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In statistics the median is a point that divides an ordered data into two equal parts. By definition if $n$ is odd the median is $\frac{n}{2}{th}$ data point that is simply the middle point of the ordered data set. When $n$ is even the median is calculated by taking the mean of the two middle observations that is median=$\frac{(\frac{n}{2}){th} + (\frac{n}{2}+1){th}}{2}$. Now taking the two middle points results in a value that does not exist actually in data set. Then how can we declare this as a median point?

Asad Ali
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  • Is there any justification? I believe that median enjoys closer kinship with mode rather than with mean. Both are mathematically intractable and both are insufficient statistics, that's both do not take the entire data set into account. – Asad Ali Apr 18 '14 at 18:51
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    Any value from the $\frac{n}{2}$th and the $\frac{n}{2}+1$th meets the requirement that at least half the data is less than or equal to it and at least half the data is greater than or equal to it. Taking the midpoint is just a sometimes-convenient way of avoiding having to have a range for the median. But for example a statement like "the median minimises the sum of the absolute deviations" would be true for any value in that range – Henry Apr 18 '14 at 19:04
  • Henry this is a brilliant answer. I did not realize the median is a range. – Vortex Mar 04 '18 at 06:23

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