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Cambridge IGCSE0607

Interpreting statistical data

International Mathematics 0607 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 charts
  2. 2Drawing inferences from diagrams
  3. 3Comparing two data sets
  4. 4Restrictions on conclusions
  5. 5Comparing medians and interquartile ranges

1. Reading tables and charts

Before you read a value from a graph, read the scale on each axis.

In a pictogram, each symbol stands for the number given in the key. A part symbol stands for a part of that number.

In a pie chart, the whole circle is 360°. Each sector's angle is its share of the total.

number in a sector = angle/360 × total

Examiner insight

“Some candidates appeared not to fully understand the key.” “Some candidates gave 6 (from 360 ÷ 60) or 40 (from 360 – 120 = 240, 240 ÷ 60 = 40, the number of girls).”

Worked example 12 marks

Do not use a calculator.

72 students are asked how they travel to school. In a pie chart of the results, the sector for bus has an angle of 150°. Find the number of students who travel by bus.

  1. 1

    number = 150/360 × 72 [1]

  2. 2

    = 5/12 × 72

  3. 3

    = 30 [1]

2. Drawing inferences from diagrams

On a scatter diagram, points that rise from left to right show positive correlation. Points that fall show negative correlation. No pattern shows no correlation.

A line of best fit passes through the mean point. Use it to estimate one value from the other, but only within the values plotted.

On a cumulative frequency diagram, each point is plotted at the upper end of its class. The number of values above a given value is found from the total.

number above x = total - reading at x

Examiner insight

“Some used words to describe the correlation, for example, decreasing or strong.” “A number of candidates omitted the subtraction from 120.”

Worked example 12 marks

The table gives the cumulative frequencies for the heights, h cm, of 60 plants. Find the number of plants taller than 30 cm.

height h cm≤ 10≤ 20≤ 30≤ 40≤ 50
cumulative frequency518445860
  1. 1

    plants of 30 cm or less = 44 [1]

  2. 2

    taller than 30 cm = 60 - 44 = 16 [1]

3. Comparing two data sets

To compare two data sets, read each one from its table or graph. Then compare an average and the range.

The average tells you which set is higher or lower. A smaller range means the values are more consistent.

Write each comparison as a sentence about the context. Give two comparisons when asked for two. Do not just repeat the numbers.

range = highest value - lowest value

mean = (sum of the values)/(number of values)

Examiner insight

“A few candidates only gave one comparison.” “The candidates appear to have trouble expressing themselves and giving reasons.”

Worked example 12 marks

The table gives the mean and the range of the times, in seconds, for two groups to run 100 m. Make two comparisons between the times of the groups.

meanrange
Group A14.63.1
Group B13.95.8
  1. 1

    Group B's average time is lower, as 13.9 < 14.6. [1]

  2. 2

    Group A's times are more consistent, as 3.1 < 5.8. [1]

4. Restrictions on conclusions

A conclusion can only be as exact as the data. A table of grouped data gives the class of each value, not the value, so you can only give an estimate of the mean.

When the first class is written like 10 < m ≤ 20, its lower boundary is not included, so the lowest value is greater than it. The highest value is at most the upper boundary of the last class. The range is therefore less than the difference between these two boundaries.

Do not use a line of best fit outside the values plotted.

Examiner insight

“The most common mistake was to try to explain why the range could not be calculated without the actual values.” “Some candidates gave answers that were much higher than 20, the largest value for t.”

Worked example 13 marks

The table groups the masses, m kg, of 50 parcels. Kai says the range of the masses is 50 kg. Explain why Kai cannot be correct.

mass m kg10 < m ≤ 2020 < m ≤ 4040 < m ≤ 60
frequency122711
  1. 1

    The lowest mass is greater than 10 kg. [1]

  2. 2

    The highest mass is at most 60 kg. [1]

  3. 3

    The range is less than 60 - 10 = 50 kg, so Kai cannot be correct. [1]

5. Comparing medians and interquartile ranges

The median is an average and the interquartile range is a measure of spread. Use both to compare two sets.

A smaller interquartile range means the middle half of the values is more consistent.

interquartile range = upper quartile - lower quartile

Examiner insight

“Many candidates managed to find the median correctly from the numbers given, but few could find the inter-quartile range correctly.” “However, very few candidates could find another correct comparison.”

