EMPIRICAL COMPARISON OF RELATIVE PRECISION OF GEOMETRIC MEASURE OF VARIATION ABOUT THE MEAN AND STANDARD DEVIATION

Mapping Intimacies ◽

10.31730/osf.io/tzkw9 ◽

2021 ◽

Author(s):

Benedict Troon

Keyword(s):

Standard Deviation ◽

Mean Squared Error ◽

Mean Deviation ◽

Efficiency Coefficient ◽

Average Deviation ◽

Statistical Tool ◽

Absolute Deviation ◽

Geometric Measure ◽

Measures Of Dispersion ◽

The Mean

Measure of dispersion is an important statistical tool used to illustrate the distribution of datasets.The use of this measure has allowed researchers to define the distribution of various datasets especially the measures of dispersion from the mean. Researchers have been able to develop measures of dispersion from the mean such as mean deviation, mean absolute deviation, variance and standard deviation. Studies have shown that standard deviation is currently the most efficient measure of variation about the mean and the most popularly used measure of variation about the mean around the world because of its fewer shortcomings. However, studies have also established that standard deviation is not 100% efficient because the measure is affected by outlier in thedatasets and it also assumes symmetry of datasets when estimating the average deviation about the mean a factor that makes it to be responsive to skewed datasets hence giving results which are biased for such datasets. The aim of this study is to make a comparative analysis of the precision of the geometric measure of variation and standard deviation in estimating the average variationabout the mean for various datasets. The study used paired t-test to test the difference in estimates given by the two measures and four measures of efficiency (coefficient of variation, relative efficiency, mean squared error and bias) to assess the efficiency of the measure. The results determined that the estimates of geometric measure were significantly smaller than those of standard deviation and that the geometric measure was more efficient in estimating the average deviation for geometric, skewed and peaked datasets. In conclusion, the geometric measure was not affected by outliers and skewed datasets, hence it was more precise than standard deviation.

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Modelling Geometric Measure of Variation About the Population Mean

10.31730/osf.io/en5ph ◽

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

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ESTIMATION OF VARIATION ABOUT THE MEAN USING GEOMETRIC MEASURE

10.31730/osf.io/jtn64 ◽

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.

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

10.31730/osf.io/avqw2 ◽

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.

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Eye Movements Simultaneously Recorded by Electrooculographic and Photoelectric Methods

Perceptual and Motor Skills ◽

10.2466/pms.1979.48.2.619 ◽

1979 ◽

Vol 48 (2) ◽

pp. 619-624 ◽

Cited By ~ 3

Author(s):

Jin Ong ◽

Gene A. Harman

Keyword(s):

Eye Movements ◽

Standard Deviation ◽

Mean Amplitude ◽

Mean Deviation ◽

Deviation Index ◽

Velocity Peak ◽

The Mean

Three types of eye movements, saccadic, reading, and pursuit, were recorded from 6 college subjects, two in each by the electrooculographic and photoelectric methods simultaneously. A deviation index (DI), which is the standard deviation divided by the mean, was devised to compare the precision of recording amplitude deflection, and a proportion index (PI), which is M1 divided by M2, was devised to compare the mean amplitude indirectly between these two methods. Results showed that the proportion indexes of three types of eye movements were comparable, and the mean index of 0.54 indicated that the amplification in the electrooculographic method was about half as much as that in the photoelectric. The mean deviation index of 0.132 vs 0.135 was, again, comparable, meaning that these two methods of recording amplitude deflections are of about the same degree of magnitude and precision. Certain qualitative differences regarding the amplitude and velocity peak deflection between these two methods were also noted.

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Classification and Statistical Trend Analysis in Detecting Glaucomatous Visual Field Progression

Journal of Ophthalmology ◽

10.1155/2019/1583260 ◽

2019 ◽

Vol 2019 ◽

pp. 1-5

Author(s):

Cristiana Valente ◽

Elisa D’Alessandro ◽

Michele Iester

Keyword(s):

Statistical Analysis ◽

Standard Deviation ◽

Visual Field ◽

Trend Analysis ◽

Open Angle Glaucoma ◽

Mean Deviation ◽

Threshold Strategy ◽

Visual Field Progression ◽

The Mean

Aim. To evaluate the agreement between different methods in detection of glaucomatous visual field progression using two classification-based methods and four statistical approaches based on trend analysis. Methods. This is a retrospective and longitudinal study. Twenty Caucasian patients (mean age 73.8 ± 13.43 years) with open-angle glaucoma were recruited in the study. Each visual field was assessed by Humphrey Field Analyzer, program SITA standard 30-2 or 24-2 (Carl Zeiss Meditec, Inc., Dublin, CA). Full threshold strategy was also accepted for baseline tests. Progression was analyzed by using Hodapp–Parrish–Anderson classification and the Advanced Glaucoma Intervention Study visual field defect score. For the statistical analysis, linear regression (r2) was calculated for mean deviation (MD), pattern standard deviation (PSD), and visual field index (VFI), and when it was significant, each series of visual field was considered progressive. We also used Progressor to look for a significant progression of each visual field series. The agreement between methods, based on statistical analysis and classification, was evaluated using a weighted kappa statistic. Results. Thirty-eight visual field series were analyzed. The mean follow-up time was 6.2 ± 1.53 years (mean ± standard deviation). At baseline, the mean MD was −7.34 ± 7.18 dB; at the end of the follow-up, the mean MD was −9.25 ± 8.65 dB; this difference was statistically significant (p<0.001). The agreement to detect progression was fair between all methods based on statistical analysis and classification except for PSD r2. A substantial agreement (κ = 0.698 ± 0.126) was found between MD r2 and VFI r2. With the use of all the statistical analysis, there was a better time-saving. Conclusions. The best agreement to detect progression was found between MD r2 and VFI r2. VFI r2 showed the best agreement with all the other methods. GPA2 can help ophthalmologists to detect glaucoma progression and to help in treatment decisions. PSD r2 was the worse method to detect progression.

