This paper recovers data from quantized and partially corrupted measurements. The data recovery is achieved through solving a constrained maximum likelihood estimation problem that exploits the low-rank property of the actual measurements. The recovery error is proven to be order optimal and decays in the same order as that of the state-of-the-art method when no corruption exists. The data accuracy is thus maintained while the data privacy is enhanced. A new application of this method for data privacy in power systems is discussed. Experiments on synthetic data and real synchrophasor data in power systems demonstrate the effectiveness of our method.