  Felsner and Reuter introduced the linear extension diameter of a partially
ordered set $\mathbf{P}$, denoted $\mbox{led}(\mathbf{P})$, as the maximum
distance between two linear extensions of $\mathbf{P}$, where distance is
defined to be the number of incomparable pairs appearing in opposite orders
(reversed) in the linear extensions. In this paper, we introduce the reversal
ratio $RR(\mathbf{P})$ of $\mathbf{P}$ as the ratio of the linear extension
diameter to the number of (unordered) incomparable pairs. We use probabilistic
techniques to provide a family of posets $\mathbf{P}_k$ on at most $k\log k$
elements for which the reversal ratio $RR(\mathbf{P}_k)\leq C/\log k$, where
$C$ is a constant. We also examine the questions of bounding the reversal ratio
in terms of order dimension and width.
