Four Formulations of Average Derived from Pythagorean Means

Dhritikesh Chakrabarty

doi:https://doi.org/10.14445/22315373/IJMTT-V67I6P512

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 67 | Issue 6 | Year 2021 | Article Id. IJMTT-V67I6P512 | DOI : https://doi.org/10.14445/22315373/IJMTT-V67I6P512

Four Formulations of Average Derived from Pythagorean Means

Dhritikesh Chakrabarty

Citation :

Dhritikesh Chakrabarty, "Four Formulations of Average Derived from Pythagorean Means," International Journal of Mathematics Trends and Technology (IJMTT), vol. 67, no. 6, pp. 97-118, 2021. Crossref, https://doi.org/10.14445/22315373/IJMTT-V67I6P512

Abstract

Four definitions / formulations of average termed here as Arithmetic-Geometric Mean (abbreviated as AGM), ArithmeticHarmonic Mean (abbreviated as AHM), Geometric-Harmonic Mean (abbreviated as GHM) and Arithmetic-GeometricHarmonic (abbreviated as Mean AGHM) respectively have been derived / developed from the three Pythagorean means namely arithmetic mean (abbreviated as Mean AM), geometric mean (abbreviated as Mean GM) and harmonic mean (abbreviated as Mean HM) with an objective of developing of more accurate measures of central tendency of data. The derivations of these four formulations of average, with numerical examples, have been presented in this paper.

Keywords

ABC4(G), GA5(G), F-polynomial, F-index, Zagreb index.

