Mathematics Developmental Continuum
Dot Plots and Stem-and-Leaf Plots: 4.5
Supporting Materials
Indicator of Progress
Success depends on students being able to produce a dot plot or stem-and-leaf plot from a given data set, making appropriate choices of scale and/or stems. At this level, they can use the resulting plots to informally discuss the distribution of the data, including the range and the mode.
At the next level, this discussion becomes more sophisticated and they identify the other measures of centre (i.e. median and mean). They then also make meaningful comparisons of 2 data sets by producing plots that can easily be compared (e.g. same scale, appropriately aligned, possibly back-to-back).
For more information see also: more about dot plots and stem-and-leaf plots.
Illustration 1: Plotting techniques
Students appreciate that they have to take care when producing dot plots and stem-and-leaf plots because the success of the representation relies on evenly spacing the numbers or symbols. Careless plotting can result in misleading plots such as those shown below. Using a word processor may not solve these problems because different numerals can have different widths. Using 5mm grid paper helps with the spacing issue for this and for many other mathematical processes.
Examples of bad plots: Box plots and stem-and-leaf plots only give a good pictorial representation of frequency when the 'dots' or 'leaves' are aligned.
Illustration 2: Choice of stems
Good stem-and-leaf plots require good choices of stems. Students may make inappropriate choices of stems for stem-and-leaf plots, resulting in too many data values being associated with each stem, or in a stem being omitted because there are no data values associated with it. For example, the two stem-and-leaf plots below depict the same data, but in the one on the left it is not obvious that there is an outlier (72 – not near other values), nor is the bi-modal (two-peaked) nature of the data revealed. Both these features are more evident in the representation on the right.
Data set {41, 41, 42, 42, 42, 43, 43, 43, 43, 46, 51, 53, 57, 57, 58, 58, 59, 59, 59, 72}
It should be noted that most students will find it easy to choose multiples of ten as stems (or multiples of 100).
A good user of these plots can make more sophisticated stem choices such as the range of 5 shown in the example on the right above (which has stems for 40-44, 45-49, 50-54, 55-59, etc.).
Illustration 3: Making the mode obvious
Both dot plots and stem-and-leaf plots produce representations that are related to bar graphs or histograms, and readily show the more frequent data. This does, however, depend on the chosen scale or stem values. For example, in the following dot plot it is obvious that the mode is 24.
The stem-and-leaf plot will reveal the modal region. The plot below shows that there are more values in the 40-44 range than any other region. It is less obvious that 43 is the mode for this data set.
Teaching Strategies
Students should be given opportunities to produce both types of plots, but beyond initial demonstrations, such tasks should not be mere mechanical exercises. Students should use real and relevant data, should discuss the choices for producing good representations (features of these plots as well as specific choice of scale) and interpret the data in terms of the real situation from which it arose.
They should also have the opportunity to use these representations and others in statistical investigations where they collect data for a purpose, analyse it and write a report on the findings.
Activity 1: Describe it encourages students to interpret the data that they are plotting.
Activity 2: Selecting stems gives students practice making appropriate choices for stem-and-leaf plots.
Activity 3: Making comparisons advocates using real relevant data for producing data plots that allow comparisons.
Activity 1: Describe it
Students need opportunities to practice the mechanics of producing dot plots and stem-and-leaf plots from data sets. After students make a plot, they should describe what the plot shows about the data.
Their descriptions will become increasingly sophisticated, as they see more features in the plots.
Mode and range and the general shape of the distribution are easiest to see. Teachers can highlight how the plots organise the data to make this information accessible.
As students become familiar with the plots and with statistical thinking, they should be encouraged to attend to range, outliers, distribution, mean, median, and so on.
For example the plot below is from data for a swimmer’s times for 50m freestyle. We can see that the swimmer seems to swim at speeds consistently within 4 seconds of 56 seconds, with a personal best time of 50 seconds.
This data below could be described as being approximately evenly distributed between 52 and 60, with both the median and the mode being 56 and with an outlier of 50 (although it is not a very distant outlier).
Activity 2: Selecting stems
Students should be given opportunities to make decisions about what numbers to use as stems for a given data set and to justify those decisions. The resource sheet 'Selecting Stems' provides some realistic data and asks the students to choose the stems for a stem-and-leaf plot. The students should discuss the choices made, and what makes an appropriate choice. The examples used in the resource sheet give an idea of the different types of data that students might encounter. Teachers could certainly come up with alternative and more relevant examples for their own students to consider, taking care to encompass the breadth of examples shown on the student resource sheet. Here is the resource sheet: 'Selecting Stems (PDF - 35Kb)'
Activity 3: Making comparisons (extension activity at level 4.75)
Both dot plots and stem-and-leaf plots are ideal representations for making comparisons between two (or more) data sets. To do this, pairs of dot plots should be constructed to have the same range and scale. They should be drawn one above the other, as shown below.
Alternatively, back-to-back stem-and-leaf plots can be constructed on either side of a central stem, as shown below.
After plotting the data, students should be able to comment on similarities and differences between the two data sets, discussing range, mode, median and other features.
Ideally, real data should be used for such activities. Here are some possibilities:
- Compare the heights of boys and girls in the class.
- Compare the heights of students in one year level with those of another.
- Compare the shoe-sizes of girls to those of boys.
- Compare the shoe-sizes of students in one year level with those of another.
- Compare the time taken to write out the alphabet for left-handers and right-handers.
- Compare the number of hours of television watched by girls with those of boys.