Maths · Statistics
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Teacher view: every answer and mark scheme set out in full.
Scatter Graphs
Describing correlation, using lines of best fit to estimate, and judging how reliable an estimate is.
Learning Objectives
- 1Plot a scatter graph and describe the correlation between two variables.
- 2Draw a line of best fit and use it to estimate values.
- 3Describe the relationship in words, interpret the gradient and spot outliers.
- 4Explain the difference between correlation and causation, and between interpolation and extrapolation.
Looking for a link
A scatter graph plots two measurements for each person or object, such as hours of revision and test score, to see whether there is a connection. The pattern of the points is called the correlation, and a line of best fit lets you make predictions. Scatter graph questions on Paper 1 are mostly reading and describing, because drawing a line by eye is awkward without more equipment, so you must be able to use a given line carefully, say how reliable an estimate is, and avoid claiming that one thing causes another.
Types of correlation
The shape of the points tells you the type of relationship.
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Positive correlation
As one variable increases, the other increases. The points slope up from left to right.
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Negative correlation
As one variable increases, the other decreases. The points slope down.
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No correlation
The points are scattered with no pattern.
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Strength
The closer the points are to a straight line, the stronger the correlation.
A scatter graph with a line of best fit
The points show positive correlation: more revision usually means a higher score. The line of best fit goes through the middle of the points, with about as many above it as below it, and it does not have to go through the origin.
Reading the graph
- Describe the correlation "As hours of revision increase, the test score increases" is a full answer.
- Outlier The point at about 9 hours and a score of 78 does not fit the pattern, and may be an error or an unusual case.
- Gradient The line rises from 8 to 80 over 18 hours, a gradient of \(\dfrac{72}{18} = 4\), so each extra hour adds about 4 marks.
- Intercept The line starts at about 8 marks, the estimated score with no revision.
Drawing and using a line of best fit
A line of best fit is a straight line that follows the trend.
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Drawing it
Use a ruler. It should pass close to as many points as possible and have roughly equal numbers of points on each side. Ignore any outliers.
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Estimating
To estimate \(y\) for a given \(x\), go up from \(x\) to the line, then across to the \(y\)-axis. Draw lines on the graph to show it.
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Using the equation
The line \(y = 4x + 8\) gives a score of \(4 \times 12 + 8 = 56\) for 12 hours of revision.
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Round sensibly
Estimates are not exact, so accept a range, and say "about".
Estimating from a line of best fit
The line of best fit on a scatter graph passes through \((0, 8)\) and \((18, 80)\). Estimate the test score of a student who revises for 12 hours.
Show the solutionHide the solution
- 1 Find the gradient \(\dfrac{80 - 8}{18 - 0} = 4\).
- 2 Write the equation \(y = 4x + 8\).
- 3 Substitute \(y = 4 \times 12 + 8 = 56\).
- 4 Check on the graph Going up from 12 hours to the line and across gives about 56.
AnswerAbout 56
How reliable is an estimate?
Examiners ask whether an estimate is reliable and why.
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Interpolation
An estimate inside the range of the data, such as 12 hours when the data runs from 2 to 16 hours. This is reasonably reliable.
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Extrapolation
An estimate outside the range, such as 25 hours of revision. It is not reliable, because the pattern may not continue. The line would give 108, which is above the maximum possible score of 100.
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Strong correlation
The closer the points are to the line, the more reliable the estimates.
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Small samples
Few points mean the line is less trustworthy.
Correlation and causation
Two things moving together does not mean that one causes the other.
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Correlation
A link between two variables.
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Causation
One variable making the other change.
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Third factor
Ice cream sales and sunburn are correlated, but hot weather causes both.
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Say it carefully
"There is a positive correlation" is safe. "More revision causes higher scores" needs more evidence.
Test yourself
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1
What type of correlation has points sloping downwards from left to right?
Show answerHide answer
Negative.
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2
What is an outlier?
Show answerHide answer
A point that does not fit the pattern of the others.
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3
What does interpolation mean?
Show answerHide answer
Estimating within the range of the data.
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4
Why is extrapolation unreliable?
Show answerHide answer
The pattern may not continue outside the range of the data.
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5
Does correlation prove causation?
Show answerHide answer
No, there may be another factor, or it may be a coincidence.
Exam technique: scatter graphs
Be clear and use the context.
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Use the variable names
"As temperature increases, ice creams sold increase" is better than "positive correlation" alone.
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Draw on the graph
Show construction lines for any reading.
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Comment on reliability
Say whether the estimate is inside or outside the range.
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Use a ruler
A line of best fit should be straight and pass through the middle of the points.
Summary and exam focus
- Correlation can be positive, negative or none, and strong or weak.
- A line of best fit passes through the middle of the points and is used to estimate values.
- Interpolation is reliable, and extrapolation is not.
- Correlation does not prove causation.
Exam focus
The scatter graph shows the age of a car and its value, and has negative correlation. Describe the relationship between the age of a car and its value. (1 mark) (1 marks)
Write "As the age of the car increases, its value decreases." Using the two quantities in the sentence is what earns the mark, and "negative correlation" on its own may not be enough.
Key terms
The words this lesson expects you to use. Each one is linked from the first place it appears above.
- Scatter graph
- A graph of plotted points showing how two variables are related.
