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Experimental scientific data sets, especially biology data, usually contain replicated measurements. The replicated measurements for the same object are correlated, and this correlation must be carefully dealt with in scientific analysis. In this paper, we propose a robust Bayesian mixture model for clustering data sets with replicated measurements. The model aims not only to accurately cluster the data points taking the replicated measurements into consideration, but also to find the outliers (i.e., scattered objects) which are possibly required to be studied further. A tree-structured variational Bayes (VB) algorithm is developed to carry out model fitting. Experimental studies showed that our model compares favorably with the infinite Gaussian mixture model, while maintaining computational simplicity. We demonstrate the benefits of including the replicated measurements in the model, in terms of improved outlier detection rates in varying measurement uncertainty conditions. Finally, we apply the approach to clustering biological transcriptomics mRNA expression data sets with replicated measurements.