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:

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. 

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:

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

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