GCSE Higher — Statistics
Statistics accounts for 7.5% of the GCSE Higher papers, and this page collects the practice for it in one place. Coverage is complete: all 6 statistics statements, each with practice that matches how the topic is actually examined. The list includes sampling and inference about populations, tables, charts and diagrams for data, histograms and cumulative frequency graphs, averages, spread and comparing distributions, describing a population with statistics and scatter graphs, correlation and lines of best fit. The Higher-only part of this area is histograms and cumulative frequency graphs. The assessment objectives split 40% AO1 against 60% AO2 and AO3 combined, which is why so many questions ask you to explain or to justify rather than simply to calculate. This area leans on the calculator papers; rounding too early is the usual way to lose an otherwise correct answer.
- S1 — Sampling and inference about populations
- S2 — Tables, charts and diagrams for data
- S3 — Histograms and cumulative frequency graphs (Higher only)
- S4 — Averages, spread and comparing distributions (part Higher)
- S5 — Describing a population with statistics
- S6 — Scatter graphs, correlation and lines of best fit
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Statistics at GCSE Higher: the 6 DfE statements
Every question on this site is filed against one of these statements. Pick one to practise it on its own.
Sample statistics questions for GCSE Higher
- The mean of 5 numbers is 8. Work out the total of the 5 numbers.(a)40(b)1.6(c)13(d)8
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40 — Method: the mean is the total divided by how many values there are, so rearranging gives total = mean × number of values. Working: the mean is 8 and there are 5 numbers, so the total is 8 × 5 = 40. Answer: 40, and checking, 40 ÷ 5 = 8, which is the mean given. The distractors: 13 comes from adding the mean and the count, 8 + 5, instead of multiplying them; 1.6 comes from dividing the mean by the count, 8 ÷ 5, which reverses the relationship; 8 comes from quoting the mean itself as the total, which is only true when there is a single number. - The mean of the numbers x, 20 and 30 is equal to the mean of the numbers 15 and 25. Work out the value of x.(a)20(b)−10(c)10(d)70
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10 — Method: work out the mean that can be found straight away, then use total = mean × number of values on the group of three to find the missing number. Working: the mean of 15 and 25 is (15 + 25) ÷ 2 = 40 ÷ 2 = 20, so the group of three must also have a mean of 20; three numbers with a mean of 20 have a total of 20 × 3 = 60, and 20 + 30 = 50 of that total is already accounted for, so x = 60 − 50 = 10. Answer: 10, and checking, (10 + 20 + 30) ÷ 3 = 20. The distractors: 20 comes from working out the mean the two groups share and writing that down as x; −10 comes from dividing the group of three by 2 instead of by 3, which gives x + 50 = 40; 70 comes from reading the total 15 + 25 = 40 as the mean of the pair, which sets the target total at 120 and leaves x = 70. - The mean of the four numbers 10, 15, 20 and x is 18. Work out the value of x.(a)27(b)72(c)45(d)18
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27 — Method: turn the mean into a total using total = mean × number of values, then subtract the numbers that are already known. Working: four numbers with a mean of 18 have a total of 18 × 4 = 72; the three known numbers give 10 + 15 + 20 = 45; so x = 72 − 45 = 27. Answer: 27, and checking, (10 + 15 + 20 + 27) ÷ 4 = 72 ÷ 4 = 18. The distractors: 72 comes from stopping at the total the four numbers must reach and never subtracting the known three; 18 comes from assuming the missing number must equal the mean; 45 comes from stopping at the total of the three known numbers. - Two classes sat the same maths test, both marked out of 100. Class A had a mean mark of 70 and a range of 30 marks. Class B had a mean mark of 70 and a range of 10 marks. Compare the marks of the two classes.(a)The means are equal, and Class B's marks are the more consistent because its range is smaller.(b)Class A's mean mark is higher, and Class A's marks are also the more consistent.(c)The means are equal, and Class A's marks are the more consistent because its range is larger.(d)Class B's mean mark is higher, and its smaller range makes its marks the more consistent.
