Box-and-whisker plots

Box-and-whisker plots exercise

• Explain how to create a box-and-whisker plot.

  Hints

  Here you see the list of scores of student one and the sorted version of this list below it.

  Above both lists you can see the resulting box-and-whisker plot.

  The median of a sorted list divides this list in two halfs with the same number of data points.

  The interquartile range $IQR=Q_3-Q_1$ is a measure for variation.

  Solution

  Here you see a complete box-and-whisker plot.

  How can we create such a plot? Well, for a given set of data, we first have to order the data points, namely the numbers.

  Next we have to determine some critical values:

  • The minimum
  • The maximum
  • The median
  • The first quartile $Q_1$
  • The third quartile $Q_3$

  Let's have a look at the ordered data list:

  $1,~2,~2,~3,~4,~4,~6,~7,~8,~9$

  The minimum of this list is the lowest or the most left value $1$, and the maximum the highest or the most right value $9$. We draw those values in a graph with a horizontal axis labeled from $1$ to $10$.

  The median of an odd data list is the middle of the list. If the number of data points in the list is even, we choose the average of both middle data points. The list above has an even number of elements. So the median is the average of $4$ and $4$, which is $4$. We also draw this value in the graph above.

  The first quartile is the median of the first half of the list. Here it's $2$. The third quartile is the median of the second half of the list, $7$. The interquartile range is given by $Q_3-Q_1=7-2=5$. We draw $Q_1$ as well as $Q_3$ in the graph above.

  But we haven't finished yet:

  • We still draw a box from $Q_1$ to $Q_3$.
  • Last we connect the minimum, $Q_1$, $Q_3$, and the maximum with whiskers.

• Find the right box-and-whisker plot.

  Hints

  Remember to first order the list.

  The minimum is the most left and the maximum the most right value of the ordered list.

  The median of an even numbered list is the average of the two middle data points.

  The median of an odd numbered list is the middle data point of the list.

  However, the median divides the list in two lists of the same size.

  $Q_1$ is the median of the first half and $Q_3$ is the median of the second half of a list of data.

  Solution

  To create the box-and-whisker plot we want, we have to do the following:

  1. Sort the list: $1,1,2,3,5,5,6,6,8,8$.
  2. The minimum is $1$ and the maximum $8$ (the most left and most right values, respectively).
  3. The median is the average of $5$ and $5$, which is $5$. Here we have to determine the average of the two middle values because the number of elements in the list is even.
  4. The median of the first half of the list, $1,1,2,3,5$, is $Q_1=2$, and the median of the second half of the list, $5,6,6,8,8$, is $Q_3=6$.
  5. We draw a box from $Q_1$ to $Q_3$.
  6. Lastly we connect the minimum $1$ and $Q_1=2$ as well as $Q_3=6$ and the maximum $8$ with whiskers.

  The resulting plot is shown beside.

• Compare the different data sets.

  Hints

  The interquartile range $IQR=Q_3-Q_1$ is a measure for variance.

  The smaller the variance the higher the consistency.

  The length of the box is the $IQR$.

  Solution

  Looking at the box-and-whisker plots for each student, we can compare the students to figure out which student is the most consistent.

  The box of student $1$ is bigger than this one of student $2$. This is a measure for variation. Or, in other words, student $2$ is more consistent.

  The minimum values are the same and the maximum of student $2$ is $1$ more than this one of student $1$.

  Also the median of student $2$ is higher than this one of student $1$.

  So we can conclude that, using the box-and-whisker plots, that student $2$ is the more consistent one.

• Find the data set(s) corresponding to the box-and-whisker plot pictured.

  Hints

  Order each set so that you can find the critical values.

  First check the minimum and maximum of the set.

  Check each data set: it's possible that two different data sets lead to the same box-and-whisker plot.

  Solution

  First exclude any list with a either a minimum or maximum different than the minimum or maximum of the box-and-whisker plot pictured.

