OCR GCSE Mathematics: Sampling, charts, cumulative frequency and histograms (J560) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover sections 12.01 and 12.02 of the OCR GCSE (9-1) Mathematics (J560) specification, Version 2.1 (August 2026): sampling, charts, time series, cumulative frequency graphs and histograms. Questions 1 to 7 are suitable for both tiers. Questions 8 to 12 are marked “(Higher tier only)” because the specification lists cumulative frequency graphs and histograms for the Higher tier only.

Links: study guide, revision notes, OCR GCSE Mathematics hub, checklist and free 10-minute diagnostics.

Questions

1. A librarian wants to know whether the 2,400 members of a library would use a new late opening. She asks the first 30 members who walk in on a Saturday morning.

(a) State the population. [1] (b) Give two reasons why her sample may be biased. [2]

2. A school has 500 pupils, numbered 001 to 500.

(a) Describe how to select a simple random sample of 25 pupils. [3] (b) In the sample, 7 pupils wear glasses. Estimate how many of the 500 pupils wear glasses. [2]

3. 80 pupils each name their favourite sport: football 28, netball 20, swimming 16, tennis 10, other 6.

(a) Work out the angle for each sport in a pie chart. [3] (b) In a pie chart for a different group, the sector for swimming has angle 90° and represents 36 pupils. How many pupils are in this group? [2]

4. A vertical line chart shows the number of goals in 30 matches. The line heights (number of matches) for 0, 1, 2, 3, 4 and 5 goals are 4, 7, 9, 5, 3 and 2.

(a) Show that the chart represents 30 matches. [1] (b) What percentage of the matches had at least 3 goals? [2]

5. The table shows museum visitors (thousands) in each quarter of two years.

	Q1	Q2	Q3	Q4
Year 1	18	34	52	22
Year 2	21	37	58	25

(a) In which quarter are visitors highest each year? Name the repeating pattern. [2] (b) Describe the trend. [1] (c) Estimate the third-quarter visitors in year 3 and give a reason. [2]

6. A survey records the year group and travel method of 160 pupils.

	Walk	Bus	Car	Total
Year 10	34	27	A	90
Year 11	B	31	18	70

(a) Find A and B. [2] (b) A questionnaire asks “How long do you spend online each day?” with boxes “0 to 1 hour”, “1 to 2 hours”, “2 to 3 hours”. Give two faults and write an improved set of boxes. [3]

7. A composite bar chart shows café sales. Monday (total 100): hot drinks from 0 to 40, cold drinks from 40 to 65, food above. Tuesday (total 120): hot drinks from 0 to 55, cold drinks from 55 to 85, food above.

(a) Find Tuesday’s cold drink sales and food sales. [2] (b) Explain whether a composite or a multiple bar chart makes it easier to compare cold drink sales on the two days. [2]

8. (Higher tier only) The heights of 80 plants are grouped: 0<h≤20: 6, 20<h≤40: 14, 40<h≤60: 25, 60<h≤80: 21, 80<h≤100: 10, 100<h≤120: 4 (cm).

(a) Complete the cumulative frequencies. [2] (b) State where the curve begins and the first two points plotted. [2] (c) Estimate the median height. [1] (d) Estimate the interquartile range. [2] (e) Estimate how many plants are taller than 90 cm. [1]

9. (Higher tier only) 60 pupils cycle to school by each of two routes. Cumulative frequencies of their journey times (minutes) at 10, 20, 30, 40 and 50 are: Route A 6, 21, 38, 52, 60; Route B 3, 11, 35, 55, 60. Both curves start at 0 minutes with 0.

(a) Estimate the median for each route. [2] (b) Estimate the interquartile range for each route. [2] (c) Compare the routes. [2]

10. (Higher tier only) Complete the table for a histogram.

Class	Frequency	Frequency density
0<x≤5	10	2
5<x≤15	35	3.5
15<x≤20	P	6
20<x≤40	Q	1.2
40<x≤60	6	R

(a) Find P, Q and R. [3] (b) Estimate how many values lie between 17 and 30. [2]

11. (Higher tier only) A histogram has frequency density bars: 0<x≤20 height 0.9; 20<x≤25 height 4.8; 25<x≤40 height 2.0; 40<x≤80 height 0.45.

(a) Find the frequency of each class. [3] (b) Sam says the 20<x≤25 bar is tallest, so it has the most values. Is he right? [1] (c) Estimate how many values are between 30 and 60. [2]

12. (Higher tier only) A histogram shows parcel masses (kg) of 80 parcels. Frequency densities: 0<m≤2: 6; 2<m≤5: 10; 5<m≤10: 4.4; 10<m≤20: 1.4; 20<m≤40: 0.1.

(a) Find the frequency of each class. [2] (b) Write the cumulative frequencies. [2] (c) Estimate the median mass. [2] (d) What percentage of the parcels weigh more than 10 kg? [2]

Answers

1. (a) All 2,400 members of the library [1]. (b) Only Saturday-morning visitors are asked, so members who visit at other times (or never) are left out [1]. The first 30 through the door are not chosen randomly and may arrive together with similar habits [1]. [3] Examiner insight: A reason earns credit when it names who is missing or why the choice is not random; “the sample is too small” on its own is weaker than a reason tied to the context.

2. (a) Generate 25 different random numbers between 1 and 500 [1]. Ignore repeats and draw again [1]. Select the pupils with those numbers [1]. (b) 7/25 of the sample wear glasses [1], so 7/25 × 500 = 140 pupils [1]. [5] Examiner insight: “Pick 25 at random” repeats the question; a method must say how random numbers are used and what happens to repeats.