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The means are equal, and Class B's marks are the more consistent because its range is smaller. — Method: comparing two distributions needs two things — a measure of average and a measure of spread — and each must be put into the context of the question. Working: both classes have a mean mark of 70, so on average the two classes scored the same; the range measures spread, and Class A's range of 30 marks is three times Class B's range of 10 marks, so Class B's marks sit closely around the mean while Class A's are far more spread out. Answer: the means are equal, and Class B's marks are the more consistent because its range is smaller. The distractors: the reply crediting Class A with more consistency reverses the meaning of the range, treating a larger range as tighter data when a larger range means more spread; the reply that Class A's mean mark is higher compares the wrong pair of figures, reading the range of 30 as an average; the reply that Class B's mean mark is higher reads the spread correctly but its claim about the means is false, since both means are 70. - The mean mass of four parcels is 17 kg. Three of the parcels have masses 12 kg, 16 kg and 18 kg. Work out the mass of the fourth parcel.(a)17 kg(b)68 kg(c)22 kg(d)5 kg
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22 kg — Method: multiply the mean by the number of parcels to rebuild the total mass, then subtract the masses that are known. Working: four parcels with a mean mass of 17 kg have a total mass of 17 × 4 = 68 kg; the three known parcels total 12 + 16 + 18 = 46 kg; so the fourth parcel has mass 68 − 46 = 22 kg. Answer: 22 kg. The distractors: 68 kg comes from stopping at the total mass of all four parcels; 17 kg comes from assuming the missing parcel must have the mean mass; 5 kg comes from multiplying the mean by 3, the number of parcels whose mass is given, leaving 51 − 46 = 5.
Frequently asked questions
How much of the GCSE Higher paper is statistics?
7.5% of the total marks, and the three papers each carry an equal share of the qualification, so it is spread across all of them rather than concentrated in one.
How many topics are there in statistics at GCSE Higher?
6 DfE content statements, coded S1 to S6 within this area. Every question in the bank is tagged to one of them.
Which statistics topics are Higher only?
Histograms and cumulative frequency graphs (S3). A further 1 statement has a Higher-only part inside otherwise shared content.
Can I use a calculator for statistics questions?
Paper 1 is non-calculator; Papers 2 and 3 allow one. Of the 6 statements in this area, 0 are naturally non-calculator and 1 naturally calculator, with the rest workable either way — so practise both.
What kind of marks does statistics carry?
The GCSE Higher split is 40% AO1 (use and apply standard techniques), 30% AO2 (reason, interpret and communicate) and 30% AO3 (solve problems in and out of context). Questions in this area are set across all three.
Are these past paper questions?
No. Every question at /gcse-higher is original, written to the DfE content statements and checked before it is published. We do not host past papers or mark schemes.
Do I need an account?
No — you can start practising straight away. Progress is saved automatically in your browser.
Is it free?
Yes. The questions, worked answers and printable worksheets are all free, with no adverts.
Tips for statistics at GCSE Higher
- In a histogram the AREA of each bar represents the frequency, not its height. The vertical axis is frequency density = frequency ÷ class width, and frequency = frequency density × class width.
- A cumulative frequency graph plots the running total against the UPPER bound of each class. Read the median at half the total, and the quartiles at a quarter and three quarters.
- The interquartile range (IQR) = upper quartile − lower quartile. It measures the spread of the middle half of the data and, unlike the range, it ignores outliers.
- A box plot shows five values: minimum, lower quartile, median, upper quartile, maximum. Compare two distributions by comparing a measure of average (the medians) and a measure of spread (the IQRs).
- For an estimated mean from grouped data use the midpoint of each class: Σ(midpoint × frequency) ÷ Σfrequency. It is an estimate because the exact values are unknown.
Common mistakes (and how to avoid them)
- Reading the height of a histogram bar as its frequency when the class widths are unequal.
- Plotting cumulative frequency at the midpoint or lower bound of a class instead of at the upper bound.
- Working out the IQR as upper quartile plus lower quartile, or confusing it with the range.
- Comparing two box plots by one number only — a full comparison needs an average and a spread, in context.
Worked examples
- Class width = 30 − 10 = 20.
- Frequency = frequency density × class width = 2.5 × 20 = 50.
- The median is at half of 80, i.e. cumulative frequency 40, which reads 45 marks.
- The lower quartile is at a quarter of 80 (20): 32 marks. The upper quartile is at three quarters (60): 58 marks.
- IQR = 58 − 32 = 26 marks.
- Multiply each score by its weight: 80 × 0.7 = 56 and 90 × 0.3 = 27.
- Add the contributions: 56 + 27 = 83.
Statistics at Higher moves from calculating an average to describing a whole distribution: histograms with unequal class widths, cumulative frequency curves, quartiles and box plots. The habits that score are precise ones — area not height in a histogram, upper bounds on a cumulative frequency graph, a comparison that names both an average and a spread. Behind every number is a question: what does it measure, how was it collected, and what does it not tell you? That critical eye is the real skill the exam is testing.
More GCSE Higher maths:
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