Cambridge O Level Statistics: Venn, box-and-whisker and stem-and-leaf diagrams, choosing and interpreting representations (4040) – 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 2.2 to 2.4 of the Cambridge O Level Statistics (4040) syllabus for examination in 2025, 2026 and 2027, version 1: Venn, box-and-whisker and stem-and-leaf diagrams, then choosing and interpreting representations. For quartiles of a small list, take the median of the lower half and of the upper half, leaving the median out of both when the number of values is odd.

Revise first with the study guide. Course pages: the 4040 hub and the printable checklist.

Questions

1. In a Venn diagram, n(A) = 23, n(B) = 18 and n(A∩B) = 7. Find n(A∪B). [2]

2. Of 50 pupils, 28 play football, 22 play hockey and 9 play neither sport.

(a) Find how many play both sports. [3] (b) Find how many play exactly one of the two sports. [1]

3. A survey of 75 homes asks which pets are kept. Dogs are kept in 35 homes, cats in 30 and fish in 22. Dogs and cats are kept together in 12 homes, cats and fish in 9, dogs and fish in 7, and all three in 5.

(a) Draw a Venn diagram showing the number in every region, including homes with none of the three. [4] (b) How many homes keep exactly two kinds of pet? [1] (c) How many homes keep cats but no dogs? [1]

4. The heights in cm of 14 seedlings are: 34, 47, 52, 41, 38, 45, 52, 29, 36, 41, 58, 43, 30, 41. Draw an ordered stem-and-leaf diagram, with a key. [3]

5. The diagram shows the points scored by 15 players.

1 | 5 8 9
2 | 0 3 3 6
3 | 1 4 4 4 7
4 | 2 8
5 | 1
Key: 1 | 5 means 15 points

(a) Find the median. [1] (b) State the mode. [1] (c) Find the range. [1] (d) How many players scored more than 30 points? [1]

6. The times, in minutes, taken by 11 Year 7 and 11 Year 9 pupils to solve a puzzle are shown.

Year 7: 4.9, 3.2, 5.7, 4.1, 6.4, 5.0, 3.8, 7.2, 4.4, 6.1, 5.3 Year 9: 5.5, 2.5, 4.8, 3.3, 5.9, 4.2, 2.9, 3.9, 5.1, 3.6, 4.4

(a) Draw a back-to-back stem-and-leaf diagram, with a key. [3] (b) Compare the two years, using a measure of average and a measure of spread. [3]

7. Thirteen test marks are: 63, 48, 71, 55, 59, 66, 52, 48, 74, 60, 57, 69, 62. Find the minimum, lower quartile, median, upper quartile and maximum. [3]

8. A box-and-whisker diagram for the marks of a class has minimum 12, lower quartile 30, median 38, upper quartile 51 and maximum 80.

(a) Find the interquartile range and the range. [2] (b) A pupil says “the mean mark is 38”. Explain why the diagram does not support this. [2]

9. State the most suitable representation for each situation, and give one reason.

(a) Comparing the exact heights of 12 pupils in class A with 12 in class B, with every value visible. [2] (b) Comparing the spread of daily sales over a year for four shops. [2] (c) Showing how 120 customers split between buying tea, coffee, both or neither. [2]

10. Two delivery shops record delivery times in minutes. Shop A: minimum 6, lower quartile 14, median 22, upper quartile 31, maximum 50. Shop B: minimum 10, lower quartile 18, median 23, upper quartile 28, maximum 36.

(a) Find the interquartile range and the range for each shop. [2] (b) Compare the delivery times of the two shops, and say which you would recommend to a customer who values reliability. [4] (c) State one limitation of using box-and-whisker diagrams for this comparison. [1]

11. The weekly exercise times, in minutes, of 20 members of club A are shown.

2 | 2 5 7
3 | 1 3 4 6 8
4 | 1 2 5 6 8
5 | 1 3 5 8
6 | 0 4 9
Key: 2 | 2 means 22 minutes

(a) Find the minimum, lower quartile, median, upper quartile and maximum. [4] (b) Describe a box-and-whisker diagram for club A on a scale from 10 to 80, giving the position of every part. [2] (c) Club B has minimum 10, lower quartile 25, median 32, upper quartile 40 and maximum 55. Compare the two clubs. [3] (d) State one thing the stem-and-leaf diagram shows that a box-and-whisker diagram does not. [1]

Answers

1. n(A∪B) = n(A) + n(B) - n(A∩B) [1] = 23 + 18 - 7 = 34 [1] [2] Examiner insight: Write the subtraction of the overlap explicitly, so your method is clear even if an arithmetic slip occurs.

2. (a) In at least one sport = 50 - 9 = 41 [1]. Football plus hockey = 28 + 22 = 50 [1]. Both = 50 - 41 = 9 [1] [3] (b) Football only 19 and hockey only 13, so 19 + 13 = 32 [1] [1] Examiner insight: The sum 28 + 22 counts the overlap twice, so start by finding the number in at least one sport, then take the difference.

3. (a) Centre = 5 [1]. Pairs only: dogs and cats 12 - 5 = 7, cats and fish 9 - 5 = 4, dogs and fish 7 - 5 = 2 [1]. Dogs only 35 - 7 - 2 - 5 = 21, cats only 30 - 7 - 4 - 5 = 14, fish only 22 - 4 - 2 - 5 = 11 [1]. Inside circles = 21 + 14 + 11 + 7 + 4 + 2 + 5 = 64, so none = 75 - 64 = 11 [1] [4] (b) 7 + 4 + 2 = 13 [1] [1] (c) Cats only 14 plus cats and fish only 4 = 18 [1] [1] Examiner insight: Label every region and add them to check the total; a missed centre subtraction usually breaks the sum.

