While working on an infinite barrier problem I was surprised to find that the location of |psi|^2 = 0.30 a while <x> is 0.32 a, where a is the width of the well. It's often said that |psi|^2 tells one about the probabilty of finding a particle in a location x. Yet the expectation of x, <x> is a slightly different value. How do I reconcile the difference between these two values? If <x> is the average of values if I observe a large number of particles prepared in the same way, then what exactly is the peak of |psi|^2 ?