Estimation of population variance using the actual ranks and values: Properties and application

Abstract

Background: Of all measures of variability, the variance is by far the most popular. This paper argues an alternative index of variability, shares many properties with the variance, but can be more informative about the properties of distributions that depart from normality. Procedures: An estimator for the population variance is proposed. The proposed estimator is based, in its calculations, on the actual variable values and their ranks. A simulation study is employed to compare the performance of the proposed estimator with that of eight other estimators appearing in the literature using the mean square error criterion. The influence of sample size, normality, and non-normality is investigated. We suggest a new measure of skewness based on the new proposed estimator for the population variance. The mean value, median, and the new estimator for the population variance are the only three summary statistics that may be effectively used to derive the suggested coefficient of skewness. A limited Monte Carlo simulation is used to examine the potential of the suggested statistic to identify skewness. The covariance between a random variable and its rankings is how the new estimator is stated. This format is used to derive new correlation coefficients. Results and conclusion: The simulation study shows that the proposed estimator, under conditions of normality, performs better than most estimators considered, whereas it fails to give better results than some others. In cases of non-normality considered, the proposed estimator was superior to eight competitors. According to preliminary results, the new coefficient of skewness performs well overall in a limited simulation. The new correlation coefficient outperforms Pearson’s correlation for the non-normal model, as demonstrated by simulation analysis. Furthermore, it has been demonstrated that the new estimator is a useful tool for extracting a straightforward method to obtain novel skewness and correlation coefficients.