We consider the effects of short orbits on the statistical properties of wave chaotic systems open to outside scattering channels. A theoretical expression for corrections to the scattering impedance is introduced and tested through a series of experiments on a thin ray-chaotic microwave cavity. The theory is found to provide a good description of non-universal effects due to short orbits in the experimental data. These results should be broadly useful in nuclear scattering, atomic physics, quantum transport in condensed matter systems, electromagnetics, acoustics, geophysics, etc.