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In a function, how should the median be denoted correctly?

Normally, one would use the tilde like this:

$$ Y = \tilde{X} $$

However, it is stated here that

"There is no widely accepted standard notation for the median, but some authors represent the median of a variable x either as x͂ or as μ1/2 sometimes also M."

Therefore my question is, if it is also possible to use the median of a set of numbers like this (or else) when referring to it in the text? Since I have a variable with a lot of indices, the tilde doesn't seem right here.

$$ Y = {\rm med}\{1,2,3,3,4,5\} $$

or

$$ Y = {\rm med}\{X^{imnkl}\} $$

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  • $\begingroup$ What do you mean by "in a function"? Is this naming for a computer program? $\endgroup$
    – AdamO
    Commented Nov 16, 2023 at 6:50

2 Answers 2

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As Wikipedia says there is no standard notation so if it is clear in the text you can use any notation you want.

For example if you state precisely that for any set of integers $A$, $\operatorname{med}(A)$ represents the median of the set $A$ then you can use this notation as in your example.

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  1. Pre-amble, not all formulas for the median are the same. Be sure the "type" of the quantile function you're using is defined and accepted by the target publication.
  2. I see "M" in a lot of elementary texts or applied papers. You can call it virtually anything if you define it, but even if you used "M", it would probably require a sentence, "Let M be the median of.." which isn't very specific
  3. Second to 2, It doesn't often require that much space to just write "median" whether in a table of results, in a text body, or just as a notation. I think a shorthand of "Med" or "Mdn" could work as well, just as we often use "Var" for the same reason.
  4. In more theoretical papers you might even see $F^{-1}(0.5)$ or $\hat{F}^{-1}(0.5)$. How, exactly, the "F" is estimated here would affect the estimator significantly. An empirical distribution function would produce the "type 1" (in R) or "type 3" (in SAS) median.
  5. A different expression of the median is the L1 minimum loss estimate of center, $\arg \min_M \| X - M \|_1 $
  6. I have never seen $\mu_{1/2}$ and I hope I never do outside of Wiki! How confusing.. If the distribution is symmetric with finite mean, then the median is the mean in which case you can refer to it by the same $\mu$.
  7. Another notation you might come across is "Q50" where Q is short for quantile. This is usually in the context of also reporting Q25, Q75, and IQR.
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    $\begingroup$ “Q2” is another option, with Q short for quartile, eg blogs.sas.com/content/iml/2017/07/19/quantile-skewness.html $\endgroup$
    – Matt F.
    Commented Nov 16, 2023 at 14:28
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    $\begingroup$ (+1) to everything and (+2) to point 6 on $\mu_{1/2}$ which is horrible. I've seen P50 too, where P may be intended to mean percentile or even probability (!). Tukey suggested a vertical bar over a variable letter. I don't recall seeing that otherwise. I can't quickly find how to do it in TeX. $\endgroup$
    – Nick Cox
    Commented Nov 16, 2023 at 15:08
  • $\begingroup$ @NickCox, maybe $X^\perp$ would be a nice and easily TeX’d alternative to Tukey’s suggestion. $\endgroup$
    – Matt F.
    Commented Nov 19, 2023 at 18:55
  • $\begingroup$ Thanks for the thought, I am still holding out for a way to do what he did. Otherwise I swing between med, $M$ and $\tilde x$. $\endgroup$
    – Nick Cox
    Commented Nov 19, 2023 at 23:55

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