich, “The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices,” WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005.

[5] C. Callender, I. Quinn and D. Tymoczko, “Generalized Voice-Leading Spaces,” Science, Vol. 320, No. 5874, 2008, pp. 346-348.

http://dx.doi.org/10.1126/science.1153021

[6] A. Papoulis, “Probabilities, Random Variables, and Stochastic Processes,” McGraw-Hill, New York, 1965.

[7] B. Gnedenko, “The Theory of Probability,” Mir, Moscow, 1978.

[8] R. Caimmi, “Il Problema della Misura,” Diade, Nuova Vita, Padova, 2000.

[9] V. Smirnov, “Cours de Mathèmatiques Supèrieures,” Tome II, Mir, Moscow, 1970.

[10] B. B. Mandelbrot, “Les Objects Fractals: Forme, Hazard et Dimensions,” 2nd Edition, Flammarion, Paris, 1986.

[11] M. R. Spiegel, “Mathematical Handbook of Formulas and Tables,” Schaum’s Outline Series, McGraw-Hill, New York, 1969.