11/5/2022 0 Comments R weighted standard deviation![]() Owing to a theorem of statistics (e.g.,, Chapter 8, Chapter 2), the explicit expression reads: Let ( ) df m m be the distribution related to the whole set of the above mentioned measure methods and statistical system, where the occurrence of the event, E, has been designed by the value of the random variable, m, which is a linear combination of the above mentioned random variables, for sake of simplicity denoted hereafter as im, via the coefficients, i Expected values and rms error estimators are known to be the arithmetic mean and the standard deviation, respectively, and the results of an earlier study well apply to each method, j, taken separately. , are the expected value, the variance, the rms error, respectively, of the distribution. Where jkm is a generic measure related to the method, j, and ( )jm The special case of Gaussian distributions, which well holds for independent measures, reads: Let ( )dj jk jkf m m be distributions related to assigned measure methods, j, 1 j n, and a specified statistical system, where the occurrence of the events, jkE, has been designed by the value of the random var- iables, jkm, 1 jk n. Further details are shown in the Appendix, including extension of analytic geometry formulation to ( )1n + - spaces, Jacobi formulae, reduction of the results to ordinary geometry, and a corrigendum to the parent paper. ![]() 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. In addition, three different attempts are exploited in Section 4. The first, second, third step are exploited in Sections 3, 4, 5, respectively. The problem is outlined in Section 2 together with three steps towards the solution. The extension of the procedure followed in the parent paper yields results which reduce to their counter- parts therein, in the special case where the weighted mean reduces to the arithmetic mean. Accordingly, the weighted mean standard deviation depends only on the uncertainties on the input data, which is not in con- tradiction with its counterpart inferred from a different approach based on the least-squares method. Mean, or in other words searching the explicit expression of the weighted mean standard deviation distribution, under the safely motivated restriction of independent measures obeying Gaussian distributions. The current attempt is aimed at extenting the above mentioned results to the general case of the Caimmi The geometrical framework of a well known statistical problem, concerning the explicit expression of the arithmetic mean standard deviation distribution, has been considered in an earlier investigation. Though further insight could be gained exploiting a geometrical framework, still pure mathematical analysis is preferred owing to a much greater difficulty in handling with hyperspace geometry. Introduction Hyperspace geometry is useful, if not essential, for the interpretation of theories involving many branches of knowledge and, in particular, general relativity (e.g., ) and superstring theory (e.g., ). Keywords Standard Deviation, n -Spaces, Direction Cosines, Quadratic Forms, Matrix Theoryġ. The reduction of some results to ordinary geometry is also considered. The distribution and related pa-rameters have the same formal expression with respect to their counterparts in the special case where the weighted mean coincides with the arithmetic mean. The semiaxes are formulated in two different ways, namely in terms of (1) eigenvalues, via the eigenvalue equation, and (2) leading principal minors of the matrix of a quadratic form, via the Jacobi formulae. To this respect, the integration domain is expressed in canonical form after a change of reference frame in the n -space, which is recognized as an infinitely thin n -cy- lindrical corona where the axis coincides with a coordinate axis and the orthogonal section is an infinitely thin, homotetic ( )1n -elliptical corona. Ībstract The current attempt is aimed to extend previous results, concerning the explicit expression of the arithmetic mean standard deviation distribution, to the general case of the weighted mean stan-dard deviation distribution. ![]() R weighted standard deviation license#This work is licensed under the Creative Commons Attribution International License (CC BY). Caimmi Physics and Astronomy Department, Padua University, Padova, Italy Email: Received 23 February 2015 accepted 11 March 2015 published 12 March 2015Ĭopyright 2015 by author and Scientific Research Publishing Inc. The Weighted Mean Standard Deviation Distribution: A Geometrical Framework R. (2015) The Weighted Mean Standard Deviation Distribution: A Geometrical Framework. Applied Mathematics, 2015, 6, 520-546 Published Online March 2015 in SciRes. ![]()
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