A Proof for Rhiel's Range Estimator of the Coefficient of Variation for Skewed Distributions

2007 ◽  
Vol 100 (1) ◽  
pp. 208-210 ◽  
Author(s):  
G. Steven Rhiel
Keyword(s):  
Standard Deviation ◽  
Coefficient Of Variation ◽  
Research Study ◽  
Skewed Distributions ◽  
The Mean ◽  
Standard Deviations

In this research study is proof that the coefficient of variation ( CVhigh-low) calculated from the highest and lowest values in a set of data is applicable to specific skewed distributions with varying means and standard deviations. Earlier Rhiel provided values for dn, the standardized mean range, and an, an adjustment for bias in the range estimator of μ. These values are used in estimating the coefficient of variation from the range for skewed distributions. The dn and an values were specified for specific skewed distributions with a fixed mean and standard deviation. In this proof it is shown that the dn and an values are applicable for the specific skewed distributions when the mean and standard deviation can take on differing values. This will give the researcher confidence in using this statistic for skewed distributions regardless of the mean and standard deviation.

Correction for Rhiel's Theory for the Range Estimator of the Coefficient of Variation for Skewed Distributions

2010 ◽  
Vol 106 (1) ◽  
pp. 93-94
Author(s):  
G. Steven Rhiel
Keyword(s):  
Standard Deviation ◽  
Correction Factor ◽  
Coefficient Of Variation ◽  
Unbiased Estimate ◽  
Correction Factors ◽  
Skewed Distributions ◽  
The Mean

In 2007, Rhiel presented a technique to estimate the coefficient of variation from the range when sampling from skewed distributions. To provide an unbiased estimate, a correction factor ( an) for the mean was included. Numerical correction factors for a number of skewed distributions were provided. In a follow-up paper, he provided a proof he claimed showed the correction factor was independent of the mean and standard deviation, making the factors useful as these parameters vary; however, that proof did not establish independence. Herein is a proof which establishes the independence.

The Effects of Ethyl Alcohol on a Driver's Driving Skill, Visual Perception, Risk Acceptance, Choice Reaction Times and Information Processing Rates

1974 ◽  
Vol 18 (2) ◽  
pp. 116-116
Author(s):  
Helmut T. Zwahlen
Keyword(s):  
Information Processing ◽  
Standard Deviation ◽  
Visual Perception ◽  
Reaction Times ◽  
Driving Skill ◽  
Alcohol Condition ◽  
The Mean ◽  
Risk Acceptance ◽  
Standard Deviations ◽  
Choice Reaction Times

Twelve subjects (20–37 years old) were tested in the laboratory and eleven out of these were also tested in a car in the field, first under a no alcohol condition and then under an alcohol condition (approximately 0.10% BAC). In the laboratory the subjects simple and choice reaction times for two uncertainty modes were measured and their information processing rates (3 bits unsertainty) were determined. In the field the subjects driving skill for driving through a gap with 20 inches total clearance at 20 MPH was measured, as well as their static visual perceptual capabilities and risk acceptance decisions for a 46 feet viewing distance using psychophysical experimental methods. Based upon the driving skill measure (standard deviation of centerline deviations in the gap), the mean of the psychometric visual gap perception function and the mean of the psychometric gap risk acceptance function, the “Safety Distance” and the “Driver Safety Index” (DSI) were obtained. Based upon a statistical analysis of the data we may conclude first that the effects of alcohol (approximately 0.10% BAC) vary widely from one subject to another (slighthly improved performance to highly impaired performance) and that the changes in the group averages of the means and standard deviations of the psychometric visual perception and risk acceptance functions, the driving skill distributions, the “Safety Distances” and the DSI's for the subjects (although all changes in the group averages are in the expected direction) are statistically not significant (α = .05). Second, the group average of the means of the choice reaction times for the subjects increased by 5% under the alcohol condition (statistically significant, α = .05), but more important the group average of the standard deviations of the choice reaction times for the subjects increased by 23% (statistically significant, α = .05). The group average of the information processing rates for the subjects decreased by 3% (statistically not significant, α = .05) under the alcohol condition. A system model in which the system demands on the driver are represented in terms of choice reaction times is used to demonstrate that the increase in performance variability (expressed by the standard deviation of choice reaction times) under the influence of alcohol provides a much better explanation for the higher accident involvement than the historically most frequently used rather small increase in average performance (expressed by the mean of choice reaction times).

