GCSE Maths: Bar charts, pictograms and line charts (Higher)
Part of the GCSE Maths statistics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Bar charts, pictograms and line charts
Charts turn a table of numbers into a picture, so patterns are easier to see. The right chart depends on the type of data. Bar charts and pictograms are used for categorical data and for discrete data. A vertical line chart (see Section 24.4) is used for discrete numerical data.
Bar charts. The bars have equal widths with gaps between them, because the categories are separate. The height of a bar shows the frequency. Both axes must be labelled and the vertical scale needs equal steps starting at 0. To read a bar, find its top and read across to the scale.
Dual bar charts show two groups side by side, and composite bar charts stack the groups on top of each other so that the total height shows the whole. Either way, a key shows what each colour means.
Pictograms use a picture or symbol to stand for a number of items. There must be a key that says what one symbol stands for. A part of a symbol stands for a part of that number: a half symbol is half the value. To find the number of symbols for a frequency, divide the frequency by the value of one symbol.
Misleading charts. Charts can give a false impression. Watch for: a vertical axis that does not start at 0 (the bars look very different when the values are almost the same); a scale that is not evenly spaced; bars of different widths; missing labels or titles; and 3-D effects. Always look at the numbers on the axis before you decide what a chart shows.
Key ideas
- Bar chart: equal-width bars with gaps; height = frequency; equal scale starting at 0; label both axes.
- Dual and composite bar charts compare groups; use the key.
- Pictogram: always use a key. Number of symbols = frequency ÷ value of one symbol.
- Vertical line chart: discrete numerical data; the height of each line is the frequency.
- Be suspicious of charts whose axes do not start at 0 or whose scales are uneven.
(a) Which fruit is the most popular? (b) How many more students chose apple than banana? (c) How many students chose pear or grape?
Method. Read the height of each bar against the vertical scale, then use the numbers.
- (a) Find the tallest bar. The tallest bar has the highest frequency, so it is the most popular.Apple is tallest, with frequency 12. The most popular fruit is apple.
- (b) Read the apple and banana bars. Read across from the top of each bar.Apple = 12, banana = 9
- Subtract. “How many more” means take away.12 − 9 = 3
- (c) Read the pear and grape bars and add. “Or” means add the two frequencies.Pear = 5, grape = 6
5 + 6 = 11
Check. The bars add to 40, the number of students: 12 + 9 + 8 + 6 + 5. ✓
Watch out. Read the height on the vertical scale, not the position along the bottom.
(a) How many books were borrowed on Thursday? (b) How many books were borrowed altogether? (c) How many more were borrowed on Tuesday than on Wednesday?
Method. Use the key: each whole symbol is 4 books and each half symbol is 2 books.
- (a) Count the symbols on Thursday. Thursday has 3 whole symbols and one half symbol.3 × 4 + 2 = 12 + 2 = 14 books
- (b) Work out each day. Do the same for each row.Monday: 3 × 4 = 12
Tuesday: 4 × 4 + 2 = 18
Wednesday: 2 × 4 + 2 = 10
Thursday: 14 - Add the days. 12 + 18 + 10 + 14 = 54 books
- (c) Compare Tuesday and Wednesday. Subtract.18 − 10 = 8 books
Check. Total symbols: 3 + 4.5 + 2.5 + 3.5 = 13.5, and 13.5 × 4 = 54. ✓
Watch out. Do not forget the key. A half symbol is half of the value of a whole symbol, here 2.
(a) How many boys chose science? (b) Which subject did more girls than boys choose by the largest amount? (c) How many students are in the year group?
Method. Read the blue bars for the boys and the orange bars for the girls. Use the key.
- (a) Read the blue bar for science. Blue stands for boys.10 boys
- (b) Find the differences girls − boys for each subject. Compare the two bars in each pair.Maths: 6 − 8 = -2
Science: 9 − 10 = -1
English: 9 − 5 = 4
Art: 6 − 3 = 3 - Pick the largest positive difference. This is the subject where girls beat boys by the most.English (4 more girls)
- (c) Add every bar. Each student chose one subject, so every bar counts different students.Boys: 8 + 10 + 5 + 3 = 26
Girls: 6 + 9 + 9 + 6 = 30
26 + 30 = 56
Check. Girls − boys: Maths −2, Science −1, English +4, Art +3. The largest is English. ✓
Watch out. Check which colour is which in the key before you read the bars.
| Transport | Frequency |
|---|---|
| Bus | 14 |
| Walk | 9 |
| Car | 7 |
| Cycle | 4 |
Method. Draw two axes, choose an even scale, then draw one bar for each category. All the bars must have the same width, with equal gaps between them.