- Correlation
- The relationship between two variables.
- Positive correlation
- Both variables increase together.
- Negative correlation
- One variable increases as the other decreases.
- Line of best fit
- A straight line drawn through the middle of the points on a scatter graph.
- Outlier
- A point that does not fit the pattern.
- Interpolation
- Estimating a value within the range of the data.
- Extrapolation
- Estimating a value outside the range of the data.
- Causation
- When one thing makes another happen.
Questions and answers
15 questions set on this lesson, with the mark schemes and model answers open.
Write down the type of correlation between (a) the age of a car and its value, (b) the heights of children and their ages. [2 marks]
Mark scheme — 2 marks available
- (a) Negative — B1
- (b) Positive — B1
Model answer
(a) Negative correlation. (b) Positive correlation.
The scatter graph shows the age and the value of some cars. A line of best fit is drawn. (a) Describe the relationship between the age of a car and its value. [1 mark] (b) Use the line of best fit to estimate the value of a car that is 4 years old. [2 marks] (c) Dev says that the line can be used to estimate the value of a car that is 12 years old. Explain why this is not sensible. [2 marks]
Mark scheme — 5 marks available
- (a) As the age increases, the value decreases — B1
- (b) A line drawn up from 4 to the line of best fit and across — M1
- (b) \(\pounds 7000\) (accept \(\pounds 6800\) to \(\pounds 7200\)) — A1
- (c) 12 years is outside the range of the data — B1
- (c) The line would give a negative value — B1
Model answer
(a) As the age of a car increases, its value decreases. (b) Going up from 4 years to the line and across gives about 7, which is \(\pounds 7000\). (c) 12 years is outside the range of the data. The line would give a negative value, \(11 - 12 = -1\), which is impossible.
A line of best fit for the hours of practice, \(x\), and the number of goals scored, \(y\), has the equation \(y = 0.5x + 4\). (a) Use the equation to estimate the number of goals for 10 hours of practice. [1 mark] (b) What does the number 0.5 represent? [1 mark] (c) Explain why the equation should not be used for 100 hours of practice. [1 mark]
Mark scheme — 3 marks available
- (a) 9 — B1
- (b) 0.5 more goals for each extra hour — B1
- (c) Outside the range of the data, so not reliable — B1
Model answer
(a) \(0.5 \times 10 + 4 = 9\). (b) For each extra hour of practice there are 0.5 more goals. (c) 100 hours is far outside the range of the data, and the pattern may not continue.
There is a positive correlation between the shoe size and the reading ability of primary school children. Does having bigger feet make a child a better reader? Give a reason for your answer. [2 marks]
Mark scheme — 2 marks available
- No — B1
- A reason such as age affecting both — B1
Model answer
No. Both shoe size and reading ability increase as a child gets older, so age explains both. Correlation does not mean that one thing causes the other.
A line of best fit on a scatter graph goes through \((0, 8)\) and \((10, 0)\). (a) Write down the type of correlation. [1 mark] (b) Use the line to estimate \(y\) when \(x = 5\). [2 marks]
Mark scheme — 3 marks available
- (a) Negative — B1
- (b) Uses the gradient \(-0.8\) or the midpoint of the line — M1
- (b) 4 — A1
Model answer
(a) Negative correlation, because the line slopes downwards. (b) The line is halfway between \(x = 0\) and \(x = 10\) at \(x = 5\), so \(y\) is halfway between 8 and 0, which is 4.
A line of best fit goes through the points \((4, 5)\) and \((12, 25)\). (a) Work out the gradient of the line. [2 marks] (b) Use the line to estimate \(y\) when \(x = 8\). [2 marks]
Mark scheme — 4 marks available
- (a) \(\dfrac{20}{8}\) — M1
- (a) 2.5 — A1
- (b) \(y = 2.5x - 5\) or the midpoint — M1
- (b) 15 — A1
Model answer
(a) \(\dfrac{25 - 5}{12 - 4} = \dfrac{20}{8} = 2.5\). (b) \(y = 2.5x - 5\), so when \(x = 8\), \(y = 2.5 \times 8 - 5 = 15\). Alternatively, \(x = 8\) is halfway between 4 and 12, so \(y\) is halfway between 5 and 25.
What type of correlation has points sloping downwards from left to right?
Why: One variable falls as the other rises.
Which statement describes positive correlation?
Why: Positive correlation slopes upwards.
What is an outlier?
Why: Outliers are far from the trend.
A line of best fit has equation \(y = 4x + 8\). Estimate \(y\) when \(x = 12\).
Why: \(4 \times 12 + 8 = 56\).
Data runs from 2 to 16 hours of revision. What do we call an estimate for 25 hours?
Why: Outside the range of the data is extrapolation.
Which estimate is likely to be more reliable?
Why: Interpolation is more reliable than extrapolation.
Ice cream sales and sunburn are positively correlated. What is the best explanation?
Why: A third factor can cause both.
The line of best fit is \(y = 4x + 8\), where \(x\) is hours of revision and \(y\) is the test score. What does the gradient mean?
Why: The gradient is the change in score for each extra hour.
In \(y = 4x + 8\), what does 8 represent?
Why: It is the \(y\)-intercept, the value when \(x = 0\).