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S6Scatter graphs and correlation

Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Statistics

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Revision notes

Scatter graphs show whether two variables are related. You need to plot points, describe the correlation, draw a line of best fit, use it to make estimates, spot outliers and explain the limits of your conclusions. These questions come up on both tiers, usually for 1 to 4 marks.

Grade by grade

What you need to be able to do, from the first marks up to the top grade.

  1. 2
    Plot points on a scatter graphPlot each pair of values as a cross, reading both scales carefully.
  2. 3
    Name the type of correlationPositive, negative or no correlation, and whether it is strong or weak.
  3. 3
    Describe the relationship in contextSay how one variable changes as the other increases, using the names of the variables.
  4. 4
    Draw a line of best fitDraw one straight ruled line that follows the trend, with the points spread evenly on either side.
  5. 4
    Identify and explain an outlierAn outlier is a point that does not fit the general pattern; leave it out when you draw the line.
  6. 5
    Make estimates using a line of best fitRead from the line inside the range of the data; estimates outside it (extrapolation) are unreliable.
  7. 5
    Explain why correlation does not prove causationTwo variables can be correlated because both depend on something else.

Notes

Scatter graphs and correlation

  • A scatter graph shows two sets of data about the same items (bivariate data), e.g. the age and the value of some cars. Each item is plotted as one point.
  • Positive correlation: as one variable increases, the other tends to increase.
  • Negative correlation: as one variable increases, the other tends to decrease.
  • No correlation: there is no clear pattern.
  • The closer the points are to a straight line, the stronger the correlation. Widely scattered points show weak correlation.

Lines of best fit

  • A line of best fit is a single straight line, drawn with a ruler, that follows the trend of the points.
  • Aim for about the same number of points on each side, spread along its length. It does not have to go through the origin or through any particular point.
  • Ignore any outlier when you draw it, and only draw one when there is correlation.

Making estimates

  • To estimate, go from the given value to the line of best fit, then across to the other axis. Use the line, not the nearest point.
  • Interpolation means estimating inside the range of the data. This is fairly reliable.
  • Extrapolation means estimating outside the range of the data. This is unreliable, because the pattern may not continue.

Outliers and causation

  • An outlier is a point that does not fit the pattern of the others. It may be an error or an unusual item.
  • Correlation does not prove that one variable causes the other. Both may depend on a third factor: ice cream sales and cases of sunburn both rise in hot, sunny weather.

Cheatsheet

  • Positive correlation: as one increases, the other increases
  • Negative correlation: as one increases, the other decreases
  • No correlation: no clear pattern
  • Line of best fit: one straight ruled line following the trend, ignoring outliers
  • Interpolation (inside the range of the data): fairly reliable
  • Extrapolation (outside the range of the data): unreliable
  • Outlier: a point that does not fit the general pattern
  • Correlation does not prove causation

How to answer each type of question

Describe the correlation and the relationship

1 to 2 marks3
  1. Look at the overall direction of the points from left to right.
  2. Name the correlation: positive, negative or none.
  3. For 'describe the relationship', use the names of the variables, e.g. 'the older the car, the lower its value'.

Example. The table shows the ages and the values of 8 cars.
Age (years): 1, 2, 3, 4, 5, 6, 7, 8
Value (£1000s): 18, 16, 15, 12, 11, 9, 8, 5
A scatter graph is drawn of these data.
(a) What type of correlation does it show?
(b) Describe the relationship between the age and the value of these cars.

Show the model answer
(a) Negative correlation (B1)
(b) The older the car, the lower its value (C1)

Draw a line of best fit and make an estimate

2 marks5
  1. Draw one straight line through the middle of the points, ignoring any outlier.
  2. Go up from the given value to your line, then across to the other axis.
  3. Read the value carefully and give the units, if there are any.

Example. The table shows the number of hours, x, that 8 students spent revising and their scores, y, in a test marked out of 80.
x: 1, 2, 3, 4, 5, 6, 7, 8
y: 28, 36, 38, 45, 48, 55, 58, 65
These data are plotted on a scatter graph.
Draw a line of best fit and use it to estimate the score of a student who revised for 2.5 hours.

Show the model answer
A single straight line of best fit drawn through the points (M1)
About 37 marks (A1, accept answers from about 34 to 40 that match your line)

Explain why an estimate may not be reliable

1 mark5
  1. Check whether the value is inside or outside the range of the data.
  2. If it is outside, say that it is outside the range of the data, so the trend may not continue.
  3. If you can, add a reason from the context.

Example. Sam uses the line of best fit for the revision data to estimate the score of a student who revised for 15 hours.
Explain why Sam's estimate may not be reliable.

