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This paper addresses the problem of the distributed delivery of correlated data sources with help of network coding. Network coding provides an alternative to routing algorithms and offers improved system performance, robustness and throughput, with no need of deploying sophisticated routing strategies. However, the performance is directly driven by the number of innovative data packets that reach the receiver. If the number of received innovative data packets is significantly small, the decoder cannot perfectly recover the transmitted information. However, we show that the correlation between the data sources can be used at decoder for effective approximate decoding. We analytically investigate the impact of the network coding algorithm, and in particular, of the size of finite fields on the decoding performance. Then, we determine an optimal field size that minimizes the expected decoding error, which represents a trade-off between quantization of the source data and probability of decoding error. The network coding with approximate decoding algorithm is implemented in illustrative multimedia streaming and sensor network applications. In both cases, the experimental results confirm the field size analysis and illustrate the effectiveness of approximate decoding of correlated data.