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

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.

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.

Eye Movements Simultaneously Recorded by Electrooculographic and Photoelectric Methods

1979 ◽  
Vol 48 (2) ◽  
pp. 619-624 ◽  
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.

scholarly journals Classification and Statistical Trend Analysis in Detecting Glaucomatous Visual Field Progression

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.

A New and Rapid Determination of Pyridinium Aldoximes in Blood and Urine

1969 ◽  
Vol 15 (1) ◽  
pp. 72-83 ◽  
Author(s):  
William A Groff ◽  
Robert I Ellin
Keyword(s):  
Standard Deviation ◽  
Transfer Rate ◽  
Rapid Determination ◽  
Accurate Method ◽  
Small Sample ◽  
Basic Solution ◽  
Pyridinium Oximes ◽  
Average Variation ◽  
The Mean

Abstract A rapid and accurate method for analyzing pyridinium oximes—N-methylpyridinium-2-aldoxime chloride, N,N'-trimethylene-(pyridinium-4-aldoxime) dihalide, and N,N'-oxydimethyl-(pyridinium-4-aldoxime) dichloride—in plasma, urine, and whole blood is described. The method is completely automated and requires small sample volumes. Concentrations ranging from 3 to 120 µEq./L. in biologic fluids can be determined at a rate of 40 samples per hour. This technic can be applied to oximes which are unstable in basic solution. The average variation of the oxime concentration used to establish calibration curves, as determined by the ratio of the standard deviation to the mean, was ± 1.5%. Plasma and albumin increase the transfer rate of the oximes through the dialyzing membrane. Theoretical concentrations to explain this phenomenon are presented.

scholarly journals Comparison of Perimetric Outcomes from Melbourne Rapid Fields Tablet Perimeter Software and Humphrey Field Analyzer in Glaucoma Patients

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Harsh Kumar ◽  
Mithun Thulasidas
Keyword(s):  
Standard Deviation ◽  
Visual Field ◽  
Rural Areas ◽  
Visual Fields ◽  
Test Time ◽  
Mean Deviation ◽  
Time Saving ◽  
Humphrey Field Analyzer ◽  
Probability Plot ◽  
The Mean

Purpose. To compare visual field results obtained using Melbourne Rapid Fields (MRF) iPad-based perimeter software and Humphrey Field Analyzer (HFA) 24-2 Swedish Interactive Threshold Algorithm (SITA) standard program in glaucoma patients. Design. A cross-sectional observational study. Methods. In this single-centre study involving patients diagnosed with glaucoma, the perimetric outcomes of MRF were compared against those returned from the HFA 24-2 SITA standard. Outcomes included mean deviation (MD), pattern standard deviation (PSD), visual field index (VFI)/visual capacity (VC), foveal threshold, test time, number of points depressed at P<5% on PSD probability plot, and glaucoma hemifield test/color coded indicator. Results. The study included 28 eyes of 28 glaucoma patients. Mean (standard deviation) test times were 342.07 (56.70) seconds for MRF and 375.11 (88.95) for HFA 24-2 SITA standard P=0.046. Mean MD was significantly lower for MRF (Δ = 3.09, P<0.001), and mean PSD was significantly higher for MRF (Δ = 1.40, P=0.005) compared with HFA. The mean foveal threshold for the MRF was significantly lower than the mean HFA foveal threshold ((Δ = 9.25, P<0.001). The number of points depressed at P<5% on the PSD probability plot was significantly less for MRF P<0.001. Other perimetric outcomes showed no significant differences between both. Bland–Altman plots showed that considerable variability existed between the programs. Conclusion. MRF is a good cost-effective, time-saving, user-friendly tool for monitoring visual fields in settings where access to traditional perimetry is limited. The lack of Internet strength in rural areas and questionable detection of early cases may be two points in MRF fields requiring an upgrade.

scholarly journals Validation of refractivity profiles derived from GRAS raw-sampling data

2011 ◽  
Vol 4 (7) ◽  
pp. 1541-1550 ◽  
Author(s):  
F. Zus ◽  
G. Beyerle ◽  
S. Heise ◽  
T. Schmidt ◽  
J. Wickert ◽  
...  
Keyword(s):  
Standard Deviation ◽  
Penetration Depth ◽  
Radio Occultation ◽  
Lower Troposphere ◽  
Research Centre ◽  
Mean Deviation ◽  
Negative Bias ◽  
Altitude Range ◽  
Full Spectrum ◽  
The Mean

Abstract. Results from GRAS (GNSS Receiver for Atmospheric Sounding) RO (Radio Occultation) data recorded in RS (Raw Sampling) mode processed at the GFZ (German Research Centre for Geoscience) Potsdam are presented. The experimental processing software POCS-X includes FSI (Full Spectrum Inversion) in order to cope with multi-path regions and enables in connection with RS data to retrieve atmospheric refractivity profiles down to the Earths surface. Radio occultation events observed between 30 September and 30 October 2007 are processed and the retrievals are validated against co-located ECMWF (European Centre for Medium-Range Weather Forecasts) profiles. The intercomparison indicates good quality of the retrieved profiles. In the altitude range 8 to 25 km the standard deviation is below 1 %. The mean deviation in this altitude range tends to be negative. At 30 km the negative bias reaches about −0.4 %. Below 8 km the standard deviation increases, reaching 2.5 % at 2 km. Below 2 km the mean deviation tends to be negative, reaching −1.9 % close to the ground. The negative bias mainly stems from the tropical lower troposphere; there, the negative bias reaches −3 %. The tropospheric penetration depth obtained from RS data shows a vast improvement compared to the tropospheric penetration depth typically obtained from CL (Closed Loop) data; 50 % of all retrieved profiles reach 720 m.

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