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Suppose you have some non-continuous data that you can bin, e.g. integer value test scores. So you go ahead and bin your data into bins of 100-90, 89-80, 79-70,...,9-0 and then you make a nice line plot over the medians of each bin e.g. over the A bin you'd put the median of the grades for values between 100-90.

Now what if you wanted to describe your binned data? Do you take the median of the binned medians? Or the median of the entire dataset?

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    $\begingroup$ Why would you want to put your data into bins rather than keep them in a more-or-less continuous form? Most analyses and descriptions work much better with continuous variables. Do you really think that a score of 79 is substantially different from one of 80? That's what binning does. $\endgroup$ – EdM Mar 25 '17 at 17:53
  • $\begingroup$ @EdM test scores were not the best example, it was just to describe the concept of the question. Replace the example of test scores with whatever suites your preference. $\endgroup$ – SumNeuron Mar 25 '17 at 18:01
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    $\begingroup$ Why would you want to bin any continuous variable? My comment was not specific to test scores. Advice on this website is almost uniformly against binning. If there is a particular situation where you think binning is helpful or necessary, please describe that in your question. $\endgroup$ – EdM Mar 25 '17 at 18:19
  • $\begingroup$ @EdM sorry, I should have specified that it was not continuous. $\endgroup$ – SumNeuron Mar 25 '17 at 18:26
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Your target is to describe the binned data by providing a median value.

Do you actually want to describe the binned data or the original data? Is there a special reason for the binning?

I imagine that in most situations binning is just a means to an end, e.g., to simplify some other part of your analysis or when using algorithms that only accept a small number of unique values per variable. In that case, you essentially sacrifice information for simplicity. However, you don't need to bin your data to provide the median of your dataset. You already aggregate your data to a single statistic by computing the median. There is no need to do another aggregation step beforehand, it would just decrease the accuracy of your median. Therefore, you should report the median of the unbinned data.

There might be situations where you would want to report the median of the binned medians: in a hypothetical situation where you collected two datasets of two populations, one where data is already binned and you don't have the original data and one where data is unbinned. If you want to compare the medians of the two datasets, you could consider first binning the yet unbinned dataset and then compute the median. However, even here it would be just throwing away information about one of the populations.

In conclusion, I would report the median of the unbinned data.

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