Maths · Statistics
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Pie Charts, Bar Charts and Stem-and-Leaf Diagrams
Drawing and reading pie charts, bar charts and stem-and-leaf diagrams, and spotting misleading charts.
Learning Objectives
- 1Read, draw and interpret pie charts, including finding angles and frequencies.
- 2Read bar charts and compare data, including dual bar charts.
- 3Construct and interpret stem-and-leaf diagrams, and find the median, mode and range from one.
- 4Spot misleading features in a chart.
Showing data clearly
A good chart lets the reader see a pattern at a glance, and examiners test whether you can both read a chart accurately and decide what it shows. Pie charts are about angles and fractions, bar charts are about heights, and stem-and-leaf diagrams keep every value while sorting them into order. All three need careful reading of the key and scale, and all the numbers are chosen to be easy: a pie chart of 60 people has 6 degrees for each person, and 360 divides neatly by 36, 40, 60, 72 and 120.
Pie charts
A pie chart shows how a whole is shared out, with the angle of each sector proportional to its frequency.
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Angle for a sector
\(\dfrac{\text{frequency}}{\text{total}} \times 360^\circ\).
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Degrees for each item
Work out \(\dfrac{360}{\text{total}}\) first. For 60 students, each is \(6^\circ\).
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Frequency from an angle
\(\dfrac{\text{angle}}{360} \times \text{total}\). In a pie chart of 36 people, \(100^\circ\) is \(\dfrac{100}{360} \times 36 = 10\) people.
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Check
The angles add up to \(360^\circ\), and so do the frequencies for the total.
A pie chart
With 60 students, each is \(\dfrac{360}{60} = 6^\circ\), so football's 24 students make \(24 \times 6 = 144^\circ\). The same method works backwards.
Using the pie chart
- Football \(24 \times 6 = 144^\circ\).
- Hockey \(9 \times 6 = 54^\circ\), the green sector.
- Fraction Netball is \(\dfrac{90}{360} = \dfrac{1}{4}\) of the students, which is 15.
- Percentage Hockey is \(\dfrac{54}{360} = 15\%\).
Drawing a pie chart
40 students choose their favourite fruit: apple 16, banana 12, pear 8, grape 4. Work out the angle for each sector.
Show the solutionHide the solution
- 1 Degrees per student \(\dfrac{360}{40} = 9^\circ\).
- 2 Multiply Apple \(16 \times 9 = 144\), banana \(12 \times 9 = 108\), pear \(8 \times 9 = 72\), grape \(4 \times 9 = 36\).
- 3 Check \(144 + 108 + 72 + 36 = 360\).
- 4 Draw Use a protractor and label each sector.
AnswerApple \(144^\circ\), banana \(108^\circ\), pear \(72^\circ\), grape \(36^\circ\)
Bar charts
A bar chart compares categories, with the height of each bar showing the frequency.
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Gaps between bars
For data in categories, such as colours or sports.
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Dual bar charts
Two sets of bars side by side to compare groups, such as boys and girls.
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Read the scale
Check what each gridline is worth, as some scales go up in 2s or 5s.
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Compare totals
To compare groups of different sizes, use fractions or percentages, not just heights.
Stem-and-leaf diagrams
A stem-and-leaf diagram keeps the exact data. The stem is the first digit or digits, and each leaf is the last digit.
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A key
It says what a value means, such as \(2 \mid 3\) means 23. A diagram without a key loses a mark.
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Ordered
The leaves go in order, smallest nearest the stem.
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Median
Count to the middle value, or the average of the middle two.
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Back to back
Two sets of data can share the same stem, with leaves going left and right.
A stem-and-leaf diagram
The ages are 12, 15, 18, 21, 23, 23, 26, 29, 30, 34, 34, 37, 42 and 45. There are 14 values, and they are already in order.
Reading the diagram
- Median With 14 values, take the mean of the 7th and 8th, 26 and 29, which gives \(\dfrac{26 + 29}{2} = 27.5\).
- Mode 23 and 34 both occur twice, so there are two modes.
- Range \(45 - 12 = 33\).
- Aged 30 or over The bottom two rows give \(4 + 2 = 6\) members.
