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Cambridge IGCSE0580

Interpreting statistical data

Mathematics 0580 Chapter Notes

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Interpreting statistical data
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Interpreting statistical data notes

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In this chapter

  1. 1Reading tables and bar charts
  2. 2Stem-and-leaf diagrams
  3. 3Pie charts and bar charts
  4. 4Comparing sets of data
  5. 5Limits on conclusions

1. Reading tables and bar charts

Read the title, the axis labels and the scale before you read any value. The height of a bar is the number in that group.

To find how many more are in one group than another, subtract the two heights. To complete a bar, use the total: the missing bar is the total minus the other bars. Draw it with correct width, the same as the other bars.

Examiner insight

“Misinterpretation of the question was seen a number of times for a height of 3, the difference between the other two bar heights.”

Worked example 13 marks

25 people own a bike. A bar chart of their bike types has two bars drawn: road bikes with height 10 and racing bikes with height 7.

(a) Find the height of the bar for mountain bikes.

(b) How many more people own a road bike than own a racing bike?

  1. 1

    (a) total = 25, so mountain = 25 - (10 + 7) [1]

  2. 2

    = 8 [1]

  3. 3

    (b) road - racing = 10 - 7 = 3 [1]

2. Stem-and-leaf diagrams

In a stem-and-leaf diagram each row has a stem and its leaves. Read the key first: it tells you what a stem and leaf stand for together.

The number of leaves is the number of values. Leaves are written in numerical order. For a percentage, write the count as a fraction of the total and multiply by 100.

Examiner insight

“Some did not appear familiar with the topic as they entered the whole numbers in the leaf section of the diagram.” “Some candidates showed the correct values but not in numerical order.”

Worked example 13 marks

This stem-and-leaf diagram gives the time, in minutes, it takes 20 students to eat lunch.

key | 3 1 represents 31 minutes

Complete these statements.

(a) ...... students took between 10 and 30 minutes to eat lunch.

(b) ...... % of the students took less than 18 minutes to eat lunch.

(c) The slowest student took ...... minutes longer than the quickest student.

stemleaf
07 8 9
11 3 5 6 9
23 4 5 5 7 8 8 9
31 2 2 9
  1. 1

    (a) leaves in rows 1 and 2: 5 + 8 = 13 students [1]

  2. 2

    (b) below 18: 7, 8, 9, 11, 13, 15, 16, so 7 students. 7/20 × 100 = 35% [1]

  3. 3

    (c) slowest = 39, quickest = 7, so 39 - 7 = 32 minutes [1]

3. Pie charts and bar charts

A pie chart shares 360° between the groups. For a total of n items, one item is 360 ÷ n degrees.

To link a pie chart to a bar chart, divide a sector's angle by the angle for one item. This gives the frequency, which is the height of the bar. A bar chart is the better choice for reading a number, because its bar heights give the frequencies.

angle for one item = 360/total

Examiner insight

“Successful candidates generally used their answer to part (a)(i) and calculated the angle sector by dividing 360 by 24 and then multiplying by 7 or/and 9.” “To be successful, the advantage given had to relate to the frequencies or number of students shown on the bar.”

Worked example 13 marks

24 students each choose a favourite subject. On a pie chart, the sector for chemistry is 120°.

(a) How many students chose chemistry?

(b) 7 students chose physics. Find the angle of the physics sector.

  1. 1

    (a) one student = 360 ÷ 24 = 15° [1]

  2. 2

    chemistry = 120 ÷ 15 = 8 students [1]

  3. 3

    (b) physics = 7 × 15 = 105° [1]

4. Comparing sets of data

To compare two data sets, compare an average and a measure of spread. The average (mean, median or mode) says which set is larger on the whole. The range says which is more spread out.

Extended: another measure of spread is the interquartile range. Compare the same measure for both sets, and say which is greater than or smaller than the other.

Examiner insight

“The comparison required more than just statements of values and the use of ‘greater than’ or ‘smaller than’ were the most appropriate phrases to use.” “Inclusion of an irrelevant statement even if accompanied by a correct statement is not acceptable.”

Worked example 14 marks

Team A scores 4, 6, 7, 7 and 11 in five games. Team B scores 5, 6, 7, 8 and 9. Compare the scores of the two teams using the mean and the range.

