# 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 ${Q}_{3}-{Q}_{1}$ .