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The paper studies distributed average consensus in sensor networks, when the sensors exchange quantized data at each time step. We show that randomizing the exchanged sensor data by adding a controlled amount of dither results in almost sure (a.s.) convergence of the protocol, if the network is connected. We explicitly characterize the mean-squared error (with respect to the desired consensus average) and show that, by tuning certain parameters associated with the protocol, the mean-squared error can be made arbitrarily small. We study the trade-offs between the rate of convergence and the resulting mean-squared error. The sensor network topology plays an important role in determining the convergence rate of the algorithm. Our approach, based on the convergence of controlled Markov processes, is very generic and can be applied to many other situations of imperfect communication. Finally, we present numerical studies, which verify our theoretical results.