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Quartiles

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

Math diagram

$Q_{1}$ (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 $1 st$ quartile, median, and $3 rd$ 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 $6 th$ 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 $29$ . 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 $Q_{3} - Q_{1}$ .