- Choose a scale for the vertical axis. The biggest frequency is 14, so the axis must go up to at least 14. Use equal steps, such as 5.Vertical axis: 0 to 15 in steps of 5
- Label both axes and give the chart a title. The reader must know what each axis shows.Horizontal axis: Method of travel. Vertical axis: Number of students.
- Draw one bar for each category. The height of each bar is its frequency, read against the vertical scale.Bus: height 14
Walk: height 9
Car: height 7
Cycle: height 4 - Check against the table. Every bar should match its frequency.
Check. Each bar’s height matches the table. ✓
Watch out. Bars must have equal widths and the vertical scale must start at 0 and have equal steps.
Explain why the chart is misleading.
Method. Compare what the chart looks like with what the numbers say.
- Look at the bars. The bar for Phone B looks much taller than the bar for Phone A. Measuring from the axis at 80 gives heights 2 and 5.Heights: Phone A = 82 − 80 = 2, Phone B = 85 − 80 = 5. The bar for B looks 2.5 times as tall.
- Look at the actual numbers. The sales are almost the same.85 ÷ 82 = 1.037: Phone B sold only about 4% more.
- Find the cause. The vertical axis does not start at 0, so the bars do not show the true sizes.The axis should start at 0.
Check. If the axis started at 0 the bars would be almost equal in height. ✓
Watch out. Always check where the scale starts.
Watch out
Always check the key and the scale before reading a chart. A pictogram symbol rarely stands for 1, and a chart with an axis that does not start at 0 can mislead.Exercise 24F Bar charts, pictograms and line charts
Non-calculator(a) How many students have a cat? (b) Which is the most common pet? (c) How many more students have a dog than a rabbit? (d) How many students are there in the class? (e) What fraction of the class have a fish? Give your answer in its simplest form.
Method. Find the cat bar and read its height.
- Find the bar labelled Cat. The second bar
- Read the height on the vertical axis. The scale goes up in steps of 5, so read between the lines carefully.12
Check. The cat bar is lower than the dog bar and higher than the fish bar. ✓
Watch out. Read the top of the bar against the vertical scale.
Method. The most common pet has the tallest bar.
- Find the tallest bar. Tallest means highest frequency.Dog, with 15
- Write the answer. Dog
Check. No other bar is as tall. ✓
Watch out. The answer is the pet, not the frequency.
Method. Read both bars, then subtract.
- Read the dog and rabbit bars. Dog = 15, rabbit = 6
- Subtract. “How many more” means take away.15 − 6 = 9
Check. 6 + 9 = 15. ✓
Watch out. Subtract the smaller from the larger.
Method. Each student owns one type of pet in this chart, so add all the bars.
- Read every bar. Dog 15 , Cat 12 , Rabbit 6 , Fish 9 , Bird 3
- Add. 15 + 12 + 6 + 9 + 3 = 45
Check. Add in a different order: 15 + 3 + 12 + 9 + 6 = 45. ✓
Watch out. Do not forget any bar.
Method. Part over whole, then simplify.
- Find the part and the whole. The part is the students with a fish, and the whole is the class.Part = 9, whole = 45
- Write the fraction. 945
- Simplify. Divide the top and bottom by 9.945 = 15
Check. 1 ÷ 5 = 0.2 and 9 ÷ 45 = 0.2. ✓
Watch out. The whole is the whole class (45), not the largest bar.
(a) How many cakes were sold on Monday? (b) How many cakes were sold on Wednesday? (c) How many cakes were sold in the five days? (d) On which day were twice as many cakes sold as on Friday?
Method. Count the symbols and use the key.
- Count the symbols. Monday has 2 whole symbols and one half symbol.2 whole symbols and a half symbol
- Use the key. A whole symbol is 10 and a half symbol is 5.2 × 10 + 5 = 25
Check. 25 ÷ 10 = 2.5 symbols. ✓
Watch out. A half symbol is half of 10, which is 5.
Method. Count the symbols and use the key.
- Count the symbols. Wednesday has 1 whole symbol and one half symbol.1 whole symbol and a half symbol
- Use the key. 1 × 10 + 5 = 15
Check. 15 ÷ 10 = 1.5 symbols. ✓
Watch out. Do not count the half symbol as a whole one.
Method. Work out each day, then add.
- Work out each day. Use the key for each row.Mon: 25, Tue: 40, Wed: 15, Thu: 30, Fri: 20
- Add. 25 + 40 + 15 + 30 + 20 = 130
Check. Total symbols: 2.5 + 4 + 1.5 + 3 + 2 = 13, and 13 × 10 = 130. ✓
Watch out. Use the key to turn symbols into numbers.
