Measures of location: Quartiles, percentiles, deciles

In a nutshell

Besides the measures of location you already know, there are others you can calculate, such as quartiles, percentiles and deciles.

Quartiles

Each data set has $3$ quartiles.

Finding quartiles in discrete data

The lower quartile ( $Q_1$ )

To find this quartile, divide $n$ , the number of values in the data set, by $4$ . If:

The number is whole, $Q_1$ is halfway between this point and the one above.
The number is not whole, round it up. That data point will be $Q_1$ .

The middle quartile ( $Q_2$ )

This is the median of the data set.

The upper quartile ( $Q_3$ )

To find this quartile, multiply $n$ by $\dfrac{3}{4}$ . Then, apply the same rule as in $Q_1$ .

Note: For continuous data, or data in a cumulative frequency table, you need to use interpolation. This means you will have to predict each quartile, assuming the data points are evenly distributed:

$Q_1=\left(\dfrac{n}{4}\right)^{th}$ data value, $Q_2=\left(\dfrac{n}{2}\right)^{th}$ data value, $Q_3=\left(\dfrac{3n}{4}\right)^{th}$ data value.

Percentiles and deciles

Percentiles, as the name says, split up a data set into $100$ parts, while deciles split it up into $10$ parts. It is possible to relate these to quartiles:

Maths; Measures of location and spread; KS5 Year 12; Measures of location: Quartiles, percentiles, deciles

This means that $25\%$ of the values are less than the $25^{th}$ percentile (i.e. $Q_1$ ) and $75\%$ are greater.

$50\%$ of the values are less than the $5^{th}$ decile (i.e. the median), and the other $50\%$ are greater.

Note: Interpolation can also be used to find percentiles and deciles.

Example 1

Consider the following table, which shows the distance ( $x$ ) students have to take to go to school.

Distance

( $\bf km$ )

0\leq x \lt 1

1\leq x \lt 2

2\leq x \lt 3

3\leq x \lt 4

4 \leq x \lt 5

5 \leq x \lt 6

6 \leq x \lt 7

7 \leq x \leq 8

Frequency

2

2

4

7

7

8

5

3

Find the upper quartile for this data set.

$Q_3=\dfrac{3\times 38}{4}=28.5^{th}$ value

This value is contained in the class $5\leq x \lt 6$ , which has a frequency of $8$ .

So, considering the data to be evenly distributed, use interpolation to find:

$\dfrac{1}{8}=0.125$

$28.5-22=6.5$

$Q_3=5+6.5\times 0.125=\underline {5.8125 \ km}$

Find the $5^{th}$ percentile.

This corresponds to $5\%$ of the data, so the $0.05\times 38=1.9^{th}$ value.

This value is contained on the class $0\leq x \lt 1$ , which has a frequency of $2$ .

So, considering the data to be evenly distributed, use interpolation to find:

$\dfrac{1}{2}=0.5$

$1.9\times 0.5=\underline{0.95 \ km}$ 