When I was in high school, I once made a survey to know how the ages of the audience at a movie theater were distributed across the weekend. For this reason, I stood at the exit of the theater as the movie ended, and politely asked each leaving person for their age.
I got very excited because I got to do my homework at the movie theater, so I forgot that I also had to buy a ruler and a protractor to make a display of the data I just gathered. Unfortunately, every stationer's shop was closed, so my mom told me to make a stem-and-leaf graph instead. What are those? you might be asking. Keep reading to find out!
Stem-and-Leaf Graphs Definition
There are many ways of displaying quantitative data. As usual, every method has its pros and cons. Here you will learn about stem-and-leaf graphs.
Single Stem-and-Leaf Graphs
Whenever you are dealing with one data set, you can use a single stem-and-leaf graph. These are typically known just as stem-and-leaf graphs, so no need to specify that it is single.
A stem-and-leaf graph is a diagram that summarizes numerical data by writing the relevant digits of each data entry.
You might also find them as stem-and-leaf displays, stem-and-leaf plots, or stem-and-leaf diagrams.
In a stem-and-leaf graph, you divide each data entry into two parts according to the digits. For example, by focusing only on the digits, you can divide the number \(8.13\) as \(8\) and \(13\), or as \(81\) and \(3\). You write the first part of this data entry in the left part of a table, known as the stem, and you write the last part at the right, known as a leaf.
In the above example, the stem is \(81\), and there is one leaf written as \(3\). Each number written on the leaves part of the graph corresponds to a data entry. If you have repeated values, they are usually divided by a comma, and it would look like this:
| \(81\) | \(3, 3, 4, 5, 5, 5\) |
Sometimes, you might find all the values clustered together, with no separation at all.
In such a case, each digit represents a data value.
From the context given before showing the diagram, the leaves in the graph mean that there are two \(81.3\) values, there is one \(8.14\) value, and there are three \(8.15\) values. Of course, you can have more rows in the graph, so a typical stem-and-leaf graph would look like this:
| Stem | Leaf |
| \(80\) | \(1, 1, 3, 6, 7\) |
| \(81\) | \(3, 3, 4, 5, 5, 5\) |
| \(82\) | \(0, 2, 2, 3, 6, 9\) |
| \(83\) | \(1, 1, 2, 3, 6, 8, 8\) |
| \(84\) | \(0\) |
Stem-and-leaf graphs require a legend telling you how to read them. One typical way is by taking an entry in the graph as an example.
| Stem | Leaf |
| \(80\) | \(1, 1, 3, 6, 7\) |
| \(81\) | \(3, 3, 4, 5, 5, 5\) |
| \(82\) | \(0, 2, 2, 3, 6, 9\) |
| \(83\) | \(1, 1, 2, 3, 6, 8, 8\) |
| \(84\) | \(0\) |
| \(81|3\) means \(8.13\) |
It is important to always indicate how to read the graph. The above graph could also be used to describe data that ranges from \(801\) to \(840\), so it is essential to know the context of the graph.
Comparative Stem-and-Leaf Graphs
It is also possible to display two data sets using a stem-and-leaf display. This is done by using a comparative stem-and-leaf graph.
A comparative stem-and-leaf graph is a comparison of two data sets using two stem-and-leaf graphs that share the stem portion of the graph.
A typical comparative stem-and-leaf graph looks like this:
| Leaf | Stem | Leaf |
| \(1,3,6,6\) | \(1\) | \(0, 5\) |
| \(2, 3, 4, 6, 9\) | \(2\) | \(1, 2, 2, 4, 4, 5, 7, 7, 9\) |
| \(0, 0, 2, 3, 4, 4, 4, 6, 7\) | \(3\) | \(0, 2, 7\) |
| \(1, 1, 2, 9\) | \(4\) | \(0, 5\) |
| \(3|7\) means \(37\) |
In this case, the stem is in the middle of the graph. The leaves to the left of the graph represent one data set, and the leaves to the right represent the other data set. It is important that these data sets have a similar range for this representation to make sense.
Please note that, even if they are to the left of the graph, you read the leaves of the left column the same way you read those to the right. That is, the first row tells you that one data set includes the values \(11\), \(13\), \(16\), and \(16\). The other data set includes the values \(10\) and \(15\).
How to Construct a Stem-and-Leaf Diagram
Constructing stem-and-leaf graphs depends on whether you are dealing with a display of one or two data sets. Here you will learn how to construct both.
Single Stem-and-Leaf Graphs
1. Select and identify the leading digits for your stems. For this step, you need to know the range of the data you are trying to display.
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 \(7|7\).
