Five-number summary
min, Q1, median, Q3, max.
Learning goals: what you will learn
- explain the central idea: Compare continuous numerical distributions using boxplots and other displays, discussing centre, spread, shape and outliers.
- choose and apply an appropriate method without relying on keyword matching
- check results using units, substitution, estimation, a second representation or contextual reasonableness
- justify a conclusion and communicate limitations where the context requires them
Prerequisite knowledge
Revise ordering numerical data, median, range, percentages, histograms and dot plots.
Concept teaching
A boxplot summarises a distribution using minimum, Q1, median, Q3 and maximum (with outlier conventions where used). IQR=Q3−Q1 measures middle-50% spread. Boxplots are excellent for comparison but hide detailed modality/frequency patterns that histograms or dot plots can show.
Spread
IQR=Q3−Q1; range=max−min.
Compare
Centre → spread → shape → outliers → context.
Outlier rule
Common convention: below Q1−1.5IQR or above Q3+1.5IQR.
Worked examples
1. Five-number summary
Data 2,4,5,7,8,9,12: min2, median7, max12; lower half 2,4,5 gives Q1=4; upper 8,9,12 gives Q3=9.
2. IQR
Q1=18,Q3=30 → IQR=12.
3. Compare medians
Class A median 72, Class B median 65: typical central result is higher for A, but spread must also be considered.
4. Compare spread
A IQR 8, B IQR 20: B's middle 50% is more variable.
5. Outlier fences
Q1=10,Q3=18 → IQR=8; fences −2 and 30. A value 35 is a potential outlier.
6. Skew from display
A longer upper tail and median closer to Q1 may suggest right skew, but a boxplot cannot show modes clearly.
7. Boxplot versus histogram
Use boxplots for compact group comparison; use histogram when distribution shape/frequency peaks matter.
8. Same summary, different shape
Different raw datasets can share a similar five-number summary. A boxplot can therefore hide clustering or multiple modes.
9. Context statement
“Group A has median 5 minutes lower and smaller IQR” is stronger than “A is better” unless the context defines lower as desirable.
10. Outliers in modelling
Do not automatically delete an outlier. Check whether it is error, valid rare case or important subgroup; handling choice can change a machine-learning model.
Common misconceptions and corrections
- Comparing only medians and ignoring spread.
- Assuming a boxplot reveals exact frequencies or modes.
- Calling every extreme value an error.
- Using range when IQR better represents middle spread.
- Describing “higher/lower” without interpreting what it means in context.
Guided practice
For the ordered data 4,5,6,7,8,9,10,12,14,18, find the median, Q1, Q3 and IQR using your school's quartile convention. Sketch a boxplot and describe centre, spread and the role of 18.
Teacher check: require a written method choice and one verification step before revealing the worked solution.
Independent practice
- Find median of 3,5,8,10,11.
- If Q1=12,Q3=21 find IQR.
- Find outlier fences for Q1=20,Q3=28.
- Is 42 a potential upper outlier for those quartiles?
- Which has more middle spread: IQR 7 or 15?
- Why use boxplots for two classes?
- Why might a histogram be better for modality?
- Can two datasets with different shapes share same five-number summary?
- Write a context-rich comparison using medians 62 and 70.
- Why should an outlier be investigated before deletion?
Check answers and explanations
- 8.
- 9.
- IQR=8; fences 8 and 40.
- Yes.
- IQR 15.
- They compactly compare centre/spread/outliers on a common scale.
- It shows frequency distribution shape and peaks.
- Yes.
- Example: Group B's typical value is 8 units higher based on medians, with spread still needing comparison.
- It may be genuine and informative rather than an error.
Reasoning and problem-solving task
Two schools have the same median exam score of 70. School A has Q1=66,Q3=75; School B has Q1=55,Q3=84 and several high outliers. Write a careful comparison of typical performance, consistency and what the outliers do and do not imply.
Important questions and answers
- Find median of 3,5,8,10,11.
- 8.
- If Q1=12,Q3=21 find IQR.
- 9.
- Find outlier fences for Q1=20,Q3=28.
- IQR=8; fences 8 and 40.
Assessment-style questions
Two classes have equal medians, but Class A IQR=8 and Class B IQR=22. Compare consistency, explain what the median alone misses, and state what additional display information you would inspect before declaring one class 'better'. [5 marks]
Marking focus: method selection, mathematically correct working, interpretation and justification.
