  In this paper, we focus on Pitman closeness probabilities when the estimators
are symmetrically distributed about the unknown parameter $\theta$. We first
consider two symmetric estimators $\hat{\theta}_1$ and $\hat{\theta}_2$ and
obtain necessary and sufficient conditions for $\hat{\theta}_1$ to be Pitman
closer to the common median $\theta$ than $\hat{\theta}_2$. We then establish
some properties in the context of estimation under Pitman closeness criterion.
We define a Pitman closeness probability which measures the frequency with
which an individual order statistic is Pitman closer to $\theta$ than some
symmetric estimator. We show that, for symmetric populations, the sample median
is Pitman closer to the population median than any other symmetrically
distributed estimator of $\theta$. Finally, we discuss the use of Pitman
closeness probabilities in the determination of an optimal ranked set sampling
scheme (denoted by RSS) for the estimation of the population median when the
underlying distribution is symmetric. We show that the best RSS scheme from
symmetric populations in the sense of Pitman closeness is the median and
randomized median RSS for the cases of odd and even sample sizes, respectively.
