Murthy and Sethi gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable. We generalise this bound to the complex case and also to the quantum case. In doing so, we make contact with several geometrical and matrix analytical concepts, such as the numerical range. Based on the quantum bound, we give a new and simplified proof for an upper bound on the Frobenius norm of commutators recently proven by Bottcher and Wenzel (B W). We also make some headway to generalise the B W bound to other norms. This is ongoing work and our proofs may still contain some pleasant gaps at the time of the presentation.