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Variability measures, such as the variance or the Gini mean difference, are widely used to summarize the dispersion of random variables. In the statistical setting, it is quite natural to assume that if a random quantity has more variability, then the estimators of its variability measures should be greater in some stochastic sense. Stochastic orders can be used to give inequalities in this regard, confirming, or not, this suitable property. This paper is devoted to the stochastic comparison of some variability measure estimators; conditions such that some of these estimators are comparable in the usual stochastic order and in the increasing convex order whenever the involved random variables have different variability are provided. Special attention is devoted to the cases of sample variance and Gini mean differences, and to the case of simple random samples. [ABSTRACT FROM AUTHOR]

The sample variance of the quality characteristic shows the variation of a product or process. When the precision is a concern, a tolerance interval (TI) for sample variances is needed. In this paper, we construct β-expectation TIs for the population of sample variances, assuming normality of the data, from the frequentist and Bayesian perspectives. The accuracy level of each TI, which is the probability that the actual coverage falls into a specified range, is calculated to evaluate the performance of the TI. Besides, the determination of the minimum number of subgroups and the minimum subgroup size is discussed to achieve the desired accuracy level. When the accuracy level, the number of subgroups and subgroup size are known, the maximum variation is also calculated to measure the goodness of the proposed TI. Comparison results show that a Bayesian TI outperforms the corresponding frequentist TI. A real example is used to demonstrate the applicability and implementations of the proposed TIs. [ABSTRACT FROM AUTHOR]

A β‐content tolerance interval (TI) is a statistical interval which contains at least some fraction (proportion) β of the population with a given confidence level. When we are interested in the precision of a quality characteristic, a TI for the sample variance is useful. In this paper, we consider an exact two‐sided β‐content TI for the sample variance from a normal distribution with a specified ratio of the tail probabilities. The proposed tolerance interval allows the practitioner more control over how the probabilities in the tails are distributed, which may be useful in certain applications. A comparison with an existing two‐sided β‐content TI shows that the proposed TI is better on the basis of expected coverage and standard deviation of the coverage. In addition, the proposed TI is shown to require fewer subgroups to achieve a specific accuracy level. Moreover, a phase II control chart with guaranteed performance is obtained from the proposed TI. Finally, a real and a simulated data are used for illustration. [ABSTRACT FROM AUTHOR]

The histogram and boxplot are effective and simple graphical tools, which are broadly used to explore the characteristics of the distribution of univariate data. In this proposed work, two statistical plots called sample variance plots (Sv-plots), are defined which illustrate squared deviations from the sample variance. Sv-plots exhibit the contribution of each data value toward the sample variability with profound insight. Besides capturing symmetry and skewness of the distribution, these plots can detect outliers in the data based on two novel bounds introduced on each plot analogous to a boxplot. In comparison with histogram and boxplot, for actual and simulated data, Sv-plots firmly lead in revealing more information. Displaying the exact data values, Sv-plots improve visual sensation for detecting masked atypical values and asymmetry of the data. Remarkably, one version of the Sv-plots can be employed to display the testing hypotheses for a single and two population means and discovers a graphical method to make the decision. It is found that the Sv-plots outperform both histogram and boxplot, and have additional benefits beyond identifying characteristics of the distribution. [ABSTRACT FROM AUTHOR]

We revisit the Helmert transformation, and provide a useful and simple derivation of the joint distribution of the sample mean and the sample variance in samples from independently and identically distributed normal random variables. Our derivation is distinguished by concreteness, very little abstractness, and should be appealing to beginning students of statistics, and to both beginning and advanced students of econometrics. We also highlight one fruitful application of the Helmert transformation in panel data econometrics. The Helmert transformation can be used to eliminate the fixed effects in the estimation of fixed effects models, and we briefly review this application of the transformation in the panel data context. [ABSTRACT FROM AUTHOR]

We derive a new expression for the covariance between sample mean and variance in terms of the moments of sample observations. Several corollaries are deduced. We also derive a general expression for the covariance between linear and quadratic functions of sample observations. [ABSTRACT FROM AUTHOR]

