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

2020 ◽  
Vol 6 (2) ◽  
pp. 23
Author(s):  
Troon John Benedict ◽  
Karanjah Anthony ◽  
Alilah Anekeya David
Keyword(s):  
Population Mean ◽  
Geometric Measure ◽  
Average Variation

scholarly journals Modelling Geometric Measure of Variation About the Population Mean

2021 ◽  
Author(s):  
Benedict Troon
Keyword(s):  
Mean Deviation ◽  
Average Deviation ◽  
Statistical Tool ◽  
Population Mean ◽  
Geometric Measure ◽  
Initial Dataset ◽  
Algebraic Laws ◽  
Average Variation ◽  
Measures Of Dispersion ◽  
The Mean

Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measureshave 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 standarddeviation 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 obeyedthe 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

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

2021 ◽  
Author(s):  
Benedict Troon
Keyword(s):  
Probability Density ◽  
Probability Density Functions ◽  
Mean Deviation ◽  
Density Functions ◽  
Average Deviation ◽  
Population Mean ◽  
Geometric Measure ◽  
Average Variation ◽  
Measures Of Dispersion ◽  
The Mean

Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measureshave 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 giveaverage 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 alsointegration 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.

scholarly journals Modelling Geometric Measure of Variation About the Population Mean

2019 ◽  
Vol 8 (5) ◽  
pp. 179
Author(s):  
Troon John Benedict ◽  
Karanjah Anthony ◽  
Alilah Anekeya David
Keyword(s):  
Population Mean ◽  
Geometric Measure

scholarly journals ESTIMATION OF VARIATION ABOUT THE MEAN USING GEOMETRIC MEASURE

2021 ◽  
Author(s):  
Benedict Troon
Keyword(s):  
Standard Deviation ◽  
Probability Distributions ◽  
Mean Deviation ◽  
Average Deviation ◽  
Statistical Tool ◽  
Geometric Measure ◽  
Average Variation ◽  
Unit Of Measurement ◽  
Probability Mass ◽  
The Mean

A measure of dispersion is a statistical tool used to define the distribution of various datasets mainly from measures of central tendency. Some notable measures of dispersion from the mean are; average deviation, mean deviation, variance, and standard deviation. However, from previousstudies, it has been established that the aforementioned measures are not absolutely perfect in estimating average variation from the mean. For instance, variance gives estimates which are of different units of measurements (squared) from the original dataset’s unit of measurement. In the case of mean deviation, it gives a large average deviation than the actual deviation due to its conformation to the triangular inequality, whereas standard deviation is affected by outliers and skewed datasets. The aim of this study was to estimate variation about the mean using a technique that would overcome the weaknesses of other global measures. The study employed the geometricaveraging technique to average deviation from the mean, which averages absolute products and not sums and it is nonresponsive to outliers and skewed datasets. The study formulated a geometric measure of variation for unweighted and weighted datasets, and probability mass and density functions. Using the formulations, the estimates of the average variation from the mean for thegiven datasets and probability distributions were computed. From the results established that the estimates obtained by the geometric measures were significantly smaller as compared to those obtained by standard deviation. In terms of efficiency, the measure was more efficient compared to standard deviation is estimating average variation about the mean for geometric, skewed and peaked datasets.

scholarly journals DETERMINING THE UNBIASED ESTIMATOR OF THE POPULATION GEOMETRIC MEASURES OF VARIATION ABOUT THE MEAN

2021 ◽  
Author(s):  
Benedict Troon
Keyword(s):  
Random Sampling ◽  
Unbiased Estimator ◽  
Simple Random Sampling ◽  
Degree Of Freedom ◽  
Estimation Technique ◽  
Averaging Technique ◽  
Geometric Measure ◽  
Replacement Technique ◽  
Average Variation ◽  
The Mean

Geometric Measures of Variation about the mean is a measure that uses the geometric averaging technique to average the deviations from the mean. From previous studies, it has been determined that the measure is more precise in estimating the average variation about the mean than the existing measures of variation about the mean. Given that the technique is a newly introduced technique of estimating the average variation about the mean, the actual sample estimator for the measure is still unknown, as a result, the study aimed at determining the unbiased estimator for the population geometric measure. The study used a mathematical estimation technique to determinethe unbiased estimator among the existing possible estimators as it assumed a simple random sampling without replacement technique. The study determined that the unbiased estimator of the population estimator was the sample estimator which did not allow one degree of freedom.

Reversion Toward the Mean Independently of Regression Toward the Mean

Methodology ◽  
2009 ◽  
Vol 5 (1) ◽  
pp. 3-6 ◽  
Author(s):  
Merton S. Krause
Keyword(s):  
Random Error ◽  
Evaluation Studies ◽  
Internal Validity ◽  
Intervention Evaluation ◽  
Population Mean ◽  
Experimental Intervention ◽  
Sample Data ◽  
The Mean ◽  
Regression Toward The Mean

There is another important artifactual contributor to the apparent improvement of persons subjected to an experimental intervention which may be mistaken for regression toward the mean. This is the phenomenon of random error and extreme selection, which does not at all involve the population regression of posttest on pretest scores but involves a quite different and independent reversion of subjects’ scores toward the population mean. These two independent threats to the internal validity of intervention evaluation studies, however, can be detected and differentiated on the sample data of such studies.

