7.05 Box plots

Exploration

• Here is a data set with $8$8 values:

First locate the median, between the $4$4th and $5$5th values:

Now there are $4$4 values in each half of the data set, so split each of the four values in half to find the quartiles. We can see the lower quartile is between the $2$2nd and $3$3rd values; there are two values on either side of the first quartile. Similarly, the upper quartile is between the $6$6th and $7$7th values:

We can see that the intervals between the quartiles each contain two values–one quarter of the total number of values in the data set.

Calculating quartiles

The lower quartile is also called the first quartile, or $Q_1$Q1​. It is the middle value between the minimum value and the median. To calculate the lower quartile, we identify the scores less than the median (which we call the lower half). Then we determine the middle value of this lower half.

The median is also known as the second quartile, or $Q_2$Q2​, which we have already learnt about and it represents the middle value in the sorted data set.

The median is the $\frac{n+1}{2}$n+12​th value in the sorted data set, where $n$n is the number of values in the data set.

The upper quartile is also called the third quartile, or $Q_3$Q3​. It is the middle value between the median and the maximum value. The upper quartile can be found by identifying the scores in the upper half (above the median). Then we determine the middle value of this upper half.

The range is the difference between the maximum value and minimum value in the data set.

The interquartile range, or IQR, is the difference between the upper quartile and the lower quartile. Half of the values in the data set lie within the interquartile range.

The interquartile range is a useful measure of the spread of data because, unlike the range, it is not affected by unusually large or small values.

Five number summary with CAS

The individual values required for a five number summary are readily obtained using the Statistics mode on a CAS calculator.

Worked examples

Example 1

Determine the five number summary and interquartile range for this data set:

$-2,10,-1,6,9,6,-6,1,7$−2,10,−1,6,9,6,−6,1,7.

ClassPad

This data can be entered in the Statistics mode, "list1", without needing to sort the values in ascending order.

Use the Calc -> One-variable menu to calculate the values for the five number summary (and other statistics).

• Minimum and maximum values are reported as $minX=-6$minX=−6 and $maxX=10$maxX=10, respectively.
• The median is reported as $Med=6$Med=6.
• Lower and upper quartiles are reported as $Q_1=-1.5$Q1​=−1.5; and $Q_3=8$Q3​=8, respectively.

Hence, the five number summary is set out below:

Note that, in this example, neither quartile is a value from the data set because the positions of the quartiles fall between values.

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

Example 2

Use class centres to determine the five number summary and interquartile range for the data represented by the histogram:

The histogram data can be represented by the frequency table below:

ClassPad

Using Statistics mode, enter class centres into "list1" and frequencies into "list2"

Use the Calc -> One-variable menu to calculate the values for the five number summary (and other statistics), using the "Freq" setting to select frequencies from "list2"

• $minX=35$minX=35 and $maxX=75$maxX=75. Note that these are the class centres, so it is possible that there are higher and lower values. However, in practice, box plots are usually used to summarise continuous data, with class intervals small enough that it is reasonably accurate to use the class centres to find estimates for summary statistics.
• The median is reported as $Med=55$Med=55.
• Lower and upper quartiles are reported as $Q_1=40$Q1​=40; and $Q_3=55$Q3​=55, respectively.

Hence, the five number summary is set out below:

In this case, median and upper quartile have the same value.

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

Practice questions

Question 1

QUESTION 2

Question 3

Box plots

Features of a box plot

We start with a number line that covers the full range of values in our data set.

We then plot the values from the five number summary above the number line, and connect them in a certain way to create a box plot. Here is an example:

The two vertical edges of the box show the upper and lower quartiles of the data range. The left hand side of the box is $Q_1$Q1​ and the right hand side of the box is $Q_3$Q3​. The vertical line inside the box shows the median.

Then there are two lines that extend from the box outwards. The endpoint of the left line is at the minimum value, while the endpoint of the right line is at the maximum value.

The box plot must be drawn parallel to a number line so that the values for the five number summary can be easily read from the graph.

The example above is represents the five number summary set out below:

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

For this example, the IQR is $87-51=36$87−51=36.

Since the marks of the box plot represent quartiles, each region represents $25%$25% of the values in the data set. Hence, in this example, we can make statements such as:

• $50%$50% of values lie in the range from $51$51 to $87$87 (the interquartile range)
• $25%$25% of values are less than $51$51
• $75%$75% of values are between $18$18 and $87$87
• the top $25%$25% of values are least spread out because this region is the smallest

Practice questions

Question 4

Question 5

Box plots with CAS

Worked examples

EXAMPLE 3

Use a CAS calculator to construct a boxplot for this data set, and determine the upper quartile:

$48,4,8,36,8,28,20,40,44$48,4,8,36,8,28,20,40,44.

ClassPad

Using the Statistics mode:

• Enter values into "list1"
• Set the graph Type to "MedBox"
• Use the Analysis -> Trace function, with the arrow keys to read values from the five-number summary

EXAMPLE 4

Use a CAS calculator to construct a boxplot for the data set represented by this frequency table:

ClassPad

Using the Statistics mode:

• Enter values into "list1"
• Enter frequencies into "list2"
• Set the graph Type to "MedBox"

For this data set the box plot shows that values are most spread out below the lower quartile.

Outliers and box plots

Sometimes a data set contains unusually high or low values. These unusual values are called outliers and may arise from data collection errors or due to the natural variation of the data.

We often want to identify the outlier values, and see the characteristics of the data without the effect of the outliers. In this case, we can construct a modified box plot to show the outlier values separately.

Worked example

EXAMPLE 6

For the data set used in Example 5, construct a box plot and determine the range with the outlier value displayed separately.

ClassPad

Using the Statistics mode:

• Enter values into "list1"
• Enter frequencies into "list2"
• Set the graph Type to "MedBox" with "Show Outliers" enabled.

The boxplot shows a dot to indicate that the value of $12$12 is an outlier.

With this outlier displayed separately, the minimum score is $14$14 and the lower $25%$25% of values no longer appears to be unusually spread out.

Practice questions

Question 6

Question 7

Parallel box plots

Parallel box plots are used to compare two sets of data visually.

We call these parallel box plots as they are presented parallel to each other along the same number line for comparison. They must therefore be in the same scale, so a visual comparison is fairly straightforward.

It is important to clearly label each box plot. Here we have plotted two sets of data, comparing the time it took two different groups of people to complete an online task.

We will see in later lessons that this format is very useful for comparing the characteristics of two (or more) data sets.

Practice question

Question 8