Show the model answer
15 hours is outside the range of the data (1 to 8 hours), so the trend may not continue. For example, the line gives close to 100 marks, which is impossible in a test marked out of 80 (C1)

Explain whether one variable causes the other

1 mark5
  1. Say that correlation does not prove causation.
  2. Suggest a third factor that could affect both variables.

Example. At a beach, there is strong positive correlation between the number of ice creams sold each day and the number of people treated for sunburn each day.
Kai says, “Eating ice cream causes sunburn.”
Is Kai correct? Give a reason for your answer.

Show the model answer
No. Correlation does not prove causation: both numbers are likely to go up on hot, sunny days (C1)

Shortcuts and memory tricks

  • Positive slopes uphill from left to right; negative slopes downhill.
  • Inter = inside the data (safe); extra = beyond the data (risky).
  • Line check: about half the points should be above your line and half below, all the way along it.
  • When you read an estimate, draw the lines: up to the line of best fit, then across. It shows your method and stops misreadings.

Where marks are lost

  • Joining the points dot to dot, or drawing a line of best fit that is not straight.
  • Forcing the line of best fit through the origin.
  • Including the outlier when you draw the line of best fit.
  • Answering 'describe the relationship' with only 'positive'. Use the context.
  • Saying that correlation proves one variable causes the other.
  • Misreading the scale: check what each small square is worth on each axis.

Exam technique

  • Use a ruler, and draw the line of best fit across the whole spread of the points.
  • Answers read from a graph are accepted within a range, so draw carefully and show your reading lines.
  • For reliability, the key phrases are 'outside the range of the data' and 'the trend may not continue'.
  • For causation, write 'correlation does not mean causation' and suggest a third factor.

Sample questions

Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.

Question 1Easy3 marks
The scatter graph shows the number of hours that each of 10 students spent revising and the mark each student scored in a test.
[object Object]
(a) Write down the type of correlation shown.[1]
(b) Liam says, 'The graph proves that revising for longer causes a student to get a higher mark.'
Explain why Liam may be wrong.[1]
(c) Explain why the graph should not be used to estimate the mark of a student who revised for 30 hours.[1]
Show the answer and mark scheme
(a) Answer: Positive
  • B1 for positive

Worked solution: As the number of hours increases, the mark tends to increase: positive correlation.

(b) Answer: Correlation does not prove causation: another factor (such as interest in the subject) could affect both.
  • C1 for stating that correlation does not show that one variable causes the other

Worked solution: A scatter graph only shows that the two variables are related. It cannot show that one causes the other.

(c) Answer: 30 hours is far outside the range of the data (1 to 10 hours), so the trend may not continue; for example, the mark cannot go above 100.
  • C1 for explaining that 30 hours is outside the range of the data (extrapolation), so the pattern may not continue

Worked solution: The data only go up to 10 hours. Continuing the trend to 30 hours would give a mark over 100, which is impossible.

Question 2Medium6 marks
The table shows the ages and trunk circumferences of some oak trees in a park.
[object Object]
(a) On the grid, draw a scatter graph to show this information.[2]
[object Object]
(b) Describe the relationship between the age and the trunk circumference.[1]
(c) Draw a line of best fit on your scatter graph.
Use your line of best fit to estimate the trunk circumference when the age is 30 years.[2]
(d) Explain why it may not be reliable to use your line of best fit to estimate the trunk circumference when the age is 100 years.[1]
Show the answer and mark scheme
(a) Answer: Points plotted at (8, 40), (12, 44), (14, 48), (18, 60), (26, 76), (36, 96), (46, 116), (56, 124).
  • B2 for all 8 points plotted correctly, each within ½ small square (B1 for at least 6 points correct)

Worked solution: Plot each pair of values from the table as a cross: (8, 40), (12, 44), (14, 48), (18, 60), (26, 76), (36, 96), (46, 116), (56, 124).

(b) Answer: Positive correlation: as the age increases, the trunk circumference tends to increase.
  • B1 for positive correlation, or for a correct description in context (as the age increases, the trunk circumference increases)
(c) Answer: 80 cm (accept 68 to 92)
  • B1 for a single ruled straight line of best fit that follows the trend of the points, with points roughly evenly spread on each side and extending across the data
  • B1 for an answer in the range 68 to 92 cm (ft from their line of best fit, if it is acceptable)

Worked solution: Draw a ruled line through the middle of the points. Read up from 30 years to the line, then across: about 80 cm.

(d) Answer: 100 years is outside the range of the data, so the estimate would be an extrapolation: the trend may not continue.
  • C1 for a correct reason, e.g. 100 years is outside the range of the data collected (extrapolation), so the trend may not continue

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