# Mean of Medians? Median of Means?

This is a question I have wondered about for a long time and have never been able to find a full mathematical explanation behind this.

Suppose there are 100 countries. As an experiment:

We give Person A the median income of each country We give Person B the mean income of each country Now, suppose the following happens:

• Person A decides to take the mean of all median incomes
• Person B decides to take the median of all mean incomes
• Person C shows up out of nowhere and decides to take the median of all median incomes

My Question: Using mathematics, can we demonstrate that perhaps some of these calculations are not very "meaningful"? For example, can we somehow show that some of these calculations lack important mathematical properties (e.g. asymptotic normality, applicability of CLT) and are basically arbitrary?

As a last resort, I am thinking of just performing a simple Bootstrap procedure - for example:

• Person A: Take a random sample of the available medians, then take the mean of these samples and repeat.
• Person B: Take a random sample of the available medians, then take the median of these samples and repeat.
• Person C: Take a random sample of the available means, then take the median of these means and repeat.

This would allow you to get both the point estimates as well as the variance of the point estimates - but I am still not sure if this correct.

Thanks!

Note: You might be wondering about "Person D" - what about the Mean of the Means? I have seen the Mean of the Means come up before quite often and seen it being used as a valid estimator, thus I decided not to mention it here.

References:

• Those are all meaningful, unbiased, asymptotically normal calculations for continuous distributions, just with higher variance than the estimator by mean of means. “Distribution of the sample median” is probably a good search term to find out more. Aug 8, 2023 at 10:24
• This question lacks an objective: what is the intention behind "taking" any of these statistics? Without such an objective, anything is arbitrary! It sounds like you are thinking of using these statistics as estimators, but of what parameter and under what distributional assumptions?
– whuber
Aug 8, 2023 at 16:37
• @whuber: i have tried to give more details here: stats.stackexchange.com/questions/623548/… Aug 10, 2023 at 13:17