We characterize the typical offset between the dark matter (DM) projected centre and the brightest cluster galaxy (BCG) in 10 000 Sloan Digital Sky Survey clusters. To place constraints on the centre of DM, we use an automated strong lensing (SL) analysis, mass-modelling technique which is based on the well-tested assumption that light traces mass. The cluster galaxies are modelled with a steep power law, and the DM component is obtained by smoothing the galaxy distribution fitting a low-order two-dimensional polynomial (via spline interpolation), while probing a whole range of polynomial degrees and galaxy power laws. We find that the offsets between the BCG and the peak of the smoothed light map representing the DM, Δ, are distributed equally around zero with no preferred direction, and are well described by a log-normal distribution with and σ = 0.501 ± 0.004 (95 per cent confidence levels) or 〈log10(Δ [arcsec])〉 = 0.564 ± 0.005 and σ = 0.475 ± 0.007. Some of the offsets originate in prior misidentifications of the BCG or other bright cluster members by the cluster finding algorithm, whose level we make an additional effort to assess, finding that ∼10 per cent of the clusters in the probed catalogue are likely to be misidentified, contributing to higher-end offsets in general agreement with previous studies. Our results constitute the first statistically significant high-resolution distributions of DM-to-BCG offsets obtained in an observational analysis, and importantly show that there exists such a typical non-zero offset, in the probed catalogue. The offsets show a weak positive correlation with redshift, so that higher separations are generally found for higher z clusters in agreement with the hierarchical growth of structure, which in turn could potentially help characterize the merger, relaxation and evolution history of clusters in future studies. In addition, the effective DM centre we adopt here, namely the peak of the smoothed light distribution representing the DM, can constitute a natural and alternative definition of cluster centres for optically selected cluster catalogues.