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Cambridge IGCSE0607

Statistical charts and diagrams

International Mathematics 0607 Chapter Notes

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Statistical charts and diagrams
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Statistical charts and diagrams notes

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In this chapter

  1. 1Bar charts
  2. 2Composite and dual bar charts
  3. 3Pictograms
  4. 4Pie charts
  5. 5Drawing a stem-and-leaf diagram
  6. 6Reading a stem-and-leaf diagram
  7. 7Simple frequency distributions

1. Bar charts

A bar chart shows how often each category or value occurs. The height of a bar is its frequency.

Rule every bar the same width and leave gaps between the bars. Label both axes.

Before you read a bar, work out what one small square on the vertical axis is worth.

Examiner insight

“A bar chart requires gaps between the bars and a few candidates omitted to provide these.” “A very small number misinterpreted the scale on the bar chart and gave the answer 4.”

Worked example 15 marks

Some runners take part in a marathon race. The bar chart shows their times, correct to the nearest hour.

(a) Write down how many runners had a time of 4 hours.

(b) Write down the most common time.

(c) Work out the total number of runners.

(d) One of the runners is chosen at random. Find the probability that this runner had a time of 5 hours. Give your answer as a fraction in its simplest form.

  1. 1

    (a) the bar at 4 hours has height 18 [1]

  2. 2

    (b) the tallest bar is at 5 hours [1]

  3. 3

    (c) 10 + 18 + 27 + 23 + 12 = 90 [1]

  4. 4

    (d) 27/90 [1]

  5. 5

    = 3/10 [1]

2. Composite and dual bar charts

A dual bar chart draws the bars side by side, one for each group. Read each bar on its own.

A composite bar chart stacks the groups in one bar. The top of the bar is the total. A segment's value is the gap between its bottom edge and its top edge. Both need a key. Shade every bar as the key says and keep the widths equal.

Examiner insight

“A small number of candidates seemed completely unfamiliar with a compound chart and simply superimposed all three bars.” “However uneven bar widths and the unshaded bar drawn for Sunday were commonly seen.”

Worked example 12 marks

The table gives the number of students in two classes who chose each sport. A dual bar chart and a composite bar chart are drawn, each with the bars grouped by sport.

(a) In the dual bar chart, write down the height of the bar for Class Q football.

(b) In the composite bar chart, write down the height of the football bar.

SportClass PClass Q
football129
tennis710
  1. 1

    (a) a dual bar shows one group alone: 9 [1]

  2. 2

    (b) a composite bar shows the total: 12 + 9 = 21 [1]

3. Pictograms

A pictogram uses symbols to show frequency. The key says how many items one symbol stands for.

Multiply the number of symbols by the key value. Half a symbol stands for half the key value.

Worked example 12 marks

The pictogram shows the numbers of different types of drink sold by a shop one lunchtime. The key shows one symbol = 4 drinks.

(a) Work out the total number of drinks sold.

(b) Work out the difference between the number of apple drinks sold and the number of mango drinks sold.

Drinklemonorangeapplemango
Symbols drawn2 and a half53 and a quarterthree quarters
  1. 1

    (a) 21/2 × 4 + 5 × 4 + 31/4 × 4 + 3/4 × 4 = 10 + 20 + 13 + 3 = 46 [1]

  2. 2

    (b) 13 - 3 = 10 [1]

4. Pie charts

A pie chart shows how a whole is shared between categories. The angle of a sector is in proportion to its frequency, and the angles add up to 360°.

To draw one, find each angle from its share of the total. Check that the angles add to 360°, measure each with a protractor and label every sector.

To read one, turn the angle back into a share of the total.

angle = frequency/total × 360^(°)

frequency = angle/(360^(°)) × total

Examiner insight

“A common wrong answer was 60° (from 360 ÷ 6) or 5 from (30 ÷ 6).” “Better answers were from those showing clear working to establish the size of the angles required followed by accurate measuring of angles and drawing and labelling of the sectors.”

Worked example 14 marks

Hanan is drawing a pie chart to show the number of each type of email she receives one day.

(a) Complete the table.

(b) Complete the pie chart to show this information.

type of emailfrequencypie chart angle
promotions (P)530°
main (M)15
work (W)30
social (S)10
  1. 1

    (a) one email is 30 ÷ 5 = 6^(°), so M = 15 × 6 = 90^(°) and W = 30 × 6 = 180^(°) [1]

  2. 2

    S = 10 × 6 = 60^(°), and 30 + 90 + 180 + 60 = 360^(°) [1]

  3. 3

    (b) draw sectors of 90^(°), 180^(°) and 60^(°) with a protractor [1]

  4. 4

    label each sector P, M, W and S [1]

5. Drawing a stem-and-leaf diagram

In a stem-and-leaf diagram the stem is the leading digit or digits and the leaf is the last digit. A value below 10 has stem 0. Write each value's leaf in its row. A repeated value is written every time it occurs.

Put the leaves in each row in order, smallest first. Then complete the key, such as 1 mid 3 = 13. Count the leaves: there must be one for every value in the data.

Examiner insight

“Some put the data in the order in correct lines but did not take off the tens digit so for example the second line was 11, 15, 17.” “Some candidates did not order their rows, and some missed out a number.”

Worked example 13 marks

The number of broken biscuits in each of 13 boxes of biscuits is counted. These are the results.

1 5 13 17 18 10 2 12 1 1 16 7 4

(a) Complete the stem-and-leaf diagram to show this information.

