The Arithmetic Mean Standard Deviation Distribution: A Geometrical Framework

R. Caimmi

Physics and Astronomy Department, Padua University, Padova, Italy.

DOI: 10.4236/am.2013.411A4001                

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.

Keywords

Standard Deviation; n-Spaces; Direction Cosines; Quadrics

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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.