Modelling Geometric Measure of Variation About the Population Mean

Troon John Benedict; Karanjah Anthony; Alilah Anekeya David

doi:doi:10.11648/j.ajtas.20190805.13

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Modelling Geometric Measure of Variation About the Population Mean

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David

Published in American Journal of Theoretical and Applied Statistics (Volume 8, Issue 5)

Received: 17 September 2019 Accepted: 28 September 2019 Published: 12 October 2019

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Abstract

Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measures have allowed researchers to define the distribution of various datasets especially the measures of dispersion from the mean. Researchers and mathematicians have been able to develop measures of dispersion from the mean such as mean deviation, variance and standard deviation. However, these measures have been determined not to be perfect, for example, variance give average of squared deviation which differ in unit of measurement as the initial dataset, mean deviation gives bigger average deviation than the actual average deviation because it violates the algebraic laws governing absolute numbers, while standard deviation is affected by outliers and skewed datasets. As a result, there was a need to develop a more efficient measure of variation from the mean that would overcome these weaknesses. The aim of this paper was to model a geometric measure of variation about the population mean which could overcome the weaknesses of the existing measures of variation about the population mean. The study was able to formulate the geometric measure of variation about the population mean that obeyed the algebraic laws behind absolute numbers, which was capable of further algebraic manipulations as it could be used further to estimate the average variation about the mean for weighted datasets, probability mass functions and probability density functions. Lastly, the measure was not affected by outliers and skewed datasets. This shows that the formulated measure was capable of solving the weaknesses of the existing measures of variation about the mean.

Published in	American Journal of Theoretical and Applied Statistics (Volume 8, Issue 5)
DOI	10.11648/j.ajtas.20190805.13
Page(s)	179-184
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Standard Deviation, Geometric Measure of Variation, Deviation About the Mean, Average, Mean, Absolute Deviation

References

[1]	Ahn, S., & Fessler, J. A., (2003). Standard Errors of Mean, Variance, and Standard Deviation Estimators. EECS Department. The University of Michigan. U.S.A.
[2]	Altman, D. G., & Bland, J. M. (2005). Standard deviations and standard errors. BMJ Volume 331.
[3]	Bhardwaj, A., (2013). Comparative Study of Various Measures of Dispersion. Journal of Advances in Mathematics. Vol 1, No 1.
[4]	Buckland, S. T., A. C. Studeny, A. E. Magurran, J. B. Illian, and S. E. Newson. (2011). The geometric mean of relative abundance indices: a biodiversity measure with a difference. Ecosphere 2 (9): 100. doi: 10.1890/ES11-00186.1.
[5]	Clark, P. L., (2012). Number Theory: A Contemporary Introduction. Available at http://math.uga.edu/~pete/4400FULL.pdf
[6]	Deshpande, S., Gogtay, N. J., Thatte, U. M., (2016). Measures of Central Tendency and Dispersion. Journal of the Association of Physicians of India. Vol. 64. July 2016.
[7]	Grechuk, B., Molyboha, A., & Zabarankin M., (2011). Mean-Deviation Analysis in The Theory of Choice. Risk Analysis.
[8]	Hu, S., (2010). Simple Mean, Weighted Mean, or Geometric Mean? Presented at the 2010 ISPA/SCEA Joint Annual Conference and Training Workshop.
[9]	Kum, S., & Lim, Y., (2012). A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions. Hindawi Publishing Corporation. Volume 2012, Article ID 836804.
[10]	Lawson, J. D., & Lim, Y., (2001). The Geometric Mean, Matrices, Metrics, and More. The American Mathematical Monthly.
[11]	Lee, D., In, J., & Lee, S., (2015). Standard deviation and standard error of the mean. Korean journal of anesthesiology. 68. 220-3. 10.4097/kjae.2015.68.3.220.
[12]	Leys, C., Klein, O., Bernard, P., & Licata, L., (2013). Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median. Journal of Experimental Social Psychology 49 (2013) 764–766.
[13]	Manikandan, S., (2016). Measures of dispersion. Journal of Pharmacology and Pharmacotherapeutics. October-December 2011. Vol 2. Issue 4.
[14]	McAlister, D., (1879). The Law of Geometric Mean. The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access to Proceedings of the Royal Society of London.
[15]	Mindlin, D., (2011). On the Relationship between Arithmetic and Geometric Returns. Cdi Advisors Research. LLC.
[16]	Mohini, P. B., &Prajakt, J. B., (2012). What to use to express the variability of data: Standard deviation or standard error of mean?. Perspectives in clinical research. July 2012.
[17]	Raymondo, J., (2015). Measures of Variation from Statistical Analysis in the Behavioral Sciences. Kendall Hunt Publishing.
[18]	Roberson, Q. M., Sturman, M. C., & Simons, T. L., (2007). Does the Measure of Dispersion Matter in Multilevel Research? A Comparison of the Relative Performance of Dispersion Indexes. Cornell University School of Hotel Administration. The Scholarly Commons.
[19]	Roenfeldt, K., (2018). Better than average: Calculating Geometric Means Using SAS. Henry. M. Foundation for the Advancement of Military Medicine.
[20]	Schuetter, J. (2007). Chapter 1. In J. Schuetter, measures of dispersation (pp. 45-54).
[21]	Thenwall, M. (2018). The precision of the arithmetic mean, geometric mean and percentiles for citation data: An experimental simulation modelling approach. Statistical Cybermetrics Research Group, School of Mathematics and Computer Science, University of Wolverhampton, Wulfruna Street, Wolverhampton, UK.