Review hints
- state centre and spread
- use IQR=Q3-Q1
- treat outliers as flags, not automatic deletions
- compare in context
Exit ticket: mastery check
- I can explain the concept without copying a formula sheet.
- I can solve a routine example and check the result.
- I can choose a method in an unfamiliar problem.
- I can explain a common error and correct it.
- I can connect the answer back to the context, including units or limitations.
Teacher and parent guidance
For teachers
Pair boxplots with dot plots or histograms of the same data so students see what summarisation hides. Mark comparison sentences for evidence + context.
For parents and carers
Ask your child what the box, median line and whiskers each mean. Then ask which group is more consistent, not just which has the higher median.
Curriculum alignment
Australian Curriculum: AC9M10ST02 — Year 10 Statistics. Compare continuous numerical distributions using boxplots and other displays, discussing centre, spread, shape and outliers.
Victoria: VC2M10ST01 — Level 10 Statistics
NSW: Stage 5 Core — Data classification, visualisation and analysis; Path — Data analysis and statistical enquiry
Alignment explanation: The explicit teaching and worked examples address the Australian Curriculum concept directly. The Victorian mapping follows the current Version 2.0 descriptor structure; where Victoria combines or extends content, that difference is stated rather than hidden. NSW uses a Stage 5 Core–Paths structure, so this page maps to the relevant content group(s) and Working mathematically processes instead of inventing a Year 10 one-to-one code.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Explicit concept teaching and worked examples | AC9M10ST02 | VC2M10ST01 — Level 10 Statistics | Stage 5 Core — Data classification, visualisation and analysis; Path — Data analysis and statistical enquiry |
| Guided and independent practice | Builds fluency and application for AC9M10ST02 | Practises the mapped Level 10/10A knowledge as applicable | Practises the mapped Stage 5 Core/Path content |
| Reasoning and assessment tasks | Applies reasoning/problem solving in the descriptor context | Supports Victorian reasoning and modelling expectations | Embeds Working mathematically: reasoning, problem solving and communication |
Australian Curriculum elaborations
AC9M10ST02: compare continuous-data distributions using displays including boxplots and discuss centre, spread, shape and outliers in context.
- E1: construct/interpret boxplots and compare datasets. Examples 1–4.
- E2: compare distribution shapes using boxplots, histograms, cumulative-frequency graphs and dot plots, including symmetry/skew/modality. Examples 6–8.
- E3: use digital tools to compare boxplots/histograms according to statistical question. Example 7.
- E4: use five-number summary/boxplot for numerical and visual centre/spread comparisons. Examples 1–4.
- E5: compare stories/information from continuous/discrete datasets displayed in different ways. Examples 7–9.
- E6: explore identifying/handling outliers in machine learning. Example 10.
Practice and teaching resources
Official curriculum references
- Australian Curriculum Version 9 — ACARA
- Victorian Curriculum Mathematics Version 2.0 — VCAA
- NSW Mathematics K–10 Syllabus (2022) — NSW Curriculum/NESA
Official wording is paraphrased on SkillrHub; use the linked curriculum sites as the source of record.
Other curriculum comparisons retained
US CCSS HSS-ID.A.1–3; UK GCSE box plots/distribution comparison; NSW Stage 5 and Victorian Level 10 statistics; comparable Canadian/NZ secondary distributions.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Example: Comparing distributions
Khan Academy — Comparing distributions through shape, centre, spread and outliers.
As you watch: Why is comparing only the averages not enough to describe two distributions?
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Try it: Compare two boxplots using median, interquartile range, overall spread and possible outliers; explain one feature a boxplot does not reveal.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Data distributions for continuous numerical variables using appropriate data displays...
Mapped skill: compare data distributions for continuous numerical variables using appropriate data displays including boxplots; discuss the shapes of these distributions in terms of centre, spread, shape and outliers in the context of the data
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M10ST02 · Year 10 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M10ST01 · Level 10 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA5-DAT-C-01 · Stage 5 |
| United States (USA) | Common Core State Standards for Mathematics | Grades 9–10 band |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 10 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 11, Key Stage 4 |
| India | NCERT / CBSE — Mathematics | Class 10 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M10ST02 — Boxplots and Comparing Continuous Distributions — AC9M10ST02
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