Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Statistics and Probability

Representing Data

Organise and show data with Venn and Carroll diagrams, two-way tables, dual and compound bar charts, pie charts, frequency diagrams, time series, scatter graphs, stem-and-leaf diagrams and infographics — and choose the right one and explain why.

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In the curriculum

Record, organise and represent categorical, discrete and continuous data, and choose and explain which representation to use: Venn and Carroll diagrams; tally charts, frequency tables and two-way tables; dual and compound bar charts; pie charts; frequency diagrams for continuous data; line graphs and time series graphs; scatter graphs; stem-and-leaf diagrams; infographics (8Ss.03)

There are many ways to show data. Checkpoint-style questions ask you to complete a table or chart, read values from one, and — at Stage 8 — choose the best representation and explain why. The key question is always: what type of data is it, and what do I want the reader to see?

Key words

Word / phrase	Meaning / use
frequency	التكرار — how many times a value or group occurs
two-way table	جدول ذو اتجاهين — a table that sorts data by two things at once (e.g. boys/girls and bus/car)
dual bar chart	أعمدة مزدوجة — two bars side by side for each category, to compare two groups
compound (stacked) bar chart	أعمدة مركّبة — the bars for two groups stacked on top of each other
sector	قطاع — one “slice” of a pie chart
class interval	فئة — a group of continuous values, e.g. 10 ≤ h < 15
time series	سلسلة زمنية — data recorded over time, plotted as points joined by lines
correlation	الارتباط — the relationship between two variables on a scatter graph
stem-and-leaf diagram	مخطط الساق والورقة — numbers split into a stem (tens) and leaves (units), in order
infographic	إنفوجرافيك — a poster that shows data with pictures, icons, short text and simple charts

1. Sorting: Venn and Carroll diagrams

A Venn diagram sorts items into sets drawn as overlapping circles inside a rectangle (the universal set, ξ). Items in the overlap belong to both sets; items outside the circles belong to neither. Below, ξ = the whole numbers 1 to 12, A = multiples of 2 and B = multiples of 3.

6 and 12 are multiples of both 2 and 3, so they go in the overlap. 1, 5, 7 and 11 are in neither set.

A Carroll diagram sorts by two properties using a grid. Each property has a column (or row) for “yes” and one for “not”. Here are the same numbers 1 to 12:

	Multiple of 2	Not a multiple of 2
Multiple of 3	6, 12	3, 9
Not a multiple of 3	2, 4, 8, 10	1, 5, 7, 11

Venn and Carroll diagrams can also show counts instead of the items themselves. If 40 students were asked, 18 play football, 15 play basketball and 6 play both, the Venn regions are: football only 18 − 6 = 12, both 6, basketball only 15 − 6 = 9, neither 40 − (12 + 6 + 9) = 13. Always fill the overlap first.

2. Tally charts, frequency tables and two-way tables

A tally chart records data as you collect it (groups of five: four lines and a line across). A frequency table gives the totals. For continuous data use class intervals with inequalities so that every value fits in exactly one class: 10 ≤ h < 15 means h is at least 10 but less than 15, so a plant of exactly 15 cm goes in the next class.

A two-way table shows two categorical variables at the same time. Rows and columns have totals, and the grand total is in the bottom-right corner. To fill gaps, use: (row total) = sum of the row, (column total) = sum of the column.

	Walk	Bus	Car	Total
Boys	12	15	8	35
Girls	14	10	9	33
Total	26	25	17	68

The bold numbers were missing and were worked out: Boys bus = 35 − 12 − 8 = 15; Girls walk = 26 − 12 = 14; Girls total = 14 + 10 + 9 = 33; Bus total = 15 + 10 = 25; Car total = 8 + 9 = 17; grand total = 35 + 33 = 68 (check: 26 + 25 + 17 = 68 ✓).

3. Dual and compound bar charts

Use bar charts for categorical or discrete data. Bars have equal widths and gaps between categories. To compare two groups there are two choices:

Dual bar chart: the bars for 8A and 8B stand side by side, so it is easy to compare the two classes in each sport (8A: football 12; 8B: basketball 9).

Compound bar chart of the same data: each bar shows the total for both classes (football 20, basketball 15, swimming 12, tennis 7), split into the two classes.

A dual bar chart is best for comparing the groups category by category. A compound bar chart is best when the totals matter as well as how each total is made up. To read the top part of a compound bar, subtract: football for 8B = 20 − 12 = 8.

4. Pie charts

A pie chart shows how a whole is shared between categories — the proportions. The full circle is 360°.

