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.

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

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