3. (a) 360 ÷ 80 = 4.5° per pupil [1]. Football 126°, netball 90° [1]. Swimming 72°, tennis 45°, other 27° [1]. (b) 360 ÷ 90 = 4, so the whole group is 4 times the swimming sector [1]: 36 × 4 = 144 pupils [1]. [5] Examiner insight: Adding your angles to 360° is a quick check, and writing it down shows the method is complete.

4. (a) 4 + 7 + 9 + 5 + 3 + 2 = 30 [1]. (b) At least 3 goals: 5 + 3 + 2 = 10 matches [1]. 10/30 × 100 = 33.3% [1]. [3] Examiner insight: “At least 3” includes 3. Writing which lines you added makes the working easy to follow.

5. (a) Third quarter (Q3) in both years [1]. This is seasonal variation [1]. (b) Upward trend, as each quarter is higher than the same quarter a year earlier [1]. (c) Q3 rose by 6 from year 1 to year 2, so about 58 + 6 = 64 thousand [1], assuming the pattern and trend continue [1]. [5] Examiner insight: A prediction needs a reason that uses the data; a different estimate close to 64 with a clear reason is also sensible.

6. (a) A = 90 − 34 − 27 = 29 [1]; B = 70 − 31 − 18 = 21 [1]. (b) The boxes overlap at 1 and 2 hours [1]. There is no box for 3 hours or more, or for none [1]. Improved: 0 up to 1, 1 up to 2, 2 up to 3, 3 or more [1]. [5] Examiner insight: An improvement must fix the faults you named, so check yours has no overlaps and no gaps.

7. (a) Cold drinks: 85 − 55 = 30 [1]. Food: 120 − 85 = 35 [1]. (b) In a composite chart the cold section starts at a different height each day, so you must subtract to compare [1]. In a multiple bar chart every bar starts at the axis, so the heights compare directly [1]. [4] Examiner insight: Reading the top of the cold section (85) as the sales is a common slip; write the subtraction.

8. (a) 6, 20, 45 [1]; 66, 76, 80 [1]. (b) The curve begins at (0, 0) [1]; first points (20, 6) and (40, 20) [1]. (c) 40th value: about 56 cm [1]. (d) Lower quartile at 20: 40; upper quartile at 60: about 74 [1]. IQR is about 74 − 40 = 34 cm [1]. (e) At 90, cumulative frequency is about 71, so 80 − 71 = 9 plants [1]. [8] Examiner insight: Plot at upper boundaries and draw reading lines; answers read from a curve are estimates, so a small tolerance is natural.

9. (a) The 30th value for route A lies between 20 and 30 minutes: 20 + (30 − 21)/(38 − 21) × 10 = 25.3 minutes [1]. Route B: 20 + (30 − 11)/(35 − 11) × 10 = 27.9 minutes [1]. (b) A: lower quartile 16, upper quartile 35, IQR 19.0 [1]. B: lower quartile 21.7, upper quartile 35, IQR 13.3 [1]. (c) Route B has the higher median, so typical journeys are longer [1]. Route B has the smaller IQR, so its times are more consistent [1]. [6] Examiner insight: A comparison needs one statement about average and one about spread, each written in the context of journey times.

10. (a) P = 5 × 6 = 30 [1]; Q = 20 × 1.2 = 24 [1]; R = 6 ÷ 20 = 0.3 [1]. (b) 17 to 20: 3 × 6 = 18 [1]; 20 to 30: 10 × 1.2 = 12, so total 30 [1]. [5] Examiner insight: Show the width in each multiplication; the width of the class is the usual place for errors.

11. (a) Frequency = class width × frequency density [1]: 20 × 0.9 = 18 and 5 × 4.8 = 24 [1]; 15 × 2.0 = 30 and 40 × 0.45 = 18 [1]. (b) No. Frequency is area, and the 25<x≤40 bar is much wider, so it has 30 against 24 [1]. (c) 30 to 40: 10 × 2.0 = 20 [1]; 40 to 60: 20 × 0.45 = 9, so 29 [1]. [6] Examiner insight: Whenever class widths differ, use area. Saying “the bar is wider” and quoting the two frequencies makes the explanation complete.

12. (a) 2 × 6 = 12; 3 × 10 = 30 [1]; 5 × 4.4 = 22; 10 × 1.4 = 14; 20 × 0.1 = 2 [1]. (b) 12, 42, 64, 78, 80 [2]. (c) The 40th value lies in 2<m≤5 [1]: 2 + (40 − 12)/30 × 3 = 4.8 kg [1]. (d) 80 − 64 = 16 parcels [1]; 16/80 × 100 = 20% [1]. [8] Examiner insight: Carrying your own frequencies into the cumulative table keeps later parts consistent, so check that the final total equals 80.

Where marks are usually lost

• Describing a sampling method without saying how random numbers are used, or giving bias reasons that do not say who is left out.
• Pie angles that do not add to 360°, or a total that was never worked out first.
• Reading the top edge of a composite bar section as its value.
• Answering a time series question with no mention of the repeating pattern or the overall direction.
• Overlapping or incomplete categories in a table or questionnaire.
• Plotting cumulative frequency at the wrong point, or missing the starting point at zero.
• Giving a histogram frequency as bar height when widths differ.

Next steps

• Re-read the revision notes and the study guide.
• Tick off topics on the checklist and visit the OCR GCSE Mathematics hub.
• Try all free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

OCR GCSE (9-1) Mathematics (J560) specification, Version 2.1 (August 2026), for first assessment in 2017, published by OCR. Sections covered: 12.01a, 12.02a and 12.02b.