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APA Style

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David. (2019). Modelling Geometric Measure of Variation About the Population Mean. American Journal of Theoretical and Applied Statistics, 8(5), 179-184. https://doi.org/10.11648/j.ajtas.20190805.13

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ACS Style

Troon John Benedict; Karanjah Anthony; Alilah Anekeya David. Modelling Geometric Measure of Variation About the Population Mean. Am. J. Theor. Appl. Stat. 2019, 8(5), 179-184. doi: 10.11648/j.ajtas.20190805.13

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AMA Style

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David. Modelling Geometric Measure of Variation About the Population Mean. Am J Theor Appl Stat. 2019;8(5):179-184. doi: 10.11648/j.ajtas.20190805.13

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@article{10.11648/j.ajtas.20190805.13,
  author = {Troon John Benedict and Karanjah Anthony and Alilah Anekeya David},
  title = {Modelling Geometric Measure of Variation About the Population Mean},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {8},
  number = {5},
  pages = {179-184},
  doi = {10.11648/j.ajtas.20190805.13},
  url = {https://doi.org/10.11648/j.ajtas.20190805.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20190805.13},
  abstract = {Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measures have allowed researchers to define the distribution of various datasets especially the measures of dispersion from the mean. Researchers and mathematicians have been able to develop measures of dispersion from the mean such as mean deviation, variance and standard deviation. However, these measures have been determined not to be perfect, for example, variance give average of squared deviation which differ in unit of measurement as the initial dataset, mean deviation gives bigger average deviation than the actual average deviation because it violates the algebraic laws governing absolute numbers, while standard deviation is affected by outliers and skewed datasets. As a result, there was a need to develop a more efficient measure of variation from the mean that would overcome these weaknesses. The aim of this paper was to model a geometric measure of variation about the population mean which could overcome the weaknesses of the existing measures of variation about the population mean. The study was able to formulate the geometric measure of variation about the population mean that obeyed the algebraic laws behind absolute numbers, which was capable of further algebraic manipulations as it could be used further to estimate the average variation about the mean for weighted datasets, probability mass functions and probability density functions. Lastly, the measure was not affected by outliers and skewed datasets. This shows that the formulated measure was capable of solving the weaknesses of the existing measures of variation about the mean.},
 year = {2019}
}

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TY  - JOUR
T1  - Modelling Geometric Measure of Variation About the Population Mean
AU  - Troon John Benedict
AU  - Karanjah Anthony
AU  - Alilah Anekeya David
Y1  - 2019/10/12
PY  - 2019
N1  - https://doi.org/10.11648/j.ajtas.20190805.13
DO  - 10.11648/j.ajtas.20190805.13
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 179
EP  - 184
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20190805.13
AB  - Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measures have allowed researchers to define the distribution of various datasets especially the measures of dispersion from the mean. Researchers and mathematicians have been able to develop measures of dispersion from the mean such as mean deviation, variance and standard deviation. However, these measures have been determined not to be perfect, for example, variance give average of squared deviation which differ in unit of measurement as the initial dataset, mean deviation gives bigger average deviation than the actual average deviation because it violates the algebraic laws governing absolute numbers, while standard deviation is affected by outliers and skewed datasets. As a result, there was a need to develop a more efficient measure of variation from the mean that would overcome these weaknesses. The aim of this paper was to model a geometric measure of variation about the population mean which could overcome the weaknesses of the existing measures of variation about the population mean. The study was able to formulate the geometric measure of variation about the population mean that obeyed the algebraic laws behind absolute numbers, which was capable of further algebraic manipulations as it could be used further to estimate the average variation about the mean for weighted datasets, probability mass functions and probability density functions. Lastly, the measure was not affected by outliers and skewed datasets. This shows that the formulated measure was capable of solving the weaknesses of the existing measures of variation about the mean.
VL  - 8
IS  - 5
ER  -

