  Depth of the Tukey median is investigated for empirical distributions. A
sharper upper bound is provided for this value for data sets in general
position. This bound is lower than the existing one in the literature, and more
importantly derived under the \emph{fixed} sample size practical scenario.
Several results obtained in this paper are interesting theoretically and useful
as well to reduce the computational burden of the Tukey median practically when
$p$ is large relative to large $n$.
