  We present a study of the statistical properties of three velocity dispersion
and mass estimators, namely biweight, gapper and standard deviation, in the
small number of galaxies regime ($N_{\rm gal} \le 75$).
  Using a set of 73 numerically simulated galaxy clusters, we characterise the
statistical bias and the variance for the three estimators, both in the
determination of the velocity dispersion and the dynamical mass of the clusters
via the $\sigma-M$ relation. The results are used to define a new set of
unbiased estimators, that are able to correct for those statistical biases with
a minimal increase of the associated variance. The numerical simulations are
also used to characterise the impact of velocity segregation in the selection
of cluster members, and the impact of using cluster members within different
physical radii from the cluster centre.
  The standard deviation is found to be the lowest variance estimator. The
selection of galaxies within the sub-sample of the most massive galaxies in the
cluster introduces a $2\,$\% bias in the velocity dispersion estimate when
calculated using a quarter of the most massive cluster members. We also find a
dependence of the velocity dispersion estimate on the aperture radius as a
fraction of $R_{200}$, consistent with previous results.
  The proposed set of unbiased estimators effectively provides a correction of
the velocity dispersion and mass estimates from all those effects in the small
number of cluster members regime. This is tested by applying the new estimators
to a subset of simulated observations. Although for a single galaxy cluster the
statistical and physical effects discussed here are comparable or slightly
smaller than the bias introduced by interlopers, they will be of relevance when
dealing with ensemble properties and scaling relations for large cluster
samples (Abridged).
