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A Level · Cambridge (CIE) · 9709

Choose a representation that answers a statistical question

Match the variable and purpose to a diagram, explaining which information it preserves and what it hides.

Section 5.1 · Representation of data

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

  • Distinguish a category from a numerical measurement.
  • Read a frequency table and identify its units.

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

Quick readiness check

Which variable is a numerical measurement suitable for grouping into intervals?

What you will learn

  • Choose representations for raw numerical and grouped data.
  • Explain a specific advantage and limitation in context.

Choose for the question and the data

For eight measured journey times, a stem-and-leaf diagram can retain every recorded time and show their ordering. A histogram suits many numerical measurements grouped into intervals, especially unequal-width classes: its area, rather than its height alone, represents frequency. A bar chart for named categories keeps the categories separate; a histogram uses a numerical scale and contiguous class boundaries.

A cumulative frequency graph answers questions about cutoffs, such as an estimated time below which 80% of journeys lie. A box-and-whisker plot compares median and quartiles on a common scale, but does not retain every individual value. A back-to-back stem-and-leaf diagram compares two modest raw datasets while preserving their values. State the variable and units before choosing a representation.

Explain what a summary cannot establish

A box plot may show the same five-number summary for different raw datasets. It cannot usually give an exact mean, mode or sample size. A grouped histogram hides the locations of values inside each class. A stem-and-leaf plot preserves recorded precision, but becomes crowded for thousands of measurements or awkwardly spread values.

For delivery times, a lower median and smaller interquartile range may describe faster and more consistent typical deliveries in the recorded data. Neither a chart nor those summaries alone proves the delivery service caused the difference, represents all future customers or is better on every criterion. Use comparable units, observation conditions and honest axis scales.

Pause and explain

Which representation retains all eight recorded numerical values with a suitable key?

Put the idea to work

Worked example

You have 12 raw journey times and want readers to recover the original values; later you want to estimate the 90th percentile from 300 grouped times. Choose a representation for each purpose and give a limitation.

Show the worked solution
  1. Use an ordered stem-and-leaf diagram with a time-unit key for the 12 raw values.
  2. Use a cumulative frequency graph for the grouped percentile question, plotting cumulative counts at upper boundaries.
  3. The stem diagram can become crowded; the grouped percentile is an estimate because locations within classes are unknown.

Answer Stem-and-leaf for recoverable raw values; cumulative frequency for a grouped percentile estimate.

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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

Choose a representation for 10 raw test scores when the reader must recover every score. Give a benefit and a limitation.

Give me a hint

Ask which diagram retains one visible entry per score.

Compare my reasoning
  1. Choose an ordered stem-and-leaf diagram and provide a score key.
  2. Each repeated leaf records a separate score and the ordering shows concentration.
  3. It retains all recorded scores, but becomes crowded for a very large dataset.

Ordered stem-and-leaf; recoverable scores, limited readability for large datasets.

Look for these in your work

  • Tie the benefit to recovering values.
  • State a genuine limitation without claiming values were grouped.
2 · Independent

Two classes have 200 grouped completion times each. Choose a display to compare typical time and middle-half spread; name information that the display cannot recover.

Give me a hint

A five-number summary puts two distributions on a shared measurement axis.

Compare my reasoning
  1. Use box-and-whisker plots on the same time scale, deriving estimated quartiles from grouped data.
  2. Compare medians for typical time and IQRs for the middle-half spread.
  3. The box plots do not recover every time or an exact mean; grouped quartiles remain estimates.

Common-scale box plots; compare median and IQR, with grouped estimates and no exact raw-value recovery.

Look for these in your work

  • Use the same time units and scale.
  • Keep grouping and summary limitations explicit.
3 · Transfer

A report uses separated bars labelled 0–5, 5–15 and 15–35 minutes, with heights equal to frequencies 10, 20 and 20. Explain how to correct it for a continuous-time histogram.

Give me a hint

Different class widths require different frequency densities.

Compare my reasoning
  1. The widths are 5, 10 and 20 minutes.
  2. Density heights are 10/5 = 2, 20/10 = 2 and 20/20 = 1 per minute.
  3. Draw adjoining rectangles on a continuous time scale; their areas are proportional to 10, 20 and 20.

Use adjoining widths 5, 10, 20 and density heights 2, 2, 1; frequency is represented by area.

Look for these in your work

  • Calculate heights from frequency divided by width.
  • Explain why using frequency height alone distorts unequal classes.

Common mistakes

  • Choose a diagram for a stated purpose, not just its appearance.
  • Grouped data do not reveal individual observations.
  • A descriptive comparison is not a causal conclusion.
Recall without your notes

What can you explain now?

Can two box plots with identical five-number summaries have different means?

Compare with the explanation

Yes; observations between the five summary points can differ.

A five-number summary does not fix the sum of all observations.

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

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Make it stick

Name the question, choose a representation, and state one piece of information it keeps and one it loses.

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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