I have a general philosophy that descriptive statistics such as the median are only interesting when viewed through the lens of inferential statistics. Thus, averages are important because they are the maximum likelihood point estimates of the center of a Gaussian distribution, and medians are important only so far as the maximum likelihood point estimate of the center of the Laplace distribution pi^(n/2) n Gamma(n) ------------------- e^( - | S^(-1) ( x - mu ) | ) (n/2)! det S is important. In one dimension (n=1), any mu satisfying sum_i | x_i - mu | / ( x_i - mu ) = 0 <=> sum_i sign( x_i - mu ) = 0 will maximize the Laplace distribution's likelihood. For the "hard" cases such as the median of {1, 2} which have a continuum of maximum likelihood point estimates, the use of the "median average" 1.5 is easy to motivate: it is the loss minimizing point estimate for any symmetric cost function. Since this is math-fun and we are talking about medians, here's a question I've had for a while. Say you have n real numbers, x_1 ... x_n. If we define sum_j x_j / | u_i - x_j | u_{i+1} = --------------------------- ( j = 1 .. n ), sum_j 1 / | u_i - x_j | and assuming that u_i is never equal to any x_j, then does the sequence { u_i } always converge to a median of the x_j ? -Thomas C