We show that standard deviation σ satisfies the Leibniz inequality σ(fg) ≦ σ(f)∥g∥ + ∥f∥σ(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 noncommutative 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.

The research reported here was supported in part by National Science Foundation grant DMS-1066368