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

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 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.

STATISTICS OF THE MEAN DEVIATION METER

1952 ◽  
Vol 30 (4) ◽  
pp. 329-341
Author(s):  
E. S. Keeping ◽  
W. W. Happ
Keyword(s):  
Probability Density ◽  
Similar Type ◽  
Mean Deviation ◽  
Density Functions ◽  
Tank Circuit ◽  
Decay Constants ◽  
The Mean ◽  
The Difference

The mean deviation is evaluated for a number of distributions of a Poisson or similar type, such as arise with counters, scaling circuits, and in particular with the mean deviation meter described by Greenberg and Happ. The mean deviation for some distributions is fairly readily obtainable, but for the charge on a single tank circuit and also for the difference of charges on two tanks with different decay constants it is necessary to obtain the probability density. Exact expressions for these density functions are given, and also approximations from which the mean deviation can be calculated.

Lagrangian measurements of inertial particle accelerations in a turbulent boundary layer

2008 ◽  
Vol 617 ◽  
pp. 255-281 ◽  
Author(s):  
S. GERASHCHENKO ◽  
N. S. SHARP ◽  
S. NEUSCAMMAN ◽  
Z. WARHAFT
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Probability Density ◽  
Isotropic Turbulence ◽  
Free Stream ◽  
Probability Density Functions ◽  
Density Functions ◽  
Inertial Particles ◽  
Free Stream Turbulence ◽  
The Mean

Two-dimensional Lagrangian acceleration statistics of inertial particles in a turbulent boundary layer with free-stream turbulence are determined by means of a particle tracking technique using a high-speed camera moving along the side of the wind tunnel at the mean flow speed. The boundary layer is formed above a flat plate placed horizontally in the tunnel, and water droplets are fed into the flow using two different methods: sprays placed downstream from an active grid, and tubes fed into the boundary layer from humidifiers. For the flow conditions studied, the sprays produce Stokes numbers varying from 0.47 to 1.2, and the humidifiers produce Stokes numbers varying from 0.035 to 0.25, where the low and high values refer to the outer boundary layer edge and the near-wall region, respectively. The Froude number is approximately 1.0 for the sprays and 0.25 for the humidifiers, with a small variation within the boundary layer. The free-stream turbulence is varied by operating the grid in the active mode as well as a passive mode (the latter behaves as a conventional grid). The boundary layer momentum-thickness Reynolds numbers are 840 and 725 for the active and passive grid respectively. At the outer edge of the boundary layer, where the shear is weak, the acceleration probability density functions are similar to those previously observed in isotropic turbulence for inertial particles. As the boundary layer plate is approached, the tails of the probability density functions narrow, become negatively skewed, and their peak occurs at negative accelerations (decelerations in the streamwise direction). The mean deceleration and its root mean square (r.m.s.) increase to large values close to the plate. These effects are more pronounced at higher Stokes number. In the vertical direction, there is a slight downward mean deceleration and its r.m.s., which is lower in magnitude than that of the streamwise component, peaks in the buffer region. Although there are free-stream turbulence effects, and the complex boundary layer structure plays an important role, a simple model suggests that the acceleration behaviour is dominated by shear, gravity and inertia. The results are contrasted with inertial particles in isotropic turbulence and with fluid particle acceleration statistics in a boundary layer. The background velocity field is documented by means of hot-wire anemometry and laser Doppler velocimetry measurements. These appear to be the first Lagrangian acceleration measurements of inertial particles in a shear flow.

scholarly journals Modeling Diameter Distributions with Six Probability Density Functions in Pinus halepensis Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)

2021 ◽  
Vol 13 (12) ◽  
pp. 2307
Author(s):  
J. Javier Gorgoso-Varela ◽  
Rafael Alonso Ponce ◽  
Francisco Rodríguez-Puerta
Keyword(s):  
Probability Density ◽  
Linear Models ◽  
Pinus Halepensis ◽  
Beta Function ◽  
Probability Density Functions ◽  
Lidar Data ◽  
Density Functions ◽  
Minimum Diameter ◽  
Location Parameters ◽  
Diameter Distributions

The diameter distributions of trees in 50 temporary sample plots (TSPs) established in Pinus halepensis Mill. stands were recovered from LiDAR metrics by using six probability density functions (PDFs): the Weibull (2P and 3P), Johnson’s SB, beta, generalized beta and gamma-2P functions. The parameters were recovered from the first and the second moments of the distributions (mean and variance, respectively) by using parameter recovery models (PRM). Linear models were used to predict both moments from LiDAR data. In recovering the functions, the location parameters of the distributions were predetermined as the minimum diameter inventoried, and scale parameters were established as the maximum diameters predicted from LiDAR metrics. The Kolmogorov–Smirnov (KS) statistic (Dn), number of acceptances by the KS test, the Cramér von Misses (W2) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·dmin). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R2 values of 0.750, 0.724 and 0.873 for dg, dmed and dmax. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

scholarly journals A synergy of the velocity gradients technique and the probability density functions for identifying gravitational collapse in self-absorbing media

2021 ◽  
Vol 502 (2) ◽  
pp. 1768-1784
Author(s):  
Yue Hu ◽  
A Lazarian
Keyword(s):  
Magnetic Fields ◽  
Probability Density ◽  
Column Density ◽  
Mass Density ◽  
Probability Density Functions ◽  
Emission Lines ◽  
Density Functions ◽  
Star Forming ◽  
Velocity Gradients ◽  
Self Gravity

ABSTRACT The velocity gradients technique (VGT) and the probability density functions (PDFs) of mass density are tools to study turbulence, magnetic fields, and self-gravity in molecular clouds. However, self-absorption can significantly make the observed intensity different from the column density structures. In this work, we study the effects of self-absorption on the VGT and the intensity PDFs utilizing three synthetic emission lines of CO isotopologues 12CO (1–0), 13CO (1–0), and C18O (1–0). We confirm that the performance of VGT is insensitive to the radiative transfer effect. We numerically show the possibility of constructing 3D magnetic fields tomography through VGT. We find that the intensity PDFs change their shape from the pure lognormal to a distribution that exhibits a power-law tail depending on the optical depth for supersonic turbulence. We conclude the change of CO isotopologues’ intensity PDFs can be independent of self-gravity, which makes the intensity PDFs less reliable in identifying gravitational collapsing regions. We compute the intensity PDFs for a star-forming region NGC 1333 and find the change of intensity PDFs in observation agrees with our numerical results. The synergy of VGT and the column density PDFs confirms that the self-gravitating gas occupies a large volume in NGC 1333.

Sign in / Sign up

Export Citation Format

Share Document