Grouped data inquiry

The prompt

Mathematical inquiry processes: Verify; test different cases; interpret and reason. Conceptual field of inquiry: Frequency tables for grouped data; estimated mean and median, modal class; linear interpolation; histograms; flat, normal, and skewed distributions; standard deviation.

The prompt is suitable for classes across the age range in secondary school. Designed for a year 8 (grade 9) class, it also holds intrigue for older students.

The year 8 students could work out averages from raw data and from discrete frequency distributions. The prompt acted as a bridge from their existing knowledge to grouping data with class intervals and using the mid-points of the intervals to estimate averages (mean and median) and identify the modal class.

The contention in the prompt - the more groups you create, the greater the accuracy of the estimates - is generally true. That is because when there are more (and, therefore, narrower) class intervals the mid-points are closer to the actual data points.

However, the number of groups is not the only factor and sometimes it is not the most important one. Often the creation of narrower class intervals in the range where the data is concentrated leads to better estimates.

In the example from the slides, the four unequal intervals (below right) give more accurate estimates than the five equal intervals because the data is concentrated in the range 20 ≤ t < 30. Students might pursue the example further by changing the intervals of the four groups to 0 ≤ t < 20, 20 ≤ t < 25, 25 ≤ t < 30, 30 ≤ t < 60.

Question, notice, and wonder

Students' contributions to the question, notice, and wonder phase of the inquiry included:

The mean, median, and mode are averages.
What do 'raw data' and 'class intervals' mean?
How do you group data?
How many groups should we use?
How do you estimate averages from grouped data?

Once students are able to estimate averages from data grouped in class intervals, they wonder whether the prompt is always true. A fast-paced inquiry develops as students test different data sets.

Guided inquiry

In the slides there are lines of inquiry based on the six regulatory cards shown above. In a guided inquiry, pairs of students suggest how the inquiry could proceed by selecting a card and justifying their selection to the class. For a detailed description of the meaining of each card in the context of the inquiry, see the lines of inquiry below.

April 2026

Lines of inquiry

The six regulatory cards form a sequence of actions that together constitute a complete mathematical inquiry.

The example in the slides involves data about the average waiting time for rides at an amusement park. The data is grouped in eight ways using three to six class intervals with equal and unequal widths.

If the topic of estimated averages is new to students, they are likely to choose this regulatory card. They will be more motivated to engage with the teacher's explanation if they have requested it.

The teacher explains how to estimate the mean, identify the modal class, and either find the class interval in which the median lies or use linear interpolation to find an estimate of the median. Ideally, some students will be able to contribute to the co-construction of the explanation as it develops.

Once the teacher has outlined the procedures, students practise with the remaining frequency tables. Pairs of students could select two tables to work on and then check the results with their partner before comparing their results to those in the slides.

At this point the teacher might orchestrate a discussion on whether the contention in the prompt is true. The provisional conclusion will be revised as the class generates more results.

As part of the discussion the class might consider the optimum number of groups for a set of data. With 21 data points in the example, three groups might be considered too few and anything above six would be too many.

The class will require more than the results from one set of data to come to a final conclusion. Students can go online to find another set of average waiting times at an amusement park to test the contention.

Alternatively they could collect primary data or find other secondary data before creating frequency tables with equal and unequal class widths, working out estimates, and deciding if the prompt is true for their new set of data.

With each data set requiring up to eight distinct frequency tables, the teacher might encourage pairs or groups to collaborate on the same data.

That the prompt is not always true becomes clear when the class reflects on the estimates from the example. The way the data is grouped seems to be more important than the number of groups. Using narrower class intervals where the data is concentrated gives better estimates.

However, estimates from other data sets might offer stronger support for the prompt. In general, more class intervals leads to more accurate estimates.

In this phase of the inquiry, attention turns to the distribution of the data. How should students group the data if the distribution is flat, normal, or skewed? Does the contention in the prompt have more relevance for one type of distribution than another?

If the inquiry moves onto students designing sets of data with different types of distribution, they will require a conceptual and procedural understanding of standard deviation. This might come from a teacher's explanation or independent research.

Finally, the teacher might take advantage of the the enthusiasm generated by the inquiry to connect the frequency tables to histograms. The 'shape' of the histogram gives a visual representation of the distribution of the data. Students can use it to group the data in a different way.

Resources

Prompt sheet

PowerPoint

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