Upper and lower bounds are obtained for the cumulative distribution function of the sample median of a sample of size $2n + 1$ drawn from a continuous population. It is shown that if the parent population is normal, then the distribution of the sample median tends "rapidly" to normality. Other kinds of parent populations are also discussed.
John T. Chu. "On the Distribution of the Sample Median." Ann. Math. Statist. 26 (1) 112 - 116, March, 1955. https://doi.org/10.1214/aoms/1177728598