2. Arrange the stems into a column.
You previously found that you will need \(8\) stems for this stem-and-leaf graph, so align them in a column.
| Stem | Leaf |
| \(0\) | |
| \(1\) | |
| \(2\) | |
| \(3\) | |
| \(4\) | |
| \(5\) | |
| \(6\) | |
| \(7\) | |
3. Fill in the data. Be sure to place each leaf with its corresponding stem!
This step is rather straightforward. Separating each value with a comma is strongly advised.
| Stem | Leaf |
| \(0\) | \(3, 7, 8\) |
| \(1\) | \(0, 1, 1, 3, 6, 6, 6, 7, 7, 7, 8, 8, 9\) |
| \(2\) | \(0, 0, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 7, 8, 9, 9, 9\) |
| \(3\) | \(0, 1, 3, 4, 4, 5, 7, 7, 9, 9\) |
| \(4\) | \(1, 4, 5, 8, 9, 9\) |
| \(5\) | \(0, 1, 6, 7\) |
| \(6\) | \(1, 3, 6,\) |
| \(7\) | \(0, 5, 7\) |
4. Add a legend specifying how to read the graph.
It is crucial to indicate how to read a stem-and-leaf graph. You can use one of the data entries as an example.
| Stem | Leaf |
| \(0\) | \(3, 7, 8\) |
| \(1\) | \(0, 1, 1, 3, 6, 6, 6, 7, 7, 7, 8, 8, 9\) |
| \(2\) | \(0, 0, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 7, 8, 9, 9, 9\) |
| \(3\) | \(0, 1, 3, 4, 4, 5, 7, 7, 9, 9\) |
| \(4\) | \(1, 4, 5, 8, 9, 9\) |
| \(5\) | \(0, 1, 6, 7\) |
| \(6\) | \(1, 3, 6\) |
| \(7\) | \(0, 5, 7\) |
| \(4|5\) means \(45\) |
You can also indicate what units represent the stem and the leaf, so rather than:
\[ 4|5 \quad \text{means} \quad 45\]
something like:
\[\begin{align} \text{Stems units: } &10 \\ \text{Leaves units: } &1 \end{align}\]
is also possible. Just try to make yourself clear.
Comparative Stem-and-Leaf Graphs
For a comparative stem-and-leaf graph, you follow the same steps as with a single stem-and-leaf display, you just write the leaves of one data set to the right, and the other data set to the left.
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.
| Leaf | Stem | Leaf |
| \(1\) | |
| \(2\) | |
| \(3\) | |
| \(4\) | |
| \(5\) | |
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.
| Saturday | | Sunday |
| Leaf | Stem | Leaf |
| \(5, 6, 8, 8\) | \(1\) | \(7, 8, 9, 9\) |
| \(1, 2, 3, 3, 7\) | \(2\) | \(3, 5, 6, 6, 7\) |
| \( 5, 8, 9\) | \(3\) | \(0, 1, 6\) |
| \(0, 1, 2\) | \(4\) | \(5, 7\) |
| \(1\) | \(5\) | |
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.
| Saturday | | Sunday |
| Leaf | Stem | Leaf |
| \(5, 6, 8, 8\) | \(1\) | \(7, 8, 9, 9\) |
| \(1, 2, 3, 3, 7\) | \(2\) | \(3, 5, 6, 6, 7\) |
| \( 5, 8, 9\) | \(3\) | \(0, 1, 6\) |
| \(0, 1, 2\) | \(4\) | \(5, 7\) |
| \(1\) | \(5\) | |
| \(3|1\) means \(31\) |
Stem-and-Leaf Graph Statistics
One big advantage of having your data displayed in a stem-and-leaf graph is that the data is already arranged numerically, so finding the 5-number summary of the data set becomes a simple task.
Remember that the 5-number summary of a data set is a collection of the following five numbers: The minimum, the lower quartile, the median, the upper quartile, and the maximum. For more information about this subject, consider reading our Box Plots article.
Consider the following stem-and-leaf graph.
| Stem | Leaf |
| \(3\) | \(0, 2, 6\) |
| \(4\) | \( 1, 3, 4, 8\) |
| \(5\) | \(0, 4, 7\) |
| \(6\) | \( 4\) |
| \(5|4\) means \(0.54\) |
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.
| 5-Number Summary |
| Minimum | \[0.3\] |
| Lower Quartile | \[0.36\] |
| Median | \[0.44\] |
| Upper Quartile | \[0.54\] |
| Maximum | \[0.64\] |
It is pretty convenient to have your data arranged into a stem-and-leaf graph, right?
Advantages of a Stem-and-Leaf Graph
Stem-and-leaf graphs are a useful way of displaying your data. Here is a list of the main advantages of using a stem-and-leaf graph.