It is well known that the independence of the sample mean and the sample variance characterizes the normal distribution. By using Anosov's theorem, we further investigate the analogous characteristic properties in terms of the sample mean and some feasible definite statistics. The latter statistics introduced in this paper for the first time are based on nonnegative, definite and continuous functions of ordered arguments with positive degree of homogeneity. The proposed approach seems to be natural and can be used to derive easily characterization results for many feasible definite statistics, such as known characterizations involving the sample variance, sample range as well as Gini's mean difference. [ABSTRACT FROM AUTHOR]

Hectolitre weight (HW) is the principal quantitative parameter used by receiving units as an indicator of the quality of wheat grain (Triticum aestivum L.). The moisture content and the equipment used can be considered important sources of variation to assess a batch, requiring correct sizing of the sample used to measure the HW. This research identified the variability between the equipment used to evaluate HW with or without correcting the grain moisture content, and to estimate the ideal number of samples for the commercial classification of wheat batches. The experiment was carried out in a randomised block design with four replications. Seven batches of wheat grain from different cultivars and/or production sites were used to measure the HW using three types of equipment (Equipment ‘A’, Equipment ‘B’ and Equipment ‘C’), with and without correcting the grain moisture content to 13%. Higher values for HW were determined with Equipment ‘A’ (77.15) compared to Equipment ‘B’ (75.08) and Equipment ‘C’ (74.69), classifying the wheat according to type. The moisture content affected the HW, but did not change the final classification of the wheat in terms of type. The ideal number of samples for HW ranged from 1 to 18 at the lowest level of precision (HW=mean±1 kg hl-1) and from 67 to 1820 samples for the highest precision (HW=mean±0.1 kg hl-1). Equipment ‘B’ requires a smaller number of samples (possibly as few as one) for the same level of precision (HW=mean±1 kg hl-1), whereas Equipment ‘A’ requires a greater number of samples for the majority of batches. Correctly classifying wheat by HW therefore depends on a larger number of samples. [ABSTRACT FROM AUTHOR]

The typical Type A evaluation of measurement uncertainty relates to the problem of approximating a constant from observations with independent errors. However, there are other realistic types of problem in which repeated observations x1 … xn might be obtained, most of which lead to different expressions for the measurement uncertainty. This paper identifies ten situations and presents relevant analyses, some of which are new. The results show that care must be taken to consider the nature of the measurand and the purpose for making the measurement.

This paper intends to study the influence of the shearing rate on the undrained monotonic behavior of marine clay under different over-consolidation ratios and the effect of sample variance. A series of undrained triaxial tests with isotropic consolidation was performed. The tests were either conducted with a constant strain rate (single-stage tests) or with different strain rates applied in succession (multi-stage tests, step-change shearing rate tests), the latter with and without relaxation phases. The tests were conducted on saturated reconstituted Macau marine clay. It has been found that a higher overconsolidation ratio tends to produce a lower excess pore pressure, while faster shearing tends to lead to a higher undrained shear strength and a lower induced excess pore pressure. Besides, the influence of sample variance, which means the differences between the undrained shear strength and excess pore water pressure derived from tests with equal OCR and shearing rate, performed as either a single-stage or a multi-stage test, was found relatively small. It can be concluded that a test with different shearing rates applied in succession to the same specimen is sufficient to study the rate-dependent undrained behavior of saturated clay. Two different approaches for the description of the rate effects on the undrained shear strength, following Mitchell and Soga or Graham et al., respectively, have been extended by the influence of OCR. Based on comparison, the extended equation of Graham et al. seems more appropriate for an application in practice. [ABSTRACT FROM AUTHOR]

