frequency weighted mean of median?

[Dennis Lund Hansen]

I am trying to estimate the age of diagnosis for a rare disease. Most of the data derive from small studies, and report a median age.
The simple question is, is it "legal" to estimate weighed average of median, to approximate the "true" value? I assume that all the studies in a way represent samples of the same population of affected patients

Exampel:

clear
input study   N   median_age
1    50    50
2    60    55
3    30    60
4    100    65
5    55    62
end 

sum median_age [fweight = N]

[Nick Cox]

The question appears to me as much clinical as statistical, namely why did people use medians in the first place? I guess at a disease that tends to manifest in later life, so one whose age distribution is left skewed, so the median of medians would be consistent with that. I don't know whether your example data are real, but they're an example. I get a median median of 62 and a mean median of about 59, so about the same which isn't a great surprise.

In all, I'd have a marginal preference to go with the median of medians -- and i would flag for the readership that while the weighted mean based on means and frequencies really is the mean you would get if the real data were all available, that is not guaranteed for the weighted median as a estimate of the median of all the data.