how to find the interquartile range

Range and box-and-whisker plots - WJEC

Calculation of range and median along with Box-and-whisker plots and Cumulative frequency tables are effective ways to compare distributions and to summarise their characteristics.

Quartiles and interquartile range

The interquartile range is another measure of spread, except that it has the added advantage of not being affected by large outlying values.

In order to calculate this value we must first understand what the lower quartile, median and upper quartile are:

the lower quartile is the median of the lower half of the data. The \(\frac{(n+1)}{4}\) value
the upper quartile is the median of the upper half of the data. The \(\frac{3{(n+1)}}{4}\) value

The interquartile range is the difference between the lower quartile and the upper quartile.

Consider the following data:

1, 3, 4, 6, 9, 14, 15, 17, 18, 22, 60

Firstly there are 11 numbers present.

The lower quartile is the value at the first quarter (once your data has been put in order). It is found using:

\(\frac{1}{4}\) of the (number of values + 1) or \(\frac{(11+1)}{4}\) = 3

So the lower quartile (LQ) is 4.

The upper quartile is the value which is three quarters of the way into our data:

\(\frac{3}{4}\) of the (number of values+ 1) or \(\frac{3{(11+1)}}{4}\) = 9

So the upper quartile (UQ) is 18.

The interquartile range (IQR) is therefore 18 - 4 = 14.

You will notice that the fact there is an outlier in this data (60) which has had no bearing on the calculation of the interquartile range. However, it would have had a massive effect on a normal range.

Sometimes we have to estimate the interquartile range from a cumulative frequency diagram.

It is acceptable when dealing with an even total cumulative frequency to use LQ = \(\frac{n}{4}\) and UQ = \(\frac{3n}{4}\) .

Line graph with Y axis of 'Cumulative frequency' and X axis of 'Length (cm)'

To work out where the quartiles are, we need to use our equations again:

\(\frac{(40)}{4}={10}\) so the LQ is the 10th number.

\(\frac{3(40)}{4}={30}\) so the UQ is the 30th number.

To find these from the graph we draw two lines across from the vertical axis, one at 10 and one at 30. We read the values for the quartiles from the graph and arrive at 38 and 47. You are allowed to be out by a little but be as accurate as you can.

Line graph with Y axis of Cumulative frequency and X axis of Length (cm) showing the lower quartile, median and upper quartile

So the interquartile range is the difference between the upper quartile and lower quartile.

Interquartile range = upper quartile - lower quartile = 47 - 38 = 9 cm.

Question

What is the interquartile range of the following data?

9, 15, 18, 22, 33, 38, 39

23. There are 7 pieces of data, so:

LQ is \(\frac{(7+1)}{4}\) = the second number.

UQ is \(\frac{3(7+1)}{4}\) = the sixth number.

The second number is 15.

The sixth number is 38.

So the interquartile range is 38 - 15 = 23.