Tolerance intervals (TIs) for variances are useful when we are interested in the precision of the quality characteristic. They can provide a deeper understanding of the process variability corresponding to process improvement and deterioration. In this paper, we consider the exact two‐sided TI for sample variances that simultaneously controls the proportions in each of the two tails of the distribution. The minimum number of subgroups required to achieve a specified level of precision is also computed. Moreover, we minimize the width of the TI through adjusting the tolerance factors, and the performance of the adjusted TI shows some improvement, especially when the amount of reference data is small. Finally, a real application is used to illustrate the construction and implementation of the proposed TIs. [ABSTRACT FROM AUTHOR]

For the first time an analytical solution for the quantification of the spatial variance of the second-order moment of correlated wind speeds was developed in this work. The spatial variance is defined as random differences in the sample variance of wind speed between different points in space. The approach is successfully verified using simulation and field data. The impact of the spatial variance on three selected applications relevant to the wind energy sector is then investigated including mitigation measures. First, the difference of the second order moment between front-row wind turbines of Lillgrund wind farm is investigated. The variance of the difference ranges between 25% and 48% for turbulence intensities ranging from 7% to 10% and a sampling period of 10 min. It is thus suggested to use the second-order moment measured at each individual turbine as input to flow models of wind farm controllers in order to mitigate random error. Second, the impact of the spatial variance of the measured second-order moment on the verification of wind turbine performance is investigated. Misalignment between the mean wind direction and the line connecting the meteorological mast and wind turbine is observed to result in an additional random error in the observed second-order moment of wind speed. In the investigated conditions the random error was up to 34 %. Such a random error adds uncertainty to the turbulence intensity-based classification of the fatigue loads and power output of a wind turbine. To mitigate the random error, it is suggested to either filter the measured data for low angles of misalignment or quantify wind turbine performance using the ensemble-averaged measurements of the same wind conditions. Third, the verification of sensors in wind farms was investigated with respect to the impact of distant reference measurements. In the case of a misalignment between the wind direction and the line connecting sensor and reference, an increased random error will hamper the comparison of the measured second-order moments. The suggested mitigation measures are equivalent to those for the verification of turbine performance. [ABSTRACT FROM AUTHOR]

The purpose of this research was to examine whether exercisers and nonexercisers are rated similarly on a variety of characteristics by a sample of randomly selected regular exercisers, nonexercisers who intend to exercise, and nonexercisers with no intention to exercise. Previous research by Martin Ginis et al. (2003) has demonstrated an exerciser stereotype that advantages exercisers. It is unknown, however, the extent to which an exerciser stereotype is shared by nonexercisers, particularly nonintenders. Following an item-generation procedure, a sample of 470 (n = 218 men; n = 252 women) people selected using random digit dialing responded to a questionnaire assessing the extent to which they agreed that exercisers and nonexercisers possessed 24 characteristics, such as "happy," "fit," "fat," and "lazy." The results strongly support a positive exerciser bias, with exercisers rated more favorably on 22 of the 24 items. The degree of bias was equivalent in all groups of respondents. Examination of the demographic characteristics revealed no differences among the three groups on age, work status, or child-care responsibilities, suggesting that there is a pervasive positive exerciser bias. [ABSTRACT FROM AUTHOR]

We present a model for the bias of free-energy differences when determined using the nonequilibrium work (NEW) formalism due to Jarzynski. Input to the model is the distribution of work values underlying the NEW calculation, and the bias is estimated by assuming that all of the inaccuracy is incurred by failure to sample work values beyond a specific point in the tail of the distribution. The model is formulated considering both small- and large-sample NEW calculations. It is then applied to the study of bias for cases in which the work distribution is Gaussian. The model is shown to give an excellent description of the bias in situations where the bias is a more significant source of error than the sample variance. A scaling law is presented and it is shown that the bias as a function of sampling can be reduced to a single universal curve, approximately valid for all Gaussian work distributions. This result is used to formulate a simple criterion that can be applied to test if a NEW calculation is effectively free of bias. The criterion is shown to be effective even though it uses the measured (and perhaps biased) free energy as an input. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Tolerance limits for variances are useful in quality assessments when the focus is on the precision of a quality characteristic. Two-sided tolerance intervals (limits) provide insight into a process degradation as well as improvement, in terms of process variability. Sarmiento, Chakraborti, and Epprecht constructed the exact two-sided tolerance intervals for the population of sample variances, assuming normality of the data. The required tolerance factors cannot be expressed in a closed-form and their computation is complex, depending on the numerical solutions of a system of three nonlinear equations. Motivated by this, from a practical point of view, we consider a simpler, approximate tolerance interval based on the approximate tolerance interval for the gamma distribution, which uses the Wilson–Hilferty approximation. The required tolerance factors for the proposed interval are readily obtained using existing tables and software and therefore can be implemented more easily in practice. The performance of the proposed tolerance interval is compared with that of the exact interval in terms of accuracy and robustness in simulation studies. In addition, the tolerance intervals are illustrated with a dataset from a real application. A summary and some conclusions are offered. It is seen that the proposed approximate tolerance intervals are fairly accurate, reasonably robust and being much simpler to calculate, can be useful in practical applications. [ABSTRACT FROM AUTHOR]

