Estimating Average Variation About the Population Mean Using Geometric Measure of Variation

Troon John Benedict; Karanjah Anthony; Alilah Anekeya David

doi:doi:10.11648/j.ijsd.20200602.11

| Peer-Reviewed |

Estimating Average Variation About the Population Mean Using Geometric Measure of Variation

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David

Received: 17 September 2019 Accepted: 16 October 2019 Published: 25 August 2020

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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 the paper was to estimate the average variation about the population mean using geometric measure of variation. The study was able to use the geometric measure of variation to estimate the average variation about the population mean for un-weighted datasets, weighted datasets, probability mass and probability density functions with finite intervals, however, the function faces serious integration problems when estimating the average deviation for probability density functions as a result of complexity in the integrations by parts involved and also integration on infinite intervals. Despite the challenge on probability density functions, the study was able to establish that the geometric measure of variation was able to overcome the challenges faced by the existing measures of variation about the population mean.

DOI	10.11648/j.ijsd.20200602.11
Published in	International Journal of Statistical Distributions and Applications (Volume 6, Issue 2, June 2020)
Page(s)	23-35
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), 2024. Published by Science Publishing Group

Keywords

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

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.

Cite This Article

Plain Text BibTeX RIS

APA Style

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David. (2020). Estimating Average Variation About the Population Mean Using Geometric Measure of Variation. International Journal of Statistical Distributions and Applications, 6(2), 23-35. https://doi.org/10.11648/j.ijsd.20200602.11

Copy | Download

ACS Style

Troon John Benedict; Karanjah Anthony; Alilah Anekeya David. Estimating Average Variation About the Population Mean Using Geometric Measure of Variation. Int. J. Stat. Distrib. Appl. 2020, 6(2), 23-35. doi: 10.11648/j.ijsd.20200602.11

Copy | Download

AMA Style

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David. Estimating Average Variation About the Population Mean Using Geometric Measure of Variation. Int J Stat Distrib Appl. 2020;6(2):23-35. doi: 10.11648/j.ijsd.20200602.11

Copy | Download

@article{10.11648/j.ijsd.20200602.11,
  author = {Troon John Benedict and Karanjah Anthony and Alilah Anekeya David},
  title = {Estimating Average Variation About the Population Mean Using Geometric Measure of Variation},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {6},
  number = {2},
  pages = {23-35},
  doi = {10.11648/j.ijsd.20200602.11},
  url = {https://doi.org/10.11648/j.ijsd.20200602.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20200602.11},
  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 the paper was to estimate the average variation about the population mean using geometric measure of variation. The study was able to use the geometric measure of variation to estimate the average variation about the population mean for un-weighted datasets, weighted datasets, probability mass and probability density functions with finite intervals, however, the function faces serious integration problems when estimating the average deviation for probability density functions as a result of complexity in the integrations by parts involved and also integration on infinite intervals. Despite the challenge on probability density functions, the study was able to establish that the geometric measure of variation was able to overcome the challenges faced by the existing measures of variation about the population mean.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Estimating Average Variation About the Population Mean Using Geometric Measure of Variation
AU  - Troon John Benedict
AU  - Karanjah Anthony
AU  - Alilah Anekeya David
Y1  - 2020/08/25
PY  - 2020
N1  - https://doi.org/10.11648/j.ijsd.20200602.11
DO  - 10.11648/j.ijsd.20200602.11
T2  - International Journal of Statistical Distributions and Applications
JF  - International Journal of Statistical Distributions and Applications
JO  - International Journal of Statistical Distributions and Applications
SP  - 23
EP  - 35
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20200602.11
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 the paper was to estimate the average variation about the population mean using geometric measure of variation. The study was able to use the geometric measure of variation to estimate the average variation about the population mean for un-weighted datasets, weighted datasets, probability mass and probability density functions with finite intervals, however, the function faces serious integration problems when estimating the average deviation for probability density functions as a result of complexity in the integrations by parts involved and also integration on infinite intervals. Despite the challenge on probability density functions, the study was able to establish that the geometric measure of variation was able to overcome the challenges faced by the existing measures of variation about the population mean.
VL  - 6
IS  - 2
ER  -

Copy | Download

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

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David. (2020). Estimating Average Variation About the Population Mean Using Geometric Measure of Variation. International Journal of Statistical Distributions and Applications, 6(2), 23-35. https://doi.org/10.11648/j.ijsd.20200602.11

Copy | Download

ACS Style

Troon John Benedict; Karanjah Anthony; Alilah Anekeya David. Estimating Average Variation About the Population Mean Using Geometric Measure of Variation. Int. J. Stat. Distrib. Appl. 2020, 6(2), 23-35. doi: 10.11648/j.ijsd.20200602.11

Copy | Download

AMA Style

Troon John Benedict, Karanjah Anthony, Alilah Anekeya David. Estimating Average Variation About the Population Mean Using Geometric Measure of Variation. Int J Stat Distrib Appl. 2020;6(2):23-35. doi: 10.11648/j.ijsd.20200602.11

Copy | Download

@article{10.11648/j.ijsd.20200602.11,
  author = {Troon John Benedict and Karanjah Anthony and Alilah Anekeya David},
  title = {Estimating Average Variation About the Population Mean Using Geometric Measure of Variation},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {6},
  number = {2},
  pages = {23-35},
  doi = {10.11648/j.ijsd.20200602.11},
  url = {https://doi.org/10.11648/j.ijsd.20200602.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20200602.11},
  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 the paper was to estimate the average variation about the population mean using geometric measure of variation. The study was able to use the geometric measure of variation to estimate the average variation about the population mean for un-weighted datasets, weighted datasets, probability mass and probability density functions with finite intervals, however, the function faces serious integration problems when estimating the average deviation for probability density functions as a result of complexity in the integrations by parts involved and also integration on infinite intervals. Despite the challenge on probability density functions, the study was able to establish that the geometric measure of variation was able to overcome the challenges faced by the existing measures of variation about the population mean.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Estimating Average Variation About the Population Mean Using Geometric Measure of Variation
AU  - Troon John Benedict
AU  - Karanjah Anthony
AU  - Alilah Anekeya David
Y1  - 2020/08/25
PY  - 2020
N1  - https://doi.org/10.11648/j.ijsd.20200602.11
DO  - 10.11648/j.ijsd.20200602.11
T2  - International Journal of Statistical Distributions and Applications
JF  - International Journal of Statistical Distributions and Applications
JO  - International Journal of Statistical Distributions and Applications
SP  - 23
EP  - 35
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20200602.11
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 the paper was to estimate the average variation about the population mean using geometric measure of variation. The study was able to use the geometric measure of variation to estimate the average variation about the population mean for un-weighted datasets, weighted datasets, probability mass and probability density functions with finite intervals, however, the function faces serious integration problems when estimating the average deviation for probability density functions as a result of complexity in the integrations by parts involved and also integration on infinite intervals. Despite the challenge on probability density functions, the study was able to establish that the geometric measure of variation was able to overcome the challenges faced by the existing measures of variation about the population mean.
VL  - 6
IS  - 2
ER  -

Copy | Download