The Arithmetic Mean Standard Deviation Distribution : A Geometrical Framework

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, 1 2 , , , n x x  x , which, by themselves, are statistically independent; 2) formulation of the arithmetic mean standard deviation distribution,  d f   , as a function of the errors, 1 2 , , , n x x  x ; 3) formulation of the arithmetic mean standard deviation distribution,  d f   , as a function of the arithmetic mean standard deviation,  , and the arithmetic mean rms error, m  . 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.


Introduction
Geometry is a branch of mathematics concerned with equations of shape, size, relative position of figures, and the properties of space.Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, surfaces, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales.In Euclid time there was no clear distinction between physical space and geometrical space.Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation.
Given   1 n  independent variables, 1 2 , , , n x x x  , 0 x , and a function, F , the equation,

 
, , x , represents a hypersurface ( dimensions) within a hyperspace ( dimensions), which enlightens the strict connection between mathematical analysis and geometry.But the great difficulty in handling with geometry, expecially with regard to hyperspaces 3 n  , makes easier dealing with mathematical analysis leaving aside geometry.On the other hand, physical theories such as general relativity (e.g., [1]) and superstring theory (e.g., [2]) need a geometrical interpretation involving hyperspaces.Accordingly, further insight could be gained exploiting the geometrical framework of the problem under consideration, regardless of the branch of knowledge.
The current attempt is aimed to the investigation of the geometrical framework related to a well known problem of statistics, concerning the explicit expression of the arithmetic mean standard deviation distribution, under the safely motivated restriction of independent measures obeying a Gaussian distribution.
The paper is organized as follows.The problem is outlined in Section 2 together with three steps towards the solution.The first, second, third step are exploited in Sections 3, 4, 5, respectively.The solution of the problem is shown in Section 6, where a number of (well known) related parameters are also inferred for sake of completeness.The conclusion is drawn in Section 7.
Useful generalizations of ordinary analytic geometry to hyperspaces are shown in the Appendix.

The Problem
Let  d f m m be the distribution related to an assigned measure method and a specified statistical system, where the occurrence of the event, E i , has been designed by the value of a random variable, i , .The special case of Gaussian distribution, which well holds for independent measures, reads: where is a generic measure and the expected value, the variance, the rms error, respectively, of the distribution.Expected value and rms error estimators are known to be the arithmetic mean, m , and the standard deviation,  , respectively, which read: ; where is the deviation from the arithmetic mean.It is worth emphasizing the bar over  means the deviation is from the arithmetic mean:  in itself is not an arithmetic mean.In addition, the following relations hold: where has to be intended in statistical sense, according to Bernoulli's theorem (e.g., [6], Chap. ; n    (7) where the bar over  means the standard deviation is related to the arithmetic mean:  in itself is not an arithmetic mean.
The substitution of Equation ( 2) into (4) yields the explicit expression of the deviation in terms of the measures, , as: where pi  is the Kronecker symbol.
The substitution of Equation ( 3) into (7) yields the explicit expression of the arithmetic mean standard deviation in terms of the deviations, 1 2 , , , n     , as: and the arithmetic mean standard deviation distribution reads: where the random variables, ,  4), (11), define an interval, centered on  , of infinitesimal amplitude equal to d .
An explicit expression of the distribution, defined by Equation (12), is difficult to be found for two orders of reasons.First, as already mentioned, the deviations, i  , are dependent random variables owing to Equation (4).Second, even   deviations could not be considered as independent, contrary to what might be suggested by an algebraic interpretation of Equation (4).Conversely, any deviation is a function of the measures, , as shown by Equation (8).Accordingly, the arithmetic mean standard deviation, , , , n m m m   , has to be expressed in terms of independent random variables.Aiming to calculate the multiple integral on the right-hand side of Equation (12), three steps shall be performed as outlined below.
1) Express the arithmetic mean standard deviation,  , as a function of the errors, 1 2 , , , n x x x  , and outline the geometrical framework.
2) Express the arithmetic mean standard deviation distribution, , , , n x x  x , and outline the geometrical framework.

