In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter $θ$. We first consider two symmetric estimators $\hatθ_1$ and $\hatθ_2$ and obtain necessary and sufficient conditions for $\hatθ_1$ to be Pitman closer to the common median $θ$ than $\hatθ_2$. 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 $θ$ 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 $θ$. 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.