  1. 1

    mean of A = 35 ÷ 5 = 7 and mean of B = 35 ÷ 5 = 7 [1]

  2. 2

    range of A = 11 - 4 = 7 and range of B = 9 - 5 = 4 [1]

  3. 3

    The means are equal. [1]

  4. 4

    The range of A is greater than the range of B, so B's scores are more consistent. [1]

5. Limits on conclusions

A conclusion must come from the data given. Check who or what the data covers, and how much of it there is.

Data about one group says nothing about other groups. A pattern in a few values does not prove the pattern carries on beyond them.

Worked example 12 marks

Flo asks 25 people who own a bike what type they own. 12 own a road bike. She says: ‘Most people in the country own a road bike.’ Give two reasons why this conclusion is not justified.

  1. 1

    12 out of 25 is less than half, so it is not most of the people she asked. [1]

  2. 2

    She asked only 25 people, all bike owners, so the data says nothing about the rest of the country. [1]

Recap

  • Read the title, axes, scale and key first.
  • Bar height = frequency. Difference = subtract two heights.
  • Stem-and-leaf: number of leaves = number of values. Read the key.
  • Pie chart: one item = 360 ÷ total degrees.
  • Compare an average and a spread, using greater than or smaller than.
  • A conclusion covers only the data given.
  • Extended: the interquartile range is a measure of spread.

End-of-chapter exercise

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

  1. Do not use a calculator. This stem-and-leaf diagram gives the masses of 9 parcels in kg. key | 2 3 represents 23 kg (a) How many parcels weigh more than 25 kg? (b) Write down the largest mass. | stem | leaf | | --- | --- | | 1 | 2 5 8 | | 2 | 0 3 3 9 | | 3 | 4 6 |2 marks
  2. Do not use a calculator. Some people each choose one favourite fruit. A bar chart of the results has bars of height 9 (apples), 4 (pears), 6 (plums) and 11 (figs). (a) How many more people chose figs than pears? (b) How many people chose a fruit altogether?2 marks
  3. Do not use a calculator. 36 pupils say how they travel to school. On a pie chart, the sector for walking is 90°. (a) How many pupils walk? (b) 15 pupils take the bus. Find the angle of the bus sector.3 marks
  4. Do not use a calculator. Class A scores 12, 15, 15, 18 and 20 in a test. Class B scores 10, 14, 16, 19 and 21. Compare the two classes using the mean and the range.4 marks
  5. Eva scores 9, 13, 14, 15 and 19. Max scores 10, 11, 13, 16 and 20. Eva says: ‘My mean is 14 and Max's median is 13, so I did better.’ Explain her mistake and compare the two using the same average.3 marks
  6. Do not use a calculator. Hana asks 40 pupils in Year 7 if they like the new canteen. 30 say yes. She says: ‘Three quarters of the pupils in the school like it.’ (a) What fraction of the pupils she asked said yes? (b) Give a reason why her conclusion is not justified.2 marks
  7. A shop records the number of ice creams it sells. Ali says: ‘Sales go up every month, so the shop will sell 500 in December.’ (a) Find the increase in sales from June to July. (b) Give a reason why Ali's conclusion is not justified. | Month | Ice creams sold | | --- | --- | | June | 120 | | July | 150 |2 marks
  8. Do not use a calculator. Set A is 2, 4, 6, 8, 10, 12, 14. Set B is 5, 6, 7, 8, 9, 10, 11. Compare the two sets using the median and the interquartile range.4 marks
  9. Some pupils each choose one favourite sport. The same results are drawn as a bar chart and as a pie chart. On the bar chart, the bar for tennis has height 9 and the bar for football has height 12. On the pie chart, the sector for tennis is 108°. The rest of the pupils chose hockey or swimming, and twice as many chose hockey as chose swimming. (a) Find the total number of pupils. (b) Find the angle of the hockey sector.5 marks
  10. A café times 7 deliveries from each of two firms. The times, in minutes, are listed in order. The mean time for firm B is 1 minute less than the mean time for firm A. (a) Find the value of k. (b) The owner says: ‘Firm B's times are more consistent.’ Use the interquartile range to support this. (c) The owner says: ‘Firm B is faster on average, so it will always be faster.’ Use the medians to give a reason why this conclusion is not justified. | Firm | Delivery times (minutes) | | --- | --- | | A | 18, 21, 22, 24, 27, 30, 40 | | B | 20, 23, 24, 25, 26, 28, k |7 marks

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