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SecTIoN 1D Describing Quantitative Data with Numbers 59

smallest to largest. Then find the quartiles, the values that divide the distribution into
four groups of roughly equal size. The first quartile Q lies one-fourth of the way
1
through the ordered list. The second quartile is the median, which is halfway through
the list. The third quartile Q lies three-fourths of the way through the list. The first
3
and third quartiles mark out the middle half of the distribution.
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DEFINITION Quartiles, First quartile Q , Third quartile Q 3
1
The quartiles of a distribution divide an ordered data set into four groups having
roughly the same number of values. To find the quartiles, arrange the data values left
to right from smallest to largest and find the median.
The first quartile Q is the median of the data values that are to the left of the
1
median in the ordered list.
The third quartile Q is the median of the data values that are to the right of the
3
median in the ordered list.

For example, here are the amounts collected each hour by a charity at a local
store:

$19 $26 $25 $37 $31 $28 $22 $22 $29 $34 $39 $31
The dotplot displays the data. Because there are 12 data values, the quartiles
divide the distribution into 4 groups of 3 values each.

Second quartile
First quartile Q 1 (median) Third quartile Q 3

18 20 22 24 26 28 30 32 34 36 38 40
Amount collected ($)
The quartiles can be used to help us describe the position of an individual data
value in a distribution of quantitative data. We’ll discuss ways to measure posi-
tion/location in Section 1E.
The interquartile range (IQR) measures variability using the quartiles. It’s
simply the range of the middle half of the distribution.

DEFINITION Interquartile range (IQR)
The interquartile range (IQR) is the distance between the first and third quartiles of
a distribution. In symbols:

IQR = Q 3 −Q 1

For the distribution of amount collected each hour at the charity store, you
can check that Q 1 = $23.50, Median = $28.50, and Q 3 = $32.50. The interquar-
tile range of the distribution is

IQR = $32.50 – $23.50 = $9.00

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