The inspiration for the mathematical description of quantum systems rests on the now-common practice of using Hilbert space operators and operator-valued functions in place of complex numbers and complex-valued functions, where this latter pair is well suited for classical systems. In particular, the mathematical description of the measurement of quantum systems is founded upon positive operator-valued measures (POVMs). If the total mass of a POVM is the identity operator, then the POVM is viewed as a quantum probability measure, leading to quantum formulations of the basic statistical concepts of averaging, variance, and Bayes' rule. In this lecture, I will survey a selection of joint work with Remus Floricel, Michael Kozdron, and Sarah Plosker that examines these statistical ideas in the quantum setting.