References

[1] Bakker Arthur., The early history of average values and implications for education, Journal of Statistics Education, 11(1)(2003) 17 ‒ 26.
[2] Christoph Riedweg ., Pythagoras: his life, teaching, and influence (translated by Steven Rendall in collaboration with Christoph Riedweg and Andreas Schatzmann, Ithaca), ISBN 0-8014-4240-0, Cornell University Press., (2005).
[3] David A. Cox., The Arithmetic-Geometric Mean of Gauss”, In J.L. Berggren; Jonathan M. Borwein; Peter Borwein (eds.). Pi: A Source Book. Springer. 481. ISBN 978-0-387-20571-7 , (first published in L'Enseignement Mathématique, t. 30(1984)(2004) 275 – 330).
[4] David W. Cantrell (……): “Pythagorean Means”, Math World.
[5] Dhritikesh Chakrabarty., Determination of Parameter from Observations Composed of Itself and Errors, International Journal of Engineering Science and Innovative Technology, 3(2)(2014) 304 – 311. Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats .
[6] Dhritikesh Chakrabarty., Analysis of Errors Associated to Observations of Measurement Type, International Journal of Electronics and Applied Research, 1(1)(2014)15 – 28. Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats .
[7] Dhritikesh Chakrabarty ., Observation Composed of a Parameter and Random Error: An Analytical Method of Determining the Parameter”, International Journal of Electronics and Applied Research, 1(2)(2014) 20 – 38. Available at http://eses.net.in/online_journal.html .
[8] Dhritikesh Chakrabarty., Observation Consisting of Parameter and Error: Determination of Parameter, Proceedings of the World Congress on Engineering, 2015, ISBN: 978-988-14047-0-1, ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) (2015). Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats .
[9] Dhritikesh Chakrabarty., Observation Consisting of Parameter and Error: Determination of Parameter," Lecture Notes in Engineering and Computer Science (ISBN: 978-988-14047-0-1), London, 2015, (2015) 680 ‒ 684. Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats.
[10] Dhritikesh Chakrabarty., Central Tendency of Annual Extremum of Surface Air Temperature at Guwahati, J. Chem. Bio. Phy. Sci. (E- ISSN : 2249 – 1929), Sec. C, 5(3) (2015) 2863 – 2877. Available at: www.jcbsc.org.
[11] Dhritikesh Chakrabarty., Central Tendency of Annual Extremum of Surface Air Temperature at Guwahati Based on Midrange and Median, J. Chem. Bio. Phy. Sci. (E- ISSN : 2249 –1929), Sec. D, 5(3)(2015) 3193 – 3204. Available at: www.jcbsc.org.
[12] Dhritikesh Chakrabarty., Observation Composed of a Parameter and Random Error: Determining the Parameter as Stable Mid Range, International Journal of Electronics and Applied Research (ISSN : 2395 – 0064), 2(1)(2015) 35 – 47. Available at http://eses.net.in/online_journal.html .
[13] Dhritikesh Chakrabarty., A Method of Finding Appropriate value of Parameter from Observation Containing Itself and Random Error”, Indian Journal of Scientific Research and Technology, (E-ISSN: 2321-9262), 3(4)(2015) 14 – 21. Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats
[14] Dhritikesh Chakrabarty., Theoretical Model Modified For Observed Data: Error Estimation Associated To Parameter, International Journal of Electronics and Applied Research (ISSN : 2395 – 0064), 2(2)(2015) 29 – 45. Available at http://eses.net.in/online_journal.html .
[15] Dhritikesh Chakrabarty., Impact of Error Contained in Observed Data on Theoretical Model: Study of Some Important Situations, International Journal of Advanced Research in Science, Engineering and Technology, (ISSN : 2350 – 0328), 3(1)(2016) 1255 – 1265. Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats .
[16] Dhritikesh Chakrabarty ., Pythagorean Mean: Concept behind the Averages and Lot of Measures of Characteristics of Data, NaSAEAST- 2016, Abstract ID: CMAST_NaSAEAST (Inv)-1601)., (2016) Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats .
[17] Dhritikesh Chakrabarty., Objectives and Philosophy behind the Construction of Different Types of Measures of Average, NaSAEAST- 2017, Abstract ID: (2017) CMAST_NaSAEAST (Inv)- 1701), Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats .
[18] Dhritikesh Chakrabarty., Theoretical Model and Model Satisfied by Observed Data: One Pair of Related Variables”, International Journal of Advanced Research in Science, Engineering and Technology, 3(2)(2017) 1527 – 1534, Available at www.ijarset.com .
[19] Dhritikesh Chakrabarty., Variable(s) Connected by Theoretical Model and Model for Respective Observed Data, FSDM2017, Abstract ID: FSDM2220, (2017). Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats.
[20] Dhritikesh Chakrabarty, Numerical Data Containing One Parameter and Random Error: Evaluation of the Parameter by Convergence of Statistic, International Journal of Electronics and Applied Research, 4(2)(2017) 59 – 73. Available at http://eses.net.in/online_journal.html .
