A Closer Look at the Definition of Median

Date: 08/14/2008 at 18:56:17
From: Jerry
Subject: The definition of median

The definition of median is the middle number and it divides the 
group into two equal parts.  What if you have an odd quantity?

If you have an odd quantity, lets say $100, $200 $300, $400, $500. 
The middle number is 300.  How does this divide the group of numbers 
into two equal groups?

Date: 08/14/2008 at 23:13:30
From: Doctor Peterson
Subject: Re: The definition of median

Hi, Jerry.

I'd like to directly answer your question, because it raises some
important issues.

The definition you quote is an informal one often used in elementary
presentations.  It's reasonably understandable when stated clearly,
but glosses over some difficulties.

Let's start with that basic definition, which is generally something
like this, including both a motivating concept and specific calculations:

  The median of a set of numbers is the number in the middle when
  the data is sorted, dividing the values into two equal parts.

  If there are an odd number 2n+1 of values, the median is the (n+1)st
  value, so that there are n values below it and n above.

  If there are an even number 2n of values, the median is the average
  (arithmetic mean) of the two middle numbers, namely the nth and the
  (n+1)st, so that there are n values below it and n above.

I have included here a little more explanation than is usually given
initially, showing in what sense we say that the median divides the
data into two equal parts.  With an odd number, the median is one of
the values themselves, so that the data EXCLUDING the median itself is
split evenly:

  o o o o|o|o o o o
  \_____/ ^ \_____/
     n    1    n

Of course, we couldn't actually divide the entire set into two equal
parts, unless we think of half of the median being in one set and half
in the other!  But it makes sense to say that it divides the REST of
the values equally; think of the median as a saw cut, with the median
itself taken out as the "kerf".

With an even number, we can actually split the entire set into two
equal parts; but there is no longer really a unique number that does
that. ANY number between the two middle numbers would divide it
evenly; it is only by convention that we take the number exactly
between them (other conventions are possible):

  o o o o|o o o o
  \_____/^\_____/
     n       n

Another difficulty that you didn't mention arises when some of the
values are equal. What about this:

  1 1 2 2 2 3 3 3 3
          ^

The median is 2; but there are two values less than 2 and four greater
than 2.  In what way does this divide the set equally??  We have to
pretend that the three 2's are different in order to make sense of the
statement, so that it is the third 2 that divides the set evenly.  But
I've never heard anyone say this.

If we want a really precise definition that covers all cases and does
not depend on your accepting different meanings for terms like "two
equal parts" in different cases, it will be something like this:

  The median of a set of data is a number such that NO MORE THAN
  half are LESS than the median, and no more than half are greater
  than the median.

(I've seen several variations of this definition; one uses "at least
half of the data are no more than the median" instead of "no more than
half are less than", and so on.)

Applying this to the two cases:

  With an even number 2n of values, any number less than the (n+1)st
  will have no more than half (n) less than it, and any number
  greater than the nth will have no more than half greater.  Note that
  if several values are equal to either of these two middle values,
  then there will definitely be fewer than half on either side, but
  it still fits the definition.  Again, by convention we choose the
  mean of the two middle values, but this is not really required by
  this definition.

  With an odd number 2n+1 of values, there are no more than n
  values less than the (n+1)st value, and n is less than (2n+1)/2.

In my bad example,

  1 1 2 2 2 3 3 3 3
          ^
we see that the number of smaller values (2) and the number of larger
values (4) are both less than half of the total (4.5), so it fits the
definition.  Any number larger than 2, say 2.1, would not, because
then there would be 5 values less than the median.  Similarly, nothing
less than 2 would work.

Of course, it would be hard to apply this precise definition in
practice, so we would use the usual formulas--not as the actual 
definition, but as a practical method of calculation:

  If there are an odd number n of values, put them in order and take
  the middle one (the (n+1)/2th) as the median.

  If there are an even number n of values, put them in order and take
  the average of the two middle ones (the n/2th and (n/2+1)st) as the
  median.

These issues become more important when you try to define quartiles or
percentiles precisely.  See the following page for a taste:

  Defining Quartiles
    http://mathforum.org/library/drmath/view/60969.html 

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


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

Date: 08/18/2008 at 20:58:58
From: Jerry
Subject: Thank you (The definition of median)

Thanks for your help.