Waveforms emerging from correlations of long seismic noise records are extensively used to investigate the crustal and upper-mantle structure of the Earth. To remove the non-stationary events that inevitably lie in seismic records, the so-called one-bit normalization is commonly applied to the noise data. This processing consists of replacing each sample of a record by its sign. Although it is a strong non-linear operation, it preserves the phase of the signal emerging from correlation. Some recent studies show that information can also be extracted from the amplitude of such correlations. In this paper, we develop a theory to understand these non-intuitive results. A statistical approach is used to get an expression of the one-bit noise correlation. This expression involves the standard deviations of coherent and incoherent noise. These two kinds of noise are precisely defined, and explicit expressions of their standard deviations are given in the case of a uniform distribution of noise sources generating surface waves on a layered half-space. In this case, we show that the one-bit noise correlation has the same phase and relative amplitude as the raw noise correlation. This is true in both elastic and anelastic media. Numerical simulations are performed to support our theory.