Find the total frequency. Example: 36 students said how they travel to school.
Find the angle for one item: 360° ÷ 36 = 10° per student.
Multiply each frequency: walk 9 × 10° = 90°, bus 15 × 10° = 150°, car 10 × 10° = 100°, bicycle 2 × 10° = 20°.
Check the angles add up to 360°: 90 + 150 + 100 + 20 = 360 ✓.
Draw a radius, measure each angle with a protractor from the last line, and label every sector (or use a key).

Pie chart of how 36 students travel to school, drawn clockwise from the top.

Reading a pie chart: number in a sector = (angle ÷ 360) × total. If 120 people are shown and a sector is 75°, it shows 75 ÷ 360 × 120 = 25 people. A pie chart shows proportions, not actual numbers — two pie charts of different-sized groups cannot be compared by sector size alone.

5. Frequency diagrams for continuous data

Continuous data (heights, masses, times) are grouped into equal class intervals and drawn as a frequency diagram: like a bar chart but with no gaps between the bars, and a continuous scale on the horizontal axis (numbers at the edges of the bars, not under the middle).

Height, h (cm)	0 ≤ h < 5	5 ≤ h < 10	10 ≤ h < 15	15 ≤ h < 20	20 ≤ h < 25
Frequency	3	7	11	6	3

Frequency diagram for the heights of 30 plants. The modal class is 10 ≤ h < 15 (the tallest bar).

6. Line graphs and time series

When data are recorded over time (each month, each year, every hour), draw a time series graph: time on the horizontal axis, plot each value and join the points with straight lines. It shows trends (going up or down overall) and seasonal patterns (repeating ups and downs). The lines between points only show the change; values between the points are not measured.

A café’s cold-drink sales each month of one year: sales rise from January to a peak in July (82), then fall again towards December.

7. Scatter graphs

A scatter graph shows whether two numerical variables are related. Each person or item gives one point (x, y), plotted as a cross. The pattern is called correlation:

Positive: as one goes up, so does the other. Negative: as one goes up, the other goes down. No correlation: no clear pattern.

Hours of revision and test score for 10 students: positive correlation — students who revised for longer generally scored higher.

Describe correlation in context: “The more hours a student revised, the higher their score tended to be.” Correlation shows a relationship; it does not prove that one thing causes the other.

8. Stem-and-leaf diagrams

A stem-and-leaf diagram keeps every data value and shows the shape of the data. For two-digit numbers, the stem is the tens digit and the leaf is the units digit. Leaves are written in order and lined up, and there must be a key. Times (seconds) of 15 students to solve a puzzle: 23, 27, 31, 34, 34, 38, 40, 42, 45, 45, 47, 51, 53, 58, 62.

Stem	Leaf
2	3 7
3	1 4 4 8
4	0 2 5 5 7
5	1 3 8
6	2

Key: 3|4 means 34 seconds. From the diagram you can read the smallest value (23), the largest (62), the mode (34 and 45 both appear twice) and count values — for example, 8 students took less than 45 seconds.

9. Infographics, and choosing a representation

An infographic combines pictures, icons, big numbers, short text and simple charts to tell a story with data quickly, e.g. a school poster: “3 out of 4 students walk or cycle” with 4 student icons, 3 of them shaded. Infographics are eye-catching, but check them: are the icons the same size? Is the scale shown? Is the source given?

You want to show…	Best choice	Why
how items sort by one or two properties	Venn or Carroll diagram	shows which items belong to which set, including overlaps
counts for two categorical variables together	two-way table	shows every combination and the totals
two groups compared category by category	dual bar chart	bars side by side are easy to compare
totals and how each total is made up	compound bar chart	each bar shows the total and its parts
proportions of a whole	pie chart	sector size shows the fraction of the total
grouped continuous data	frequency diagram	continuous scale, no gaps between bars
change over time	time series / line graph	shows trends and seasonal patterns
relationship between two numerical variables	scatter graph	shows correlation
a small set of numbers, keeping every value	stem-and-leaf diagram	shows shape and keeps the raw data; easy to find median and range
key facts for the general public	infographic	quick and eye-catching

بالعربي: أختار أنهي رسم؟

بيانات وصفية (categorical) أو عدّ (discrete): أعمدة بينها مسافات — مزدوجة للمقارنة، مركّبة لو المجموع مهم.
نِسَب من الكل: دائرة (pie chart) — زاوية الشخص الواحد = 360 ÷ العدد الكلي.
بيانات متصلة (continuous) مجمّعة في فئات: frequency diagram — الأعمدة لازقة في بعض من غير مسافات.
تغيّر مع الزمن: time series. علاقة بين متغيرين: scatter graph. ولازم مفتاح (key) في الساق والورقة.

