Comparing Data Distributions

Overview

Students analyse and compare box plots using the median and interquartile range (IQR), then investigate how adding a constant to every value shifts a distribution. The lesson develops precise statistical communication and higher-order reasoning through “what changed, what stayed the same?” problems.

Learning intentions

• WALT read and interpret the five-number summary represented by a box plot.
• WALT calculate and use the IQR to describe spread.
• WALT compare two distributions using evidence from their medians and IQRs.
• WALT explain the effect of shifting every data value by the same amount.

Success criteria

• I can identify the minimum, lower quartile, median, upper quartile and maximum.
• I can calculate IQR using (Q3-Q1).
• I can compare distributions using numerical evidence and appropriate statistical language.
• I can explain that adding or subtracting a constant changes the location but not the spread.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: applying statistical ideas through relevant, connected challenge.
• Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners: reasoning, justification and interpretation beyond routine calculation.

Lesson structure (60 minutes)

1. 0–7 min · Hook and retrieval. Display the opening comparison and retrieval questions: two box plots showing travel times for two groups. Ask, “Which group is more consistent? Which usually takes longer? What evidence would convince you?” Students write an individual response, then compare with a partner. Do not confirm answers yet; listen for use of median and range versus IQR.

2. 7–17 min · Explicit teaching. Use the box plot teaching sequence to review the five-number summary. Model reading the minimum, (Q1), median, (Q3) and maximum from a labelled box plot, then calculate (IQR=Q3-Q1). Emphasise that the median describes the middle of the data and the IQR describes the spread of the middle 50%. Students annotate a quick example and answer hinge questions: “Where are the middle 50%?” and “Which value is not needed to calculate IQR?”

3. 17–30 min · Guided comparison. Distribute the box plot analysis worksheet. Complete the first comparison together: students identify the medians and IQRs of two distributions and construct a response using the frame, “Group A has a higher/lower median…, while its IQR is…, so…”. Pairs complete the next two questions, including a case where one group has a higher median but a larger IQR. Pause halfway for students to justify whether “higher” means “better” in a particular context.

4. 30–43 min · Shift investigation. Return to the shift investigation instructions. Give each pair the data set (4, 7, 8, 10, 12, 15, 18). Students record its five-number summary, then add 5 to every value and find the new summary. They repeat mentally for a subtraction of 3. Pairs discuss: Which summary values changed? Did the IQR change? Teacher records the generalisation: adding (k) shifts every location measure by (k), while the range and IQR remain unchanged.

5. 43–53 min · Independent reasoning. Students complete the remaining the shift and comparison questions independently. Include questions requiring them to determine an unknown shift from two medians, predict a new IQR without recalculating, and explain whether a claim is always true. Circulate and check that students distinguish a shift from a change in spread. Invite early finishers to create two different data sets with the same median and IQR but different ranges.

6. 53–60 min · Plenary and exit check. Show the final challenge and plenary prompts. Students respond to: “A data set has median 24 and IQR 8. Every value is increased by 6. What are the new median and IQR? Explain.” Students then share one sentence comparing two distributions using both centre and spread. Collect responses to identify misconceptions for the next lesson.

Resources

• One slide deck: the complete box plot comparison deck
• One worksheet: the box plot analysis worksheet
• Projector or interactive display
• Calculators, optional for checking arithmetic
• Mini-whiteboards or scrap paper
• Rulers and coloured pens
• Exercise books

Assessment

• Listen during the hook and guided comparison for correct use of median, range and IQR; address the common error of treating the box length as the full range.
• Check pair work and independent answers, especially whether students use numerical evidence rather than claims such as “the graph is bigger”.
• Use the final response to assess whether students understand that a constant shift changes the median but leaves the IQR unchanged.

Differentiation

• Support students with a colour-coded box plot, a five-number-summary word bank and sentence stems: “The median is…”, “The IQR is…”, and “This means…”. Read all instructions aloud and allow students to verbalise reasoning before writing.
• Provide a partially completed table on the worksheet for students who need reduced writing or calculation load. Use clear sans-serif text, generous spacing, high contrast and avoid presenting essential information through colour alone for dyslexic learners.
• Pair students deliberately and allow calculators for checking, while requiring the statistical explanation to be completed independently.
• Extension: students create two box plots with equal medians but different IQRs, then write a justified recommendation about which distribution is preferable in a realistic context. They should also explain why a shift cannot change the IQR.