Comparing Data Distributions

Overview

Students revisit measures of centre and spread before applying them to five-point summaries and box plots. They begin collaboratively with a differentiated starter worksheet, then use the slide deck to develop accurate calculations and explain which statistics best represent different data sets.

Learning intentions

• WALT calculate and interpret mean, median, mode, range and interquartile range.
• WALT determine and use a five-point summary.
• WALT read, compare and interpret box plots.
• WALT justify statistical conclusions using evidence from data.

Success criteria

• I can order data and correctly find the minimum, quartiles, median and maximum.
• I can calculate range and IQR and explain what each measure describes.
• I can identify the median and IQR from a box plot.
• I can explain how an outlier may affect the mean, median and range.
• I can use numerical evidence to compare two data sets.

Curriculum links

• Mathematics and Statistics — statistical investigation, including describing, representing and interpreting numerical data.
• Mathematics and Statistics — selecting and using measures of centre and spread to make comparisons.
• Mathsteasers — higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathsteasers / Alignment — relevant challenge connected to students’ existing statistics knowledge.

Lesson structure (60 minutes)

1. 0–3 min · Settle and hook. Open with the athlete consistency hook slide and ask, “If two athletes have the same mean, do they perform identically?” Students make a quick individual prediction, then share one reason with a partner.

2. 3–18 min · Differentiated starter. Distribute the differentiated five-point summaries and box plots starter worksheet to groups of three or four. Students complete the key-concept check, order the quiz scores, calculate the five-point summary, and interpret the box plot. Explain that students should begin at the section matched to their confidence: supported recall and guided calculations, core calculations, or the challenge questions requiring explanation. Circulate, listen for misconceptions, and ask students to explain how they know rather than simply supplying answers.

3. 18–25 min · Diagnose and teach. Use the objectives and key vocabulary slides to review central tendency, spread, quartiles and outliers. Invite groups to compare answers for the starter, then address common errors, especially treating Q1 and Q3 as individual data values without ordering the set, or confusing range with IQR. Model the quiz data in order and show that the median for ten values is the mean of the fifth and sixth values.

4. 25–37 min · Explicit modelling. Present the worked calculation and choosing-a-centre slides. Model the mean, median and mode for the study-minutes data, including the effect of the value 90. Then model the five-point summary for the quiz scores: minimum 6, Q1 11, median 11.5, Q3 15, maximum 19, IQR 4 and range 13. Students annotate their worksheet and use mini-whiteboards or fingers to show answers to short checks.

5. 37–50 min · Compare and interpret. Display the box-plot comparison and discussion slides. In pairs, students read the median, quartiles, IQR and range from two box plots and decide which data set is more consistent. They must give a complete statement such as, “Data set A is more consistent because its IQR is smaller.” Pairs then consider: “If the maximum increases by 20, which statistics must change, and which might stay the same?” Groups justify their responses using the five-number summary and report one conclusion.

6. 50–56 min · Independent application. Return to the challenge interpretation questions. Students independently answer one question about an outlier and one about comparing range with IQR, showing calculations and a written justification. Students who have completed the core work may construct an ordered data set that could produce a given box plot.

7. 56–60 min · Plenary and exit check. Use the final reflection and exit-question slide. Students respond on the back of the worksheet: “For the data set 4, 5, 5, 6, 20, which measure of centre is most useful, and why?” They also record one difference between range and IQR. Collect responses to identify the next teaching focus.

Resources

• the complete statistics slide deck
• the differentiated five-point summaries and box plots starter worksheet
• Projector or interactive whiteboard
• Mini-whiteboards and pens
• Calculators, used only after students show the statistical method
• Rulers and pencils for reading or drawing box plots
• Board space for the worked five-point summary

Assessment

• During the starter, check whether students can order data, locate quartiles and distinguish range from IQR. Use targeted questions and note students needing reteaching.
• During modelling and pair work, use mini-whiteboard responses and explanations to assess calculation accuracy and interpretation of spread.
• Use the final written response to assess whether students understand the effect of an outlier and can justify the choice of median over mean.

Differentiation

• Support learners with the scaffolded worksheet section, the word bank, partially completed ordered data, a five-point-summary table and the sentence starter: “The ___ is more suitable because…”.
• Provide a mixed-attainment group structure with assigned roles: reader, calculator, checker and explainer. Allow students to use a number line or physically mark ordered positions before calculating quartiles.
• For EAL learners and students requiring additional support, pre-teach visual vocabulary through the slides, read instructions aloud, chunk multi-step questions and accept oral explanations before written recording.
• Extend advanced learners with the critique questions: evaluate whether increasing the maximum always changes the median, and construct a plausible ordered data set for a specified box plot. Require justification using precise statistical language.