Median, Quartiles

Date: 1/09/98 at 16:26:05
From: Dano
Subject: Find median, upper and lower quartiles

22,18,9,26,14,15,6,19,28

Date: 01/22/98 at 17:07:13
From: Doctor Sonya
Subject: Re: Find median, upper and lower quartiles

Hi there, Dano.

To find the median of a group of values, e.g., 

  1 3 4 55 6 82 2

begin by sorting the values in increasing order:

  1 2 3 4 6 55 82

If you start pairing up the values, starting at the 
extremes, one of two things will happen.  First, you 
might end up with one unpaired value in the middle:

  1 2 3 4 6 55 82
  | | |___|  |  |
  | |________|  |       median = 4
  |_____________|

In this case, the median is just the unpaired value.  
Second, you might end up with all the values paired:

  1 2 3 4 5 6 55 82
  | | | |_| |  |  |
  | | |_____|  |  |     median = (4+5)/2 = 4.5
  | |__________|  |
  |_______________|

In this case, the median is the mean of the innermost pair.  
Note that in this case, the median is not actually an 
element of the data set!

Now, if you know how to find a median, then you also know 
how to find a quartile, because:

  The lower quartile is the median of the lower half 
  of the data set.

  The upper quartile is the median of the upper half 
  of the data set. 

In case like the first one above - where we have an odd 
number of elements in the data set - we include the median 
value in both halves:

  1 2 3 4 <--> 4 6 55 82
  | |_| |      | |__|  |
  |_____|      |_______|

      2+3            6+55
  Q = ---        Q = ----
       2               2

In cases where we have an even number of elements in the 
data set, the median isn't included in either half:

  1 2 3 4 5   6 9 11 55 82
  | |___| |   | |_____|  |
  |_______|   |__________|

    Q = 3        Q = 11

You might also want to take a look at 

  Box and Whisker Plots
  http://mathforum.org/library/drmath/view/52792.html

for another point of view. 

-Doctors Sonya, Celko, and Ian, The Math Forum
http://mathforum.org/dr.math/   

For more on the meanings of "quartile" and mathematicians' 
disagreements about them, see

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

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