Least squares minimization is by nature global and, hence, vulnerable to distortion by outliers. We present a novel technique to reject outliers from an m-dimensional data set when the underlying model is a hyperplane (a line in two dimensions, a plane in three dimensions). The technique has a sound statistical basis and assumes that Gaussian noise corrupts the otherwise valid data. The majority of alternative techniques available in the literature focus on ordinary least squares, where a single variable is designated to be dependent on all others - a model that is often unsuitable in practice. The method presented here operates in the more general framework of orthogonal regression, and uses a new regression diagnostic based on eigendecomposition. It subsumes the traditional residuals scheme and, using matrix perturbation theory, provides an error model for the solution once the contaminants have been removed.