13.06 Quartiles

Whilst the range is very simple to calculate, it is based on only two numbers in the data set, it does not tell us about the spread of data within these two values. To get a better picture of the internal spread in a data set, it is often more useful to find the set's quartiles, which can be used for a measure of spread called interquartile range (IQR).

Quartiles are scores at particular locations in the data set-similar to the median, but instead of dividing a data set into halves, they divide a data set into quarters. Let's look at how we would divide up some data sets into quarters now.

Make sure the data set is ordered before finding the quartiles or the median.

Here is a data set with 8 scores.

First locate the median, between the 4\text{th} and 5\text{th} scores:

Now there are four scores in each half of the data set, so split each of the four scores in half to find the quartiles.

We can see the first quartile, Q_{1} is between the 2\text{nd} and 3\text{rd} scores. Similarly, the third quartile, Q_{3} is between the 6\text{th} and 7\text{th} scores.

Now let's look at a situation with 9 scores:

This time, the 5\text{th} term is the median. There are four terms on either side of the median. So Q_{1} is still between the 2\text{nd} and 3\text{rd} scores and Q_{3} is between the 6\text{th} and 7\text{th} scores.

Finally, let's look at a set with 10 scores:

For this set, the median is between the 5\text{th} and 6\text{th} scores. This time, however, there are 5 scores on either side of the median. So Q_{1} is the 3\text{rd} term and Q_{3} is the 8\text{th} term.

Each quartile represents 25\% of the data set. The least score to the first quartile is approximately 25\% of the data, the first quartile to the median is another 25\%, the median to the third quartile is another 25\%, and the third quartile to the greatest score represents the last 25\% of the data. We can combine these sections together, for example, 50\% of the scores in a data set lie between the first and third quartiles.

These quartiles are sometimes referred to as percentiles. A percentile is a percentage that indicates the value below which a given percentage of observations in a group of observations fall. For example, if a score is in the 75\text{th} percentile in a statistical test, it is higher than 75\% of all other scores. The median represents the 50\text{th} percentile, or the halfway point in a data set.

Examples

The interquartile range (IQR) is the difference between the third quartile and the first quartile. 50\% of scores lie within the IQR because it contains the data set between the first quartile and the median, as well as the median and the third quartile.

Since it focuses on the middle 50\% of the data set, the interquartile range often gives a better indication of the internal spread than the range does, and it is less affected by individual scores that are unusually high or low, which are the outliers.

Subtract the first quartile from the third quartile. That is, \text{IQR} = Q_{3}-Q_{1}

Examples

Example 2

Answer the following given the frequency table:

Score	Frequency
5	1
14	1
18	3
24	2
32	1
38	2
50	5

a

Find the number of scores.

Worked Solution

Create a strategy

Add all the frequencies.

Apply the idea

\displaystyle \text{Number of scores}	\displaystyle =	\displaystyle 1+1+3+2+1+2+5	Find the sum
	\displaystyle =	\displaystyle 15	Evaluate the addition

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b

Find the median.

Worked Solution

Create a strategy

Use the formula: \left(\dfrac{n+1}{2}\right)\text{th} where n is the total number of scores, to find the position of the median.

Apply the idea

There are 15 scores in the list.

\displaystyle \text{Position of median}	\displaystyle =	\displaystyle \dfrac{15+1}{2}	Substitute n=15
	\displaystyle =	\displaystyle 8\text{th}	Evaluate

Using the frequency table we can see that the 8\text{th} score is 32.\text{Median}=32

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c

Find the lower quartile of the set of scores.

Worked Solution

Create a strategy

Find the median of the lower half of the scores.

Apply the idea

There are 7 scores in the lower half excluding the median. The lower quartile will be the 4th score from the frequency table.\text{Lower quartile}=18

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d

Find the upper quartile of the set of scores.

Worked Solution

Create a strategy

Find the median of the upper half of the scores.

Apply the idea

There are 7 scores in the upper half excluding the median. The upper quartile will be the 4th in the upper half, which is the 4+8=12th score overall.\text{Upper quartile}=50

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e

Find the interquartile range.

Worked Solution

Create a strategy

Use the interquartile range formula: \text{IQR} = Q_{3} - Q_{1}

Apply the idea

\displaystyle \text{IQR}	\displaystyle =	\displaystyle 50-18	Substitute the quartiles
	\displaystyle =	\displaystyle 32	Evaluate

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Example 3

Consider the dot plot below:

a

Find the total number of scores.

Worked Solution

Create a strategy

Count the number of dots in the dot plot.

Apply the idea

\displaystyle \text{Number of scores}

\displaystyle =

\displaystyle 15

Count the dots

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b

Find the median.

Worked Solution

Create a strategy

Use the formula: \left(\dfrac{n+1}{2}\right)\text{th} where n is the total number of scores, to find the position of the median.

Apply the idea

\displaystyle \text{Position of median}	\displaystyle =	\displaystyle \dfrac{15+1}{2}	Substitute n=15
	\displaystyle =	\displaystyle 8\text{th}	Evaluate the division

The 8\text{th} score in the dot plot is on 15.\text{Median}=15

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c

Find the lower quartile of the set of scores.

Worked Solution

Create a strategy

Find the median of the lower half of the scores.

Apply the idea

There are 7 scores in the first half excluding the median. The lower quartile will be the 4th score in the dot plot.\text{Lower quartile}=9

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d

Find the upper quartile of the set of scores.

Worked Solution

Create a strategy

Find the median of the upper half of the scores.

Apply the idea

There are 7 scores in the second half excluding the median. The upper quartile will be the 8+4=12th score in the dot plot.\text{Upper quartile}=16

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e

Find the interquartile range.

Worked Solution

Create a strategy

Use the interquartile range formula: \text{IQR} = Q_{3} - Q_{1}

Apply the idea

\displaystyle \text{IQR}	\displaystyle =	\displaystyle 16-9	Substitute the quartiles
	\displaystyle =	\displaystyle 7	Evaluate

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