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
 C. W. Misner, J. A. Wheeler and K. S. Thorne, “Gravitation,” W.H. Freeman & Company, 1973.
 B. Greene, “The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory,” W.W. Norton, 1999.
 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
 A. M. Kosevich, “The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices,” WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005.
 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
 A. Papoulis, “Probabilities, Random Variables, and Stochastic Processes,” McGraw-Hill, New York, 1965.
 B. Gnedenko, “The Theory of Probability,” Mir, Moscow, 1978.
 R. Caimmi, “Il Problema della Misura,” Diade, Nuova Vita, Padova, 2000.
 V. Smirnov, “Cours de Mathèmatiques Supèrieures,” Tome II, Mir, Moscow, 1970.
 B. B. Mandelbrot, “Les Objects Fractals: Forme, Hazard et Dimensions,” 2nd Edition, Flammarion, Paris, 1986.
 M. R. Spiegel, “Mathematical Handbook of Formulas and Tables,” Schaum’s Outline Series, McGraw-Hill, New York, 1969.