A solid grasp of the probability distributions for measuring physical observables is central to
connecting the quantum formalism to measurements. However, students often struggle with the
probability distributions of measurement outcomes for an observable and have difficulty expressing
this concept in different representations. Here we first describe the difficulties that upper-level
undergraduate and PhD students have with the probability distributions for measuring physical
observables in quantum mechanics. We then discuss how student difficulties found in written surveys
and individual interviews were used as a guide in the development of a quantum interactive learning
tutorial (QuILT) to help students develop a good grasp of the probability distributions of
measurement outcomes for physical observables. The QuILT strives to help students become proficient
in expressing the probability distributions for the measurement of physical observables in Dirac
notation and in the posi…