Alum Production from some Nigerian Kaolinite Deposits

2012 ◽  
Vol 7 ◽  
pp. 13-19 ◽  
Author(s):  
L.C. Edomwonyi-Otu ◽  
B.O. Aderemi ◽  
O. Edomwonyi-Otu ◽  
A. Simo ◽  
M. Maaza
Keyword(s):  
Standard Deviation ◽  
Surface Area ◽  
Science And Technology ◽  
Mineral Resources ◽  
Kaolinite Clay ◽  
Processing Technologies ◽  
Commercial Alumina ◽  
Bet Analysis ◽  
The Mean ◽  
Standard Deviations

The Development of Sustainable Processing Technologies for the Vast Mineral Resources Available in Nigeria and their Varied Applications Is a Major Pursuit by the Federal Ministry of Science and Technology. in this Work, Alum Was Produced from Three Different Kaolin Deposits in Nigeria Namely Kankara Brown, Bauchi and Kankara White by Acid Dealumination of the Metakaolin Obtained by Calcination of the Beneficiated Kaolinites and the Yields Were Measured to Ascertain the Process Repeatability. the Reproducibility Studies Carried Out on Samples from each Deposit Showed a Mean Yield of 80 %, 92 % and 87 % and Standard Deviation of 2.50 %, 1.063 % and 1.296 %, for Kankara Brown, Bauchi and Kankara White Respectively. the Values from the Three Deposits Fall within 3 Standard Deviations of the Mean in Accordance with the 68-95-99.7/three-Sigma Rule. the Alum Quality Also Compares Well with Available Commercial Alums in the Market. BET Analysis, of the Alumina Obtained by Calcination of the Alum (Kankara White), Gave a Surface Area of 192.2441m2/g Comparable to Commercial Alumina. these Results Suggest/establishes the Huge Possibility of Commercial Alum Production, Including Alumina, Using Kaolinite Clay from these Deposits as Starting Materials.

scholarly journals Properties of the Standard Deviation that are Rarely Mentioned in Classrooms

2016 ◽  
Vol 38 (3) ◽  
Author(s):  
Mohammad Fraiwan Al-Saleh ◽  
Adil Eltayeb Yousif
Keyword(s):  
Standard Deviation ◽  
Coefficient Of Variation ◽  
Sample Variance ◽  
Mean Absolute Deviation ◽  
Hidden Information ◽  
Absolute Deviation ◽  
Sample Mean ◽  
The Mean ◽  
Vague Concept ◽  
Chebyshev's Inequality

Unlike the mean, the standard deviation ¾ is a vague concept. In this paper, several properties of ¾ are highlighted. These properties include the minimum and the maximum of ¾, its relationship to the mean absolute deviation and the range of the data, its role in Chebyshev’s inequality and the coefficient of variation. The hidden information in the formula itself is extracted. The confusion about the denominator of the sample variance being n ¡ 1 is also addressed. Some properties of the sample mean and varianceof normal data are carefully explained. Pointing out these and other properties in classrooms may have significant effects on the understanding and the retention of the concept.

scholarly journals The GRIMMER test: A method for testing the validity of reported measures of variability

2016 ◽  
Author(s):  
Jordan Anaya
Keyword(s):  
Standard Deviation ◽  
Sample Size ◽  
Standard Error ◽  
Standard Errors ◽  
Rounding Error ◽  
Small Variance ◽  
The Mean ◽  
Standard Deviations ◽  
Mathematical Foundations ◽  
Repetitive Pattern