Memory type control charts have built using logarithmic transformations for the normalization of unbiased sample variance for monitoring of changes in the dispersion of process. The intention of this study is to give a new hybrid exponentially weighted moving average (HEWMA) control chart for monitoring theprocess dispersion. The logarithmic transformation is used to normalize the distribution of sample variance. The execution of the proposed HEWMA control chart is evaluated in accordance with the average run length and standard deviation run length estimated with the help of Monte Carlo method. The proposed HEWMTn control chart has been compared with the corresponding existing charts to detect increase and decrease in the process variability. It showed that the proposed HEWMTn control chart is uniformly better than its parallels for detection of shifts in the process variability. A numerical study is also given to understand the operationalization of the proposed control chart. [ABSTRACT FROM AUTHOR]

The problem of the sample variance computation for epistemic interval-valued data is, in general, NP-hard. Therefore, known efficient algorithms for computing variance require strong restrictions on admissible intervals like the no-subset property or heavy limitations on the number of possible intersections between intervals. A new asymptotic algorithm for computing the upper bound of the sample variance in a feasible time is proposed. Conditions required for its application with finite samples are discussed and some properties of the algorithm are also given. It appears that our new algorithm could be effectively applied in definitely more situations than methods used so far. [ABSTRACT FROM AUTHOR]

Some approaches to evaluating the predictive validity of models of consumer choice behavior are developed for the situation in which predictions are made from one sample of consumers and are validated on the basis of a second sample of consumers. This situation is becoming more common in marketing. [ABSTRACT FROM AUTHOR]

The analysis of marketing data is frequently complicated by factorial designs with unequal cell sizes. This article discusses the marketing research advantages and problems of the General Linear Hypothesis approach to such analysis and illustrates the General Linear Hypothesis in the context of a new product experiment. [ABSTRACT FROM AUTHOR]

This paper is concerned with generalizability issues in experimental consumer research. In particular, it discusses how the usefulness of experimental results is affected by a consumer researcher's treatment of unmanipulated "background" factors in designing and analyzing the experiment. [ABSTRACT FROM AUTHOR]

This study analyzes correlates of adherence to a physical fitness regimen. The sample consisted of adults who had volunteered to take a physical fitness stress test at a sports physiology laboratory. A self-administered questionnaire was mailed to persons who had taken the stress test over the past 5 years. The rationale for the study was an attempt to develop a profile of the prospective dropout from a physical tuners program in order to permit special intervention to enhance the probability of adherence. The findings showed that 31% of the persons in our sample were basically inactive subsequent to the stress test. Our set of 11 predictor variables was able to explain 33% of the variance in physical activity. [ABSTRACT FROM AUTHOR]

Marketing researchers frequently face constraints on the size of the sample they can assemble, and may feel that quantitative analysis cannot be undertaken. This article seeks to illustrate approaches that improve the performance of regression type techniques with small samples. The article focuses on areas where small samples can lead to problems, particularly test power, violation of assumptions, collinearity and validation. Sample reuse validation techniques such as jackknife and bootstrap are explained, and some general guidelines are given. [ABSTRACT FROM AUTHOR]

The article discusses the alternative tests for the independence of the financial time series. The previous research on the financial time series has used the four tests for the independence of successive changes of net income and earnings per share, the runs test, serial correlations, Von Neumann ratios and partial adjustment models. The analysis of the independence recognizes that the sample variance for a randomly selected sample of firms will be considerably less than the variance of observations expected for a single representative firm.

