The main philosophical lesson of holographic gravity is that spatiotemporal descriptions of a field theory may not be respected by observational reality. We apply this lesson to the correspondence principle, and then later justify the lesson (holographic correspondence) without necessitating such spatiotemporal deviations via the law of large numbers in statistics, applied to an average of n quantum measurements. Classical observables are considered to emerge as n approaches infinity. The averaging approach can be interpreted both as a generalization of Bohr's correspondence principle (large number limit) and Planck's correspondence principle (Planck's constant approaching zero). Holographic correspondence and the averaging approach are then re-analyzed in light of preceding works in the correspondence principle. A key consequential lesson is that we should not conflate the nature of fundamental theory observables with that of emergent theory observables, and once this confusion is avoided, the correspondence principle has a solid standing.