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Are there terms for measures of variation around the median? I know that we use "variance" for means of data but how can variance be defined for the median? Is there a similar statistic for median?

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    $\begingroup$ See en.wikipedia.org/wiki/Median_absolute_deviation, then read web.ipac.caltech.edu/staff/fmasci/home/statistics_refs/… (which is also discussed at en.wikipedia.org/wiki/Robust_measures_of_scale). $\endgroup$
    – whuber
    Nov 27, 2012 at 23:28
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    $\begingroup$ Mark, You have received an answer that focuses on assessing the sampling distribution of the median, whereas in my previous comment I originally understood your question to ask about summarizing data. Could you please indicate which interpretation you intended? $\endgroup$
    – whuber
    Nov 28, 2012 at 17:30
  • $\begingroup$ @whuber your comment completely resolved my issue. Would you mind adding it as an answer please? $\endgroup$
    – MarkSAlen
    Nov 28, 2012 at 21:15
  • $\begingroup$ Which comment, Mark--the first or second? $\endgroup$
    – whuber
    Nov 28, 2012 at 21:17
  • $\begingroup$ @whuber the first one, sorry for not being clear $\endgroup$
    – MarkSAlen
    Nov 28, 2012 at 23:09

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Since the median is a statistic estimated from sample data, it has an associated sample standard error which can give confidence intervals and tests of location for that value.

The variance of a normally distributed random variable can be used directly to compute exact confidence intervals for sample means of IID random normal variables. Practically all distributions give sampling distributions for sample means which are approximately normal for reasonably large $n$. The standard deviation of the sampling distribution of the sample mean is what's called the standard error. The relationship between the sample standard error and the standard deviation of the sample data's probability distributions are related. Anything you estimate from data, whether the minimum, maximum, median, (etc.) has a sampling distribution and hence an associated standard error. This means you have an associated standard error for sample medians. This value is computed by using the inverse quantile function for that data, practical example here.

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