The expectation value of an ordinary die roll is 7/2. But what if we keep adding more and more faces? As faces are added, the probability of any one face collapses toward 0, but the probability density, the probability divided by the face width, survives. The sum then turns into an integral — and that probability density is |ψ|². Finally, we compare the eigenvalue, which comes out the same no matter how many times you measure, with the expectation value, the center of the spread.
Mochizuki Research Group, Tokyo University of Science / Based on lecture notes “(19) Expectation Values (Means)” and Lecture 2, “How to Calculate Physical Quantities in Quantum Mechanics,” of the course “Computational Materials Science.” The derivation of the formulas, and why the average in quantum mechanics is exactly the expectation value, are on the explanation page (in Japanese).