Measures of Central Tendency

Active learning works because constructing cumulative frequency graphs and box plots requires spatial reasoning and pattern recognition that static examples cannot provide. When students physically plot points or measure class boundaries, they develop an intuitive sense of how data accumulates and spreads, which is difficult to grasp from formulas alone.

• Year 10
• Mathematics
• 3 activities
• 15 min–45 min

Learning Objectives

1. Calculate the mean, median, and mode for a given set of raw data.
2. Construct and interpret frequency tables to find the mean, median, and mode.
3. Compare the strengths and weaknesses of mean, median, and mode for different data distributions.
4. Evaluate the most appropriate measure of central tendency for a given dataset, justifying the choice.
5. Explain how the presence of outliers impacts the mean, median, and mode.

Inquiry Circle

⏱ 45 min·Small Groups

Inquiry Circle: Comparing Datasets

Groups are given two sets of data (e.g., test scores from two different classes). They must calculate the cumulative frequencies, draw the graphs, and create box plots to decide which class performed 'better' and why.

Prepare & details

Compare the strengths and weaknesses of mean, median, and mode as measures of central tendency.

Facilitation Tip: During the Collaborative Investigation, circulate and ask groups to explain how their cumulative frequency graph changes as they add each new interval.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Gallery Walk

⏱ 25 min·Pairs

Gallery Walk: Spot the Outlier

Box plots are displayed around the room, some with deliberate errors or extreme outliers. Students move in pairs to identify the outliers using the 1.5 x IQR rule and discuss whether they should be kept or removed from the data.

Prepare & details

Evaluate which measure of central tendency is most appropriate for a given dataset.

Facilitation Tip: For the Gallery Walk, place a large copy of each box plot on the wall so students can compare their own plots to the correct versions during the discussion.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: Skewness and Spread

Students are shown three different box plots representing different distributions. They must individually describe the 'story' the data tells (e.g., 'most people scored high but a few did very poorly') before sharing their interpretation with a partner.

Prepare & details

Explain how outliers can affect different measures of central tendency.

Facilitation Tip: In the Think-Pair-Share, provide a dataset with a clear skew and ask students to estimate the median and quartiles before calculating, to build intuition.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teachers should model the precise plotting of cumulative frequency at upper class boundaries, not midpoints, to avoid the common misconception. Use real-world data sets that students can relate to, such as test scores or sports times, to keep engagement high. Avoid rushing to formulas; let students discover relationships between the median, quartiles, and box plot whiskers through guided construction.

What to Expect

Successful learning looks like students accurately plotting cumulative frequency at upper class boundaries, correctly identifying quartiles from their graphs, and constructing box plots that show the spread and skew of datasets. They should confidently explain why the median or IQR is more representative than the mean in skewed distributions.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation, watch for students plotting cumulative frequency at midpoints instead of upper class boundaries.

What to Teach Instead

Ask students to trace the step-by-step accumulation of frequencies and point out that the cumulative total 'up to' a boundary includes all data before it, so the point must align with the upper class limit.

Common Misconception

During Gallery Walk, watch for students interpreting longer whiskers as larger frequencies within that quartile.

What to Teach Instead

Have students measure the length of each section on their box plot and confirm that each quartile always represents 25% of the data, regardless of the whisker length.

Assessment Ideas

Quick Check

After Collaborative Investigation, present students with two datasets: one with a clear outlier and one without. Ask them to calculate the mean, median, and IQR for both, then explain which measure best represents the 'typical' value and why.

Exit Ticket

After Gallery Walk, provide a simple frequency table showing hours spent on homework. Ask students to calculate the mean, median, and mode, then justify which measure best describes the typical homework time.

Discussion Prompt

During Think-Pair-Share, pose the question: 'Would you report the mean or median salary for a company where the CEO earns significantly more than other employees? Have students discuss in pairs before sharing their reasoning with the class.

Extensions & Scaffolding

• Challenge: Provide a dataset with overlapping or ambiguous class intervals and ask students to adjust the intervals before plotting.
• Scaffolding: Give students a partially completed cumulative frequency table to fill in before constructing the full graph.
• Deeper exploration: Ask students to compare two datasets using both box plots and cumulative frequency curves, writing a paragraph explaining which visual tool is more informative for their specific comparison.

Key Vocabulary

Mean	The average of a dataset, calculated by summing all values and dividing by the number of values.
Median	The middle value in a dataset when the data is ordered from least to greatest. If there is an even number of values, it is the average of the two middle values.
Mode	The value that appears most frequently in a dataset. A dataset can have one mode, more than one mode, or no mode.
Frequency Table	A table that displays the frequency of various outcomes in a sample. Each entry shows the frequency (count) for a particular category or value.
Outlier	A data point that differs significantly from other observations in a dataset. Outliers can distort statistical measures.