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GCSEMathsAQARevision Notes6. StatisticsStatistical DiagramsWorking with Statistical Diagrams

Working with Statistical Diagrams (AQA GCSE Maths)

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Paul

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Paul

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Reading & Interpreting Statistical Diagrams

How do I read and interpret statistical diagrams?

  • Rather than present you with a list of values (raw data), questions may present information using a statistical diagram
    • Such diagrams may or may not be the ones that are already familiar
    • Unfamiliar diagrams will be reasonably straightforward to read and interpret
      • you won't need any new skills for such diagrams
  • Reading and interpreting statistical diagrams requires gathering any required information from a diagram
    • this enables meaningful statistics like the mean, median, mode, range and interquartile range to be calculated
    • from these, conclusions about the data can be made
  • Important things to look for in diagrams include
    • a key, and/or shading, that indicate what certain parts of the diagram mean
      • e.g. a dual bar-chart may show year 7 data in solid shading and year 8 data in striped shading
    • information given through the labels on the axes
    • key words on diagrams such as frequency
    • anything unusual or unexpected mentioned in words, whether they come before or after a diagram
    • anomalies (outliers)
  • You may also be asked to comment on a given statistical diagram
    • this may be to point out something that could be misleading or incorrect
      • e.g.  uneven gaps in axes values
      • e.g.  a missing key

Worked example

The dual line chart below shows the number of books loaned from a library by male and female adults each day for a week.

cie-igcse-4-5-3-dual-bar-line-chart

a)

Work out the mean number of books loaned per day by males during this particular week.

We need the number of books from the males (red line) for each of the five days and then find their mean.

Mean space equals space fraction numerator 10 plus 12 plus 12 plus 15 plus 15 over denominator 5 end fraction equals 64 over 5 equals 12.8

Mean number of books loaned per day by males is 12.8

b)

Work out the median number of books loaned by women per day by females during this particular week.

We need the number of books from the females (blue line) for each of the five days.

12, 15, 12, 16, 12

To find the median, these will need to be put into ascending order, then the middle value found.

12, 12, circle enclose 12, 15, 16

The median number of books loaned per day by females is 12

c)

Determine whether the males or females had the greater range of books loaned per day during this particular week.

Using the lists/values from the parts a) and b) ...

table row cell Male space range space end cell equals cell space 15 space minus space 10 space equals space 5 end cell row cell Female space range end cell equals cell 16 minus 12 equals 4 end cell end table

The males had the greater range of books loaned per day
(5 compared to 4 so their range was 1 book higher)

Comparing Statistical Diagrams

What is meant by comparing statistical diagrams?

  • Some questions may present you with data as a diagram rather than as a list of values
  • You may then be asked to compare one diagram with another that represent different characteristics
      • e.g.  one diagram/table may be for 'dogs' data, the other for 'cats'
        • so dogs and cats data/results can be compared;
      • e.g.  one diagram/table may be taken at one point in time, the other at a later date
        • this would allow comparisons that may reveal a change (improvement) over time
  • When data is presented as a diagram, it may be an unfamiliar diagram that you have not seen elsewhere on the course
    • in such cases the diagram will be explained in the question or through a key
    • such diagrams tend to be straightforward and fairly easy to interpret and read

How do I compare statistical diagrams?

  • By commenting on differences or similarities using some of the following
    • averages - mean, median and mode
    • spread - range and interquartile range
    • unusual data values (anomalies or outliers)
  • You should aim to make at least two pairs of comments when asked to compare data
    • The first pair of comments should use an average - mean or median, rather than mode
      • a comparison of an average mentioning the numbers involved
        e.g.  class A's median of 11 was higher than class B's median of 6
      • what that comparison means in the context of the question
        e.g.  on average class A scored higher marks on the test than class B
    • The second pair of comments should use a measure of spread - range or interquartile range
      • a comparison of a measure of spread mentioning the numbers involved
        e.g.  class A's interquartile range of 5 was higher than class B's interquartile range of 3
      • what that comparison means in the context of the question
        e.g.  the test scores in class A showed more variation than the scores in class B
  • The mode can be mentioned if appropriate - for example with non-numerical data
    • the mode is a relatively simple average so there are not always many marks available for using it
  • Before doing any comparisons of data or diagrams you may have to calculate averages and spread
  • You may also be asked to suggest assumptions, or problems with the data that could affect the reliability of results and comparisons
    • e.g.  do we assume that the test class A and class B took were the same?
    • e.g.  were class A and class B of similar ability/age?
    • It may be we cannot tell from the information given in the question
      • but they are considerations that would influence how valid comparisons are

Examiner Tip

  • When asked to compare data or diagrams consider how many marks are available
    • aim to write something different for each mark
      • 1 mark is often be for comparing the numbers involved
      • 1 mark is often for explaining what that then means in the context of the question

Worked example

The diagram below shows the temperatures, measured in °C, for each day of one week in the summer in the UK.

cie-igcse-4-5-3-time-series-graph

a)

Find

i)

the modal temperature,

ii)

the mean temperature for the week,

iii)

the range of temperatures for the week.

i)

The modal (mode) temperature is the temperate that occurred during the week more than any other. Look for the temperature that occurred on the most number of days. Notice this is not the same as the highest temperature.

The modal temperature is 9 °C

ii)

The mean needs calculating.

table row cell Mean space end cell equals cell space fraction numerator 12 plus 10 plus 9 plus 9 plus 11 plus 10 plus 9 over denominator 7 end fraction end cell row blank equals cell 70 over 7 equals 10 end cell end table

Mean temperature for the week is 10 °C

iii)

The range also needs working out.

table row cell Range space end cell equals cell space Hi space minus space Lo end cell row blank equals cell 12 space minus space 9 end cell row blank equals 3 end table

Range of temperature for the week is 3 °C

b)

For a different week in the UK, the following data is available.

Mean temperature (°C) Range of temperatures (°C)
11 2.5

Compare the temperatures for the two different weeks.

Two pairs of comments - first pair comparing the means, second pair comparing the range.

The mean in the second week (11 °C) was 1 °C higher than the mean in the first week (10 °C)
On average, the temperature in the second week was higher than in the first week

The range in the first week (3 °C) was 0.5 °C higher the range in the second week (2.5 °C)
The temperatures were more spread out in the first week

c)

State one assumption that should be made about the two weeks' temperatures so that comparisons are valid.

Consider the information - or lack of - given in the question.  We are told when the first weeks' temperatures were collected but are not told anything in this respect for the second week.

The two weeks' temperatures were recorded at a similar time of year
e.g.  both in the spring, or both in the same month

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.