Cambridge O Level Statistics: Venn, box-and-whisker and stem-and-leaf diagrams, choosing and interpreting representations (4040)

This guide follows the Cambridge O Level Statistics (4040) syllabus for examination in 2025, 2026 and 2027, version 1. It covers sections 2.2 (Venn diagrams, box-and-whisker diagrams and stem-and-leaf diagrams, including back-to-back), 2.3 (purpose, use, advantages and disadvantages of representations) and 2.4 (interpreting data in tabular, pictorial or diagrammatic form). The course is not tiered. Tables, bar-type charts, pie charts and pictograms are taught on separate pages.

Other pages for this course: the 4040 hub, the printable checklist, the topic 2 overview, revision notes for this unit and the practice set. To find your weak spots first, try the free 10-minute diagnostics.

What this unit covers

Section	What you must be able to do
2.2	Draw and read Venn diagrams, box-and-whisker diagrams, and stem-and-leaf diagrams including back-to-back ones
2.3	Say what each representation is for, and give its advantages and disadvantages
2.4	Read values and draw conclusions from tables, pictures and diagrams, including comparisons

Venn diagrams

A Venn diagram shows how a group of items falls into overlapping categories. The rectangle is the whole group being studied. Each circle is one category, and where circles overlap an item belongs to both.

Notation used in this guide:

• n(A) is the number of items in A.
• A∩B means in both A and B.
• A∪B means in A or B or both.
• A′ means not in A.

Each item sits in exactly one region of the diagram, so the numbers in all regions add up to the total.

Method

1. Draw the rectangle and the circles, with overlaps.
2. Fill the centre first (the “both” or “all three” region).
3. Work outwards: subtract what you have already placed from each given total.
4. Put the “neither” figure outside the circles. If it is not given, find it from the total.
5. Check that all regions add up to the total.

Worked example: two sets

Of 40 students, 19 are in the chess club, 17 are in the debate club and 6 are in both. Find how many are in neither.

Both = 6
Chess only  = 19 - 6 = 13
Debate only = 17 - 6 = 11
In at least one club = 13 + 6 + 11 = 30
Neither = 40 - 30 = 10

Answer: 10 students. As a check, n(C∪D) = 19 + 17 - 6 = 30, which matches.

Worked example: three sets

A group of 60 pupils: 24 take art, 27 take music, 20 take drama. Also 9 take art and music, 8 take music and drama, 6 take art and drama, and 4 take all three. Find the number in each region.

The pair totals include the centre, so remove it first.

Art and music only   = 9 - 4 = 5
Music and drama only = 8 - 4 = 4
Art and drama only   = 6 - 4 = 2
Centre               = 4
Art only   = 24 - 5 - 2 - 4 = 13
Music only = 27 - 5 - 4 - 4 = 14
Drama only = 20 - 4 - 2 - 4 = 10
Total in a circle = 13 + 14 + 10 + 5 + 4 + 2 + 4 = 52
None = 60 - 52 = 8

Now answer questions from the finished diagram. Exactly one subject: 13 + 14 + 10 = 37 pupils. Exactly two subjects: 5 + 4 + 2 = 11. Music or drama (or both): 27 + 20 - 8 = 39; checking by regions, 14 + 10 + 5 + 4 + 2 + 4 = 39.

The common trap is to read “9 take art and music” as the region without the centre. Always ask whether a given number includes the middle.

Stem-and-leaf diagrams

A stem-and-leaf diagram lists every value, grouped by its leading digits. The stem is the leading part. The leaf is the final digit. Rules:

• Order the leaves from smallest to largest in each row.
• Space the leaves evenly so the row length shows the frequency.
• Include every stem from lowest to highest, even if a row is empty.
• Write a key, for example 2 | 3 means 23.
• If the values have decimals, the key must say so, for example 3 | 4 means 3.4.

Worked example: building one

Journey times in minutes for 20 pupils: 31, 23, 45, 12, 28, 40, 21, 36, 23, 58, 15, 31, 47, 30, 25, 34, 18, 27, 42, 38.

Step 1: sort them: 12, 15, 18, 21, 23, 23, 25, 27, 28, 30, 31, 31, 34, 36, 38, 40, 42, 45, 47, 58.

Step 2: stems 1 to 5, leaves in order.

1 | 2 5 8
2 | 1 3 3 5 7 8
3 | 0 1 1 4 6 8
4 | 0 2 5 7
5 | 8
Key: 2 | 1 means 21 minutes

Count the leaves: 3 + 6 + 6 + 4 + 1 = 20, matching the data.

Reading it

• Mode: 23 and 31 each appear twice, so there are two modes.
• Range: 58 - 12 = 46 minutes.
• Median: with 20 values, take the mean of the 10th and 11th, which are 30 and 31. Median = 30.5 minutes.
• Shape: most values lie in the twenties and thirties, with one value (58) well above the rest.

Back-to-back diagrams

A back-to-back diagram compares two sets that share the same stems. The stem is in the middle. One set has leaves to the left, the other to the right. Both sets are ordered outwards from the stem, so the left-hand rows read smallest leaf nearest the stem.

