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What rules are available for deciding which quantile a particular value would fall into when the boundaries of more than one quantile are identical?

My data set includes score ranging between 1 and 20. As there are a lot of scores of 15, my quartiles look like this:

  • 1st: 1 to 14
  • 2nd: 15 to 15
  • 3rd: 15 to 15
  • 4th: 16 to 20

So which quartile does a score of exactly 15 fall into?

As it falls into both the second and third quartiles, which quartile should be selected? For example - if this were used to rank a student's score, where should they fall?

Are there multiple ways of resolving this tie and if so, are there any that are statistically 'fairer'? Or is fairness an irrelevant concept here?

charliefortune
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  • That not totally clear: is $20$ the maximum value or the $75%$ quantile? In any case, it seems that $15$ falls into more than one quartile, at least the second and third, and possibly the fourth. – Henry Sep 28 '21 at 15:16
  • Edited the question – charliefortune Sep 28 '21 at 15:23
  • With you edit $15$ falls into the second and third quartiles – Henry Sep 28 '21 at 15:26
  • So if this were a piece of software giving out grades to students, should it award second or third quartile for a score of 15? – charliefortune Sep 28 '21 at 15:38
  • Think of it it another way: if every student got the same score (some of the questions were very easy and the others impossible), would you want to say they were top or halfway? Here the choice for those with $15$ is three-quarters of the way up (possibly minus $\frac1{2n}$) or halfway up. – Henry Sep 28 '21 at 15:50

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