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If I know the coefficient of dispersion and the median of a data set, is it possible for me to then calculate the percent of data points that are above the median?

For example, what if the median is 1 but the dataset is all 1's? Would standard deviation help for this?

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    $\begingroup$ Do you know what the definition of the median is? $\endgroup$
    – jbowman
    May 31, 2018 at 20:40
  • $\begingroup$ If the data are constant you can report that fact directly. There is no reason to report any measure of variability, but the SD is 0 (or indeterminate if $n = 1$ and $n - 1$ appears in the divisor). $\endgroup$
    – Nick Cox
    May 31, 2018 at 23:31

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It's independent of the coefficient of dispersion.

The median is always exactly in the middle of all data points. By definition, if there are $N$ data points there will always be $\frac{N-1}{2}-1$ data points above the median for an uneven number of elements in the dataset and $\frac{N}{2}-1$ above the median for an even number.

Therefore, the percentage of numbers above the median is thus given by $\frac{1}{2} -\frac{3}{2N}$ (for uneven number of datapoints) and $\frac{1}{2}-\frac{1}{N}$ (for even number of datapoints).

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    $\begingroup$ Only in some situations. Your statements require that the value of the median occurs only once if the number of values is odd and also make assumptions about distinct values if the number of values is even. Trivial counter-example: if all values are 42, or any other constant, then so is the median, but no values are above and none below. That may seem pedantic, but your statements aren't part of the definition and they don't follow from it without extra conditions. $\endgroup$
    – Nick Cox
    May 31, 2018 at 23:27

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