Box and Whisker Plots

Date: 05/18/2000 at 14:19:59
From: Drew Ashley
Subject: Quartiles of box-and-whisker plots

Dear Dr. Math,

My 7th-grade students and I are drawing box and whisker plots. I am 
looking for confirmation on the placement of the first and third 
quartiles.

If there is an odd number of data, is the median considered to be 
part of the subgroup used to find the upper (or lower) quartile? It 
seems reasonable to do so if that makes the subgroup an odd number of 
data, and not to do so otherwise. Is that right?

     Example: 1, 2, 3, 4, 5

The median is 3; should the lower quartile be 2 or 1.5?

Thanks for your help,
D. Ashley

Date: 05/19/2000 at 12:13:57
From: Doctor TWE
Subject: Re: Quartiles of box-and-whisker plots

Hi Drew - thanks for writing to Dr. Math.

According to _Statistics for Engineering and the Sciences_, W. 
Mendenhall and T. Sincich, 1995: For small data sets, given n data 
points A_1 to A_n, the lower and upper quartiles are calculated as 
follows:

   1. Calculate l = (1/4)(n+1), round to the nearest integer
        (if l falls halfway between two integers, round UP)

   2. A_l is the lower quartile.

   3. Calculate u = (3/4)(n+1), round to the nearest integer
        (if u falls halfway between two integers, round DOWN)

   4. A_u is the upper quartile.

So for your example, n = 5 and

     l = (1/4)(5 + 1) = 1.5 -> 2, thus LQ = A_2 = 2

     u = (3/4)(5 + 1) = 4.5 -> 4, thus UQ = A_4 = 4

Note that with this definition, the upper and lower quartiles are 
always one of the data points (thus, they could not be 1.5 and 4.5 for 
your example). This differs from the median, which is the average of 
the middle two if n is even, and thus might not be one of the data 
points.

The reason for the "inconsistency" in rounding quartiles that fall 
halfway between two integers (i.e. they end in .5) is to achieve 
symmetry. If we rounded up on both quartiles when they ended in .5, 
for your example data set we'd have LQ = 2 and UQ =5, and these are 
not equidistant (in terms of number of data points) from the median or 
the extreme data points.

You might also want to take a look at   

  Median, Quartiles
  http://mathforum.org/library/drmath/view/58399.html

for another point of view.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, 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/