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Quartiles and Interquartile Range


Date: 10/27/1999 at 11:17:39
From: Newton 
Subject: What is interquartile range?

What is interquartile range? How do I calculate Q1, Q2, and Q3?


Date: 11/12/1999 at 13:16:28
From: Doctor Johnny
Subject: Re: What is interquartile range?

Newton,

I hope that I can give you some insight into your question. Let's 
begin by looking at some definitions to help better explain the answer 
to your question.

Median (Q2): the number that divides the data into two equal halves. 
The piece of data that is exactly in the middle, when arranged in 
numerical order.

Lower Quartile (Q1): the number that divides the lower half of the 
data into two equal halves.

Upper Quartile (Q3): the number that divides the upper half of the 
data into two equal halves.

These three pieces are called quartiles because they divide the data 
into four equal portions.

Interquartile range: the difference between the upper and lower 
quartiles.

Let's look at a set of data and explain these four points of emphasis:

     25, 26, 27, 28, 29, 30, 40, 41, 42

The median is to be found first. A general formula used to find the 
piece that represents the median is (n+1)/2, where n is the number of 
pieces of data that you have. Since there are nine pieces of data 
listed, you would say (9+1)/2 = 5. This says that the 5th piece of 
data is the median.

     25, 26, 27, 28,     29,     30, 40, 41, 42
                          ^
                       median

Now we find the upper and lower quartiles. These are actually the 
medians of the upper and lower halves of data. Since there are 
four pieces of data in the upper and lower halves, you can use 
(4+1)/2 = 2.5. This means that the median is halfway between the 
second and third pieces of data in each half.

     Lower Quartile: 26 + 27 = 53/2 = 26.5
     Upper Quartile: 40 + 41 = 81/2 = 40.5

   25, 26,    (26.5)    27, 28,    29,    30, 40,    (40.5)  41, 42
                 ^                  ^                   ^
                Q1              median (Q2)            Q3

The interquartile range can be found by taking the difference between 
the upper and lower quartiles (40.5 - 26.5 = 14.) This means that 
there are fourteen units between the upper and lower quartiles.

I hope that this will help you better understand the concept of 
interquartile range. If you have any further questions, feel free to 
write and ask.

- Doctor Johnny, 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/   
    
Associated Topics:
High School Statistics

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