# Quartiles

A
**
Quartile
**
is a percentile measure that divides the total of
$100\%$
into four equal parts:
$25\%,50\%,75\%\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}100\%$
. A particular quartile is the border between two neighboring quarters of the distribution.

*
${Q}_{1}$
(quartile
$1$
) separates the bottom
$25\%$
of the ranked data (Data is ranked when it is arranged in order.) from the top
$75\%$
.
${Q}_{2}$
(quartile
$2$
) is the mean or average.
${Q}_{3}$
(quartile
$3$
) separates the top
$25\%$
of the ranked data from the bottom
$75\%$
. More precisely, at least
$25\%$
of the data will be less than or equal to
${Q}_{1}$
and at least
$75\%$
will be greater than or equal
${Q}_{1}$
. At least
$75\%$
of the data will be less than or equal to
${Q}_{3}$
while at least
$25\%$
of the data will be greater than or equal to
${Q}_{3}$
.
*

**
Example 1:
**

Find the $1\text{st}$ quartile, median, and $3\text{rd}$ quartile of the following set of data.

$24,26,29,35,48,72,150,161,181,183,183$

There are $11$ numbers in the data set, already arranged from least to greatest. The $6\text{th}$ number, $72$ , is the middle value. So $72$ is the median.

Once we remove $72$ , the lower half of the data set is

$24,26,29,35,48$

Here, the middle number is $\text{29}$ . So, ${Q}_{1}=29$ .

The top half of the data set is

$150,161,181,183,183$

Here, the middle number is $181$ . So, ${Q}_{3}=181$ .

The interquartile range or IQR is the distance between the first and third quartiles. It is sometimes called the H-spread and is a stable measure of disbursement. It is obtained by evaluating ${Q}_{3}-{Q}_{1}$ .