In this explainer, we will learn how to use a stem-and-leaf plot to organize data sets.
When you have a set of data, one of the first steps in understanding it is to try and form a picture of it. A stem-and-leaf plot is one way of doing this, giving us a picture of how often the different values within a data set occur. It is called a stem-and-leaf plot because each data value is split into a “stem,” which is the first digit or digits, and a “leaf,” which is the last digit or the units.
Let us look at an example.
Example 1: Stem-and-Leaf Plot
Write the stems, in ascending order, for the following data set and, hence, make a stem-and-leaf plot from the data: .
We are first asked to write the stems for the plot in ascending order, and to do this we will need to write the data itself in ascending order. Fourteen is the smallest number, so we put that first; 21 is the second largest, so we put that second; and so on:
Looking at the data, they are all two-digit numbers. So we will use the “tens” digits as the stems. For example, the stem for the number 14 is 1, and the stem for 36 is 3. Our stems in ascending order are then 1, 2, 3, and 4.
If we place these in a vertical column, we can then begin to draw our stem-and-leaf diagram.
Now, on the right-hand or “leaf” side of the plot, we can fill in the leaves. For example, for the number 14, the stem is 1 and the leaf is 4, so we put 4 on the right-hand side.
Notice that the leaves are also placed in ascending order from left to right. Our completed stem-and-leaf plot then looks like this.
For this data set, since the smallest value is 14 and the largest is 42, we have chosen stems that can cover all possible values from 10 to 49.
The next example also uses “tens” as the stems. But since the number of values in the data set is small, with a wide range of values, it gives us a slightly strange stem-and-leaf plot.
Example 2: A Stem-and-Leaf Plot for a Small Data Set
Write the stems, in ascending order, of a stem-and-leaf plot for the following data set: 67, 81, 74, 10, and 36. Then, complete the stem-and-leaf plot.
To write the stems, we first put the data itself in ascending order, starting with the lowest value 10 so that we have 10, 36, 67, 74, and 81. The data are all two-digit numbers, so we will use the “tens” as the stems. Therefore, we can list the stems in order as 1, 3, 6, 7, and 8. To create a stem-and-leaf plot, we list all the possible stems, as follows.
Starting with the data value 10, the stem is 1 (the tens) and the leaf is 0 (the units). So we place a “0” leaf next to the “1” stem in our diagram. This is shown in the “key” for this plot.
We can now complete our stem-and-leaf plot by filling in the remaining leaves in the same way.
This plot looks a little odd because many of the stems have no leaves and those that do have only one leaf. But it is necessary to list all the possible stems within the range of the data to get a complete picture of how the data looks. In this case, since our smallest value is 10 and our largest is 81, we have chosen stems to cover all possible values from 0 up to 99.
If you are solving a stem-and-leaf problem, it is a good idea to include a “key” to indicate how the stems and leaves are formed.
Now let us look at a slightly more complicated example.
Example 3: Stem-and-Leaf Plot: Bugs
A class of students were asked to count and record the number of bugs they could find in their yards in 30 minutes. The results were as follows:
- Write the stems, in ascending order, of a stem-and-leaf plot for the data.
- Display the data in a stem-and-leaf plot.
- Use your stem-and-leaf plot to find out how many students counted bugs in their yards.
The easiest way to put the stems in ascending order is first to find the smallest number, which in this case is 11. We then strike it out in our list of data values:
and we begin a new list with the number 11.
The next smallest number is 12, so strike that out and add it to the new list. 13 is next, so do the same with that:
and our new ordered list begins as follows:
Continuing until the new list is complete and all of the old lists have been struck out, we now have the data in ascending order:
All of the data values are two-digit numbers, so we can use the “tens” as our stems. We list the stems in our plot in ascending order.
Notice that since the smallest value is 11 and the largest is 91, we use stems that can cover all possible data values between 0 and 99. (We could also have used “1” as the first stem.)
Given our newly ordered data list, and starting from the left, one by one we put the leaves into the plot. So, for example, inputting the data ranging from 11 to 22, we have
And, finally, we complete the plot with the remaining data.
Notice that in this data set there are two instances of the number 14 and that both of these instances are represented in the stem-and-leaf plot. The same applies to the number 31 here. We must include all of the data within the plot.
To find the number of students from the stem-and-leaf plot, we count the number of leaves in the completed plot.
There are 10 leaves associated with stem 1, 5 leaves with stem 2, 4 leaves with stem 3, 1 leaf with each of stems 4 and 5, 2 leaves with stem 7, and 1 with stem 9. So, in total, there are leaves, which means that 24 students counted bugs.
