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Interquartile Range
The range from the minimum to maximum data values
can be distorted by one or two unusual observations. To avoid this problem,
we often focus on the range of the data values in the middle, particularly on
the middle 50 percent of the data values. To find the middle 50 percent,
first find the median
of the bottom half of the data values. In the data sets below, the data values in the
bottom half
(i.e., those observations below the median displayed in red)
are displayed in blue. It is then easy to find the
median (the boldface value) of the blue numbers. This observation is the middle or half-way
number of the bottom half of the numbers, so it marks the boundary between the first and
second quarters of the data. This number is called the first quartile
or Q1.
| Set A: |
7 |
8 |
9 |
10 |
11 |
11 |
12 |
12 |
12 |
13 |
13 |
13 |
14 |
14 |
14 |
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Min |
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Q1 |
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Q2 |
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Q3 |
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Max |
| Set B: |
1 |
2 |
3 |
4 |
5 |
5 |
6 |
12 |
18 |
19 |
19 |
19 |
20 |
20 |
20 |
Similarly, the data values in the top half (i.e., those observations above the median) are displayed
in green. The middle number of the top half of the numbers
(boldfaced) is the third quartile or Q3 because it marks the boundary between the third
and fourth quarters of the data. Note that the median, because it divides the second
quarter of the observations from the third quarter, is the second quartile or Q2.
The range between Q3 and Q1 is the range for the middle 50 percent of the observations.
Because this range is between the two quartiles, it is known as the
interquartile range (IQR).
In the case of Set A, the IQR = 13 - 10 = 3 while for Set B, the IQR = 19 - 4 = 15.
As did the range for these data sets, the IQR indicates that the data in Set B are
more spread out away from the center than are the data in Set A.
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If there are many observations, then the Q1 data value will be greater than or
equal to 25 percent of the data values. Similarly, the median Q2 will be greater than
or equal to 50 percent of the data values and Q3 will be greater than or equal to 75 percent
of the data values. However, when there are small numbers of observations, it
is impossible to hit those proportions exactly. For example, in the above data sets,
the Q3 value is greater than or equal to 12 of the 15 data values (or 80 percent). In such
cases, some computer programs calculate Q1 and Q3 as above, but some instead try
to split the difference, or "interpolate," between data values. This means that
the same value for the quartiles and the interquartile range are not always reported
by all computer programs. Click on the "Computer" icon to the left to see a
comparison of the numbers reported by several different statistical software packages.
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© 1999, Duxbury Press.
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