4.7.2 Box-and-whisker Plots
Box-and-whisker plots are graphical representations of statistical data that display the distribution of a dataset along with its median, quartiles, and outliers.
They are often used to summarize the distribution of a dataset and provide a visual summary of its key statistical measures.
A box plot typically consists of a rectangular box, which represents the interquartile range \( (IQR) \) of the dataset, and two whiskers, which extend from the box to the minimum and maximum values that fall within \( 1.5 \) times the \( IQR \).
Any data points that fall outside the whiskers are considered outliers and are plotted as individual points.
The box itself is divided into three sections that represent the first quartile \( (Q_1) \), median, and third quartile \( (Q_3) \) of the data set.
The median is the value that falls in the middle of the dataset
The first quartile represents the value that is greater than or equal to \( 25 \)% of the data points
The third quartile represents the value that is greater than or equal to \( 75 \)% of the data points.
Box plots can be used to compare the distribution of multiple datasets side by side, making it easy to identify differences in the center, spread, and shape of each dataset.
They are commonly used in statistical analysis, data visualization, and machine learning.
Drawing a box plot:
To draw a box-and-whisker plot
Collect your data and organize it into a dataset
Calculate the minimum value, maximum value, median, and quartiles of the dataset.
Calculate the interquartile range \( (IQR) \)
Which is the difference between the third quartile and the first quartile.
Identify any outliers in the dataset.
Outliers are data points that fall more than \( 1.5 \) times the \( IQR \) below the first quartile or above the third quartile.
Draw a number line that spans the range of the dataset.
Draw a box that spans the first quartile to the third quartile, with a line inside the box marking the median.
Draw whiskers that extend from the box to the minimum and maximum values within \( 1.5 \) times the \( IQR \).
If there are outliers, plot them as individual points beyond the whiskers.
Add a title and labels for the \( x \) and \( y \) axis.
Worked example:
Here is some information about the masses of potatoes in a sack:
The largest potato has a mass of \( 174 \) g.
The range is \( 69 \)g.
The median is \( 148 \) g.
The lower quartile is \( 121 \) g.
The interquartile range is \( 38 \)g.
On the grid below, draw a box and whisker plot to show this information.
The range is the difference between the largest mass and the smallest mass.
The largest mass and the range are given calculate smallest mass by
\( Range=Largest-Smallest \)
\( 69=174-smallest \ mass \)
\( Smallest \ mass=174-69 \)
\( Smallest \ mass=105 \)
The interquartile range is the difference between upper quartile and lower quartile.
The interquartile range and lower quartile are given, calculate upper quartile by
\( IQR=UQ-LQ \)
\( 38=UQ-121 \)
\( UQ=121+38 \)
\( UQ=159 \)
Draw the vertical lines for smallest mass, lower quartile, median, upper quartile and largest mass.
Form a box by using the lines for the upper and lower quartiles as two sides of a rectangle.
For the whiskers draw horizontal lines from the middle of the vertical lines
For the smallest and largest masses to the box.
Comparing box plots:
Comparing box plots can be a useful way to identify differences and similarities in the distribution of different datasets.
To compare box plots
Plot the box plots side by side, with the same scale on the \( y \)-axis.
This will make it easy to visually compare the center, spread, and shape of each dataset.
Compare the medians of each dataset.
A higher median may indicate that one dataset has a higher central tendency than another dataset.
Look at the boxes of each dataset.
A wider box may indicate a larger spread of the data, while a narrow box may indicate a more tightly clustered dataset.
Look at the whiskers of each dataset.
Longer whiskers may indicate a larger range of the data, while shorter whiskers may indicate a more concentrated dataset.
Identify any outliers in each dataset.
Outliers may indicate extreme values in the data or errors in the data collection process.
Worked example:
The box and whisker plots show the times spent exercising in one week by a group of women
and a group of men.
Below are two statements comparing these times.
For each one, write down whether you agree or disagree, giving a reason for your answer.
For the first statement, compare the medians
For the second statement, compare the interquartile ranges
\( IQR \) for men\( =160-40=120 \)
\( IQR \) for women \( =210-50=160 \)
\( IQR \) for men\( < \) (\( IQR \) for women)
Test yourself
Question 1:
The average speeds, in \( km \)/h of cars traveling along a road are recorded.
Find
the lowest speed is recorded,
[1]
the median,
[1]
the interquartile range.
[1]
Question 2:
Sue works for a company that delivers parcels.
One day the company delivered \( 80 \) parcels.
The table shows information about the weights, in \( kg \), of these parcels.
Complete the cumulative frequency table.
[1]
On the grid opposite, draw a cumulative frequency graph for your table.
[2]
Sue says,
\( 75 \)% of the parcels weigh less than \( 3.4 \)kg.
Is Sue correct?
You must show how you get your answer.
[3]
Question 3a:
Use the cumulative frequency diagram to find an estimate for
the median,
[1]
the interquartile range,
[2]
the number of students who took more than \( 40 \) minutes.
[2]
Question 3b:
Roberto records the value of each of the coins he has at home.
The table shows the results.
Find the range.
[1]
Find the mode.
[1]
Find the median.
[1]
Work out the total value of Roberto’s coins.
[2]
Work out the mean.
[1]
Question 3c:
Calculate an estimate of the mean.
[6]
Question 4:
The cumulative frequency graph shows the times, in minutes, each of \( 100 \) children spent
exercising in one week.
Use the cumulative frequency diagram to find an estimate of
the \( 60^{th} \)percentile,
[1]
the number of children who spent more than \( 3 \) hours exercising.
[2]
Question 5:
\( 100 \) students were each asked how much money, \( $m \), they spent in one week.
The frequency table shows the results.
Complete the frequency table below.
[2]
On the grid, draw the cumulative frequency diagram.
[3]
Use your cumulative frequency diagram to find an estimate for
the median,
[1]
the interquartile range,
[2]
the number of students who spent more than $\( 25 \).
[2]