We systematically analyze the prompt emission and the early afterglow data of a sample of 31 GRBs detected by Swift before 2005 September and estimate the GRB radiative efficiency. BAT's narrow band inhibits a precise determination of the GRB spectral parameters, and we have developed a method to estimate these parameters with the hardness ratio information. The shallow decay component commonly existing in early X-ray afterglows, if interpreted as continuous energy injection in the external shock, suggests that the GRB efficiencies previously derived from the late-time X-ray data were not reliable. We calculate two radiative efficiencies using the afterglow kinetic energy E sub(K) derived at the putative deceleration time (t sub(dec)) and at the break time (t sub(b)), when the energy injection phase ends, respectively. At t sub(b) XRFs appear to be less efficient than normal GRBs. However, when we analyze the data at t sub(dec), XRFs are found to be as efficient as GRBs. Short GRBs have similar radiative efficiencies to long GRBs despite of their different progenitors. Twenty-two bursts in the sample are identified to have the afterglow cooling frequency below the X-ray band. Assuming e sub(e) = 0.1, we find g(t sub(b))usually <10% and g(t sub(dec)) varying from a few percent to >90%. Nine GRBs in the sample have the afterglow cooling frequency above the X-ray band for a very long time. This suggests a very small e sub(B) and/or a very low ambient density n.