Mathematics 9709 · syllabus 5.1
Cambridge A Level Mathematics 9709 — Representation of data topical past papers
Choosing and drawing statistical diagrams, and calculating and comparing averages and spreads, including mean and standard deviation from coded totals.
- Questions
- 148
- Years
- 2009–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 44
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Representation of data covers
Representation of data opens Paper 5, Probability & Statistics 1 (section 5.1). You draw and interpret stem-and-leaf diagrams, including back-to-back ones, box-and-whisker plots, histograms and cumulative frequency graphs, and choose a diagram that suits the data, saying what it shows well and what it hides. Histograms use frequency density, so the area of each bar, not its height, represents the frequency.
The calculations cover measures of central tendency, the mean, median and mode, and measures of spread, the range, interquartile range and standard deviation. You find the mean and standard deviation from raw data, from grouped data using mid-points, or from summary totals such as Σx and Σx² or coded totals such as Σ(x − a) and Σ(x − a)², and you may combine two data sets. Comparison questions want a statement about both average and spread, in the context of the data.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 5.1 Representation of data
What you should be able to do
- Draw and interpret stem-and-leaf diagrams, box-and-whisker plots, histograms and cumulative frequency graphs
- Choose a suitable diagram and explain its advantages or disadvantages for given data
- Estimate the median, quartiles and percentiles from a cumulative frequency graph
- Calculate the mean and standard deviation from raw, grouped or summarised data, including coded totals
- Combine summary statistics from two data sets
- Compare two distributions using an average and a measure of spread, in context
Key ideas
- Frequency density = frequency ÷ class width, so the area of a histogram bar is proportional to its frequency
- Mean x̄ = Σx/n; variance = Σx²/n − x̄²; the standard deviation is the square root of the variance
- Coded data: if y = x − a, then x̄ = ȳ + a, and x and y have the same standard deviation
- Interquartile range = upper quartile − lower quartile, which extreme values do not affect
- With grouped data, mid-points stand in for the values, so the mean and standard deviation are estimates
How Representation of data has been examined
From 2022 to 2026, Representation of data was on Paper 5 · Probability & Statistics 1 in 14 of 14 exam series, worth about 18% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 5 · Probability & Statistics 1 | 44 | 18% | 14 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 8.
Filed in Momentum under: Representation of Data.
A worked example
For 20 values of x, Σ(x − 50) = 36 and Σ(x − 50)² = 410. Find the mean and the standard deviation of x.
- Let y = x − 50, so Σy = 36 and Σy² = 410, with n = 20.
- ȳ = 36/20 = 1.8, so x̄ = 50 + 1.8 = 51.8.
- Var(y) = Σy²/n − ȳ² = 410/20 − 1.8² = 20.5 − 3.24 = 17.26.
- Subtracting 50 from every value does not change the spread, so the variance of x is also 17.26.
- Standard deviation = √17.26 = 4.154.
Answer: Mean 51.8; standard deviation 4.15 (3 s.f.)
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Drawing histogram bars with height equal to frequency when the class widths differ: plot frequency density so that the areas are right.
- Ignoring the gaps between classes: for data recorded to the nearest whole number, a class written 10–19 has boundaries 9.5 and 19.5, and both its width and its mid-point come from those.
- Forgetting to subtract x̄², or to take the square root, in the standard deviation formula, which gives Σx²/n or the variance instead.
- Adding a back on to the standard deviation of coded data: shifting every value changes the mean but not the spread.
- Comparing two data sets with numbers alone: the comment must say which group has the higher average and which is more spread out, in the question's context.
Exam technique
- Show a frequency-density column in a table before drawing a histogram, and label the vertical axis 'frequency density'.
- Write down Σx and Σx², or the coded totals, clearly before substituting into the formulae, and keep full accuracy until the final answer.
- Read a cumulative frequency graph at n/4, n/2 and 3n/4 for the quartiles and median, and show the lines you drew.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.