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Wireless sensor networks (WSNs) enable the collection of physical measurements over a large geographic area. It is often the case that we are interested in computing and tracking the spatial-average of the sensor measurements over a region of the WSN. Unfortunately, the standard average operation is not robust because it is highly susceptible to sensor faults and heterogeneous measurement noise. In this paper, we propose a computational efficient method to compute a weighted average (which we will call robust average) of sensor measurements, which appropriately takes sensor faults and sensor noise into consideration. We assume that the sensors in the WSN use random projections to compress the data and send the compressed data to the data fusion centre. Computational efficiency of our method is achieved by having the data fusion centre work directly with the compressed data streams. The key advantage of our proposed method is that the data fusion centre only needs to perform decompression once to compute the robust average, thus greatly reducing the computational requirements. We apply our proposed method to the data collected from two WSN deployments to demonstrate its efficiency and accuracy.