Is there one word to describe the sections of data on either side of the median? I'd guess "half", but it seems like there ought be something better…
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…as half to median. "Quarter" and "half" are names for parts of something, "quartile" and "median" are quantiles, so the points that divide the data ranges into equal parts. So "half" is absolutely fine. You can say "data-points greater/lower then median" etc., but this is rather a discussion on English language rather then statistics.
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