Re: Median: Oasis and Fast Sorting Algorithm

John Machin wrote:
So who cares? The median value is 1. Is your alternative going to return some value other than 1 ????

Please define mathematically the middle value! It is NOT trivial as my definitions showed. Anything else would be ambiguous. This should be a standard, so make a better definition.

Well, I could have used a much shorter definition: the median is the value that halves the list so that there are two sets of equal size with numbers in the first set being higher than the median and numbers in the second set being lower. As noted, this definition avoids the sorting, too. (One could extend this definition for even and odd number of elements. Or even a much shorter definition: the 50th percentile. BUT all these definitions are ambiguous, see later.)

The one thing that I do NOT agree at all with the OASIS definition is, that it includes the wording "sorting". Sorting is definitely NOT necessary to calculate the median. You can take any array, even one that is NOT sorted, and determine the median without first sorting it. This is much to often stated wrongly in so many textbooks, BUT sorting is really not necessary.

So, this is NOT a prerequisite that should enter a standard definition.

May I even point out, that for even number of elements, one may define/have an upper median and a lower median. Alternatively, in serious mathematical uses, the median is usually calculated using a weighted approach. Therefore, the median of 1,2,2,3,4,5 is NOT (2+3)/2 = 2.5, BUT rather (2+2+3)/3 = 2.66. So, it does make sense to have a very strong and unambiguous definition in a standard.

The *weighted median* may be introduced later into the standard and then the ambiguity would be complete. So, let's do a good job now and not when it is too late.


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