The distributed estimation of the average value of the sensors initial measures is one of the most popular issues in the Wireless Sensor Networks (WSN) area. In WSNs, broadcasting data seems natural to exchange information quickly because of the broadcast nature of the Wireless channel. Nevertheless, although broadcast-based algorithms converge faster than pairwise algorithms, the obtained consensus is not necessarily the true average. By the means of additional side-information exchange, we propose a broadcast-based algorithm converging rapidly to the true average. The convergence of this new algorithm is established and its convergence speed is exhibited. We remark that the proposed algorithm outperforms the existing ones.