  We show that standard deviation $\s$ satisfies the Leibniz inequality $\s(fg)
\leq \s(f)\|g\| + \|f\|\s(g)$ for bounded functions f, g on a probability
space, where the norm is the supremum norm. A related inequality that we refer
to as "strong" is also shown to hold. We show that these in fact hold also for
non-commutative probability spaces. We extend this to the case of matricial
seminorms on a unital C*-algebra, which leads us to treat also the case of a
conditional expectation from a unital C*-algebra onto a unital C*-subalgebra.
