Median Calculation – How to Calculate Median of Medians

median

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.

Best Answer

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.

Related Solutions

Solved – K-medians, formula to compute the median

The definition of the geometric median is that of the $L_1$ optimum.

There seem to be two common approximations in use:

component-wise medians, optimizing each dimension independently
medoids, taking only the data samples into account

It's not clear to me why the component-wise median is not the same as the geometric median.

Solved – When does the sum of the medians = the median of the sum

Actually my comment is not entirely correct, allow me to clear up;

The median of a series of numbers $X$ is calculated by ordering all the numbers from smallest to largest, then finding the number in the middle. This means that when you change the numbers in $X$ you also change the ordering, hence the median changes. Therefore (in general) you can almost always assume that: $$ \text{MED}(X + Y) \neq \text{MED}(X) + \text{MED}(Y) $$ However there is at least one exception, whenever the ordering of $X$ (after adding $Y$ to $X$) does not change neither does the median. For instance if all numbers in $X$ and $Y$ are the same, see this example (written in R):

set.seed(42)
n <- 100 
x <- rnorm(n)
c <- x
y <- rnorm(n)

median(x+y)           # 0.0767433
median(x) + median(y) # 0.02050838
median(x + c)         # 0.1795935
median(x) + median(c) # 0.1795935

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Related Solutions

Solved – K-medians, formula to compute the median

Solved – When does the sum of the medians = the median of the sum

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