# Finding median without raw data?

I have only summary statistics for each state in the United States. I have the mean and median prices for each state and that’s it.

How can I estimate an “overall” median price for the nation? I understand whatever I calculate won’t be a “true” median due to the lack of raw data, but I still need a number.

Can I somehow estimate percentiles with this information and find the 50th percentile? A weighted “average” is not that reliable I feel So I would really appreciate some guidance.

*Edit: I am currently using a weighted median and I would like to improve upon that and make it more complex/exact to avoid large differences in prices.

• Yes, sorry, I meant the United States.
– Didi
Oct 16, 2017 at 17:10
• Are there other distributional summaries of the groups aside from mean/median? Oct 16, 2017 at 17:27
• The only other pieces I have are similar to a min/max although not the actual min/max...just the typical prices on the lower-end and higher-end of the spectrum. It's bizarre but i repy on other organizations reporting this data. We usually calculate this "weighted-avg" median and find percentiles from there.
– Didi
Oct 16, 2017 at 17:34
• oop, that was for @AdamO
– Didi
Oct 16, 2017 at 17:34
• Please don't vandalize your posts. Oct 17, 2017 at 16:14

Your least-bad option is likely to be a weighted median of the state medians, where you would weight each separate state's median by the number of data points that went into calculating it, e.g., the number of house sales if you are looking at house prices.

If you don't have this raw number, you might be able to use state population as a proxy for weighting.

This is still systematically wrong, since some states will likely have systematically higher or lower prices than others. So you could definitely improve matters by collecting more data and/or setting up a more complex model.

• What about using two different weighting approaches? One as main analysis and the other (or additional others) as sensitivity analysis? Oct 16, 2017 at 17:13
• Hello @Joe_74. I apologize I am not sure what you mean, I am sort of a novice at this stuff. Could you please elaborate on these analyses?
– Didi
Oct 16, 2017 at 17:16
• Hello @stephan, that is precisely what I have done. However I do need a more complex problem to avoid the exact issue you mentioned (I have some wildly different prices). Do you have any pointers/ideas for a more complex model?
– Didi
Oct 16, 2017 at 17:18
• @Didi Basically, you can choose a weighting approach and run the main analysis. Then, use a different weighting approach, and see whether the results hold or change systematically in direction or magnitude. Have a look here for a detailed explanation of what a sensitivity analysis is: en.wikipedia.org/wiki/Sensitivity_analysis Oct 16, 2017 at 17:20
• To be honest, all the smart ideas that come to my mind (e.g., sampling data points randomly between each state's min and max) have probably about as much chance to further bias your result as they have to improve it. So all you can probably do is collect more data. Are there any additional sources you could get data from? Also: if prices differ systematically between states, how useful is an overall median, anyway? Oct 16, 2017 at 18:05