Quantum uncertainty is a fundamental concept in quantum mechanics, but challenging for students to master. In this article, we describe student difficulties with visual and conceptual understanding of quantum uncertainty in the context of discrete probability distributions such as those for a spin 1/2 particle. We collected written responses from students at two institutions to a homework activity focusing on uncertainty of spin measurement outcomes, as well as written responses to a test question from one of the institutions. We also conducted interviews with six students to gain further insight into difficulties found. Common incorrect ideas found included a depiction of uncertainty as the error around each of the individual measurement outcomes,  not depicting the uncertainty region from the expectation value outwards, and the idea that quantum uncertainty of an observable can never be zero. These ideas may indicate a confusion between quantum uncertainty and errors due to instrumental imperfections of the measurement apparatus, a lack of conceptual understanding of quantum uncertainty as the standard deviation of the probability distribution with respect to its mean, and an incorrect interpretation of the uncertainty relation between two incompatible observables to deduce that quantum uncertainty can never be zero. The results of this study show the importance of supporting students in visual and conceptual understanding of quantum uncertainty.