Worked example 13 marks

The heights of tomato plants in greenhouse P have median 31 cm, lower quartile 27 cm and upper quartile 36 cm. In greenhouse Q the median is 26 cm and the interquartile range is 14 cm.

(a) Work out the interquartile range for greenhouse P.

(b) Make one comparison using the medians and one using the interquartile ranges.

  1. 1

    (a) 36 - 27 = 9 cm [1]

  2. 2

    (b) Greenhouse P has the greater median height, as 31 > 26. [1]

  3. 3

    Greenhouse P's heights are more consistent, as 9 < 14. [1]

Recap

  • Read the scale of a graph and the key of a pictogram first.
  • Pie chart: number = angle/360 × total.
  • Scatter diagram: positive, negative or no correlation. A line of best fit passes through the mean point.
  • Cumulative frequency: number above x = total minus the reading at x.
  • Compare two sets with an average and the range. Smaller range = more consistent.
  • Grouped data gives an estimate, not the exact value. Stay within the values plotted.
  • Extended: compare medians and interquartile ranges. Interquartile range = upper quartile − lower quartile.

End-of-chapter exercise

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

  1. Do not use a calculator. In a pictogram, one symbol stands for 6 books. Monday has 3 whole symbols and half a symbol. How many books is that?2 marks
  2. Do not use a calculator. In a pie chart for 80 students, the sector for swimming has an angle of 45°. How many chose swimming?2 marks
  3. Do not use a calculator. Of 50 students, 8 have a height of at most 140 cm, 30 of at most 150 cm and 46 of at most 160 cm. Find the number taller than 140 cm and at most 160 cm.2 marks
  4. On a scatter diagram of the age and value of 12 cars, the points lie close to a line with a negative gradient. The oldest car is 10 years old. (a) State the type of correlation. (b) Georgia uses the line of best fit to estimate the value of a 15-year-old car. Why should she not rely on this estimate?2 marks
  5. Class A's puzzle times have mean 14.2 minutes and range 6 minutes. Class B's have mean 11.8 minutes and range 15 minutes. Make two comparisons between the classes' times.2 marks
  6. Do not use a calculator. Five marks in each of two tests are P: 9, 4, 7, 9, 6 and Q: 14, 3, 10, 5, 8. Work out the mean and range of each. Which test is more consistent?4 marks
  7. The times, t minutes, of 40 journeys are grouped: 6 with 0 < t ≤ 10, 20 with 10 < t ≤ 20 and 14 with 20 < t ≤ 30. Can the exact mean time be found from this? Give a reason.2 marks
  8. Do not use a calculator. Team A's scores have median 52 and interquartile range 8. Team B's have median 47 and interquartile range 15. Make two comparisons between the teams' scores.2 marks
  9. The lifetimes, in hours, of 11 brand A batteries are 42, 35, 51, 38, 47, 33, 44, 40, 55, 37, 46. The table gives readings from the cumulative frequency curve for 40 brand B batteries. (a) Find the median and the interquartile range for each brand. (b) Make two comparisons between the lifetimes of the two brands. | lifetime t hours | ≤ 30 | ≤ 36 | ≤ 41 | ≤ 50 | ≤ 60 | | --- | --- | --- | --- | --- | --- | | cumulative frequency | 4 | 10 | 20 | 30 | 40 |7 marks
  10. The table gives cumulative frequencies for the delivery times, t minutes, of 80 deliveries by each of two firms. (a) Find the median and the interquartile range for each firm. (b) A delivery of more than 60 minutes is late. Find the percentage of each firm's deliveries that are late. (c) Firm Q's manager says, ‘Our median is the same as firm P's, so the firms are equally good.’ Give two reasons why this conclusion is not justified. | time t min | ≤ 20 | ≤ 30 | ≤ 40 | ≤ 50 | ≤ 60 | ≤ 90 | | --- | --- | --- | --- | --- | --- | --- | | firm P | 6 | 20 | 40 | 60 | 74 | 80 | | firm Q | 20 | 32 | 40 | 48 | 60 | 80 |8 marks

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