  The location of a galaxy cluster's centroid is typically derived from
observations of the galactic and/or gas component of the cluster, but these
typically deviate from the true centre. This can produce bias when observations
are combined to study average cluster properties. Using data from the BAHAMAS
cosmological hydrodynamic simulations we study this bias in both two and three
dimensions for 2000 clusters over the $10^{13} - 10^{15} ~\mathrm{M_{\odot}}$
mass range. We quantify and model the offset distributions between
observationally-motivated centres and the `true' centre of the cluster, which
is taken to be the most gravitationally bound particle measured in the
simulation. We fit the cumulative distribution function of offsets with an
exponential distribution and a Gamma distribution fit well with most of the
centroid definitions. The galaxy-based centres can be seen to be divided into a
mis-centred group and a well-centred group, with the well-centred group making
up about $60\%$ of all the clusters. Gas-based centres are overall less
scattered than galaxy-based centres. We also find a cluster-mass dependence of
the offset distribution of gas-based centres, with generally larger offsets for
smaller mass clusters. We then measure cluster density profiles centred at each
choice of the centres and fit them with empirical models. Stacked, mis-centred
density profiles fit to the Navarro-Frenk-White dark-matter profile and
Komatsu-Seljak gas profile show that recovered shape and size parameters can
significantly deviate from the true values. For the galaxy-based centres, this
can lead to cluster masses being underestimated by up to $10\%$.