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Robust Confidence Intervals for PM2.5 Concentration Measurements in the Ecuadorian Park La Carolina

Sensors ◽

10.3390/s20030654 ◽

2020 ◽

Vol 20 (3) ◽

pp. 654 ◽

Cited By ~ 4

Author(s):

Wilmar Hernandez ◽

Alfredo Mendez ◽

Rasa Zalakeviciute ◽

Angela Maria Diaz-Marquez

Keyword(s):

Air Pollution ◽

Standard Deviation ◽

Confidence Intervals ◽

Quality Index ◽

Urban Park ◽

Absolute Deviation ◽

The Mean ◽

Concentration Measurements ◽

Pm2.5 Concentration

In this article, robust confidence intervals for PM2.5 (particles with size less than or equal to 2.5 μ m ) concentration measurements performed in La Carolina Park, Quito, Ecuador, have been built. Different techniques have been applied for the construction of the confidence intervals, and routes around the park and through the middle of it have been used to build the confidence intervals and classify this urban park in accordance with categories established by the Quito air quality index. These intervals have been based on the following estimators: the mean and standard deviation, median and median absolute deviation, median and semi interquartile range, a -trimmed mean and Winsorized standard error of order a , location and scale estimators based on the Andrew’s wave, biweight location and scale estimators, and estimators based on the bootstrap- t method. The results of the classification of the park and its surrounding streets showed that, in terms of air pollution by PM2.5, the park is not at caution levels. The results of the classification of the routes that were followed through the park and its surrounding streets showed that, in terms of air pollution by PM2.5, these routes are at either desirable, acceptable or caution levels. Therefore, this urban park is actually removing or attenuating unwanted PM2.5 concentration measurements.

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Identification of Daily Activites and Environments Based on the AdaBoost Method Using Mobile Device Data: A Systematic Review

Electronics ◽

10.3390/electronics9010192 ◽

2020 ◽

Vol 9 (1) ◽

pp. 192 ◽

Cited By ~ 3

Author(s):

José M. Ferreira ◽

Ivan Miguel Pires ◽

Gonçalo Marques ◽

Nuno M. Garcia ◽

Eftim Zdravevski ◽

...

Keyword(s):

Systematic Review ◽

Standard Deviation ◽

Mobile Devices ◽

Magnetic Sensors ◽

Daily Activities ◽

Motion Sensors ◽

Accurate Identification ◽

Absolute Deviation ◽

Environment Recognition ◽

The Mean

Using the AdaBoost method may increase the accuracy and reliability of a framework for daily activities and environment recognition. Mobile devices have several types of sensors, including motion, magnetic, and location sensors, that allow accurate identification of daily activities and environment. This paper focuses on the review of the studies that use the AdaBoost method with the sensors available in mobile devices. This research identified the research works written in English about the recognition of daily activities and environment recognition using the AdaBoost method with the data obtained from the sensors available in mobile devices that were published between 2012 and 2018. Thus, 13 studies were selected and analysed from 151 identified records in the searched databases. The results proved the reliability of the method for daily activities and environment recognition, highlighting the use of several features, including the mean, standard deviation, pitch, roll, azimuth, and median absolute deviation of the signal of motion sensors, and the mean of the signal of magnetic sensors. When reported, the analysed studies presented an accuracy higher than 80% in recognition of daily activities and environments with the Adaboost method.

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Cleaning of Matched License Plate Data

Transportation Research Record Journal of the Transportation Research Board ◽

10.3141/1804-01 ◽

2002 ◽

Vol 1804 (1) ◽

pp. 1-7 ◽

Cited By ~ 11

Author(s):

Stephen D. Clark ◽

S. Grant-Muller ◽

Haibo Chen

Keyword(s):

Standard Deviation ◽

Road Traffic ◽

Statistical Test ◽

License Plate ◽

Mean Absolute Deviation ◽

Test Results ◽

Absolute Deviation ◽

Traffic Conditions ◽

The Third ◽

The Mean

Three methods for identifying outlying journey time observations collected as part of a motorway license plate matching exercise are presented. Each method is examined to ensure that it is comprehensible to transport practitioners, is able to correctly classify outliers, and is efficient in its application. The first method is a crude method based on percentiles. The second uses a mean absolute deviation test. The third method is a modification of a traditional z- or t-statistical test. Results from each method and combinations of methods are compared. The preferred method is judged to be the third method alone, which uses the median rather than the mean as its measure of location and the inter-quartile range rather than the standard deviation as its measure of variability. This method is seen to be robust to both the outliers themselves and the presence of incident conditions. The effectiveness of the method is demonstrated under a number of typical and atypical road traffic conditions. In particular, the method is applied to a different section of motorway and is shown to still produce useful results.

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