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The prompt
The prompt
Mathematical inquiry processes: Interpret; compare; reason. Conceptual field of inquiry: Averages and range; sampling; representations of data.
Bryn Jones, a Principal Teacher of mathematics at Kirkcaldy High School (Fife, Scotland), devised the prompt for students in lower secondary school. He has used the prompt as a 'think, pair, share' activity.
In the question, notice, and wonder phase of the inquiry, students have described the distribution of the data (a cluster between zero and three, a gap at four and five, and outliers at six and seven), posed questions about how the data was collected, and speculated about what it represents.
The teacher should establish at the outset that dot plots, which use a number line or scale, are used to represent quantitative data.
Context
Context
The prompt invites students to suggest a context for the data. Could it represent the number of siblings of 20 students? Or the number of times 20 people exercised last week? Which of these two alternatives is more realistic?
The teacher could decide to close down the start of the inquiry by providing a context. Bryn suggests labeling the dot plot 'Number of dentist visits in a year (n = 20)'.
Design of the prompt
Design of the prompt
Bryn lists the key questions behind the design of the prompt:
What type of data does the dot plot represent?
Is this a sample or the whole population?
If it is a sample, how were the 20 data points chosen? How large would the population have been if the sample is representative?
What is the mean, mode, and median for the data set? Which average is most appropriate in this case?
What other statistical diagrams could be used to represent the data? What would be gained and what would be lost?
Each question could form the basis of a line of inquiry (see the suggestions below).
September 2025
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Bryn has created the Applying Maths website to support teachers in Scotland who teach the Higher Applications of Mathematics course. He presented the dot plot prompt in his session Not your average lesson at the Tayside Maths conference held at the University of Dundee (Scotland) in October 2023.
Lines of inquiry
Lines of inquiry
- Create contexts
There are more dot plots in the slides for which pairs of students can suggest contexts. The class can then decide which suggestions seem feasible and which one is the most realistic.
To create more diagrams, use the Dot plot generator.
2. Represent data in a dot plot
2. Represent data in a dot plot
A class could collect its own sets of discrete data to represent in dot plots. The teacher should consider the type of data carefully in order to avoid categories that embarrass or disadvantage members of the class.
The box plots can be compared with box plots drawn using data from other populations. For example, a department might collect data from the other classes in the same year group.
Alternatively, students might compare data from different sub-populations in the same class. The sub-populations should not be based on personal details but on factors over which students have no control, such as membership of a tutor group or house.
3. Compare dot plots with other representations
3. Compare dot plots with other representations
One of the teacher's aims in using the prompt might be to introduce students to other representations of the data. A pie chart, which best represents qualitative data, serves as a good comparison.
Students learn to draw a pie chart from the information in the dot plot (see the slides) before comparing the advantages and disadvantages of the two representations. The pie chart below has been drawn from the data used to create the dot plot in the prompt.
4. Calculate averages
4. Calculate averages
Another of the teacher's aims might be to introduce or consolidate knowledge of averages. Students work out the mean, median, and mode for the set of data represented in the box plot. Presenting data in a frequency table makes calculating the mean easier for large sets of data. The frequency table below uses the data from the prompt.
This line of inquiry might end with students deciding which average is the most appropriate to represent the data. In the case of the prompt, the outliers at six and seven skew the distribution of the data. They are included in the calculation of the mean but do not affect the median. For that reason, the median is the most appropriate.
Mathematician or statistician?
Mathematician or statistician?
During Bryn's session at the Tayside Maths conference, he argued that a mathematician and a statistician define a good data task for school students differently.
For the mathematician, Don Steward's Reverse averages questions (from Don's Median website) constitute a creative task that challenges students to reason about averages and range. (Sample questions are illustrated below left.)
However, for the statistician, the task lacks context and focuses exclusively on calculating averages. Bryn listed the criteria for a good data task:
The task should be set in a real-life context.
It should get students thinking about which average to use, rather than merely thinking about the calculation.
It should encourage students to think about whether it is appropriate to calculate or estimate an average for the population.
The data should be messy.
An example of a good data task from Bryn's presentation is illustrated below right. The slide shows the salaries of the people who work in a business. Is the claim about the average salary correct? Which average is representative of the population? In this case, the median is the most appropriate average to use because the data is skewed by an outlier.
Bryn created the Which average? worksheet with contextualised questions. There are more questions of the same type on The 'best' average worksheet.
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