Predictive regressions are subject to two small sample biases: the coefficient estimate is biased if the predictor is endogenous, and asymptotic standard errors in the case of overlapping periods are biased downward. Both biases work in the direction of making t-ratios too large so that standard inference may indicate predictability even if none is present. Using annual returns since 1872 and monthly returns since 1927 we estimate empirical distributions by randomizing residuals in the VAR representation of the variables. The estimated biases are large enough to affect inference in practice, and should be accounted for when studying predictability. [ABSTRACT FROM AUTHOR]

The article presents information about determining dependable samples for marketing surveys. The author explains the feasibility and desirability of using probability samples and stresses the importance of thoughtful review and consideration of whether a probability sample is the best in a particular circumstance. Details related to the planning and cost of obtaining a probability sample are presented. A discussion is presented about the use of probability samples in prediction and the reliability of prediction based on these samples.

Sample estimates of predictive power in N-way discriminant analysis are likely to be subject to a strong upward bias. This bias occurs because the discriminant technique tends to fit the sample data in ways that are systematically better than would be expected by chance, even if the underlying populations are identical (i.e., no predictive power truly exists). Sample tests of predictive power against chance models are often invalid, and no simple methods of adjusting for the bias are available, as in the analogous case of multiple regression. The nature and causes of sample bias are discussed and two validation procedures are presented and illustrated that can be used to obtain realistic estimates of predictive power in discriminant analysis. [ABSTRACT FROM AUTHOR]

Sequential estimation is proposed as a rigorous, asymptotically valid technique in cases where it is administratively feasible to draw sample units one at a time. A special two-sample procedure is described as an alternative when if is more practical to conduct the sampling in large batches. The validity of neither technique requires prior knowledge of assumptions about the variance of the response parameter. [ABSTRACT FROM AUTHOR]

Conventional telephone directory sampling techniques fall short of producing a perfect sample of domestic telephone subscribers. Near the end of the 1963-64 local telephone directory year, when the number of new, but as yet unlisted, telephones was at its maximum, 18 percent of private telephones in use in the Greater Cincinnati area were not listed in the current directory. By means of a new technique for obtaining numbers, one large source of sampling error can now be virtually eliminated in sampling private telephone subscribers, whether listed or not. [ABSTRACT FROM AUTHOR]

City directory and block supplement techniques are used as the sample design for a scattered metropolitan population. The problems arising out of varying selection rates within dwelling units are handled by the use of a screening phase in the field and subsequent stratification of dwelling units by number of eligible respondents. The results of this design point up the feasibility of using a city directory technique to sample widely scattered groups with relatively small increases in sample variances. [ABSTRACT FROM AUTHOR]

The article discusses predetermining sample sizes in marketing studies. Market analysts say sample size is critical in accurately measuring market characteristics. Analysts use statistical methods including the arithmetic mean, or arithmetic average and the percentage proportion of a total. These methods present problems of "true" measurement for analysts. A method of obtaining an arithmetic average of a desired accuracy is presented. The equation incorporates the desired number of cases to ensure accuracy, the accuracy necessary for the arithmetic average, and the range of data in which all data will fall.

