Modelling Geometric Measure of Variation About the Population Mean

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.

Determining the Unbiased Estimator of the Population Geometric Measures of Variation About the Mean

“…As newly formulated measures of variation about the mean, the unbiased sample estimator of the population parameter still unknown. As a result, the aim of this study was to determine the unbiased estimator of the population geometric measure of variation, which will assist in the precise estimation of the measure for various samples [15,16].…”

“…Therefore, the population deviation vector will be given by   Hence, by definition, the geometric measure of variation about the mean G for the population will be given by [15,16] 1 0 00…”

“…Based on the above sample, we can estimate the population geometric measure of variation using two methods. First, we can average the sample average variation about the mean using geometric average as follows [15,16].…”

“…Secondly, we can estimate the population geometric measure of variation about the mean by allowing a 1 degree of freedom in the sample estimation process, this will result into a new measure of variation about the mean q g which is given by [15,16]…”

“…Geometric Measures of Variation about the mean is a measure that uses the geometric averaging technique to average the deviations from the mean [15,16]. Unlike other measures of variation about the mean such as mean deviation, variance, and standard deviation [1,2,5].…”