  This means that we exclude the fourth list, as it does not have a minimum of $1$, and the first list, as it does not have a maximum of $6$.

  Next let's have a look at the median: the median of the fourth list is $5$ and for all the remaining lists it's $4$.

  So we calculate $Q_1$:

  • $Q_1=2$ for the third list
  • $Q_1=1$ for the last list

  Those lists can also be excluded, as $Q_1=3$ for the box-and-whisker plot pictured.

  There are two lists left: calculating $Q_3$ for those lists, we get:

  • The second list has the third quartile $Q_3=5$.
  • The fifth list has the third quartile $Q_3=5$.

  So those lists are the lists which correspond to the box-and-whisker plot pictured.

• Label the values in a box-and-whisker plot.

  Hints

  The minimum is the lowest and the maximum the highest value of a data list.

  The median lies in the middle of a sorted data list. It divides the list in two halfs of the same size.

  The quartiles are the medians of the halves of a list:

  $Q_1$ ($Q_3$) is the median of the first (second) half of the list.

  minimum $\le$ $Q_1$ $\le$ median $\le$ $Q_3$ $\le$ maximum

  Solution

  Here you see the solution pictured.

  The data list as well as the sorted data list is already given.

  The minimum, $1$, is the most left and the maximum, $8$, the most right value.

  The median lies in the middle of the list. The given list has an even number of entries, so the median is the average of the two middle data points, $4$ and $4$. Thus the median is $4$.

  The quartiles are medians as well, each time for lists with an odd number of entries:

  • $Q_1=2$ is the median of the first half $1,1,2,3,5$.
  • $Q_3=2$ is the median of the second half $5,6,6,8,8$.

• Determine the interquartile range $(IQR)$.

  Hints

  The interquartile range $IQR=Q_3-Q_1$ is a measure of variance for a data set.

  First sort the list and determine the median.

  If the list has an odd number of entries, then the median is the middle data point of the list.

  If the list has an even number of entries, then the median is the average of both middle data points.

  The first quartile $Q_1$ is the median of the first half of a data set and the third quartile $Q_3$ is the median of the second half of a data set.

  The interquartile range can also be a rational number.

  Solution

  To determine the interquartile range you first have to establish the first and the third quartile.

  For this you need the median of the list.

  The median of an odd numbered list is the middle data point of the list. If the list is even numbered the median is the average of both middle data points.

  Let's start with $3,~3,~4,~7,~10,~11,~8,~2,~5,~4,~9$.

  1. The sorted list is $2,~3,~3,~4,~4,~5,~7,~8,~9,~10,~11$.
  2. The median is the middle of this list: $5$.
  3. The median of the first half $2,~3,~3,~4,~4$ is $Q_1=3$ and the median of the second half $7,~8,~9,~10,~11$ is $Q_3=9$.
  4. So $IQR=9-3=6$.

  Now let's consider $5,~3,~4,~3,~12,~7,~8,~4$.

  1. The sorted list is $3,~3,~4,~4,~5,~7,~8,~12$.
  2. The median is the average of the two middle data points $4$ and $5$, namely $\frac{4+5}2=\frac92=4.5$.
  3. The median of the first half $3,~3,~4,~4$ is $Q_1=\frac{3+4}2=\frac72=3.5$ and the median of the second half $5,~7,~8,~12$ is $Q_3=\frac{7+8}2=\frac{15}2=7.5$.
  4. So $IQR=7.5-3.5=4$.

  Finally, we have $7,~5,~8,~5,~7,~6,~12,~3,~5$.

  1. The sorted list is $3,~5,~5,~5,~6,~7,~7,~8,~12$.
  2. The median is the middle of this list: $6$.
  3. The median of the first half $3,~5,~5,~5$ is $Q_1=5$ and the median of the second half $7,~7,~8,~12$ is $Q_3=\frac{7+8}2=\frac{15}2=7.5$.
  4. So $IQR=7.5-5=2.5$.