It is shown that under certain regularity conditions, the central moments of the sample median are asymptotically equal to the corresponding ones of its asymptotic distribution (which is normal). A method of approximation, using the inverse function of the cumulative distribution function, is obtained for the moments of the sample median of a certain type of parent distribution. An advantage of this method is that the error can be made as small as is required. Applications to normal, Laplace, and Cauchy distributions are discussed. Upper and lower bounds are obtained, by a different method, for the variance of the sample median of normal and Laplace parent distributions. They are simple in form, and of practical use if the sample size is not too small.