Has the Time Come to Employ Population and Individual Bioequivalence for the Evaluation of Generics?

2020 ◽  
Vol 21 (2) ◽  
pp. 112-125
Author(s):  
Francis Micheal ◽  
Mohanlal Sayana ◽  
Rajendra Prasad ◽  
Balamurali Musuvathi Motilal
Keyword(s):  
Therapeutic Index ◽  
Bioequivalence Study ◽  
Open Label ◽  
Statistical Parameters ◽  
Population Mean ◽  
Average Bioequivalence ◽  
Individual Bioequivalence ◽  
Population Bioequivalence ◽  
Enteric Coated ◽  
Bioequivalence Assessment

Background: Bioequivalence studies are a vital part of drug development. The average bioequivalence approach is the standard method of assessment to conclude whether the generic product is bioequivalent to the innovator product. Of late, debates are on whether the average bioequivalence approach adequately addresses drug interchangeability as it considers only population mean for the evaluation especially when highly variable drug products and narrow therapeutic index drugs are dealt with. Hence, the alternative approaches like population bioequivalence and individual bioequivalence assessment approaches emerge as they consider inter/intra-subject variance and subject- by-formulation variance along with population mean. Objectives: The objective of the study was to apply different bioequivalence assessment approaches in a replicate bioequivalence study to evaluate the drug interchangeability. Methods: This was an open-label, single-dose, randomized, balanced, two-treatment, three-period, three-sequence, partial replicate crossover bioequivalence study of omeprazole enteric-coated tablet 20 mg conducted on 48 normal healthy subjects under fed conditions. The plasma concentration of omeprazole was analyzed by a validated bioanalytical method to determine the pharmacokinetic and statistical parameters to assess average bioequivalence, population bioequivalence, and individual bioequivalence. Results: In this study, test formulation was shown to be bio-inequivalent to the reference formulation by average bioequivalence, population bioequivalence, and individual bioequivalence approaches. Conclusion: The outcome of the evaluation clearly states that the bioequivalence outcome of all these approaches are the same. Obviously, it does not mean that these three approaches provide the same outcome though the consideration of variances varies. Certainly, population bioequivalence and individual bioequivalence approach will be more accurate for the assessment of drug interchangeability.

scholarly journals Understanding variability in greenhouse gas emission estimates of smallholder dairy farms in Indonesia

2021 ◽  
Author(s):  
Titis Apdini ◽  
Windi Al Zahra ◽  
Simon J. Oosting ◽  
Imke J. M. de Boer ◽  
Marion de Vries ◽  
...  
Keyword(s):  
Greenhouse Gas ◽  
Rainy Season ◽  
Greenhouse Gas Emission ◽  
Dairy Farms ◽  
Dry Season ◽  
Enteric Fermentation ◽  
Tropical Regions ◽  
Population Mean ◽  
Smallholder Dairy ◽  
Smallholder Dairy Farms

Abstract Purpose Life cycle assessment studies on smallholder farms in tropical regions generally use data that is collected at one moment in time, which could hamper assessment of the exact situation. We assessed seasonal differences in greenhouse gas emissions (GHGEs) from Indonesian dairy farms by means of longitudinal observations and evaluated the implications of number of farm visits on the variance of the estimated GHGE per kg milk (GHGEI) for a single farm, and the population mean. Methods An LCA study was done on 32 smallholder dairy farms in the Lembang district area, West Java, Indonesia. Farm visits (FVs) were performed every 2 months throughout 1 year: FV1–FV3 (rainy season) and FV4–FV6 (dry season). GHGEs were assessed for all processes up to the farm-gate, including upstream processes (production and transportation of feed, fertiliser, fuel and electricity) and on-farm processes (keeping animals, manure management and forage cultivation). We compared means of GHGE per unit of fat-and-protein-corrected milk (FPCM) produced in the rainy and the dry season. We evaluated the implication of number of farm visits on the variance of the estimated GHGEI, and on the variance of GHGE from different processes. Results and discussion GHGEI was higher in the rainy (1.32 kg CO2-eq kg−1 FPCM) than in the dry (0.91 kg CO2-eq kg−1 FPCM) season (P < 0.05). The between farm variance was 0.025 kg CO2-eq kg−1 FPCM in both seasons. The within farm variance in the estimate for the single farm mean decreased from 0.69 (1 visit) to 0.027 (26 visits) kg CO2-eq kg−1 FPCM (rainy season), and from 0.32 to 0.012 kg CO2-eq kg−1 FPCM (dry season). The within farm variance in the estimate for the population mean was 0.02 (rainy) and 0.01 (dry) kg CO2-eq kg−1 FPCM (1 visit), and decreased with an increase in farm visits. Forage cultivation was the main source of between farm variance, enteric fermentation the main source of within farm variance. Conclusions The estimated GHGEI was significantly higher in the rainy than in the dry season. The main contribution to variability in GHGEI is due to variation between observations from visits to the same farm. This source of variability can be reduced by increasing the number of visits per farm. Estimates for variation within and between farms enable a more informed decision about the data collection procedure.

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