Two-dimensional lead (Pb) islands of varying heights have been grown on the Si(111)-(7 x 7) surface at low temperature. Individual islands are investigated concurrently with real-space and local-probe scanning tunneling microscopy and spectroscopy. Quantum size effects, manifested in the formation of new electronic bound states, redistribution of surface charge density, and oscillatory relaxations in island thickness are found to be perfectly correlated to each other.