GRIMMER (Granularity-Related Inconsistency of Means Mapped to Error Repeats) builds upon the GRIM test and allows for testing whether reported measures of variability are mathematically possible. GRIMMER relies upon the statistical phenomenon that variances display a simple repetitive pattern when the data is discrete, i.e. granular. This observation allows for the generation of an algorithm that can quickly identify whether a reported statistic of any size or precision is consistent with the stated sample size and granularity. My implementation of the test is available at PrePubMed (http://www.prepubmed.org/grimmer) and currently allows for testing variances, standard deviations, and standard errors for integer data. It is possible to extend the test to other measures of variability such as deviation from the mean, or apply the test to non-integer data such as data reported to halves or tenths. The ability of the test to identify inconsistent statistics relies upon four factors: (1) the sample size; (2) the granularity of the data; (3) the precision (number of decimals) of the reported statistic; and (4) the size of the standard deviation or standard error (but not the variance). The test is most powerful when the sample size is small, the granularity is large, the statistic is reported to a large number of decimal places, and the standard deviation or standard error is small (variance is immune to size considerations). This test has important implications for any field that routinely reports statistics for granular data to at least two decimal places because it can help identify errors in publications, and should be used by journals during their initial screen of new submissions. The errors detected can be the result of anything from something as innocent as a typo or rounding error to large statistical mistakes or unfortunately even fraud. In this report I describe the mathematical foundations of the GRIMMER test and the algorithm I use to implement it.

Adaptation of an Alkaline Phosphatase Method for Automatic Colorimetric Analysis

1959 ◽  
Vol 5 (2) ◽  
pp. 119-126 ◽  
Author(s):  
Walton H Marsh ◽  
Benjamin Fingerhut ◽  
Elaine Kirsch
Keyword(s):  
Alkaline Phosphatase ◽  
Enzyme Activity ◽  
Standard Deviation ◽  
Enzyme Concentration ◽  
Mean Value ◽  
Colorimetric Analysis ◽  
Manual Method ◽  
Automated Method ◽  
The Mean ◽  
Standard Deviations

Abstract The alkaline phosphatase method of Kind and King was adapted to an automated recording colorimeter. The precision of the automated method (1 standard deviation as per cent of the mean value) was ±1.7 and for the manual method ±3.6 per cent. The color produced was proportional to the enzyme concentration by both methods, and recoveries of added phenol were satisfactory. In more than 150 serum specimens surveyed for enzyme activity, over 95 per cent of the results (2 standard deviations) of the 2 methods in the range 3.4-129 agree to within ±2.8 King-Armstrong units/1OO ml.

scholarly journals The precision of temporal judgement: milliseconds, many minutes, and beyond

2009 ◽  
Vol 364 (1525) ◽  
pp. 1897-1905 ◽  
Author(s):  
P.A. Lewis ◽  
R.C. Miall
Keyword(s):  
Standard Deviation ◽  
Coefficient Of Variation ◽  
Human Subjects ◽  
Short Interval ◽  
Time Measurement ◽  
Scalar Property ◽  
Timing Mechanisms ◽  
Timing Data ◽  
The Mean ◽  
Different Levels

The principle that the standard deviation of estimates scales with the mean estimate, commonly known as the scalar property, is one of the most broadly accepted fundamentals of interval timing. This property is measured using the coefficient of variation (CV) calculated as the ratio between the standard deviation and the mean. In 1997, John Gibbon suggested that different time measurement mechanisms may have different levels of absolute precision, and would therefore be associated with different CVs. Here, we test this proposal by examining the CVs produced by human subjects timing a broad range of intervals (68 ms to 16.7 min). Our data reveal no evidence for multiple mechanisms, but instead show a continuous logarithmic decrease in CV as timed intervals increase. This finding joins other recent reports in demonstrating a systematic violation of the scalar property in timing data. Interestingly, the estimated CV of circadian judgements fits onto the regression of decreasing CV, suggesting a link between short interval and circadian timing mechanisms.

Statistics of spontaneous electrical activity of supra- and ectosylvian gyri of the dog

1963 ◽  
Vol 204 (1) ◽  
pp. 51-59
Author(s):  
Archie R. Tunturi
Keyword(s):  
Standard Deviation ◽  
Electrical Activity ◽  
Negative Correlation ◽  
Evoked Potential ◽  
Correlation Coefficients ◽  
Moderate Correlation ◽  
Spontaneous Electrical Activity ◽  
Suprasylvian Gyrus ◽  
The Mean ◽  
Standard Deviations

