Stem and Leaf Graph

Suppose you made a survey on the ages of the customers at a mall on a particular day. From the observations you made, the youngest one was \(3\) years old, while the eldest one was \(77\) years old.

Since your data values are numbers with up to \(2\) digits, a natural way of dividing your data values is by choosing the leftmost digit as the stem, and the rightmost digit as the leaves. However, what to do with toddlers? In this case, you can simply place a \(0\) to the left, so \(3\) years would be \(03\) years. This way, you can represent \(3\) years as \(0|3\), and \(77\) years as \(7|7\).

You previously found that you will need \(8\) stems for this stem-and-leaf graph, so align them in a column.

This step is rather straightforward. Separating each value with a comma is strongly advised.

It is crucial to indicate how to read a stem-and-leaf graph. You can use one of the data entries as an example.

Sarah, one of your friends, happens to work in a boutique at the mall. When you tell her that you were doing a stem-and-leaf graph, she tells you that she is doing one as well. Sarah shows you a list of the ages of the customers of the last weekend.

\[ \begin{gathered} \textbf{Saturday} \\ 15, 16, 18, 18, 21, 22, 23, 23, 27, 35,38, 39, 40,41, 42, 51 \end{gathered}\]

\[ \begin{gathered} \textbf{Sunday} \\ 17, 18, 19, 19, 23, 25, 26, 26, 27, 30, 31, 36, 45, 47 \end{gathered}\]

Use the above data to make a comparative stem-and-leaf graph of the ages of the customers on the different days of the weekend.

Solution:

1. Select and identify the leading digits for your stems.

Here, the lowest data value is \(15\) and the highest is \(51\). Since all the data consists of two-digit numbers, the easiest way to select the stems is by choosing them as the leftmost digit, so the rightmost digit will be your leaves.

2. Arrange the stems into a column.

Here you will have five stems to write the data, so arrange them into a column. Remember that, since you are doing a comparative stem-and-leaf display, the stems will be in the middle.

3. Fill in the data.

Here it is advised that you add a header to identify each data set. Note that there will be no Sunday data on the \(5\) row.

4. Add a legend specifying how to read the graph.

An example is enough to illustrate how to read the graph, so add it at the bottom of the graph.

Consider the following stem-and-leaf graph.

Find the 5-number summary of the data set that is being represented by the graph.

Solution:

The data is already arranged numerically in a stem-and-leaf graph, so the first value, \(3|0\), corresponds to the minimum, while the last value, \(6|4\), corresponds to the maximum. Considering the directions of the graph, this means that:

\[ \text{Min} = 0.3\]

and

\[ \text{Max} = 0.64\]

For the median, note that there are \(11\) leaves in total, which means that there are \(11\) data values, so the median corresponds to the \(6\)^th value, \( 4|4\), which represents \(0.44\), so:

\[\text{Med} = 0.44\]

For the lower quartile note that the first half of the data corresponds to \(5\) numbers, so the lower quartile is the third leaf, \(3|6\), so:

\[ Q_1 = 0.36\]

Find the upper quartile by looking at the last half of the data. There are five values that come after the median, so the third value after the median, which corresponds to the \(9\)^th leaf, \(5|4\), is the upper quartile. This means that:

\[ Q_3=0.54\]

With the above information, you can now write the 5-number summary of the data set.

The following list is a record of the prices of the items from a local bakery.

\[ \$ 1.00, \, \$ 0.75, \, \$ 1.25, \, \$ 0.50, \, \$ 1.25, \, \$ 1.50, \, \$ 1.00, \$ 0.50, \, \$ 0.75, \]

\[ \$ 2.00, \, \$ 0.50, \, \$ 0.75, \, \$ 1.00, \, \$ 1.75, \, \$ 2.50, \, \$ 1.25, \$ 1.50, \, \$ 2.00. \]

Make a stem-and-leaf graph of the data.

Solution:

1. Select and identify the leading digits for your stems.

The prices range from \( \$ 0.50\) up to \( \$2.50,\) so the leading digit for your stems can be the whole dollars. This means that you will have three stems: \(0\), \(1\), and \(2\).

2. Arrange the stems into a column.

This stem-and-leaf display will consist of three rows.

3. Fill in the data

This step is rather straightforward. In this case, your leaves will correspond to the cents of the prices.

4. Add a legend specifying how to read the graph.

Here the stems represent whole dollars, while the leaves represent cents. Add this information at the bottom of the graph.

The following is a stem-and-leaf graph of the Biology grades of a particular high school classroom.

Find the 5-number summary of the data.

Solution:

By looking at the first value of the stem-and-leaf graph, you can find that the lowest value is \((5|5)\), which corresponds to \(55\). Similarly, the highest is \( (10|0) \), so the highest grade is \(100\). This means that:

\[\text{Min} = 55\]

and

\[\text{Max}=100\]

Next, find the median. Begin by noting that there are \(22\) grades, which is an even number. This means that the median will be the average of the \(11\)^th and \(12\)^th values.

so:

\[\begin{align} \text{Med} &= \frac{85+85}{2} \\ &= 85.\end{align}\]

To find the quartiles, note that the data can be divided into two equal halves. Find the lower quartile by looking for the median of the first \(11\) values of the data, that is, find the \(6\)value.