The problem of selecting a prior distribution when it comes to Bayes estimation often constitutes a choice between conjugate or noninformative priors, since in both cases the resulting posterior Bayes estimator (PBE) can be solved analytically and is therefore easy to calculate. Nevertheless, some of the implicit assumptions made by choosing a certain prior can be difficult to justify when a concrete sample of small size has been drawn. For example, when the underlying distribution is assumed to be normal, there is no reason to expect that the true but unknown location parameter is located outside the range of the sample. So why should a distribution with a non-compact domain be used as a prior for the mean? In addition, if there is some skewness in a sample of small size due to outliers when a symmetric distribution is assumed, this finding can be used to correct the PBE when determining the hyperparameters. Both ideas are applied to an empirical Bayes approach called plausible prior estimation (PPE) in the case of estimating the mean of a normal distribution with known variance in the presence of outliers. We propose an approach for choosing a prior and its respective hyperparameters, taking into account the above mentioned considerations. The resulting influence function as a frequentistic measure of robustness is simulated. To conclude, several simulation studies have been carried out to analyze the frequentistic performance of the PPE in comparison to frequentistic and Bayes estimators in certain outlier scenarios.