 d
3) Express the arithmetic mean standard deviation distribution,  d f   , as a function of the standard deviation,  , the arithmetic mean rms error, m  , and outline the geometrical framework.
In dealing with the geometrical framework, for sake of simplicity, the formalism has to be specified in the following way.Hyperspaces with dimensions, hyperplanes with dimensions, hyperlines with 1 n  dimensions, hereafter shall be quoted as -spaces, -planes, -lines, respectively.Hypervolumes with mensions, hypersurfaces with n dimen- sions, hyperlengths with   shall be quoted as   When the denomination of a solid is preserved, it shall be intended the symmetry is also preserved.For instance, a n -cylinder is intended as hibiting a single symmetry axis similarly to an ordinary cylinder.The extension of usual formulation of analytic geometry to -spaces, which shall be needed in the following, is outlined in Appendix., , , n x x  x , can be expressed as:

Expression of  in Terms of
; according to the general definition of error.It is apparent the error of the arithmetic mean equals the arithmetic mean of the errors.The substitution of Equation (15) into (13) yields: which shows the deviation of a measure from the arithmetic mean of the measures equals the deviation of the related error from the arithmetic mean of the errors.The right-hand side relation appearing in Equation ( 16) represents a n-plane passing through the origin within a -space described by the reference frame, The substitution of Equation ( 16) into (11) after some algebra yields:  which represents a one-sheet n-hyperboloid where the symmetry axis coincides with the coordinate axis, x , the equatorial semiaxis reads: and the equator is the intersection between the n- hyperboloid and the principal n-plane,   . The asymptotes of the n-hyperboloid are generatrixes of a   1 n  -cone where the symmetry axis coincides with the coordinate axis, x , the vertex coincides with the origin, O, and the lateral n-surface reads: which may be considered as the equation of the The generatrixes lying on the principal plane,   O i x x , are expressed as: which can be extended to a generic direction, Using general formulation of analytic geometry extended to   1 n  -spaces, Equations ( 73) and (76), it can be seen the angle,  , formed by the coordinate axis, x , and the n-plane, expressed by Equation ( 16), equals the angle,  , formed by the coordinate axis, x , and the generatrixes of the   1 n  -cone, expressed by Equation (20).Accordingly, the n-plane, expressed by Equation ( 16), is tangent to the   1 n  -cone, expressed by Equation (19), along a generatrix, t g , which can be determined via the condition that the generic generatrix, g , lies on the n-plane, expressed by Equation ( 16).
Keeping in mind the   1 n  -cone has vertex on the origin and symmetry axis coinciding with the coordinate axis, x , the equation of the generic generatrix reads: that is equivalent to: where, in the case under discussion of generatrixes, the square coefficient, 2  , equals the arithmetic mean of the Open Access AM R. CAIMMI 4 square coefficients, .Finally, the substitution of Equation ( 23) into (21) yields: and the generatrix of interest, t g g  , can be determined via the condition of parallelism between g and the n-plane, expressed by Equation ( 16).
Owing to Equation (79), the result is: where, in the case under discussion of the generatrix, t g , the coefficient,  , equals the arithmetic mean of the coefficients, 1 2 . The further condition, expressed by Equation ( 23), necessarily implies 1 2 , as is needed to define a straight line in the , , , n     space under the validity of Equations ( 23) and ( 25).
Accordingly, the generatrix of the   1 n  -cone, defined by Equation ( 19), where the n-plane, defined by Equation ( 16), is tangent, can be expressed as: which is the -sector1 of the first and 2 n+1 th 2 n+1 -ant2 of the reference frame, . An explicit expression can be obtained erasing the additional coordinate, x , from the definition of t g , Equation (26).The result is: which is the n-sector of the first and th -ant of the reference frame, x , from Equation ( 16).The result is: (28) where passes through the origin, as expected., , , n x x x  , and their arithmetic mean, x , via Equation (17), which represents a one-sheet n-hyperboloid where the symmetry axis coincides with the coordinate axis, x , and the asymptotes are the generatrixes of a  

