A New Opinion Polarization Index Developed by Integrating Expert Judgments

Opinion polarization is increasingly becoming an issue in today’s society, producing both unrest at the societal level, and conflict within small scale communications between people of opposite opinion. Often, opinion polarization is conceptualized as the direct opposite of agreement and consequently operationalized as an index of dispersion. However, in doing so, researchers fail to account for the bimodality that is characteristic of a polarized opinion distribution. A valid measurement of opinion polarization would enable us to predict when, and on what issues conflict may arise. The current study is aimed at developing and validating a new index of opinion polarization. The weights of this index were derived from utilizing the knowledge of 58 international experts on polarization through an expert survey. The resulting Opinion Polarization Index predicted expert polarization scores in opinion distributions better than common measures of polarization, such as the standard deviation, Van der Eijk’s polarization measure and Esteban and Ray’s polarization index. We reflect on the use of expert ratings for the development of measurements in this case, and more in general.

The Mathematics of Statistics

This chapter looks at how the application of error theory and of probability models to social statistics was pursued with growing success in Germany during the last third of the nineteenth century. The most successful and influential of those mathematical writers on statistics was the economist and statistician Wilhelm Lexis. The chapter then studies the index of dispersion that Lexis introduced in 1879. Lexis's writings on dispersion and the distribution of human physical attributes were influential within German anthropometry, which began to make interesting use of the analytical techniques associated with the error law during the last quarter of the nineteenth century. Meanwhile, Francis Edgeworth, the poet of statisticians, was led to probability in the context of his campaign to introduce advanced mathematics into the moral and social sciences. He hoped through analogies to bring the same rigor and elegance to economics and ethics.

A note on size– biased new quasi Poisson– Lindley distribution

In this paper some important properties including coefficients of variation, skewness, kurtosis and index of dispersion of size–biased new quasi Poisson–Lindley distribution (SBNQPLD) have been discussed and their behaviors have been explained graphically for varying values of parameters. Some applications of SBNQPLD have also been discussed.

A zero-truncated discrete Akash distribution with properties and applications

This study proposes and examines a zero-truncated discrete Akash distribution and obtains its probability and moment-generating functions. Its moments and moments-based statistical constants, including coefficient of variation, skewness, kurtosis, and the index of dispersion, are also presented. The parameter estimation is discussed using both the method of moments and maximum likelihood. Applications of the distribution are explained through three examples of real datasets, which demonstrate that the zero-truncated discrete Akash distribution gives better fit than several zero-truncated discrete distributions.

A generalization of Poisson-Sujatha distribution and its applications to ecology

A generalization of Poisson Sujatha distribution (AGPSD), which includes Poisson-Lindley distribution (PLD) and Poisson-Sujatha distribution (PSD) as particular cases, has been proposed and studied. Its moments and moments-based measures including coefficient of variation, skewness, kurtosis and index of dispersion have been obtained and their behaviors have been discussed. The estimation of its parameters has been discussed with maximum likelihood estimation. The applications of the proposed distribution has been explained through two examples of count data from ecology and the goodness of fit of the distribution has been compared with Poisson distribution, PLD and PSD.

A Dispersion Test for the Modified Borel-Tanner Distribution

Dispersion tests based on the second order component of smooth test statistics are related to Fisher’s Index of Dispersion test, used for testing for the Poisson distribution when there are no covariates present. Such tests have been recommended in [1] to test for the Poisson distribution when covariates are present. The modified Borel-Tanner (MBT) distribution seems suited to data with extra zeroes, a monotonic decline in counts and longer tails. Here we recommend a dispersion test for the MBT distribution for both when covariates are absent and when they are present.

The index of dispersion as a metric of quanta – unravelling the Fano factor

In statistics, the index of dispersion (or variance-to-mean ratio) is unity (σ2/〈x〉 = 1) for a Poisson-distributed process with variance σ2for a variablexthat manifests as unit increments. Wherexis a measure of some phenomenon, the index takes on a value proportional to the quanta that constitute the phenomenon. That outcome might thus be anticipated to apply for an enormously wide variety of applied measurements of quantum phenomena. However, in a photon-energy proportional radiation detector, a set ofMwitnessed Poisson-distributed measurements {W1,W2,…WM} scaled so that the ideal expectation value of the quantum is unity, is generally observed to give σ2/〈W〉 < 1 because of detector losses as broadly indicated by Fano [Phys. Rev.(1947),72, 26]. In other cases where there is spectral dispersion, σ2/〈W〉 > 1. Here these situations are examined analytically, in Monte Carlo simulations, and experimentally. The efforts reveal a powerful metric of quanta broadly associated with such measurements, where the extension has been made to polychromatic and lossy situations. In doing so, the index of dispersion's variously established yet curiously overlooked role as a metric of underlying quanta is indicated. The work's X-ray aspects have very diverse utility and have begun to find applications in radiography and tomography, where the ability to extract spectral information from conventional intensity detectors enables a superior level of material and source characterization.