  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.