4. Stems 2 to 5 all present [1], leaves in order in each row [1], key given [1]:

2 | 9
3 | 0 4 6 8
4 | 1 1 1 3 5 7
5 | 2 2 8
Key: 2 | 9 means 29 cm

[3] Examiner insight: Sort the data first, then check that the leaves number 14 and the key has the unit.

5. (a) 8th value of 15 = 31 points [1] [1] (b) 34 appears three times, so the mode is 34 [1] [1] (c) 51 - 15 = 36 points [1] [1] (d) 31, 34, 34, 34, 37, 42, 48, 51 gives 8 players [1] [1] Examiner insight: Read leaves back with the stem attached (3 | 1 is 31, not 1), because quoting the leaf alone is a frequent slip.

6. (a) Stems 2 to 7 in the middle [1], both sets of leaves in order outwards from the stem [1], key for each side [1]:

Year 7        | Year 9
              | 2 | 5 9
        8 2   | 3 | 3 6 9
      9 4 1   | 4 | 2 4 8
      7 3 0   | 5 | 1 5 9
        4 1   | 6 |
          2   | 7 |
Key: Year 7: 4 | 1 means 4.1 minutes. Year 9: 4 | 2 means 4.2 minutes.

(b) Median Year 7 = 5.0 minutes, median Year 9 = 4.2 minutes [1]. Range Year 7 = 7.2 - 3.2 = 4.0, range Year 9 = 5.9 - 2.5 = 3.4 [1]. Year 9 pupils typically solved the puzzle faster, and their times were slightly less spread out [1] [6] Examiner insight: A comparison earns more when each number is paired with a statement in context; “Year 9 is quicker” says more than two bare medians.

7. Ordered: 48, 48, 52, 55, 57, 59, 60, 62, 63, 66, 69, 71, 74 [1]. Median = 7th value = 60; lower half 48, 48, 52, 55, 57, 59 gives 53.5; upper half 62, 63, 66, 69, 71, 74 gives 67.5 [1]. Minimum 48, lower quartile 53.5, median 60, upper quartile 67.5, maximum 74 [1] [3] Examiner insight: Order the data before finding any quartile, and show the halves so the method you used is visible.

8. (a) IQR = 51 - 30 = 21 [1]. Range = 80 - 12 = 68 [1] [2] (b) The line in the box marks the median [1], and a box-and-whisker diagram does not contain enough information to find the mean, which depends on every value [1] [2] Examiner insight: Say what 38 actually is (the median) instead of only saying the pupil is wrong.

9. (a) Back-to-back stem-and-leaf diagram [1], because it keeps every value and puts the two classes side by side [1] [2] (b) Box-and-whisker diagrams on one scale [1], because they compare median and spread for large data sets quickly [1] [2] (c) Venn diagram [1], because it shows the overlap (both) and the neither group as separate regions [1] [2] Examiner insight: Tie each reason to the purpose in the question; “it is clear” is too general.

10. (a) Shop A: IQR = 31 - 14 = 17, range = 50 - 6 = 44 [1]. Shop B: IQR = 28 - 18 = 10, range = 36 - 10 = 26 [1] [2] (b) The medians are close, with 22 minutes for A and 23 for B [1]. Shop B has the smaller IQR (10 against 17) and smaller range (26 against 44) [1]. So Shop B’s times are more consistent, whereas Shop A can be very quick but also very slow (up to 50 minutes) [1]. For reliability, recommend Shop B [1] [4] (c) The diagrams do not show individual times, so one rare late delivery cannot be told apart from a pattern of lateness [1] [1] Examiner insight: End with a recommendation that follows from your comparison; numbers alone leave the question unanswered.

11. (a) Minimum 22, maximum 69 [1]. Median = mean of 10th and 11th values, (42 + 45) ÷ 2 = 43.5 [1]. Lower quartile = mean of 5th and 6th, (33 + 34) ÷ 2 = 33.5 [1]. Upper quartile = mean of 15th and 16th, (53 + 55) ÷ 2 = 54 [1] [4] (b) Box from 33.5 to 54 with a line at 43.5 [1]. Whiskers from 22 to 33.5 and from 54 to 69 [1] [2] (c) Median: club A 43.5, club B 32, so members of club A exercise more in a typical week [1]. IQR: club A 54 - 33.5 = 20.5, club B 40 - 25 = 15, so club B is more consistent [1]. Conclusion: club A is more active typically but with greater variation between members [1] [3] (d) It shows every individual value, such as the actual minutes of each member [1] [1] Examiner insight: In long comparisons, give average, spread and conclusion as three separate points.

Where marks are usually lost

• Leaving leaves unordered or omitting the key in a stem-and-leaf diagram.
• Missing an empty stem, which changes the shape.
• Putting back-to-back leaves in the wrong order, with the smallest furthest from the stem.
• Forgetting the region outside the circles in a Venn diagram.
• Using a pair total as an “only” region without removing the centre.
• Finding quartiles from an unordered list.
• Quoting a median or IQR without comparing it with the other group.
• Describing the median as the mean.
• Giving comparison answers that are all numbers, with no conclusion in context.

Next steps

• Re-read the revision notes or the study guide.
• Visit the 4040 hub and the printable checklist.
• Try the free 10-minute diagnostics to choose your next unit.
• Book a free trial class if you want help with a section.

Official syllabus

Cambridge O Level Statistics 4040, syllabus for examination in 2025, 2026 and 2027 (version 1), published by Cambridge Assessment International Education, part of Cambridge University Press & Assessment. Sections 2.2, 2.3 and 2.4.