We present an analysis of 16 galaxy clusters, one group, and one galaxy drawn from the Chandra Data Archive. These systems possess prominent X-ray surface brightness depressions associated with cavities or bubbles that were created by interactions between powerful radio sources and the surrounding hot gas. The central galaxies in these systems harbor radio sources with luminosities ranging between ∼2 × 1038 and 7 × 1044 ergs s-1. The cavities have an average radius of ∼10 kpc, and they lie at an average projected distance of ∼20 kpc from the central galaxy. The minimum energy associated with the cavities ranges from pV ∼ 1055 ergs in galaxies, groups, and poor clusters to pV ∼ 1060 ergs in rich clusters. We evaluate the hypothesis that cooling in the hot gas can be quenched by energy injected into the surrounding gas by the rising bubbles. We find that the instantaneous mechanical luminosities required to offset cooling range between 1pV and 20pV per cavity. Nearly half of the systems in this study may have instantaneous mechanical luminosities large enough to balance cooling, at least for a short period of time, if the cavities are filled with a relativistic gas. We find a trend or upper envelope in the distribution of central X-ray luminosity versus instantaneous mechanical luminosity, with the sense that the most powerful cavities are found in the most X-ray–luminous systems. Such a trend would be expected if many of these systems produce bubbles at a rate that scales in proportion to the cooling rate of the surrounding gas. Finally, we use the X-ray cavities to measure the mechanical power of radio sources over six decades of radio luminosity, independently of the radio properties themselves. We find that the ratio of the instantaneous mechanical (kinetic) luminosity to the 1.4 GHz synchrotron luminosity ranges typically between a few and roughly a few thousand for luminous radio sources but can be several thousand for weaker sources. This wide range implies that the 1.4 GHz synchrotron luminosity is an unreliable gauge of the mechanical power of radio sources.