Finding the Median with Ties

Date: 06/16/2009 at 10:54:13
From: Henrik
Subject: To find the median when there are ties in the set

Find the median for the set {2,3,5,5,5,10}.

I understand how to calculate the median when there are odd or even
number of elements in a set.  However, I am confused about situations
when there are ties.  For the set given, if I use the traditional
method, it would be 5.  But 5 would not be a correct median since only
one value (10) is above 5, and two values are below 5 (2 and 3).  It
is therefore not a true central tendency.  Is there an alternative way
to calculate the correct median in these instances?  Thanks.

Date: 06/16/2009 at 12:53:47
From: Doctor Peterson
Subject: Re: To find the median when there are ties in the set

Hi, Henrik.

This is a very perceptive question!

With small data sets, there are cases, like this one, where there is 
no number you could choose that fits the common, naive definition of 
median, "the value that divides the ordered data into two equal 
halves".  This forces us to make a more careful definition of the 
term:

  The median of a data set is a value such that at most half the
  population have values less than the median, and at most half
  have values greater than the median.

Note two things: First, it is _A_ value fitting the condition; we 
commonly take it as the average of two middle values, but really any 
number between them would work!  Second, we don't say that EXACTLY 
half are on each side, but only that AT MOST half are on each side. 
This deals with your issue.

In your example, 2, 3, 5, 5, 5, 10, there are 2 values less than 5 
and 1 value greater than 5, which fits the definition: no more than 
3 are in either part.  If we chose anything greater than 5, more than 
half the data would be less than our "median", and if we chose 
anything less than 5, more than half would be greater than that.  So the 
only possible choice is 5.  The traditional method is in fact an 
efficient way to find a median that fits the definition.

Note also that the meaning of "measure of central tendency" is far 
more broad than you are imagining.  All that means is "any statistic 
that is guaranteed to be between the lowest and highest values"!  It 
does not have to be in the "exact center" in any sense; the mean, 
median, and midrange all define "center" in different ways, and the 
mode can't really be said to have anything to do with the center. 
Yet they are all SOMEWHERE in the middle, and that is enough.

For more on these ideas, see

  A Closer Look at the Definition of Median
    http://mathforum.org/library/drmath/view/72726.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/