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A Level · Cambridge (CIE) · 9709

Draw and interpret common-scale box-and-whisker plots

Place five summary values on one numerical axis and compare centre and middle-half variation without inferring hidden raw values.

Statistics 1 · Representation of data · 5.1
Exam scope and assessment notes

Probability & Statistics 1 (Paper 5, AS & A Level) · Mathematics 9709, 2026–2027, section 5.1. Label variables, units, class boundaries and diagram keys. In these raw-data quartile exercises, use the median of each ordered half, excluding the overall median when the number of observations is odd; follow any convention supplied in an examination question. Cumulative-frequency quartiles use ranks n/4 and 3n/4, with a median rank n/2. Grouped-data interpolation and midpoint moments are estimates, not recovered raw measurements. Histogram area represents frequency, with height proportional to frequency density. Use descriptive variance with divisor n, not the sample-estimator divisor n − 1. Coded and combined totals must refer to the same units and compatible coding before addition. Retain unrounded working and distinguish variance units from standard-deviation units. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

  • Calculate raw median and quartiles using a stated convention.
  • Read a linear measurement axis.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which endpoints define the box itself?

What you will learn

  • Draw a five-number box plot with the correct locations.
  • Compare box plots and explain what they cannot determine.
Two basic minimum-to-maximum box plots use a shared time scale. A summary is minimum 2, Q1 3, median 6, Q3 9.5, maximum 11 minutes. B summary is 1, 4, 5, 7, 13 minutes. B has a lower median and smaller IQR, 3 versus 6.5 minutes, but a larger range, 12 versus 9 minutes. Numerical distances are drawn to the common scale; whisker lengths are not frequency counts.Compare centre and named spreadA2369.511B14571302468101214Time / minutesIQR: A 6.5; B 3 minutes
Two basic minimum-to-maximum box plots use a shared time scale. A summary is minimum 2, Q1 3, median 6, Q3 9.5, maximum 11 minutes. B summary is 1, 4, 5, 7, 13 minutes. B has a lower median and smaller IQR, 3 versus 6.5 minutes, but a larger range, 12 versus 9 minutes. Numerical distances are drawn to the common scale; whisker lengths are not frequency counts.

Use a scale rather than equally spaced summary marks

For the eight-minute dataset 2, 3, 3, 4, 8, 9, 10, 11, the five-number summary is minimum 2, Q1 3, median 6, Q3 9.5 and maximum 11. Draw the box from 3 to 9.5 and a median line at 6; extend whiskers to 2 and 11. Distances follow the numerical time axis, so summary points are not equally spaced.

This lesson uses the basic minimum-to-maximum box plot. Some software uses an outlier convention with whiskers ending at different values. Do not automatically apply a 1.5 × IQR rule or alter endpoints unless the question specifies it. Label the unit and use a common axis when comparing groups.

Read centre and spread separately

A second group with summary 1, 4, 5, 7, 13 minutes has median 5 and IQR 7 − 4 = 3. It has a lower median and smaller middle-half spread than the first group, whose median is 6 and IQR 6.5. However, the second group’s range is 12, larger than the first group’s 9. The words ‘less variable’ need the chosen measure of variation.

The five marks do not give a unique exact mean, mode or sample size. Unequal whiskers can suggest tail asymmetry but do not justify treating lengths as counts. A box covering the middle half is a summary of ordered positions; it does not show that measurements are uniformly distributed inside it. Compare like observations in context.

Pause and explain

B has IQR 3 and range 12; A has IQR 6.5 and range 9. Which claim is justified?

Put the idea to work

Worked example

Draw box plots for A summary (2, 3, 6, 9.5, 11) and B summary (1, 4, 5, 7, 13), in minutes, and compare median and IQR.

Show the worked solution
  1. Plot both summaries on a single linear time axis and draw each box from Q1 to Q3.
  2. A median is 6 and IQR 6.5; B median is 5 and IQR 3 minutes.
  3. B has lower typical time and smaller middle-half spread; its larger range does not contradict that IQR comparison.

Answer B has lower median (5 versus 6) and smaller IQR (3 versus 6.5 minutes), but larger range (12 versus 9).

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Give the box, median line and whisker endpoints for summary (2, 3, 6, 9.5, 11) minutes under the basic min-to-max rule.

Give me a hint

Identify the role of each ordered summary entry.

Compare my reasoning
  1. Draw the box from Q1 = 3 to Q3 = 9.5 minutes.
  2. Draw the median line at 6 minutes inside the box.
  3. Extend whiskers to minimum 2 and maximum 11 on the same linear axis.

Box 3–9.5; median 6; whiskers 2 and 11 minutes.

Look for these in your work

  • Use the actual numerical spacing.
  • Keep minimum and maximum outside the box endpoints.
2 · Independent

Compare A summary (2, 3, 6, 9.5, 11) with B summary (1, 4, 5, 7, 13) for completion times in minutes.

Give me a hint

Calculate IQR and range separately before writing a contextual comparison.

Compare my reasoning
  1. A median 6, IQR 6.5 and range 9; B median 5, IQR 3 and range 12.
  2. B has a smaller typical completion time and more concentrated middle-half times.
  3. B’s overall range is larger, so do not claim it has smaller spread by every measure.

B median 5 versus 6; IQR 3 versus 6.5; range 12 versus 9 minutes.

Look for these in your work

  • Support each comparison with values and units.
  • Specify which spread measure supports consistency.
3 · Transfer

A report claims that summary (1, 4, 5, 7, 13) determines an exact mean of 6 and that the long upper whisker contains more values. Evaluate both claims.

Give me a hint

Ask whether the five summary positions specify the sum or individual frequencies.

Compare my reasoning
  1. The five-number summary does not fix all raw observations or their sum.
  2. Therefore an exact mean of 6 cannot be recovered from it alone.
  3. Whisker length is a value interval, not a frequency count; a longer interval does not show more observations.

Neither claim follows: the exact mean is undetermined, and whisker length is not frequency.

Look for these in your work

  • Explain the missing information for the mean.
  • Separate numerical distance from observation count.

Common mistakes

  • Place points according to a common numerical scale.
  • Name IQR or range when describing variation.
  • Do not recover an exact mean or count from the five-number summary alone.
Recall without your notes

What can you explain now?

Does a wider right whisker mean it contains more observations than the left whisker?

Compare with the explanation

No; length measures numerical spread, not a larger observation count.

Whisker lengths are not histogram areas or frequencies.

After trying it yourself, choose your next review. This is your self-assessment.

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Make it stick

Draw five values to scale; compare median, IQR and range by name; keep hidden means and counts undetermined.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.

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