A Level · Cambridge (CIE) · 9709

Compare distributions using suitable centre and variation

Use context and numerical evidence to distinguish typical performance, middle-half consistency and extremes.

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

Find raw centre, range and IQR.
Read a common-scale box plot.

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

Quick readiness check

Which measure directly uses every recorded value in its calculation?

What you will learn

Choose mean or median and an appropriate spread measure.
Write qualified comparisons without causal or universal claims.

Centre and spread are separate features

Class A completion times have median 6, IQR 6.5 and range 9 minutes; B has median 5, IQR 3 and range 12. B is typically faster by the median and its middle half is more concentrated. Its extremes span a larger interval. State the chosen measure whenever you say ‘more consistent’ or ‘less variable’.

The mean uses all observations and suits questions about total or average resource use. For a skewed dataset with an extreme delay, the median may describe a typical journey better. This does not make the extreme irrelevant: a service manager may need to investigate it. A mode can identify the most common discrete score or recorded value, while grouped data may identify only a modal class.

Support comparisons and preserve uncertainty

Datasets C = 4, 4, 6, 6 and D = 2, 4, 6, 8 both have mean 5, but their ranges are 2 and 6. Using medians of halves, their IQRs are 2 and 4. Equal centres can coexist with different spreads; judging consistency from mean alone is inadequate.

Different sample sizes, units, observation periods or task difficulties can undermine a direct comparison. A graph of recorded exam scores does not by itself establish teaching effectiveness. Grouped means and quartiles remain estimates even when their numerical working is precise. Summaries describe the supplied datasets; predictions and causal claims need additional evidence.

Pause and explain

Two groups have mean 5 but ranges 2 and 6. What follows?

Put the idea to work

Worked example

Compare C = 4, 4, 6, 6 and D = 2, 4, 6, 8 minutes by mean, range and IQR using medians of halves.

Show the worked solution

Both sums are 20 with count 4, so both means are 5 minutes.
C range is 2 and IQR is 6 − 4 = 2; D range is 6 and IQR is 7 − 3 = 4.
C is less spread by both measures despite the equal means.

Answer Both means 5 minutes; C range/IQR 2/2, D range/IQR 6/4 minutes.

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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

Compare C = 4, 4, 6, 6 and D = 2, 4, 6, 8 minutes by mean and range.

Give me a hint

Calculate the totals and the difference between extremes separately.

Compare my reasoning

Both totals are 20 and counts 4, so both means are 5 minutes.
C range = 6 − 4 = 2; D range = 8 − 2 = 6 minutes.
C has smaller overall spread while the means are equal.

Equal means 5 minutes; ranges C 2 and D 6 minutes.

Look for these in your work

Calculate both features.
Avoid inferring spread from the mean.

2 · Independent

A has median 6, IQR 6.5, range 9; B has median 5, IQR 3, range 12 minutes. Write a typical-time and consistency comparison.

Give me a hint

Use median and IQR, then acknowledge what the range adds.

Compare my reasoning

B’s median is lower by 1 minute, suggesting a smaller typical time in these observations.
Its IQR is lower by 3.5 minutes, so its middle half is more concentrated.
Its range is larger by 3 minutes, so it is not smaller in every measure of spread.

B is lower by median and more concentrated by IQR, but has a larger range.

Look for these in your work

Name the statistics behind each statement.
Keep the claim within the recorded data and selected measure.

3 · Transfer

Journey times are 4, 5, 5, 6, 80 minutes. Find mean and median, choose a typical-time summary and state why the large delay still matters.

Give me a hint

The extreme raises the sum but is not the middle position.

Compare my reasoning

Total = 100 and count 5, giving mean 20 minutes.
The ordered third value is 5, so median is 5 minutes and describes a typical recorded journey more robustly.
The 80-minute delay remains relevant to reliability or worst-case experience; median alone does not dismiss it.

Mean 20, median 5 minutes; median for typical time, with the 80-minute delay still relevant to reliability.

Look for these in your work

Support the choice with both numerical summaries.
Do not confuse robustness with ignoring important extremes.

Common mistakes

Equal means do not imply equal variation.
Explain what ‘consistent’ measures.
Do not turn a descriptive comparison into proof of causation.

Recall without your notes

What can you explain now?

Can a group have both a smaller IQR and a larger range?

Compare with the explanation

Yes.

The central half may be more concentrated while extremes are farther apart.

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Make it stick

Compare one centre and one named spread in context, then state any limitation or conflicting extreme behaviour.

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

Check the official syllabus 2026–2027

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.