When stimulated by tones, the ear appears to emit tones of its own, stimulus-frequency otoacoustic emissions (SFOAEs). SFOAEs were measured in 17 chinchillas and their group delays were compared with a place map of basilar-membrane vibration group delays measured at the characteristic frequency. The map is based on Wiener-kernel analysis of responses to noise of auditory-nerve fibers corroborated by measurements of vibrations at several basilar-membrane sites. SFOAE group delays were similar to, or shorter than, basilar-membrane group delays for frequencies >4 kHz and <4 kHz, respectively. Such short delays contradict the generally accepted "theory of coherent reflection filtering" [Zweig and Shera, J. Acoust. Soc. Am. 98, 2018-2047 (1995)], which predicts that the group delays of SFOAEs evoked by low-level tones approximately equal twice the basilar-membrane group delays. The results for frequencies higher than 4 kHz are compatible with hypotheses of SFOAE propagation to the stapes via acoustic waves or fluid coupling, or via reverse basilar membrane traveling waves with speeds corresponding to the signal-front delays, rather than the group delays, of the forward waves. The results for frequencies lower than 4 kHz cannot be explained by hypotheses based on waves propagating to and from their characteristic places in the cochlea.