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In this work, we report an analytical and experimental investigation of the signal-to-noise ratio for a recently developed method called x ray dark-field imaging. Our approach is based on the propagation of signal and noise through the reconstruction algorithm. We find that the statistical nature of the dark-field images can be understood by a Rician distribution. The analysis shows that, for high flux, the noise in the dark-field images is proportional to the noise in the raw data. In the limit of low flux and, thus, low signal-to-noise ratio, the dark-field signal exhibits a breakdown of the signal transmission, which can be described by an asymptotic behavior of the underlying noise distribution. In this limit, the dark-field signal is no longer connected to the coherence degradation, but rather to the attenuation in the sample. The model is verified in exemplary test measurements using a compact laboratory setup with a polychromatic source and a photon counting detector.