Finding the median
A stem-and-leaf diagram shows the masses of 13 parcels in kilograms. Stem 1 has leaves 3 and 7. Stem 2 has leaves 0, 4, 4, 5 and 8. Stem 3 has leaves 1, 3, 6 and 9. Stem 4 has leaves 2 and 5. The key says that stem 2 with leaf 4 means 24 kg. Find the median mass.
Show the solutionHide the solution
- 1 Count the values \(2 + 5 + 4 + 2 = 13\).
- 2 Find the position The median is the \(\dfrac{13 + 1}{2} = 7\)th value.
- 3 Count along the rows Stem 1 gives the 1st and 2nd values, and stem 2 gives the 3rd to 7th values: 20, 24, 24, 25, 28.
- 4 Read the 7th value The 7th value is 28, so the median is 28 kg.
Answer28 kg
Misleading charts
Examiners like to ask what is wrong with a chart.
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A scale that does not start at zero
It makes small differences look large.
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Uneven scales or bars
Bars with different widths, or scales that jump, are misleading.
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A 3D effect
It distorts the sizes of bars and pie sectors.
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Missing labels or a missing key
The reader cannot tell what the chart shows.
Test yourself
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1
How do you find the angle for a sector?
Show answerHide answer
Frequency divided by total, times 360.
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2
For 40 people, how many degrees represent each person?
Show answerHide answer
\(\dfrac{360}{40} = 9^\circ\).
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3
What must every stem-and-leaf diagram have?
Show answerHide answer
A key.
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4
How do you find the median from a stem-and-leaf diagram with 11 values?
Show answerHide answer
The 6th value.
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5
Why might a bar chart be misleading?
Show answerHide answer
The scale may not start at zero, or the bars may be uneven.
Exam technique: charts
Be exact with angles and keys.
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Measure angles accurately
Within 2 degrees is usually accepted.
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Show your working
Write \(\dfrac{360}{40} = 9\) before multiplying.
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Write the key
Stem and leaf diagrams must show it.
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Say "compare" correctly
Use both averages and spread, and give figures.
Summary and exam focus
- A sector's angle is \(\dfrac{\text{frequency}}{\text{total}} \times 360^\circ\), and a frequency is \(\dfrac{\text{angle}}{360} \times \text{total}\).
- Bar charts compare categories, and a dual bar chart compares two groups.
- A stem-and-leaf diagram has ordered leaves and a key, and gives the median, mode and range.
- Check scales and labels for misleading features.
Exam focus
A pie chart shows the favourite drinks of 40 pupils. The angle for tea is \(135^\circ\). How many pupils chose tea? (2 marks) (2 marks)
Each pupil is \(\dfrac{360}{40} = 9^\circ\), so tea is \(\dfrac{135}{9} = 15\) pupils. Alternatively, \(\dfrac{135}{360} \times 40 = 15\). Either way, show the method.
Key terms
The words this lesson expects you to use. Each one is linked from the first place it appears above.
- Pie chart
- A circular chart where the angle of each sector represents its share of the total.
- Sector
- A slice of a pie chart.
- Bar chart
- A chart that uses bars of different heights to show frequencies.
- Dual bar chart
- A bar chart with two sets of bars side by side.
- Stem-and-leaf diagram
- A diagram that shows data by splitting each value into a stem and a leaf.
- Stem
- The first digit or digits of each value in a stem-and-leaf diagram.
- Leaf
- The last digit of each value in a stem-and-leaf diagram.
- Key
- A note that explains what the values in a chart or diagram mean.
- Frequency
- The number of times a value or category occurs.
Questions and answers
15 questions set on this lesson, with the mark schemes and model answers open.
Yasmin asks 30 people which fruit they like best. 12 choose apple, 9 choose banana, 6 choose pear and 3 choose grape. She draws a pie chart. Work out the angle for each sector.
Mark scheme — 3 marks available
- \(360 \div 30 = 12\) — M1
- At least two angles correct — A1
- \(144^\circ, 108^\circ, 72^\circ, 36^\circ\) — A1
Model answer
Each person is \(\dfrac{360}{30} = 12^\circ\). Apple: \(12 \times 12 = 144^\circ\). Banana: \(9 \times 12 = 108^\circ\). Pear: \(6 \times 12 = 72^\circ\). Grape: \(3 \times 12 = 36^\circ\).
