  We characterise the typical offset between the Dark Matter (DM) projected
centre and the Brightest Cluster Galaxy (BCG) in 10,000 SDSS clusters. To place
constraints on the centre of DM, we use an automated strong-lensing 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 2D 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, \Delta, are
distributed equally around zero with no preferred direction, and are well
described by a log-normal distribution with <log_{10}(\Delta [h^{-1}
Mpc])>=-1.895^{+0.003}_{-0.004}, and \sigma=0.501\pm0.004 (95% confidence
levels), or <log_{10}(\Delta [\arcsec])>=0.564\pm0.005, and
\sigma=0.475\pm0.007. 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% 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 help characterise the
merger, relaxation and evolution history of clusters, in future studies.
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