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I know that the distribution of sample mean of a number of iid random variables tends to a Gaussian distribution if the sample size is at least 30, where the mean value of the resultant distribution turns out to be the same as the population mean of the iid random variables. I need to know whether any such results exist about sample median of a number of iid random variables? Can anything be said about the distribution of the sample median of $n$ iid random variables where $n \ge 30$?

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  • $\begingroup$ Yes. Under various broad circumstances, the median is also approximately normally distributed with mean equal to the population median, and variance inversely proportional to the sample size. Let me see if I can hunt up the proper reference. $\endgroup$
    – Brian Tung
    Jun 20 '20 at 6:32
  • $\begingroup$ stats.stackexchange.com/a/76096/119261 $\endgroup$ Jun 20 '20 at 6:33
  • $\begingroup$ Also see the Wikipedia plot summary for Median: Medians for Samples. $\endgroup$
    – Brian Tung
    Jun 20 '20 at 6:35
  • $\begingroup$ One of my old professors, Tom Ferguson, wrote this paper that may be relevant to your interests. $\endgroup$
    – Brian Tung
    Jun 20 '20 at 6:37
  • $\begingroup$ This Q&A quotes relevant theorem. Illustrates for median of exponential dist'n. $\endgroup$
    – BruceET
    Jun 21 '20 at 21:31
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The sample median, if you are lucky (if your sample size is odd), can be an order statistic. So given that you know the distribution of your original sample you can order it as the Wikipedia article describes and then the middle observation has a specific cumulative distribution function and potentially a probability density function. Practically even if your sample size is even you can calculate the CDF of the middle value (call it $m$) and the $m+1$ and average them out for practical purposes it should be fine.

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