This should be the same as the number of data points in our original data set, which it is!
It is always a good idea to count the number of entries in your stem-and-leaf plot and to compare it with the number of points in the data set to ensure you have included all the data and that you have not mistakenly counted one value twice.
Let us see how the stem-and-leaf plot can be used to illustrate data that is in decimal number form.
Example 4: Stem-and-Leaf Plot: Rainfall
The monthly rainfall was measured (in inches) in Minneapolis, MN, over one year with the following results.
Illustrate the rainfall data in a stem-and-leaf plot.
First, we must put the data in ascending order from smallest to largest. Ordering the data, we have
Since the data values are to two significant figures and range from 0.8 to 4.3, it makes sense to choose our stems as 0, 1, 2, 3, and 4. These cover the units for each number and every possibility for data with values between 0.0 and 4.9. So, for example, 1.2 will have stem “1.” And if the leaves are the “tenths,” that is, the numbers after the decimal points, 1.2 will have leaf “2.” We can fill in our stem-and-leaf plot using the ordered data as follows.
Our final example demonstrates how to gain information from stem-and-leaf plots and how they can be used to compare data sets.
Example 5: Comparing Stem-and-Leaf Plots
A survey was done to collect data on the number of hours per month that students in a class spent doing three activities: (1) playing computer games, (2) watching TV, and (3) playing sports. The data was put into three stem-and-leaf plots, as shown below.
- How many students played computer games for more than 60 hours a month?
- What were the greatest and the least number of hours a student played sports for?
- Comparing the stem-and-leaf diagrams for computer games and watching TV, what can you say about the number of hours students spent on these two activities?
To find how many students played computer games for more than 60 hours a month, we count the number of leaves next to the stems representing 60 and 70 on the “computer games” plot, that is, next to the 6 and 7 stems. (There are no leaves next to the stem for 80, so nothing to count there.)
There are 5 leaves next to the 6 stem, representing 62, 64, 64, 66, and 68 hours, and 7 leaves next to the 7 stem, representing hours. Hence, the number of students who played computer games for more than 60 hours a month is the total number of leaves next to the 6 and 7 stems, which is .
Had there been any leaves with value 0 next to the 6 stem (indicating a value of 60 hours), we would not have counted these as we are looking only for students who played more than 60 hours per month.
To find the greatest and least number of hours a student played sports for, we look for the largest and the smallest stem values with any leaves next to them on the “sports” plot.
The largest stem value with a leaf next to it is 5. This stem relates to potential data values from 50 to 59.
There is only one leaf next to this stem, indicating that only one student played sports for 50 or more hours. The leaf value is a 0 and, as it is next to the “5” stem, it represents 50 hours playing sports. This is the highest value on the plot, so the largest number of hours per month a student spent playing sports was 50.
Similarly, for the smallest number of hours, we look for the stem with the smallest value, which is the “0” stem. There are two leaves next to this stem with values 0 and 4.
A stem of 0 with a leaf also equal to 0 means the student this data point represents played 0 hours of sports. A stem of 0 with a leaf of value 4 means the student this data point represents played 4 hours of sports per month. The least of these is 0; hence, the least number of hours per month played by any student was 0; that is, there was one student who played no sports at all.
To compare the stem and leaf diagrams, we can actually plot them back-to-back as in the diagram below. This makes it easier to compare the data for the two activities “computer games” and “watching TV”.
The diagram shows that the number of hours spent watching TV by students is almost completely opposite to that of the number of hours spent playing computer games. Students spent much more time playing computer games than watching TV.
Specifically, we can say that very few students played computer games for less than 30 hours per month, and the vast majority played for more than 50 hours per month. We can also say that most students watched TV for less than 30 hours per month, although one student watched TV for 72 hours per month!
Let us summarize what we have learned about how to use a stem-and-leaf plot to organize data sets.
- List the data in ascending order of size.
- Decide on the form the stems will take. For example, for two-digit numbers, the stems could be the “tens.”
- Put the stems in a vertical column and separate them from the leaves with a vertical line.
- Using the ordered list of data, determine what the value of the leaf is for each item of data (e.g., in a two-digit number, the leaf will be the unit), and put the leaf in the row across from the associated stem. Continue until all the data has a leaf in the table.
Remember: You can find out the number of items in a data set represented in a stem-and-leaf plot by counting the number of leaves in the plot.