Scores out of 50 for 15 pupils in group P and 15 in group Q:

        P           |   | Q
          7 4       | 1 | 8
    9 8 6 5 3 1     | 2 | 2 4 7
      8 6 4 3 1     | 3 | 0 2 2 5 7 7 9
          5 1       | 4 | 0 2 3 4
Key: P: 1 | 2 means 21     Q: 2 | 2 means 22

Group P: median 29, range 45 - 14 = 31. Group Q: median 35, range 44 - 18 = 26. Group Q has the higher median and the smaller range. In context: pupils in group Q scored higher in the middle, and their scores were more consistent.

Box-and-whisker diagrams

A box-and-whisker diagram is drawn from five numbers: the minimum, lower quartile, median, upper quartile and maximum. The box runs from the lower to the upper quartile, with a line at the median. The whiskers run to the minimum and maximum. The interquartile range (IQR) is the upper quartile minus the lower quartile, the width of the box. The range is the maximum minus the minimum.

Quartile method used here: order the data, find the median, then take the median of the lower half and of the upper half (leave the median out of both halves when the number of values is odd). Methods for small lists differ between courses, so follow the one you are taught and show your working.

Worked example

Battery life in hours: phone X (11 values) 18, 21, 24, 26, 27, 29, 31, 33, 35, 38, 44; phone Y (12 values) 22, 25, 27, 28, 30, 31, 32, 33, 34, 36, 37, 40.

Phone X: median is the 6th value, 29. Lower half 18, 21, 24, 26, 27 has median 24. Upper half 31, 33, 35, 38, 44 has median 35. Five numbers: 18, 24, 29, 35, 44.

Phone Y: median is the mean of the 6th and 7th values, (31 + 32) ÷ 2 = 31.5. Lower half 22, 25, 27, 28, 30, 31 has median (27 + 28) ÷ 2 = 27.5. Upper half 32, 33, 34, 36, 37, 40 has median (34 + 36) ÷ 2 = 35. Five numbers: 22, 27.5, 31.5, 35, 40.

   15   20   25   30   35   40   45
   +    +    +    +    +    +    +
X     |-----[====#=====]--------|
Y         |----[===#===]----|

The # is the median, the brackets mark the quartiles, and the bars are the minimum and maximum.

Comparison: median X = 29, Y = 31.5. IQR X = 35 - 24 = 11, Y = 35 - 27.5 = 7.5. Range X = 26, Y = 18. Phone Y typically lasts longer and is more consistent. Phone X has the longest single battery life (44 hours), but also the shortest (18).

The box contains about half of the data, and each whisker section and each half of the box contains about a quarter. A box plot cannot give the mean or any individual value.

Choosing a representation (2.3)

A good answer names the purpose first, then the advantage that fits it, then a limitation.

Representation	Purpose	Advantage	Disadvantage
Venn diagram	Show overlap between categories	Counts “both”, “either” and “neither” at a glance	Gets crowded with more than three sets; shows counts, not values of a measurement
Stem-and-leaf	Show every value and the shape of a small data set	Keeps the raw data; ordered, so median and mode are easy	Awkward for large data sets or widely spread values
Back-to-back	Compare two small sets	Direct comparison with all values kept	Only two sets; same stems needed
Box-and-whisker	Compare median and spread	Fast comparison of several data sets on one scale; works for large sets	Individual values lost; shape inside the box hidden

Use these questions to decide:

• Do you need the exact values? Use a stem-and-leaf diagram.
• Do you need to compare the spread of two or more groups? Use box-and-whisker diagrams.
• Are you sorting people or items into overlapping groups? Use a Venn diagram.
• Is the data set large? Prefer the box-and-whisker diagram, because a stem-and-leaf diagram becomes long.

Interpreting and comparing (2.4)

A comparison needs at least one measure of average, one measure of spread, and a conclusion in the context of the question. Use comparison words such as “higher”, “more consistent” and “whereas”.

Worked comparison

Two delivery shops record delivery times in minutes. Shop A: minimum 8, lower quartile 15, median 20, upper quartile 26, maximum 40. Shop B: 12, 17, 21, 24, 30. Which shop is the better choice?

Median: A = 20, B = 21
IQR:    A = 26 - 15 = 11,   B = 24 - 17 = 7
Range:  A = 40 - 8 = 32,    B = 30 - 12 = 18

Model answer: The median delivery time is about the same, with Shop A one minute shorter. Shop B is more consistent, since its IQR (7) and range (18) are smaller than Shop A’s (11 and 32). Shop A is sometimes very quick (8 minutes) but is also sometimes very slow (40 minutes). If reliability matters, choose Shop B.

Reading tables

For tables, check the headings, units and whether numbers are counts or percentages. Compare like with like: a percentage of 20% of a small group can be fewer people than 10% of a large group.

Common errors

• Leaving leaves unordered, or leaving out the key.
• Forgetting an empty stem, which distorts the shape.
• In a back-to-back diagram, ordering the left-hand leaves with the smallest furthest from the stem.
• Taking the median of the whole list as a quartile boundary without splitting into halves first.
• Drawing box plots on different scales when comparing.
• Giving only numbers in a comparison, with no sentence about what they mean.
• Treating a number in a Venn pair total as the “only” region.
• Describing a box plot’s median as the mean.

Next steps

Condense this into the revision notes, then try the practice set. The free diagnostics show which topic to work on next.

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.