Interquartile Range
The interquartile range (IQR) is a measure of variability,
based on dividing a data set into
quartiles.
Quartiles divide a rank-ordered data set into four equal parts. The values that
divide each part are called the first, second, and third quartiles;
and they are denoted by Q1, Q2, and Q3, respectively.
- Q1 is the "middle" value in the first half of the rank-ordered
data set.
- Q2 is the
median
value in the set.
- Q3 is the "middle" value in the second half of the rank-ordered
data set.
The interquartile range is equal to Q3 minus Q1.
For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11.
Q1 is the middle value in the first half of the data set. Since there are an
even number of data points in the first half of the data set, the middle value
is the average of the two middle values; that is,
Q1 = (3 + 4)/2 or Q1 = 3.5. Q3 is the middle value in the second half
of the data set. Again, since the second half of the data set has an even
number of observations, the middle value is the average of the two
middle values; that is, Q3 = (6 + 7)/2 or Q3 = 6.5. The interquartile
range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3.
An Alternative Definition for IQR
In some texts, the interquartile range
is defined differently. It is defined as the difference between
the largest and smallest values in the middle 50% of a set of
data.
To compute an interquartile range using this definition, first remove
observations from the lower quartile. Then, remove observations
from the upper quartile. Then, from the remaining observations,
compute the difference between the largest and smallest values.
For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11.
After we remove observations from the lower and upper quartiles,
we are left with: 4, 5, 5, 6. The interquartile range (IQR) would
be 6 - 4 = 2.
