# Median of Medians calculation

I do a bunch of Real Estate Reporting and the Median price is often reported. Particularly by the NAR (National Association Of Realtors) ,,, How can this be,, As best I can figure they only get the Medians from each area,, so how should they calculate a national median, a median of medians a a simple average of medians or weighted average of medians or....??? How valid can it be,, I know that the NAR is not getting the total transaction table so can a reasonable representation of Median (nationally be done? Particularly since regional density and prices and market variances are so large now?

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Can you please work on the style and clarity of your question? –  mbq Nov 11 '10 at 23:13
if I understand this correctly, dartdog wants to know whether the median of medians is the can be used to calculate the median of the whole sample, i.e. if $M=M_1 \cup M_2 \cup \ldots \cup M_k$ then median($M$)=median(median$(M_1)$,median$(M_2)$,$\ldots$,median$(M_k)$)... –  psj Nov 13 '10 at 19:23

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.

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drknexus You get the issue I was driving at,. The issue is that the national housing stats often report national medial house prices,, but I'm pretty sure they at best get the median values for a state so I believe that the "national" median house price statistic is pretty bogus. Particularly when one realizes that the mix of demand by state probably fluctuates quite a bit as well due to local economic conditions and local foreclosure trends (which vary widely). So, Is there a semi valid way to do this is is it as bogus as I suspect?? –  dartdog Jan 19 '11 at 22:33
<shrug> It is 'valid' to whatever extent it represents the information you are interested in/the extent to which it represents what it purports to represent. If the people doing these analyses have the raw data that allows them to find the median by state, I see no compelling reason they would opt to report the median of state medians rather than just the overall median, because (as you suggest) the overall median has a clear interpretation whereas the median of medians does not. –  rpierce Jan 20 '11 at 0:57
P.S. You can spot check their technique if you can find the median by state. If their reported median national house selling price matches the median of reported state medians - then they are doing what you fear. If not, then you can probably assume that they are using the simple median of all houses sold. –  rpierce Jan 20 '11 at 0:59
If you had the medians by state (and not actual sale data) AND you had the number of sales per state, you could do a weighted median to get an approximate of the 'true' median e.g. (MedianState1 * NumberofSalesState1 + MedianState2*NumberofSalesState2 + ... MedianState50*NumberofSalesState50)/TotoalNumberOfSales. –  rpierce Jan 20 '11 at 1:02