Worked examples

Worked example 1 — Two-way table [2 marks] In a group of 50 students, 28 are girls. 19 of the girls and 12 of the boys have a pet. How many students do not have a pet?

Boys = 50 − 28 = 22.
Have a pet = 19 + 12 = 31.
No pet = 50 − 31 = 19. (Check by parts: girls with no pet 28 − 19 = 9; boys with no pet 22 − 12 = 10; 9 + 10 = 19 ✓)

Answer: 19 students. Common mistake: subtracting 31 from 28 or 22 instead of from the grand total.

Worked example 2 — Pie chart angle [2 marks] 60 people chose their favourite drink. 18 chose tea. Work out the angle of the sector for tea.

360° ÷ 60 = 6° per person. 18 × 6° = 108°. Answer: 108°. (1 mark for 6° per person or 18/60 × 360 seen.) Common mistake: using 100 instead of 360, giving 30 — that is the percentage, not the angle.

Worked example 3 — Reading a pie chart [1 mark] A pie chart shows the favourite fruit of 120 people. The sector for mango has an angle of 75°. How many people chose mango?

75 ÷ 360 × 120 = 75 ÷ 3 = 25. Answer: 25 people.

Worked example 4 — Venn diagram [2 marks] Of 40 students, 18 play football, 15 play basketball and 6 play both. How many play neither?

Football only = 18 − 6 = 12. Basketball only = 15 − 6 = 9. At least one sport = 12 + 6 + 9 = 27. Neither = 40 − 27 = 13. Answer: 13. Common mistake: 40 − 18 − 15 = 7. This counts the 6 students who play both sports twice.

Worked example 5 — Choosing a representation [2 marks] Amira recorded the height and the arm span of 20 students. She wants to see whether taller students have longer arm spans. Which diagram should she draw? Explain why.

Answer: a scatter graph (1 mark), because it shows the relationship (correlation) between two sets of numerical data from the same students (1 mark). Common mistake: choosing a bar chart or pie chart; these show one variable only.

Worked example 6 — Stem-and-leaf [2 marks] Draw an ordered stem-and-leaf diagram for these marks: 34, 52, 41, 38, 45, 29, 52, 47, 33, 40.

Order the marks: 29, 33, 34, 38, 40, 41, 45, 47, 52, 52.
Stems 2, 3, 4, 5. Leaves: 2 → 9; 3 → 3 4 8; 4 → 0 1 5 7; 5 → 2 2.
Add a key, for example 4|5 means 45 marks.

Answer: as above (1 mark for correct leaves, 1 mark for ordered with a key). Common mistake: forgetting the key, or not putting the leaves in order (an unordered diagram usually loses a mark).

Common mistakes

Leaving gaps between the bars of a frequency diagram for continuous data — or no gaps on a bar chart for categories.
Writing class intervals that overlap (10–15, 15–20). Use 10 ≤ h < 15, 15 ≤ h < 20.
Pie chart angles that do not add up to 360°; or measuring each angle from the same starting line instead of from the previous sector.
Comparing two pie charts of different-sized groups as if the sectors showed numbers.
Reading the top part of a compound bar from zero instead of subtracting.
Forgetting the key or the ordering in a stem-and-leaf diagram.
Saying scatter-graph correlation proves that one variable causes the other.
In a Venn diagram, filling the circles before the overlap, so the “both” items are counted twice.

How Checkpoint asks this

“Complete the two-way table” or “Complete the Venn/Carroll diagram” — fill the given totals first, then subtract.
“Complete the pie chart” — show 360 ÷ total; draw angles accurately (±2°) and label each sector.
“Draw a frequency diagram” or “complete the time series graph” on a printed grid — plot exactly, join time series points with straight lines.
“What type of correlation …?” and “Describe the relationship” — use the context words.
“Which diagram would you use? Give a reason.” — name the diagram and say what it shows (proportion, comparison, change over time, relationship).
“Draw an ordered stem-and-leaf diagram” — include a key.

Practice

Remember

Categorical/discrete data: bar charts with gaps — dual to compare groups, compound to show totals and parts.
Pie charts show proportions: angle = frequency × (360 ÷ total); angles add up to 360°.
Continuous data: equal class intervals with inequalities and a frequency diagram with no gaps.
Time series show change over time; scatter graphs show correlation (positive, negative, none) — not cause.
Venn and two-way tables: fill the overlap or given totals first, then subtract. Stem-and-leaf: ordered leaves and a key.
When asked to choose, name the diagram and say what it shows in the context.

All pages in this course

Start here

Stage 8 Maths: The Tests and How to Prepare

Number

Algebra

Geometry and measure

Statistics and probability

Practice tests