As part of ongoing research on student thinking about quantum mechanical concepts and formalism, we explored how students defined and made sense of expectation values. Previous research has focused on student difficulties when defining the expectation value for a generic operator and found that students conflate other quantum mechanical ideas with expectation values. In this study, we analyzed survey data collected from two universities over a number of years and interviews carried out at two points during the semester. With a focus on underlying student thinking, we used a concept image perspective to categorize students' concept definitions of the expectation values in the context of measuring spin in a spin-1/2 system and in the contexts of measuring energy and position for a particle in an infinite square well potential. Analysis of interview data showed that students invoke many different ideas when explaining their reasoning. The two most common definitions for the expectation value were weighted average and most probable value. In interviews, students' definitions were influenced by whether the problem context involves continuous or discrete observables.