Expression of
, and Related Geometrical Framework , , , , , for which the sum of deviations from the arithmetic mean is null, lie on a n-plane, defined by Equation ( 16).The combination of Equations ( 15)-(17), yields: where the equatorial semiaxis of the   In terms of the errors, 1 2 n , , , x x  x , Equation (30) represents the intersection between the above mentioned   1 n  -hyperboloid and n-plane, projected on the principal plane, 0 x  , as: where, with regard to the middle side, the single sum is made of n square terms and the double sum of the domain of the distribution,  d f   , depending on the arithmetic mean standard deviation,  , via the er- . The related expression, Equation (31), is a -quadric where the coefficients of the firstdegree terms are null and the symmetry axis coincides with the n-sector, tp g , defined by Equation ( 27).
The canonical form of the above mentioned   1 n quadric can be attained changing the reference frame from   via rigid rotation around the origin, where the resulting coordinate axes, 1 2 n , , , X X  X , coincide with the principal axes of the n-volume bounded by the   1 n  -quadric and, without loss of generality, 1 X may be chosen as symmetry axis.To this aim, the direction cosines must be determined where, in general, k   is the cosine of the angle formed by the resulting coordinate axis, X  , and the starting coordinate axis, k x , 1 The extension of standard relations involving direction cosines to n-spaces yields: ; and the condition of parallelism and orthogonality between coordinate axes read: with regard to the starting reference frame,   , and: with regard to the resulting reference frame, The validity of Equations ( 34)-(37) implies the orthogonality of the Jacobian determinant: where the positive value relates to a rigid rotation of the starting reference frame around the coordinate axis, x , i.e. within the principal n-plane, 0 x  , while the negative value relates, in addition, to an odd number of rigid rotations by an angle,  , each around a different coordinate axis, i x , , i.e. outside the principal n-plane, Owing to Equation (27), the symmetry axis, 1 X , coincides with the n-sector of the first and 2 n th 2 n -ant of the starting reference frame.Accordingly, related direction cosines are equal and can be inferred from Equation (76) particularizing the straight lines, and r r  , to the n-sector, tp g , and the coordinate axis, i x , 1 i n   , respectively, which implies 1; 0; 1; 0, 1 ; ; and Equation (76) reduces to: which, in turn, implies: with regard to the direction cosines involving the coordinate axis, 1 .The remaining coordinate axes, , can be arbitrarily selected, according to Equations ( 34) and ( 35), in that they are related to the  principal axes of the  -circle, centered on the origin and normal to the coordinate axis, 1  2 n  X .For this reason, the starting and the resulting reference frame are not needed to be congruent provided the Jacobian determinant is orthogonal according to Equation (38).
Following the above mentioned procedure with regard to the resulting reference frame,   , Equation (32) takes the explicit expression: where the power,   1 1 n  , ensures congruence (not needed, as mentioned above) between the starting and the resulting reference frame.
The substitution of Equations ( 40) and ( 42) into (38), after some determinant algebra, yields the explicit expression of the Jacobian determinant, as: Open Access AM R. CAIMMI 6 which, after additional determinant algebra, takes the expression: in agreement with Equation (38).
Particularizing Equation ( 41) to , respectively, and performing the sum on the left and righthand side, after some algebra yields: on the other hand, the invariance of the norm by changing the reference frame implies the following: and the substitution of Equations ( 45) and ( 46) into (30) yields: which, using Equation (18), after some algebra produces: that is the locus of -circles normal to the coordinate axis,  2 n   1 X , centered therein, where the radius is: in other terms, Equation (48) defines a n-cylinder where the symmetry axis coincides with the coordinate axis, 1 X , and the radius equals R .In summary, the arithmetic mean standard deviation distribution can be expressed as a function of the errors, 1 2 , , , n x x  x , as: where n C is a normalization constant, n D the integration domain, expressed by Equation (31) which, turned into canonical form, Equation (48), represents a n-cylinder of infinite height, symmetry axis coinciding with the coordinate axis, 1 X , and radius defined by Equation (49). Finally, , are error distributions   (1) as: where m  is the related rms error.The su algebra yields:

Expression of
the special case, 2 n  , is shown in Figure 1. the re With regard to sulting reference frame, , after a change of variables (e ; [8], Chap.4) by use of Equation (46), Equation (52) translates into: .g., [9], Chap.III, § 4.10 where the integration domain, n  , is an infinitely thin try n-cylindrical corona with symme axis, 1 X , and radius defined by Equation (49).
Then the substitution of Equation (38) into (5 and (48) 3) yields: where s the is the Euler Gamma function, which satisfie following relations (e.g., [11], Chap.39, § 39.6): which is independent of the reference frame.In summary, the arithmetic mean standard deviation distribution,  d f   , may be expressed as in a multiple tegral where the integration domain, n  , is an infinitely thin n cy indrical corona where the symmetry axis coincides with the coordinate axis, -l 1 X , and the radius is defined by Equation (49).The result, expressed by Equation (54), after some algebra takes th form: e where the integration domain of the ordinary and the multiple integral are the symmetry axis and the     .

Conclusions
The arithmetic mea related parameters procedure where the geometrical framework is clearly shown using typical formulation generalized to n-spaces.The integration has been performed after a change of reference frame, where the integration domain turns out to be an infinitely thin n-cylindrical corona that is symmetric with respect to a coordinate axis.
Although alternative approaches grounded on mathematical analysis and statistics could appear shorter and less coumbersome, still the geometrical features are lost.A geometrical interpretation is essential in modern physical, cosmological and elementary particle theories, such as general relativity, superstring theory and supersymmetric theory.In this view, an investigation of the geometrical framework related to any field e.g., mechanics, statistics, crystallography, music, could be of some utility.

Acknowledgements
A more extended version of in the last edition (in Italian,   (67) by use of Equations ( 65) and (66).

f
  , as a function of the errors, the straight p  1  -line, intersection between the n-plane, , defined by Equation (16), and the principal n-plane, p 0 x  .The expression of can be obtained by erasing the additional coordinate, 0 p , be the angle formed by the straight line, tp g , and the straight -line, 0 .The particularization of Equation (73) to the case under discussion ( the arithmetic mean standard deviation, p  , can be expressed as a function of the errors, 1 2 a fixed value of the arithmetic mean standard deviation,  , lie on a one-sheet   1 n  -hyperboloid, defined by Equation (17), and 2) the points,   1 2

Figure 1 .D 2 ,
Figure 1.The integration domain, n D , with regard to the reference frame,   n 1 2 x x x O  , in the special case, n 2  .

Figure 2 .
Figure 2. The integration domain, n  , with regard to the reference frame,   n X X X O  1 2 56e)  and the particularization of Equation (55) to the sim casespoints, segments, circles, sph pectively.eres, res-In particular, 1 n  implies a single deviation from the mean, 1 0   according to Equation (4), then 1 0  R   via E ions (11) and (49).Accordingly, the 0-circle coin es with the origin of the reference frame, 0-surface of which is clearly null.For this reason, the undetermined expression, a aring in Equation (55), may safely be put equal to 0, hence ppe 0 0 S  , in agreement w tion (57).On the other hand, Equations (17), (19), (20), lose their validity surface of an infinitely thin   1 n circular corona can be determined by differentiating both sid f E uatio es o n (55).The result is: q re erstimated according toIn summary, the arithmetic mean standard deviation distribution is explicitly expressed by Equation (60) and lated expectation values, expressed by Equations (65)-(67), respectively.

Finally
distribution and have been determined following a the current attempt appears unpublished) of the quoted FERENCES [1] C. W. Misner, J. A. Wheeler and K. S. Thorne, "Gravita-ation (70), where the value is understimated.
the coordinate axis, x .The condition of parallelism between the straight line, , and the n-plane, , reads: by the author., be the angle formed by the straight line and the n-plane.Related trigonometric functions can be inferred from the explicit expression of the sine, as: