Abstract Let X 1, …, Xn be identically distributed according to a symmetric distribution F (satisfying suitable regularity conditions). An approximation to the variance of the median is computed up to terms of order 1/n 2 and a corresponding approximation to the efficiency of relative to the average up to terms of order 1/n. [These are compared with the actual efficiencies when F is normal or rectangular and in these cases are shown to give a much closer approximation to the exact efficiency than that given by the usual asymptotic efficiency.] It is pointed out that to the accuracy of this approximation, one should not use the median based on an odd number of observations since the median based on the next smaller even number is equally accurate. These results are extended to other averages of two symmetrically placed order statistics, suggesting the possibility of a further reduction in sample size without loss of accuracy.