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Author Information

Troon John Benedict

Department of Mathematics and Physical Science, Maasai Mara University, Narok, Kenya
Karanjah Anthony

Department of Mathematic, Multimedia University, Nairobi, Kenya
Alilah Anekeya David

Department of Mathematics and Statistics, Masinde Muliro University of Science and Technology, Kakamega, Kenya

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Plain Text BibTeX RIS

APA Style

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David. (2019). Modelling Geometric Measure of Variation About the Population Mean. American Journal of Theoretical and Applied Statistics, 8(5), 179-184. https://doi.org/10.11648/j.ajtas.20190805.13

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ACS Style

Troon John Benedict; Karanjah Anthony; Alilah Anekeya David. Modelling Geometric Measure of Variation About the Population Mean. Am. J. Theor. Appl. Stat. 2019, 8(5), 179-184. doi: 10.11648/j.ajtas.20190805.13

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AMA Style

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David. Modelling Geometric Measure of Variation About the Population Mean. Am J Theor Appl Stat. 2019;8(5):179-184. doi: 10.11648/j.ajtas.20190805.13

Copy | Download

@article{10.11648/j.ajtas.20190805.13,
  author = {Troon John Benedict and Karanjah Anthony and Alilah Anekeya David},
  title = {Modelling Geometric Measure of Variation About the Population Mean},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {8},
  number = {5},
  pages = {179-184},
  doi = {10.11648/j.ajtas.20190805.13},
  url = {https://doi.org/10.11648/j.ajtas.20190805.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20190805.13},
  abstract = {Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measures have allowed researchers to define the distribution of various datasets especially the measures of dispersion from the mean. Researchers and mathematicians have been able to develop measures of dispersion from the mean such as mean deviation, variance and standard deviation. However, these measures have been determined not to be perfect, for example, variance give average of squared deviation which differ in unit of measurement as the initial dataset, mean deviation gives bigger average deviation than the actual average deviation because it violates the algebraic laws governing absolute numbers, while standard deviation is affected by outliers and skewed datasets. As a result, there was a need to develop a more efficient measure of variation from the mean that would overcome these weaknesses. The aim of this paper was to model a geometric measure of variation about the population mean which could overcome the weaknesses of the existing measures of variation about the population mean. The study was able to formulate the geometric measure of variation about the population mean that obeyed the algebraic laws behind absolute numbers, which was capable of further algebraic manipulations as it could be used further to estimate the average variation about the mean for weighted datasets, probability mass functions and probability density functions. Lastly, the measure was not affected by outliers and skewed datasets. This shows that the formulated measure was capable of solving the weaknesses of the existing measures of variation about the mean.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Modelling Geometric Measure of Variation About the Population Mean
AU  - Troon John Benedict
AU  - Karanjah Anthony
AU  - Alilah Anekeya David
Y1  - 2019/10/12
PY  - 2019
N1  - https://doi.org/10.11648/j.ajtas.20190805.13
DO  - 10.11648/j.ajtas.20190805.13
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 179
EP  - 184
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20190805.13
AB  - Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measures have allowed researchers to define the distribution of various datasets especially the measures of dispersion from the mean. Researchers and mathematicians have been able to develop measures of dispersion from the mean such as mean deviation, variance and standard deviation. However, these measures have been determined not to be perfect, for example, variance give average of squared deviation which differ in unit of measurement as the initial dataset, mean deviation gives bigger average deviation than the actual average deviation because it violates the algebraic laws governing absolute numbers, while standard deviation is affected by outliers and skewed datasets. As a result, there was a need to develop a more efficient measure of variation from the mean that would overcome these weaknesses. The aim of this paper was to model a geometric measure of variation about the population mean which could overcome the weaknesses of the existing measures of variation about the population mean. The study was able to formulate the geometric measure of variation about the population mean that obeyed the algebraic laws behind absolute numbers, which was capable of further algebraic manipulations as it could be used further to estimate the average variation about the mean for weighted datasets, probability mass functions and probability density functions. Lastly, the measure was not affected by outliers and skewed datasets. This shows that the formulated measure was capable of solving the weaknesses of the existing measures of variation about the mean.
VL  - 8
IS  - 5
ER  -

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