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This paper presents frameworks to extend the mean and median filtering schemes in image processing to smoothing noisy 3D shapes given by triangle meshes. The frameworks consist of the application of the mean and median filters to face normals on triangle meshes and the editing of mesh vertex positions to make them fit the modified normals. We also give a quantitative evaluation of the proposed mesh filtering schemes and compare them with conventional mesh smoothing methods such as Laplacian smoothing flow and mean curvature flow. The quantitative evaluation is performed in error metrics on mesh vertices and normals. Experimental results demonstrate that our mesh mean and median filtering methods are more stable than conventional Laplacian and mean curvature flows. We propose thee new mesh smoothing methods as one possible solution of the oversmoothing problem.