Observable

An observable represents a measurable quantity in quantum mechanics; in the standard projective description it is a self-adjoint operator on the system's Hilbert space. Its spectrum gives the possible measurement values, and its spectral projections give their probabilities through the Born rule.

For a qubit, the Pauli matrix $Z$ observable has outcomes $+1$ and $-1$. In the state $\alpha |0⟩+\beta |1⟩$, their probabilities are $|\alpha {|}^2$ and $|\beta {|}^2$, so the expectation value is $|\alpha {|}^2-|\beta {|}^2$. That average need not be an allowed individual outcome. For unbounded observables such as position and momentum, operator domains matter. More general measurement procedures use a positive operator-valued measure and need not correspond to one projective observable alone.