Modern versions of the Single Kernel Characterization System allow force profiles recorded for individual kernels during crushing to be retrieved. The current understanding of the meaning of the averaged Crush Response Profile (aCRP) is challenged in the current work. When bran layers are removed, the features of the aCRP that have been interpreted as the bran layers' elastic response and subsequent collapse are retained. By contrast, when the crease is removed, these features are largely eliminated. The crease depth correlates with the first peak force, providing further evidence that wheat kernels tend initially to break across the crease. The initial slope of the force-deformation curve depends on the inherent strength of the endosperm material opposite the crease, while the magnitude of the initial peak force depends on the strength and thickness of this material, and hence on the crease depth. It is concluded that through the averaging process, aCRPs give a misleading picture of wheat kernel breakage in the SKCS. The Hardness Index currently reported by the SKCS, which is based on identifying breakage events as indicated by the jaggedness of the force profile, would appear to give a more fundamentally sound basis for quantifying wheat properties relevant to breakage.