[21] Dhritikesh Chakrabarty., Observed Data Containing One Parameter and Random Error: Evaluation of the Parameter Applying Pythagorean Mean, International Journal of Electronics and Applied Research, 5(1)(2018) 32 – 45. Available at http://eses.net.in/online_journal.html .
[22] Dhritikesh Chakrabarty ., Derivation of Some Formulations of Average from One Technique of Construction of Mean, American Journal of Mathematical and Computational Sciences, 3(3)(2018) 62 – 68. Available at http://www.aascit.org/journal/ajmcs.
[23] Dhritikesh Chakrabarty., One Generalized Definition of Average: Derivation of Formulations of Various Means, Journal of Environmental Science, Computer Science and Engineering & Technology, Section C, (E-ISSN: 2278 – 179 X), 7(3)(2018) 212 – 225. Available at www.jecet.org.
[24] Dhritikesh Chakrabarty., fH -Mean: One Generalized Definition of Average”, Journal of Environmental Science, Computer Science and Engineering & Technology, Section C, 7(4)(2018) 301 – 314. Available at www.jecet.org .
[25] Dhritikesh Chakrabarty., Generalized fG - Mean: Derivation of Various Formulations of Average, American Journal of Computation, Communication and Control, 5(3) (2018) 101 – 108. Available at http://www.aascit.org/journal/ajmcs .
[26] Dhritikesh Chakrabarty., General Technique of Defining Average, NaSAEAST- 2018, Abstract ID: CMAST_NaSAEAST -1801 (I)(2018), Available at https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats .
[27] Dhritikesh Chakrabarty., Observed Data Containing One Parameter and Random Error: Probabilistic Evaluation of Parameter by Pythagorean Mean, International Journal of Electronics and Applied Research, 6(1)(2019) 24 – 40. Available at http://eses.net.in/online_journal.html .
[28] Dhritikesh Chakrabarty., One Definition of Generalized fG - Mean: Derivation of Various Formulations of Average, Journal of Environmental Science, Computer Science and Engineering & Technology, Section C, (E- ISSN : 2278 – 179 X), 8(2)(2019) 051 – 066. Available at www.jecet.org.
[29] Dhritikesh Chakrabarty ., One General Method of Defining Average: Derivation of Definitions/Formulations of Various Means, Journal of Environmental Science, Computer Science and Engineering & Technology, Section C, (E-ISSN : 2278 – 179 X), 8(4)(2019) 327 – 338. Available at www.jecet.org .
[30] Dhritikesh Chakrabarty., A General Method of Defining Average of Function of a Set of Values, Aryabhatta Journal of Mathematics & Informatics {ISSN (Print) : 0975-7139, ISSN (Online) : 2394-9309}, 11(2)(2019) 269 – 284. Available at www.abjni.com .
[31] Dhritikesh Chakrabarty., Pythagorean Geometric Mean: Measure of Relative Change in a Group of Variables, NaSAEAST- 2019, Abstract ID: CMAST_NaSAEAST -1902 (I). Available at (2019) , https://www.researchgate.net/profile/Dhritikesh_Chakrabarty/stats .
[32] Dhritikesh Chakrabarty ., Definition / Formulation of Average from First Principle, Journal of Environmental Science, Computer Science and Engineering & Technology, Section C, (E-ISSN : 2278 – 179 X), 9(2)(2020) 151 – 163. Available at www.jecet.org .
[33] Hazewinkel, Michiel ed., Arithmetic–geometric mean process, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 ., (2001).
[34] Foster D. M. E. and Phillips G. M., The Arithmetic-Harmonic Mean, Journal of American Mathematical Society, 42(165)183-191.
[35] Kolmogorov Andrey., On the Notion of Mean, in “Mathematics and Mechanics., (Kluwer 1991), (1930)144 – 146.
[36] Kolmogorov Andrey., Grundbegriffe der Wahrscheinlichkeitsrechnung (in German), Berlin: Julius Springer., (1933).
[37] Miguel de Carvalho., Mean, what do you Mean?”, The American Statistician, 70(2016) 764 ‒ 776.
[38] Plackett, R. L., Studies in the History of Probability and Statistics: VII. The Principle of the Arithmetic Mean., Biometrika. 45 (1/2)(1958) 130–135. doi:10.2307/2333051. JSTOR 2333051.
[39] Youschkevitch A. P., A. N. Kolmogorov: Historian and philosopher of mathematics on the occasion of his 80th birfhday, Historia Mathematica, 10(4)(1983) 383 – 395.
[40] Weir, Alan J.,The Convergence Theorems, Lebesgue Integration and Measure, Cambridge: Cambridge University Press. 93–118. ISBN 0-521-08728-7.
[41] Weisstein, Eric W. ( )., Harmonic-Geometric Mean., From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Harmonic-GeometricMean.html.
[42] Weisberg H. F., Central Tendency and Variability, Sage University Paper Series on Quantitative Applications in the Social Sciences, ISBN 0-8039-4007-6 (1992) 2.
[43] Williams R. B. G., Measures of Central Tendency, Introduction to Statistics for Geographers and Earth Scientist, Softcover ISBN978-0-333-35275-5, eBook ISBN978-1-349-06815-9 , Palgrave, London, (1984) 51 – 60.