Algebraic Inequalities for Measures of Dispersion

Abstract

Some upper and lower bounds for sample standard deviation are established in terms of sample mean, median, range, the smallest and the largest order statistics. Upper bounds for variance are also derived for odd and even sample sizes whenever the sample observations are of the same sign. They are used to find bounds for some well-known sample statistics: z-scores, coefficient of variation, coefficient of skewness and the least squares estimator of the slope parameter in the context of a simple linear regression. Statistical inference of related parameters can be improved on the basis of these fixed sample properties.