The Arithmetic Mean Standard Deviation Distribution: A Geometrical Framework

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.