The pie chart shows how 36 students travel to school. The angle for cycle is not shown. (a) Work out how many students walk. (2 marks) (b) Work out the angle for cycle. (1 mark) (c) Work out how many students cycle. (1 mark)
Mark scheme — 4 marks available
- (a) \(\dfrac{100}{360} \times 36\) or \(100 \div 10\) — M1
- (a) 10 — A1
- (b) \(60^\circ\) — B1
- (c) 6 — B1 (follow through from (b))
Model answer
(a) Each student is \(\dfrac{360}{36} = 10^\circ\), so the number who walk is \(\dfrac{100}{10} = 10\). (b) \(360 - 140 - 100 - 60 = 60^\circ\). (c) \(\dfrac{60}{10} = 6\).
The stem-and-leaf diagram shows the times, in minutes, that 14 runners took to finish a race. (a) Work out the median time. (2 marks) (b) Work out the range of the times. (2 marks)
Mark scheme — 4 marks available
- (a) 7th and 8th values 16 and 18 identified — M1
- (a) 17 — A1
- (b) \(36 - 5\) — M1
- (b) 31 — A1
Model answer
(a) There are 14 values, so the median is halfway between the 7th and 8th values, which are 16 and 18, giving 17 minutes. (b) \(36 - 5 = 31\) minutes.
A newspaper shows a bar chart of the sales of a phone. The vertical axis starts at 98 instead of 0, and the headline says “Sales soar”. Explain why the chart may be misleading.
Mark scheme — 2 marks available
- States that the scale does not start at zero — B1
- States that a small difference looks large or exaggerated — B1
Model answer
The vertical axis does not start at zero, so a very small difference in sales looks like a large change.
A stem-and-leaf diagram shows the ages, in years, of some members of a club. The stem 1 has leaves 2, 5 and 8. The stem 2 has leaves 1, 4, 4 and 9. The stem 3 has leaves 0 and 3. The key says that 2 bar 1 means 21 years. For these ages, find (a) the number of members, (b) the median age, (c) the range.
Mark scheme — 3 marks available
- (a) 9 — B1
- (b) 24 — B1
- (c) 21 — B1
Model answer
(a) \(3 + 4 + 2 = 9\) members. (b) In order the ages are 12, 15, 18, 21, 24, 24, 29, 30, 33, so the 5th value is 24. (c) \(33 - 12 = 21\).
Pie chart A shows the travel to school of 40 students. The angle for cycle is \(90^\circ\). Pie chart B shows the travel to school of 120 students. The angle for cycle is \(60^\circ\). Kim says, “The cycle sector is bigger on chart A, so more students cycle in school A.” Is Kim correct? You must show your working.
Mark scheme — 3 marks available
- \(\dfrac{90}{360} \times 40 = 10\) — M1
- \(\dfrac{60}{360} \times 120 = 20\) — M1
- Kim is not correct, with a reason using the totals — C1
Model answer
In school A, \(\dfrac{90}{360} \times 40 = 10\) students cycle. In school B, \(\dfrac{60}{360} \times 120 = 20\) students cycle. Kim is not correct. A bigger fraction cycle in school A, but more students cycle in school B, because there are more students.
60 students choose a sport. 24 choose football. What is the angle for football on a pie chart?
Why: Each student is \(\dfrac{360}{60} = 6^\circ\), so \(24 \times 6 = 144^\circ\).
A pie chart shows 40 people. How many degrees represent each person?
Why: \(\dfrac{360}{40} = 9\).
A pie chart shows 36 people. A sector is \(100^\circ\). How many people does it represent?
Why: \(\dfrac{100}{360} \times 36 = 10\).
What fraction of a pie chart is a \(90^\circ\) sector?
Why: \(\dfrac{90}{360} = \dfrac{1}{4}\).
What must every stem-and-leaf diagram have?
Why: Without a key the values cannot be read.
A stem-and-leaf diagram has 14 ordered values. The 7th is 26 and the 8th is 29. What is the median?
Why: \(\dfrac{26 + 29}{2} = 27.5\).
The values in a stem-and-leaf diagram run from 12 to 45. What is the range?
Why: \(45 - 12 = 33\).
Which of these makes a bar chart misleading?
Why: A scale that does not start at zero exaggerates differences.
On a pie chart for 60 people, a sector is \(54^\circ\). What percentage is that?
Why: \(\dfrac{54}{360} = 0.15\).