If a data set of values is arranged
in ascending order of magnitude, then:

The median is the middle value of the data set.

The lower quartile (Q_{1}) is the median of
the lower half of the data set.

The upper quartile (Q_{3}) is the median of the upper half of the data set.

The interquartile range (IQR) is the spread of the middle 50%
of the data values. So:

The interquartile range is a more useful measure ofspread than
the range as it describes the middle 50% of the data values.

Example 8

Find the median, lower quartile, upper quartile and interquartile range
of the following data set of scores:

18 20
23 20
23 27
24 23 29

Solution:

Arrange the values in ascending order of magnitude:

18 20
20 23
23 23
24 27 29

There are 9 values in the data set.

This means the middle 50% of the data values range from 20 to 25.5.

Note:

The quartiles divide the set of measurements into four equal
parts. Twenty-five per cent of the measurements are less
than the lower quartile, fifty per cent of the measurements are
less than the median and seventy-five per cent of the
measurements are less than the upper quartile. So, fifty per
cent of the measurements are between the lower quartile and the upper
quartile.

The lower quartile, median and upper quartile are often denoted by Q_{1}, Q_{2} and Q_{3 }respectively.