Question #32237

I assume you are asking the question in context of quantum or statistical mechanics.

So, for a system you can have certain observables, for example considering a single particle a system, it's position can be an observable or it can be any other dynamic variable.

So you measure the position of the particle (within limits of uncertainty principle) and you find it at a position at an instant. After a few moments, you measure it again and you find it somewhere else. So you make repeated measurements after certain intervals and you obtain a table of observations of position of the particle at various instants of time.

The mean or expectation value is simply the average of the measurements.

In Quantum mechanics, if the wavefunction is known, you can calculate the mean or expectation value of any dynamic variable theoretically and such predictions fit well with experimental observations as well (ofcourse a single particle system is a purely theoretical one for simplicity and is not realized in experiment).

Within limits of statistical error, the expectation value agrees well between theory and experiment.