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Just a note

An alternate way around this problem is to exclude the offending value, and use this definition:

The pth quantile of a set of values (Y) is a number (yp), such that the proportion of Y that are less than yp is ≤ p - and the proportion of Y that are greater than yp is ≤ 1-p.

Where more than one number satisfies this criterion, then the pth quantile is the average of the smallest and largest numbers that do so.

The reason this definition is rather complicated is because it allows for the fact that, in heavily-tied sets, the offending value may have more than one sequential rank. Given the complexities this definition introduces many people prefer to just exclude one rank.