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Highly concentrated distributed crystallographic orientation measurements within individual crystalline grains are analysed by means of ordinary statistics neglecting their spatial reference. Since crystallographic orientations are modelled as left cosets of a given subgroup of SO(3), the non-spatial statistical analysis adapts ideas borrowed from the Bingham quaternion distribution on {\bb S}^3. Special emphasis is put on the mathematical definition and the numerical determination of a `mean orientation' characterizing the crystallographic grain as well as on distinguishing several types of symmetry of the orientation distribution with respect to the mean orientation, like spherical, prolate or oblate symmetry. Applications to simulated as well as to experimental data are presented. All computations have been done with the free and open-source texture toolbox MTEX.

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