How to measure home value distribution in a city? I would like to know how I can measure and compare the distributions of home values across cities. I have collected the following data for each city:

a) median price in the city;

b) number of housing units by value range, such as 
500 units between \$100k-200k, 
300 units between \$200k-300k, 
600 units between \$300k-400k, etc;

c)Lower value quartile of housing units;

d)higher value quartile of housing units.

I know some of these cities have a lot high-end homes but few low-end homes while others have a lot of low-end homes but few high-end homes. Can I develop an index to measure that?
When it comes to constructing indices, the axiomatic approach is the standard approach in the economics literature.

An axiom is a property you would like your index to satisfy. For instance, in your example, you would probably wan your index to increase when the distribution in b) changes from $X$ to $Y$ and $Y$ first order stochastically dominates $X$.

You can require your index to satisfy the above property, as well as others, and hope that as you impose more and more desirable properties, you will narrow down the class of acceptable indices to a single index.

Now, such axiomatic characterization (i.e. proving that index $I$ is the only index that satisfies a set of properties) is typically not an easy task. As a matter of fact, some very fine researchers have made their whole career out of these kinds of characterization results.

These characterizations are also very much context dependent. Although there are some general results out there characterizing price indices (see https://scholar.google.be/scholar?q=axiomatic+price+index&amp;btnG=&amp;hl=nl&amp;as_sdt=0%2C43) they might not be directly amenable to your problem .

(I don’t know enough about the topic to tell you whether something like an axiomatic theory of housing price indices already exists. It’s not impossible that it does, but I just don’t know.)

Depending on how much time you want to put into this, here is what I would do if I were you:


Learn about axiomatic price index theory ( again see https://scholar.google.be/scholar?q=axiomatic+price+index&amp;btnG=&amp;hl=nl&amp;as_sdt=0%2C43 as well as Afriat, S. (2014). The index number problem: construction theorems (1st ed). Oxford ; New York: Oxford University Press. and
Diewert, W. (1988). The early history of price index research. Retrieved from http://www.nber.org/papers/w2713)
If you learn about an already existing theory of housing price indices, just pick the index you like best among the ones in the literature.
Otherwise, try to see if you can adapt some of the general price indices in the literature to your housing case.
In last resort, develop an axiomatic theory of your own. 


If you have the time and ability, you can try to obtain characterization and publish nice research papers on the topic. 
Otherwise, you can still use the axiomatic approach to craft an index that satisfies some of the properties you think are the most important. Just list those properties, and start looking for an index that does the job. 
If you can’t find one, don’t disappear. It may be due to a lack of creativity, but it may also be that you have imposed so many properties that there does not even exist an index satisfying all of them (this is called an “impossibility result” in the axiomatic literature). 
Just start relaxing or dropping properties one at the time up to the point where you find an index that works. Then if you still have the time and energy, start re-adding some of the properties you dropped, one at the time, and see if you can find a new index that works.