Method. Friday had 20 cakes. Twice as many is 40.
- Find twice the Friday sales. Friday has 2 symbols, which is 20 cakes.2 × 20 = 40
- Find the day with 40 cakes. Look for 4 whole symbols.Tuesday
Check. Tuesday has 4 symbols and Friday has 2, so Tuesday is twice Friday. ✓
Watch out. Compare the numbers of cakes, not just the symbols.
Method. In a pictogram each symbol stands for a fixed number of items. Divide each frequency by the number each symbol stands for.
- Decide what one symbol stands for, and write a key. Choose a value that divides nicely into the data. Here one symbol stands for 10 apples, and a half symbol stands for 5.Key: one symbol = 10 apples
- Work out the number of symbols for each row. Divide the frequency by the value of one symbol.Mon: 30 ÷ 10 = 3 symbols
Tue: 45 ÷ 10 = 4.5 symbols (4 whole and one half)
Wed: 20 ÷ 10 = 2 symbols
Thu: 35 ÷ 10 = 3.5 symbols (3 whole and one half) - Draw the symbols in each row. Draw them the same size and in line, using a half symbol where there is a half.
Check. Multiply the number of symbols by 10 to get each frequency back. ✓
Watch out. Symbols must be the same size and evenly spaced, and there must be a key.
| Sport | Frequency |
|---|---|
| Football | 11 |
| Netball | 8 |
| Rugby | 5 |
| Tennis | 9 |
| Hockey | 7 |
Method. Draw two axes, choose an even scale, then draw one bar for each category. All the bars must have the same width, with equal gaps between them.
- Choose a scale for the vertical axis. The biggest frequency is 11, so the axis must go up to at least 11. Use equal steps, such as 2.5.Vertical axis: 0 to 12.5 in steps of 2.5
- Label both axes and give the chart a title. The reader must know what each axis shows.Horizontal axis: Favourite sport. Vertical axis: Number of students.
- Draw one bar for each category. The height of each bar is its frequency, read against the vertical scale.Football: height 11
Netball: height 8
Rugby: height 5
Tennis: height 9
Hockey: height 7 - Check against the table. Every bar should match its frequency.
Check. Each bar’s height matches the table. ✓
Watch out. Bars must have equal widths and the vertical scale must start at 0 and have equal steps.
(a) How many girls joined in Year 8? (b) In which year group did more boys than girls join? (c) How many students joined altogether?
Method. Find the Year 8 pair of bars and read the girls’ bar.
- Use the key. Orange stands for girls.Look at the orange bar
- Read the height. 11
Check. The Year 8 girls’ bar is higher than the boys’ bar. ✓
Watch out. Read the right colour.
Method. Compare the two bars in each pair.
- Compare each year. Look for the year where the blue bar is higher.Year 7: boys 10, girls 12
Year 8: boys 14, girls 11 (more boys)
Year 9: boys 9, girls 8 (more boys)
Year 10: boys 6, girls 10 - Write the answer. Year 8 and Year 9
Check. In Years 7 and 10 the girls’ bars are taller. ✓
Watch out. Check the key.
Method. Add all the bars.
- Add the boys. 10 + 14 + 9 + 6 = 39
- Add the girls. 12 + 11 + 8 + 10 = 41
- Add the totals. 39 + 41 = 80
Check. Add year by year: 22 + 25 + 17 + 16 = 80. ✓
Watch out. Count both colours.
(a) How many students are in Year 8? (b) How many Year 9 students come by car? (c) Which year group has the most students?
Method. The whole bar shows all the Year 8 students, so read the top of the bar, or add the three sections.
- Read the three sections. Walk, bus and car are stacked in the Year 8 bar.Walk 9, bus 14, car 5
- Add them. The whole bar is all the Year 8 students.9 + 14 + 5 = 28
Check. The top of the Year 8 bar is at 28. ✓
Watch out. In a composite bar chart, the sections are on top of each other, so you add them.
Method. Read the height of the section for car, not the top of the bar.
- Find the car section in the Year 9 bar. The car section is the top part (green).From 17 up to 25
- Subtract to find its height. The section starts where the bus section ends.25 − 17 = 8
Check. Walk 6 + bus 11 + car 8 = 25. ✓
Watch out. The height of a section is the difference between its top and bottom.
Method. Work out the total for each year group.
- Year 7. 12 + 10 + 3 = 25
- Year 8. 9 + 14 + 5 = 28
- Year 9. 6 + 11 + 8 = 25
- Compare. The tallest whole bar is the biggest group.Year 8
Check. The tallest bar on the chart is Year 8. ✓
Watch out. Compare whole-bar heights.
