This paper addresses the problem of outlier detection and trimmed-average state estimation in an LTI network system. We consider that only some states are measured and there exists an outlier among the unmeasured states, which is so different from the remaining states that it affects the average value significantly. The goal of this paper is both to detect the outlier and to estimate the average state excluding the outlier (trimmed-average). Moreover, we also investigate the case where the system matrices are partially unknown since the outlier results from an unknown localized fault in the system. Finally, we illustrate the method on a thermal diffusion system.