  1. 1

    leaves as they come: row 0 is 1 5 2 1 1 7 4 and row 1 is 3 7 8 0 2 6, which is 7 + 6 = 13 leaves [1]

  2. 2

    leaves in order: 0 mid 1 1 1 2 4 5 7 and 1 mid 0 2 3 6 7 8 [1]

  3. 3

    key: 1 mid 3 = 13 [1]

6. Reading a stem-and-leaf diagram

Rebuild a value by joining its stem to its leaf, as the key says.

To count the values, count the leaves. Do not add them up.

The most common value has the leaf that appears most often in one row. Give the whole value, not just the leaf.

Examiner insight

“A few added up all the numbers after the stem to arrive at 54 and some added up all the numbers to arrive at 354.” “Occasionally an answer of 3 was seen, indicating that the candidates had omitted the stem.”

Worked example 11 mark

The age of each of 15 doctors is shown in the stem-and-leaf diagram.

(a) Write down the most common age.

  1. 1

    the leaf 8 appears twice in the row for stem 2, and no other value repeats, so the age is 28 [1]

7. Simple frequency distributions

A frequency table lists each value with the number of times it occurs. Tally the data, then write each count as a number in the frequency column. A tally alone is not enough.

Add the frequencies. The total must equal the number of values in the data.

The most common value, the mode, is the value with the highest frequency. It is not the frequency itself.

Examiner insight

“A few gave only a tally and there were some examples of miscounting.” “In part (iii), some quoted the frequency rather than the temperature.”

Worked example 13 marks

The number of goals scored in each of 12 matches is

2 0 1 3 2 1 2 0 2 1 4 2

Complete a frequency table and state the most common number of goals.

  1. 1

    frequencies for 0, 1, 2, 3, 4 goals: 2, 3, 5, 1, 1 [2]

  2. 2

    check: 2 + 3 + 5 + 1 + 1 = 12, the number of matches

  3. 3

    most common: 2 goals (5 is its frequency, not the answer) [1]

Recap

  • Bar chart: equal widths, gaps between bars, height is frequency.
  • Dual bars sit side by side. In a composite bar the top is the total.
  • Pictogram: number of symbols times the key value.
  • Pie chart: angle = frequency ÷ total × 360°. Label every sector.
  • Stem-and-leaf: leaves in order, repeats included, key given.
  • Count the leaves to count the values.
  • Mode: the value with the highest frequency, not the frequency.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. State two things that must be included in a stem-and-leaf diagram so that its values can be read correctly.2 marks
  2. Do not use a calculator. A pictogram shows the pets owned by a class. The key is one symbol = 4 pets. How many symbols are needed for each pet? | Pet | Number of pets | | --- | --- | | cat | 12 | | dog | 9 | | fish | 3 |3 marks
  3. The sizes of shoes sold in a shop one morning are 5 6 5 7 6 5 8 6 5 7 Complete the frequency table and write down the most common size. | Shoe size | Frequency | | --- | --- | | 5 | | | 6 | | | 7 | | | 8 | |3 marks
  4. The vertical axis of a bar chart is labelled 0, 10, 20 and 30. There are five equal squares between each pair of labels. The top of one bar is three squares above the 10 line. Write down the frequency of that bar.2 marks
  5. Do not use a calculator. The number of pages in 12 books is recorded in a stem-and-leaf diagram. Key: 1 mid 2 = 12 pages. (a) How many books are there? (b) Write down the most common number of pages. (c) How many books have more than 25 pages? | Stem | Leaf | | --- | --- | | 1 | 2 5 5 9 | | 2 | 0 3 4 4 4 6 | | 3 | 1 8 |3 marks
  6. A group of eight students scored these marks in a quiz. 23 31 25 38 23 40 31 27 Draw an ordered stem-and-leaf diagram. Include a key.3 marks
  7. Do not use a calculator. 24 students are asked their favourite sport. 5 choose swimming. Work out the angle of the swimming sector in a pie chart.2 marks
  8. The table shows the students in three classes. A composite bar chart has one bar for each class, with boys at the bottom and girls on top. (a) Write down the height of the bar for class 9B. (b) The girls' segment of the 9C bar starts at one height and ends at another. Write down both heights. (c) In a dual bar chart, write down the height of the boys' bar for class 9A. | Class | Boys | Girls | | --- | --- | --- | | 9A | 11 | 14 | | 9B | 13 | 9 | | 9C | 12 | 16 |3 marks
  9. Do not use a calculator. A pie chart is drawn for the vehicles on a ferry. The car sector has angle 150^(°) and stands for 60 cars. The van sector has angle 45^(°) and the lorry sector is a right angle. All the other vehicles are motorbikes. (a) Work out the total number of vehicles. (b) Work out the number of motorbikes. (c) 16 more cars drive on and no vehicle leaves. A new pie chart is drawn. Work out the new angle of the car sector. (d) Work out the new angle of the motorbike sector.7 marks
  10. Do not use a calculator. The masses, in kg, of the 18 parcels on a van are recorded in a stem-and-leaf diagram. Key: 0 mid 7 = 0.7 kg. (a) Write down the most common mass. (b) A parcel with a mass of more than 2.5 kg is called heavy. How many parcels are heavy? (c) A pie chart with two sectors, heavy and not heavy, is drawn for the 18 parcels. Work out the angle of the heavy sector. (d) Two more parcels, of mass 3.0 kg and 0.7 kg, are put on the van. Write out the rows for stem 0 and stem 3 as they are now. | Stem | Leaf | | --- | --- | | 0 | 7 9 | | 1 | 2 4 4 8 | | 2 | 0 5 5 5 6 | | 3 | | | 4 | 1 3 3 9 | | 5 | 0 2 6 |7 marks

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