What order statistic is the median for a even-numbered sample?

I am calculating a Chakraborti-Desu test statistic from information available in Nonparametric Statistical Inference, 5th Edition by Gibbons and Chakraborti (hmm...). This statistic is for determining if at least one of k-1 samples is stochastically larger than the control sample.

Part of the calculation requires counting the number of elements in each sample that are smaller than the $i$th order statistic of the control sample. Now, suppose that the control sample has 24 elements and we want to check for elements that are smaller than the median.

The median is the average of the 12th and 13th order statistic. Does that make median the 12.5th order statistic? I've never seen a non-integer order statistic. What is the correct value of $i$?

• Just to be clear: In the first part you say "Part of the [Chakraborti-Desu] calculation requires ... the ith order statistic". This should be well-defined in all cases. For an even-numbered sample $N=2m$, the $m$ and $m+1$ order statistics are well-defined, and the "median" is just not needed. So the "setup" you give seems irrelevant to your final question. FWIW in some definitions the "median" could be any number in $[x_m,x_{m+1}]$. (See also a discrete case here.) Commented Dec 8, 2016 at 6:33
• Quantiles are defined for any value of $0<p<1$. Commented Dec 8, 2016 at 7:08
• There is a literature on fractional order statistics going back decades. See e.g. Stigler, S. 1977. Fractional order statistics, with applications. Journal of the American Statistical Association 72: 544-550. doi:10.2307/2286215 That paper includes the nice comment (independently thought up by Roger Koenker) that the two middle order statistics in a sample of even size may be thought of as comedians. (In applied statistics, averaging those is explained as a rule, and in mathematical statistics as just a convention.) Commented Dec 8, 2016 at 9:27
• @Xi'an I don't understand how what you said applies to this question. Commented Dec 8, 2016 at 13:56