Re: Median: Oasis and Fast Sorting Algorithm

On Sat, 2007-10-02 at 02:41 +0200, Leonard Mada wrote:
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.

Contrary to your claims, there is nothing ambiguous. Any non-decreasing
list of the same values has the same middle value(s). 

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.

The OpenFormula standard does not prescribe any method used to find the
value. It only prescribes what the value is.

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. 

MEDIAN is not intended to implement a weighted median. None of the
current spreadsheet implementation uses that name for a weighted

Gnumeric for example does also provide a function for a weighted median,
namely SSMEDIAN. That function may at some time also be introduced in
the Standard but would in no way make other definition ambiguous.

Prof. Dr. Andreas J. Guelzow
Dept. of Mathematical & Computing Sciences
Concordia University College of Alberta

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