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Cambridge IGCSE0580

Scatter diagrams

Mathematics 0580 Chapter Notes

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Scatter diagrams
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Scatter diagrams notes

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In this chapter

  1. 1Drawing a scatter diagram
  2. 2Positive, negative and zero correlation
  3. 3Drawing a line of best fit
  4. 4Using the line to estimate

1. Drawing a scatter diagram

A scatter diagram plots pairs of values, one point for each pair. Each axis is for one of the two variables.

Mark each point with a small cross (×). Check the scale of each axis before you plot, then plot the point where the two readings meet.

Examiner insight

“Many candidates were able to correctly plot the points. Errors were largely due to incorrect use of the scale.” “A few candidates plotted the point (85, 41) at (80, 41).”

Worked example 11 mark

The scatter diagram is of the number of hot drinks sold by a café each day against the average temperature for that day.

These are the results for two more days. Plot this information on the scatter diagram.

temperature (°C)1828
number of hot drinks sold4021
  1. 1

    Each pair is (temperature, hot drinks): the temperature is the horizontal value.

  2. 2

    Mark (18, 40) and (28, 21) with small crosses. Check the scale on each axis first. [1]

2. Positive, negative and zero correlation

Correlation describes the link between the two variables. When one increases as the other increases, the correlation is positive. When one increases as the other decreases, it is negative.

When the points show no pattern, there is zero correlation. Give the type of correlation only. Do not describe the trend with words such as ‘increasing’.

Examiner insight

“The most common incorrect response was to indicate direct correlation or directly proportional.” “Incorrect answers included negative and no correlation or described the trend of the points as increasing.”

Worked example 12 marks

The table gives the age of six cars and the value of each car. A scatter diagram of these values is drawn. What type of correlation does it have?

age (years)124578
value (£)900082006500560041003300
  1. 1

    As the age increases, the value decreases. [1]

  2. 2

    The correlation is negative. [1]

3. Drawing a line of best fit

A line of best fit is a single ruled line drawn by inspection. It follows the trend of the points.

The line should extend across the full data set. It does not need to pass through any point, but there should be a roughly even distribution of points either side of it along its whole length.

Examiner insight

“The majority of candidates were able to correctly draw the line of best fit although a significant number of candidates incorrectly thought that it had to pass through the origin and a small number were seen that were not a single straight ruled line.” “Some just joined the points in order, in a zig-zag fashion.”

Worked example 12 marks

The table shows the time 9 students each spent studying for a test and the number of errors each student made in the test.

Draw a line of best fit on the scatter diagram.

Another student studies for 13 minutes. Use your line of best fit to estimate the number of errors this student is expected to make in the test.

study time (minutes)5811151821252731
number of errors3534302524201396
  1. 1

    The points fall from top left to bottom right. Rule one straight line from about (5, 37) to (31, 6), with points on both sides. [1]

  2. 2

    Read up from 13 minutes to the line, then across: about 27 errors. Any reading from their own line is accepted. [1]

4. Using the line to estimate

To estimate a value, read from the line of best fit. Go up (or across) from the given value to the line, then across (or down) to the other axis.

Read the scale on the axis you are reading from. An estimate is not a guess: it is the value read from your line.

Examiner insight

“Again, ‘estimate’ caused a problem in part (d). This does not mean guess, it means ‘use a mathematical method to find out’ and in this case the method is reading from the line of best fit when the time is 5.5 hours.” “Others who had a ruled line with a positive gradient were able to score by giving an accurate value from their line of best fit.”

Worked example 13 marks

The line of best fit for a scatter diagram of ice creams sold against temperature passes through (10, 30) and (30, 70). Estimate the number of ice creams sold when the temperature is 24 °C.

  1. 1

    Gradient = (70 - 30)/(30 - 10) = 2 [1]

  2. 2

    At 24 °C: 30 + 2 × (24 - 10) [1]

  3. 3

    = 58 ice creams [1]

Recap

  • A scatter diagram plots one point, marked ×, for each pair of values.
  • Positive correlation: both increase together. Negative: one increases as the other decreases.
  • Zero correlation: no pattern in the points.
  • Line of best fit: a single ruled line, drawn by inspection, across the full data set.
  • Roughly even numbers of points lie either side of the line.
  • The line need not pass through the origin or through any point.
  • To estimate, read a value from the line of best fit.

End-of-chapter exercise

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

  1. On a scatter diagram, one variable decreases as the other increases. State the type of correlation.1 mark
  2. Do not use a calculator. The heights of eight plants and the lengths of their roots are recorded. Taller plants have longer roots. State the type of correlation.1 mark
  3. Do not use a calculator. The horizontal axis of a scatter diagram is numbered 0, 10, 20, 30, 40, 50. Each small square on it is 2 units wide. A point has a horizontal value of 34. Starting from the line for 30, how many small squares to the right is the point?2 marks
  4. The points on a scatter diagram of shoe size against test score are spread at random with no pattern. State the type of correlation and what it tells you.2 marks
  5. Write down three features of a correct line of best fit.3 marks
  6. Do not use a calculator. A line of best fit passes through (10, 22) and (30, 62). Estimate the value of y when x = 25.3 marks
  7. Do not use a calculator. Ravi draws a line of best fit through a scatter diagram of 12 points. 9 points are above his line and 3 are below it. Is it a good line? Give a reason.2 marks
  8. Do not use a calculator. A line of best fit for the price y (£) of a used car against its age x (years) passes through (3, 800) and (7, 400). (a) Estimate the price of a car that is 5 years old. (b) Estimate the age of a car with a price of £500.4 marks
  9. For some delivery vans, the load x (kg) and the fuel used y (litres per 100 km) are plotted on a scatter diagram. The line of best fit passes through (200, 9.5) and (800, 12.5). (a) A van carries a load of 0.65 tonnes. Estimate the fuel it uses in litres per 100 km. (b) Another van carries 500 kg and uses 10.4 litres per 100 km. Is the point for this van above or below the line of best fit? Show your working.5 marks
  10. In ten experiments a scientist records the temperature x (°C) of some water and the time y (seconds) a tablet takes to dissolve in it. The temperatures are between 20 °C and 60 °C. A line of best fit for the scatter diagram passes through (20, 95) and (60, 35). (a) State the type of correlation. (b) Estimate the time taken to dissolve at 44 °C. (c) Estimate the temperature at which the tablet dissolves in 1 minute 5 seconds. (d) Kai draws his line from (0, 0) to (60, 35). Give one reason why Kai’s line is not a line of best fit.6 marks

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