- It is easy to do. If you know how to arrange numbers, then you are ready to make a stem-and-leaf graph.
- It is made of raw data. There is no need to do operations to display data in a stem-and-leaf graph, unlike a pie chart for instance, where you have to calculate the size of each region.
- The data is arranged numerically. This particular advantage makes it easier to find things like the 5-number summary.
- You can visualize the distribution of data. By looking at which stem has the most leaves, you can have a quick idea of where the data is clustered around.
No data display is perfect. Here are some disadvantages of using stem-and-leaf graphs.
- You cannot use them to display large amounts of data values. If you are working with a data set that consists of hundreds or thousands of entries, this display becomes impractical.
- If the range of data is small, it might not be possible to choose stems that are representative enough. There is no point in doing a stem-and-leaf graph if your data values range from \(50\) to \(59\).
- You have to count each leaf to know the frequency of each observation. Finding frequencies in histograms and bar charts is easier, as you can simply take a look at the heights of the bars.
Examples of Stem-and-Leaf Graphs
As usual, you should practice building a stem-and-leaf graph. Here is a quick example.
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.
| Stem | Leaf |
| \(0\) | \(50, 50, 50, 75, 75, 75,\) |
| \(1\) | \(00, 00, 00, 25, 25, 25, 50, 50, 75,\) |
| \(2\) | \(00, 00, 50\) |
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.
| Stem | Leaf |
| \(0\) | \(50, 50, 50, 75, 75, 75\) |
| \(1\) | \(00, 00, 00, 25, 25, 25, 50, 50, 75\) |
| \(2\) | \(00, 00, 50\) |
| \(1|25\) means \( \$ 1.25\) |
How about finding the 5-number summary given a stem-and-leaf graph?
The following is a stem-and-leaf graph of the Biology grades of a particular high school classroom.
Stem | Leaf |
\(5\) | \(5\) |
\(6\) | \(0, 2, 7\) |
\(7\) | \(0, 2, 6, 7, 9\) |
\(8\) | \(0, 5, 5, 5, 8\) |
\(9\) | \(0, 2, 5, 7, 8, 9\) |
\(10\) | \(0, 0\) |
\(8|5\) means \(85\) |
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.
Stem | Leaf |
\(5\) | \(5\) |
\(6\) | \(0, 2, 7\) |
\(7\) | \(0, 2, 6, 7, 9\) |
\(8\) | \(0, \underline{5}, \underline{5}, 5, 8\) |
\(9\) | \(0, 2, 5, 7, 8, 9\) |
\(10\) | \(0, 0\) |
\(8|5\) means \(85\) |
so:
\[\begin{align} \text{Med} &= \frac{85+85}{2} \\ &= 85.\end{align}\]
You can omit to find the average if both numbers are equal.
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.
Stem | Leaf |
\(5\) | \(5\) |
\(6\) | \(0, 2, 7\) |
\(7\) | \(0, \underline{2}, 6, 7, 9\) |
\(8\) | \(0, 5, 5, 5, 8\) |
\(9\) | \(0, 2, 5, 7, 8, 9\) |
\(10\) | \(0, 0\) |
\(8|5\) means \(85\) |
So, the lower quartile is:\[ Q_1 = 72\]For the upper quartile, find the median of the second half of the data, which corresponds to the \(17\)
th value.
Stem | Leaf |
\(5\) | \(5\) |
\(6\) | \(0, 2, 7\) |
\(7\) | \(0, 2, 6, 7, 9\) |
\(8\) | \(0, 5, 5, 5, 8\) |
\(9\) | \(0, 2, \underline{5}, 7, 8, 9\) |
\(10\) | \(0, 0\) |
\(8|5\) means \(85\) |
So, the upper quartile is:\[Q_3 = 95\]With this information, you can write the 5-number summary of the data.
| 5-Number Summary |
| Minimum | \[55\] |
| Lower Quartile | \[72\] |
| Median | \[85\] |
| Upper Quartile | \[95\] |
| Maximum | \[100\] |
Stem-and-Leaf Graphs - Key takeaways
- A stem-and-leaf graph is a data display that summarizes numerical data by writing the relevant digits of each data value.
- You can compare two data sets by using a comparative stem-and-leaf graph. In these graphs, you write the data values of the other data set as leaves to the left of the stem.
- To draw a stem-and-leaf graph, you should follow these steps:
- Select and identify the leading digits for your stems.
- Arrange the stems into a column.
- Fill in the data.
- Add a legend specifying how to read the graph.
- Stem-and-leaf graphs have advantages such as:
- They are easy to do.
- They do not involve calculations.
- The data is arranged numerically.
- You can easily visualize the distribution of data.
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