Let $A$ be an observable, then, is there a way to experimentally determine the mean of $\hat A$, namely $\langle \hat A \rangle $?
I mean, for example, consider the position operator; it is intuitively plausible that if a particle can be in $x =0$ (for a fixed reference frame), then another particle can also be in $x = 10, 100, 1000$, or for any other value, there is nothing special about $x=0$, or another value. However, we never went to the edge of our galaxy and tested whether a particle can indeed exists there.
(Of course astronomical observations can be made for lots regions in space, I'm sure there are some regions in space where nobody has ever observed another matter there; Addition to that, even we observed any matter, we still wouldn't know whether is made of matter or antimatter - there might be some update on that topic which I'm now aware of the I'm digressing).
This is, in a sense, a trivial example, but, in general, how can we measure $\langle \hat A \rangle$ ?