A rapidly spinning, strongly magnetized neutron star (magnetar) has been proposed as one possible candidate of the central engine of gamma-ray bursts (GRBs). We systematically analyze the Swift/XRT light curves of long GRBs detected before 2013 August, and characterize them into four categories based on how likely they may harbor a magnetar central engine: Gold, Silver, Aluminum, and Non-magnetar. We also independently analyze the data of short GRBs with a putative magnetar central engine. We then perform a statistical study of various properties of the magnetar samples and the non-magnetar sample, and investigate whether the data are consistent with the hypothesis that there exist two types of central engines. By deriving the physical parameters of the putative magnetars, we find that the observations of the Gold and Silver samples are generally consistent with the predictions of the magnetar model. For a reasonable beaming factor for long GRBs, the derived magnetar surface magnetic field $B_p$ and initial spin period $P_0$ fall into the reasonable range. Magnetar winds in short GRBs, on the other hand, are consistent with being isotropic. No GRB in the magnetar sample has a beam-corrected total energy exceeding the maximum energy budget defined by the initial spin energy of the magnetar, while some non-magnetar GRBs do violate such a limit. With beaming correction, on average the non-magnetar sample is more energetic and luminous than the magnetar samples. Our analysis hints that millisecond magnetars are likely operating in a good fraction, but probably not all, GRBs.