Provides statistical analysis of the results of testing of students involved in the experiment conducted by the Department of Higher Mathematics at the Samara State University of Railways, since the resulting test data (rates of assimilation of educational information), which are taken as random variables are a set of numbers, which is difficult to detect any pattern of change (variation). Built interval variation series calculated the most important numerical characteristics of random variable - the sample mean (arithmetic mean value of the flag of sample), the sample variance (the arithmetic average of the squared deviations of the observed values of the trait from their average values) and sample standard deviation was given the opportunity to build a histogram of relative frequencies - a stepped shape consisting of rectangles whose bases are partial intervals, and the height equal to the density of the relative frequency. The area of the histogram is equal to the sum of all the relative frequencies, ie, unit. Combining adjacent mid-upper side of the rectangle of the histogram line segments, we received a broken line, called line empirical density. By type of line empiric density statistic put forward the hypothesis of normal distribution of the random variable. To test this hypothesis using one of the criteria for approval - a specially selected random variable, exact or approximate distribution, is known. Pearson , which consists in comparing empirical and theoretical frequencies falls into the "region of acceptance of the hypothesis", is therefore considered a random variable subject to the normal distribution law. This makes it possible to determine its unknown parameters to estimate the unknown expectation (average rate of assimilation of educational information) found using a sample according to the sample average. Assessment of the probability that a random value within the range of learning, characterized by a deficiency in the assimilation of educational material, leads to the conclusion that about 30% of students will require more self-educational activity to achieve a satisfactory invariant form of self-competence. Selection of the normal distribution curve also allows you to build a scale of success of training, that is practiced standard estimates.

Provide statistical processing and analysis of results of pedagogical experiment on introduction of innovative methods based on the matrix model of cognitive activity carried out in Samara state University of railway engineering teachers of the Department "Higher mathematics". To confirm the effectiveness of the proposed method investigates four stochastic variables that characterize the coefficient of mastering of educational material by the individual student resulting from two control tests in experimental and control groups. As the result of an experiment, data is a set of numbers, which are difficult to capture any pattern of change (variation), they are subjected to statistical processing. Built for this interval variational series; the calculated basic numerical characteristics: sample average (arithmetic average value of the attribute of the sample), sample variance (average of squared deviations of observed values from their average values) and sample standard deviation (square root of sample variance). Determined the observed value of goodness-of-fit Pearson that allows you to install, after comparing them with critical values that all of the investigated random variable subject to normal distribution law. Built line of empirical densities of the results obtained in control and experimental groups after the first and after the second test, allowing, to estimate dynamics of process of mastering of educational material. The calculated coefficients of variation characterizing the dispersion of the variational series. Thus it is shown that in the experimental group by the end of training not only increased the average (10% compared with the control group), but decreased the dispersion of results about the mean (16,55 to 13,67 %), which undoubtedly confirms the effectiveness of the proposed method.

While unbiased central moment estimators of lower orders (such as a sample variance) are easily obtainable and often used in practice, derivation of unbiased estimators of higher orders might be more challenging due to long math and tricky combinatorics. Moreover, higher orders necessitate calculation of estimators of powers and products that also amount to these orders. We develop a software algorithm that allows the user to obtain unbiased estimators of an arbitrary order and provide results up to the sixth order, including powers and products of lower orders. The method also extends to finding pooled estimates of higher central moments of several different populations (e.g. for two-sample tests). We introduce an R package Umoments that calculates one- and two-sample estimates and generates intermediate results used to obtain these estimators. [ABSTRACT FROM AUTHOR]

This article presents a reflection and a methodological proposal on the use of the typological paragon in mixed sequential designs. The paragon is defined as the average or individual profile of a particular cluster that is obtained through multivariate techniques (correspondences analysis and cluster analysis). Here we propose four applications of the typological paragon: first, as a descriptive element, which aims to account for the content; second, as an element of sampling; third, as a link or connecting element between databases with the aim of facilitating matching; and finally, as an element that allows the convergence of results. Two studies provide the empirical basis of this innovative procedure and show the practical implementation of the four uses of paragon. The first deals with the analysis of the social stratification of households in Argentina (Fachelli, 2009). The second focuses on the study and identification of employment trajectories and their mutual influence with social capital in Spain and Catalonia (Alcaide, 2013). The four uses of the paragon that derive from both investigations allow us to reflect on the potentialities and limitations of the proposed model, the typological paragon. [ABSTRACT FROM AUTHOR]