The standard deviations of the spontaneous electrical activity (SEA) of the suprasylvian gyrus (SSG) ranged between 57–131 µv and for the middle ectosylvian (MES) gyrus, 88–175 µv. Correlation coefficients, r, served to distinguish three regions of the SSG. The rostral showed low correlation with the middle, high correlation with the caudal, and low to negative correlation with the MES. The middle showed moderate correlation with the MES, and the caudal showed zero to negative correlation with the MES. Within the SSG, correlation was low and in the MES high, for spacings of 2 mm. Cocaine applied to both areas sharpened the boundaries at the sulci, reduced standard deviations, did not affect the correlation between the caudal SSG and the MES area, and increased r between all locations in the MES but not in the SSG. Cocaine on the SSG had no effect on the mean and standard deviation of the evoked potential in the MES, but decreased r of the SEA significantly.

scholarly journals On the Formulation and Universality of Monin–Obukhov Similarity Functions for Mean Gradients and Standard Deviations in the Unstable Surface Layer: Results from Surface-Layer-Resolving Large-Eddy Simulations

2017 ◽  
Vol 74 (4) ◽  
pp. 989-1010 ◽  
Author(s):  
Björn Maronga ◽  
Joachim Reuder
Keyword(s):  
Surface Layer ◽  
Standard Deviation ◽  
Large Eddy Simulations ◽  
Specific Humidity ◽  
Horizontal Wind ◽  
Convective Conditions ◽  
Similarity Functions ◽  
Large Eddy ◽  
The Mean ◽  
Standard Deviations

Abstract Surface-layer-resolving large-eddy simulations (LESs) of free-convective to near-neutral boundary layers are used to study Monin–Obukhov similarity theory (MOST) functions. The LES dataset, previously used for the analysis of MOST relationships for structure parameters, is extended for the mean vertical gradients and standard deviations of potential temperature, specific humidity, and wind. Also, local-free-convection (LFC) similarity is studied. The LES data suggest that the MOST functions for mean gradients are universal and unique. The data for the mean gradient of the horizontal wind display significant scatter, while the gradients of temperature and humidity vary considerably less. The LES results suggest that this scatter is mostly related to a transition from MOST to LFC scaling when approaching free-convective conditions and that it is associated with a change of the slope of the similarity functions toward the expected value from LFC scaling. Overall, the data show slightly, but consistent, steeper slopes of the similarity functions than suggested in literature. The MOST functions for standard deviations appear to be unique and universal when the entrainment from the free atmosphere into the boundary layer is sufficiently small. If entrainment becomes significant, however, we find that the standard deviation of humidity no longer follows MOST. Under free-convective conditions, the similarity functions should reduce to universal constants (LFC scaling). This is supported by the LES data, showing only little scatter, but displaying a systematic height dependence of these constants. Like for MOST, the LFC similarity constant for the standard deviation of specific humidity becomes nonuniversal when the entrainment of dry air reaches significant levels.

Firing Variability Is Higher than Deduced from the Empirical Coefficient of Variation

2011 ◽  
Vol 23 (8) ◽  
pp. 1944-1966 ◽  
Author(s):  
Susanne Ditlevsen ◽  
Petr Lansky
Keyword(s):  
Experimental Data ◽  
Standard Deviation ◽  
Coefficient Of Variation ◽  
Estimation Methods ◽  
Interspike Intervals ◽  
Estimation Errors ◽  
Empirical Standard Deviation ◽  
Receptor Neurons ◽  
The Mean ◽  
Empirical Standard

A convenient and often used summary measure to quantify the firing variability in neurons is the coefficient of variation (CV), defined as the standard deviation divided by the mean. It is therefore important to find an estimator that gives reliable results from experimental data, that is, the estimator should be unbiased and have low estimation variance. When the CV is evaluated in the standard way (empirical standard deviation of interspike intervals divided by their average), then the estimator is biased, underestimating the true CV, especially if the distribution of the interspike intervals is positively skewed. Moreover, the estimator has a large variance for commonly used distributions. The aim of this letter is to quantify the bias and propose alternative estimation methods. If the distribution is assumed known or can be determined from data, parametric estimators are proposed, which not only remove the bias but also decrease the estimation errors. If no distribution is assumed and the data are very positively skewed, we propose to correct the standard estimator. When defining the corrected estimator, we simply use that it is more stable to work on the log scale for positively skewed distributions. The estimators are evaluated through simulations and applied to experimental data from olfactory receptor neurons in rats.

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