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The aim of the present research work is to investigate algorithms to compute empirical averages of finite sets of sample-points over the Stiefel manifold by extending the notion of Pythagoras' arithmetic averaging over the real line to a curved manifold. The idea underlying the developed algorithms is that sample-points on the Stiefel manifold get mapped onto a tangent space, where the average is taken, and then the average point on the tangent space is brought back to the Stiefel manifold, via appropriate maps. Numerical experimental results are shown and commented on in order to illustrate the numerical behaviour of the proposed procedure. The obtained numerical results confirm that the developed algorithms converge steadily and in a few iterations and that they are able to cope with relatively large-size problems.