I do a bunch of real estate reporting and the median price is often reported, particularly by the NAR (National Association Of Realtors). As best I can tell, they only get the medians of real estate prices from each area. My question is, how should the national median be calculated, given the data restrictions? As a median of medians, as a simple average of medians, or weighted average of medians or something entirely different? Second, how valid would these estimates be? I know that the NAR is not getting the total transaction table, so can a reasonably accurate representation of the median still be estimated at the national level? I ask, in particular, because regional density and prices and market variances are so large.
The median of medians is not the same as the median of the raw scores. A simple case of this is that when you have an odd number of sales, the median is the middle value; when you have an even number of sales, the median is commonly taken as the average between those two values. A more "real world" challenge to this is that states will sell differing numbers of houses and thus the median of their medians is a poor guess as to the median of all home sales. Though it also will to be precise, a good first pass estimation would be to find the median of values where each state's median is reflected a number of times proportional to the number of sales in that state. Thus, I am essentially suggesting a weighted median.