Representing Data: Charts and Diagrams

Visual representations make data easier to interpret. At GCSE you need to draw and interpret bar charts, dual/composite bar charts, pie charts, stem-and-leaf diagrams and pictograms.

---

Key Vocabulary

Term	Definition
Bar chart	Uses bars of equal width; the height represents frequency
Dual bar chart	Two sets of bars side by side for comparison
Composite bar chart	Bars stacked on top of each other
Pie chart	A circle divided into sectors; the angle of each sector is proportional to the frequency
Stem-and-leaf diagram	Displays data using stems (leading digits) and leaves (trailing digits) in order
Pictogram	Uses symbols to represent frequencies; includes a key

---

Bar Charts

Drawing a Bar Chart

• Bars must be the same width with equal gaps between them.
• The vertical axis (frequency) must be labelled and scaled evenly.
• Bars should be labelled on the horizontal axis.

Dual Bar Charts

Used to compare two sets of data on the same axes. Each category has two bars side by side, usually with a key.

Worked Example 1

A survey asks students in Year 7 and Year 11 about their favourite sport. Describe what a dual bar chart of this data would show.

Sport	Year 7	Year 11
Football	12	8
Tennis	6	10
Swimming	9	7
Cricket	3	5

Solution: For each sport, two bars are drawn next to each other — one for Year 7 (e.g. blue) and one for Year 11 (e.g. red). The horizontal axis shows the four sports; the vertical axis shows frequency (0 to say 14). A key shows which colour is which year. From the chart we can see:

• Football is most popular in Year 7 (12 > the other Year 7 bars) but Tennis is most popular in Year 11.
• Football's popularity falls with age (12 → 8) while Tennis's rises (6 → 10).

Composite (Stacked) Bar Charts

The bars for each category are stacked. The total height shows the combined frequency. Useful for showing totals as well as individual contributions.

Worked Example 2

A dual bar chart shows sales of vanilla and chocolate ice creams by month. In June, vanilla sales were 60 and chocolate sales were 40. In July, vanilla sales were 90 and chocolate sales were 55.

(a) Which flavour increased the most in absolute terms from June to July? (b) Which flavour increased the most in percentage terms?

Solution: (a) Vanilla: 90 − 60 = 30. Chocolate: 55 − 40 = 15. Vanilla increased the most in absolute terms (by 30). (b) Vanilla: 30 ÷ 60 = 50%. Chocolate: 15 ÷ 40 = 37.5%. Vanilla also increased the most in percentage terms (50%).

---

Pie Charts

Drawing a Pie Chart

There are 360° in a full circle. To find the angle for each sector:

Angle = (frequency ÷ total frequency) × 360°

Worked Example 3

60 students were asked their favourite fruit. Find the angles needed to draw a pie chart.

Fruit	Frequency	Angle
Apple	18	(18 ÷ 60) × 360 = 108°
Banana	12	(12 ÷ 60) × 360 = 72°
Orange	15	(15 ÷ 60) × 360 = 90°
Grape	9	(9 ÷ 60) × 360 = 54°
Other	6	(6 ÷ 60) × 360 = 36°
Total	60	360°

Check: 108 + 72 + 90 + 54 + 36 = 360° ✓

Use a protractor to measure each angle from the previous sector's boundary. Label each sector clearly.

Worked Example 4

In a survey of 180 people, 55 chose football, 40 chose tennis, 25 chose swimming and the rest chose "other." Calculate the pie chart angles for each category.

Solution:

• "Other" frequency = 180 − 55 − 40 − 25 = 60
• Football: (55 ÷ 180) × 360 = 110°
• Tennis: (40 ÷ 180) × 360 = 80°
• Swimming: (25 ÷ 180) × 360 = 50°
• Other: (60 ÷ 180) × 360 = 120°
• Check: 110 + 80 + 50 + 120 = 360° ✓

Worked Example 5 (Reverse Direction)

A pie chart shows the favourite colours of 90 students. The "Blue" sector has an angle of 140°. The "Red" sector has an angle of 80°. Green has a frequency of 15.

(a) How many students chose blue? (b) How many students chose red? (c) What angle does the green sector have?

Solution: (a) (140 ÷ 360) × 90 = 35 students (b) (80 ÷ 360) × 90 = 20 students (c) Angle = (15 ÷ 90) × 360 = 60°

Comparing Pie Charts

Two pie charts may have different totals. You CANNOT just compare sector sizes visually — you must calculate actual frequencies.

Worked Example 6

Pie chart A (total = 120) shows "football" at 90°. Pie chart B (total = 200) shows "football" at 54°. Which has more football fans?

Solution:

• A: (90 ÷ 360) × 120 = 30 students
• B: (54 ÷ 360) × 200 = 30 students
• Both have the same number of football fans, even though sector A looks much bigger than sector B.

---

Box Plot Anatomy (Preview)

Before we look at stem-and-leaf diagrams, here is the anatomy of a box plot — a diagram you'll learn to draw in Lesson 7. It summarises a distribution using five numbers.

graph LR
    A["Minimum"] --- B["Q1<br/>(25%)"]
    B --- C["Median<br/>(50%)"]
    C --- D["Q3<br/>(75%)"]
    D --- E["Maximum"]

    style B fill:#3498db,color:#fff
    style C fill:#e74c3c,color:#fff
    style D fill:#3498db,color:#fff

• Left whisker: minimum to $Q_1$Q1​
• Box: $Q_1$Q1​ to $Q_3$Q3​ (width = IQR)
• Line inside box: median
• Right whisker: $Q_3$Q3​ to maximum

---

Stem-and-Leaf Diagrams

A stem-and-leaf diagram organises data while retaining every individual value.

Rules:

• Each data item is split into a stem (leading digits) and a leaf (final digit).
• Leaves MUST be in numerical order.
• A key is essential.

Worked Example 7

Draw an ordered stem-and-leaf diagram for: 23, 15, 31, 27, 45, 38, 22, 19, 34, 41, 27, 33, 16, 29, 37. Then find the mode, median and range.

Solution:

Stem	Leaves
1	5 6 9
2	2 3 7 7 9
3	1 3 4 7 8
4	1 5

Key: 2 | 3 means 23

From this diagram:

• Smallest value: 15
• Largest value: 45
• Mode: 27 (appears twice)
• Median: n = 15 → 8th value = 29
• Range: 45 − 15 = 30

Back-to-Back Stem-and-Leaf Diagrams

Used to compare two data sets. One set's leaves go to the left, the other to the right.

Worked Example 8

A back-to-back diagram shows test scores for boys and girls:

Boys	Stem	Girls
8 5	1	2 4 7
9 6 3 1	2	0 5 8
7 4 2	3	1 3

Key: 5 | 1 means 15 (boys); 1 | 2 means 12 (girls)

(a) How many boys took the test? (b) Which gender has the higher median?