This work sheds some light on the relationship between a distribution's standard deviation and its range, a topic that has been discussed extensively in the literature. While many previous studies have proposed inequalities or relationships that depend on the shape of the population distribution, the approach here is built on a family of bounded probability distributions based on skewing functions. We offer closed-form expressions for its moments and the asymptotic behavior as the support's semi-range tends to zero and $\infty$. We also establish an inequality in which the well-known Popoviciu's one is a special case. Finally, we provide an example using US dollar prices in four different currencies traded on foreign exchange markets to illustrate the results developed here.

翻译：本文揭示了分布标准差与其支撑区间之间的关系，该主题已在文献中得到广泛讨论。以往研究多提出依赖于总体分布形状的不等式或关系，而本文的方法基于偏斜函数构建有界概率分布族。我们给出了该分布族各阶矩的闭式表达式，并分析了支撑区间半长趋近于零和无穷大时的渐近行为。同时建立了新的不等式，其中著名的波波维丘不等式为其特例。最后，通过四个外汇市场交易的不同货币计价的美元价格实例，验证了本文所提出的结果。