Method. Look at the numbers and the scale on the vertical axis.
- Read the values. The bars show sales of 84 thousand and 80 thousand.84 − 80 = 4 thousand
- Compare the sizes. Our firm has only about 5% more than its rival.84 ÷ 80 = 1.05, only 5% more
- Look at the axis. The vertical scale starts at 78, so it cuts off the bottom of the bars and makes the difference look huge.The bars look in the ratio 6 : 2, a factor of 3.
Check. If the axis started at 0, the two bars would be almost the same height. ✓
Watch out. A truncated axis (one that does not start at 0) exaggerates differences.
Method. Think about the type of data first.
- Decide the data type. Lunch choices are categories (words).Categorical data.
- Choose a chart. A bar chart (or a pictogram) shows how many chose each category.Bar chart
Check. Bars with gaps suit separate categories. ✓
Watch out. Do not join categorical data with a line.
Method. Think about the type of data first.
- Decide the data type. Goals are counted, so they are discrete numbers.Discrete numerical data.
- Choose a chart. A vertical line chart shows the frequency for each value.Vertical line chart
Check. Each goal total has its own line. ✓
Watch out. A vertical line chart is for discrete numbers.
Method. Think about whether time is involved.
- Decide what is shown. The data is measured over time.Time series data.
- Choose a chart. A line graph shows the trend as time passes.Line graph
Check. Joining the points shows how the temperature changes between readings. ✓
Watch out. Use a line graph when time is on the horizontal axis.
Method. Each whole symbol is 8 students, and a half symbol is half of 8.
- Work out the whole symbols. 3 whole symbols of 8 students each.3 × 8 = 24
- Work out the half symbol. Half of 8.8 ÷ 2 = 4
- Add. 24 + 4 = 28
Check. 3.5 symbols × 8 = 28. ✓
Watch out. Half a symbol is half of the value of a whole symbol.
Method. Compare the fractions (or percentages) who chose pizza, because the classes are different sizes.
- Write each as a fraction of its class. Part over whole.Class A: 1025
Class B: 816 - Write them as percentages. Make the denominator 100, or divide.Class A: 1025 = 40100 = 40%
Class B: 816 = 50% - Compare. Class B has the larger percentage.Pizza is more popular in Class B, so Ana is wrong.
Check. 10 ÷ 25 = 0.4 and 8 ÷ 16 = 0.5. ✓
Watch out. When groups are different sizes, compare fractions or percentages, not the raw numbers.
Method. Count the symbols in total, then divide the total number of items by that number of symbols.
- Count the symbols for each day. A half symbol counts as 0.5.Mon = 2.5, Tue = 4, Wed = 3.5
- Find the total number of symbols. Add them.2.5 + 4 + 3.5 = 10
- Find what one symbol stands for. The total is 100 items, shared across 10 symbols.100 ÷ 10 = 10 items per symbol
- Work out Tuesday. Tuesday has 4 symbols.4 × 10 = 40 items
Check. 2.5 × 10 + 40 + 3.5 × 10 = 25 + 40 + 35 = 100. ✓
Watch out. Divide the total by the number of symbols, including half symbols as 0.5.
(a) Which drink sold least? (b) How many more cups of juice than coffee were sold? (c) What percentage of the drinks sold were tea?
Method. The shortest bar has the smallest frequency.
- Find the shortest bar. Water, with 10 sold
- Write the answer. Water
Check. All other bars are taller. ✓
Watch out. Name the drink, not the number.
Method. Read the two bars and subtract.
- Read the bars. Juice = 40, coffee = 20
- Subtract. 40 − 20 = 20
Check. 20 + 20 = 40. ✓
Watch out. “More than” means subtract.
Method. Find the total, then write tea as a fraction of it and as a percentage.
- Find the total. Add all four bars.30 + 20 + 40 + 10 = 100
- Write tea as a fraction. Part over whole.30100
- Change to a percentage. The total is 100, so the number sold is already the percentage.30 out of 100 = 30%
Check. 30% of 100 = 30. ✓
Watch out. The whole is the total number sold.
Method. Check each feature of the chart against the rules for bar charts.
- Check the widths. All bars should have equal widths so that only the height shows the frequency.A wider chess bar looks bigger, although only its height matters.
- Check the vertical axis. A scale that starts at 8 cuts off the bottoms of the bars, so small differences look big.The axis should start at 0.
- State two reasons. Unequal bar widths, and a vertical axis that does not start at 0.
Check. Both faults change how big the bars look without changing the data. ✓
Watch out. Look at the widths and at the scale.