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Either paper 2 marks
A shop recorded the highest temperature and the number of bottles of cold water sold on each of nine days.
(a)Describe the correlation shown by the scatter graph.
(b)Write down what this tells you about the number of bottles of cold water the shop sold.
Show answer
(a) Positive correlation
(b) The warmer the day, the more bottles of cold water the shop soldRead the graph from left to right and watch what the crosses do.
As the temperature rises the points rise as well.
So the graph shows positive correlation.
A correlation is only worth naming if you can say it back in the words of the question.
The warmer the day, the more bottles of cold water the shop sold.
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Either paper 4 marksA school records the time each of eight students spends gaming each week and their mark in a test, out of 50. The table shows the results.
The scatter graph shows six of the eight results.Gaming time (hours) 1.5 2.5 3.5 4 5.5 6.5 7.5 8 Test mark 21 39 26 28 31 14 24 13 (a)Complete the graph by adding the missing label on the vertical axis and plotting the two missing points.
(b)Describe the correlation shown by the completed graph.
Write down the type of correlation, positive or negative.
(c)Write down the strength of the correlation, strong or weak.
You cannot draw on this page. Draw (a) on paper, then show the answer and check your drawing against the model.
Show answer
(a) Vertical axis labelled Test mark; crosses plotted at (3.5, 26) and (5.5, 31)
(b) Negative
(c) Weak(a)The horizontal axis is labelled Gaming time (hours), so the vertical axis is the other row of the table: Test mark.
The two results not yet plotted are the ones at \(3.5\) and \(5.5\) on the horizontal axis.
Their heights, from the second row of the table, are \(26\) and \(31\). On the vertical scale \(26\) is between \(25\) and \(30\), and \(31\) is between \(30\) and \(35\), so each cross goes between two horizontal gridlines.
Plot crosses at \((3.5, 26)\) and \((5.5, 31)\).
With all eight crosses plotted, the crosses fall as you move from left to right.
The type of correlation is negative.
The crosses are spread widely around any straight line.
The strength of the correlation is weak.
- (a) B1vertical axis labelled Test mark
- (a) B1both crosses plotted at (3.5, 26) and (5.5, 31), each within half a square
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Either paper 2 marks
The scatter graph shows the height and the mass of each of the nine students in Zainab's tutor group.
A line of best fit has been drawn.
(a)Use the line of best fit to estimate the mass of a student of height 170 cm.
(b)Use the line of best fit to estimate the height of a student of mass 57 kg.
Show answer
(a) \(63\ \text{kg}\)
(b) \(158\ \text{cm}\)Go across to \(170\) cm on the height axis, straight up to the line of best fit, then straight across to the mass axis.
The line rises \(1\) kg for every \(2\) cm of height, so from \(164\) cm to \(170\) cm it rises \(63 - 60 = 3\) kg.
The estimate is about \(63\) kg. Anything within about a kilogramme of that is a fair reading.
This time start on the mass axis. Go up to \(57\) kg, straight across to the line of best fit, then straight down to the height axis.
The estimate is about \(158\) cm. Anything within about a centimetre or two of that is a fair reading.
The line of best fit is used both ways round: down from a height to a mass, and across from a mass to a height.
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Either paper 3 marks
Nine students each sat two maths papers. Each paper was marked out of \(60\).
The scatter graph shows their marks.
(a)One of the points is an outlier.
Write down the coordinates of that point.
(b)Write down what the outlier tells you about that student's marks.
(c)Ignore the outlier.
Describe the correlation shown by the other eight points.
Show answer
(a) \((42, 20)\)
(b) This student did far worse on Paper 2 than their Paper 1 mark suggests.
(c) Positive correlationAn outlier is the point that does not follow the pattern the rest of them make. Here eight of the crosses lie in a narrow band and one sits well below it.
Read across to the Paper 1 axis and down to the Paper 2 axis: \((42, 20)\).
This student did far worse on Paper 2 than their Paper 1 mark suggests: \(42\) on Paper 1 but \(20\) on Paper 2.
An outlier is not automatically a mistake. It may be a real result with a reason behind it — illness, a missed section, or a topic that suited that student.
Without the outlier the crosses rise steadily from left to right: students who scored well on Paper 1 also scored well on Paper 2.
That is positive correlation.
An outlier drags a line of best fit towards itself, which is why it is usually left out before the line is drawn.
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Either paper 2 marks
Here are four scatter graphs.
Graph A
Graph B
Graph C
Graph D
(a)For which graph is a straight line of best fit appropriate? Circle your answer.
A B C D
(b)Which graph has one outlier? Circle your answer.
A B C D
Show answer
(a) A
(b) DA straight line of best fit only suits crosses that lie close to a line.
Graph A: close to a straight line. Graph B: no pattern at all. Graph C: a U shape. Graph D: a curve.
The crosses in graph A follow a rising line very close to it, so the answer is graph A.
An outlier is a single cross that sits far away from an otherwise tidy pattern.
Only graph D has one: it is the curved graph, where one cross sits far from the curve the others follow.
The other graphs have either no stray cross or no pattern for a cross to stray from, so the answer is graph D.
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Either paper 4 marks
A cafe records its sales of cups of tea and its sales of cups of coffee each month.
The scatter graph shows the sales for 10 months.
(a)Write down the type of correlation shown by the graph.
(b)Draw a line of best fit on the graph. Plot two or more points of your line, then rule it.
(c)The manager expects sales of cups of tea next month to be £600.
Use your line of best fit to estimate the sales of cups of coffee next month.
You cannot draw on this page. Draw (b) on paper, then show the answer and check your drawing against the model.
Show answer
(a) Negative correlation
(b) a straight line close to the mean point \((500, 473)\), for example from \((370, 538)\) to \((630, 408)\)
(c) \(\text{£}423\)(b)The crosses run from top left to bottom right, so as sales of cups of tea rise, sales of cups of coffee tend to fall.
This is negative correlation.
The mean sales of cups of tea are about £500 and the mean sales of cups of coffee are about £473, so a good line passes close to \((500, 473)\).
Draw a ruled straight line through that point, along the direction of the crosses, so that about as many crosses are above it as below it. It should run from about £370 to about £630 on the horizontal axis.
For example, it passes through \((370, 538)\) and \((630, 408)\).
Start at £600 on the horizontal axis and go straight up to your line. Then go straight across to the vertical axis.
The line of best fit gives about £423. Any value from £403 to £443 is a fair reading from a line drawn close to the mean point.
- (b) B1a straight line sloping the same way as the crosses, within one square of the line of best fit at both ends of the data
- (b) B1the line runs through the middle of the crosses, with about as many above it as below it: within £20 of \((370, 538)\) and \((630, 408)\) at the ends of the data
- (c) B1a reading from £403 to £443, or a correct reading from their own line of best fit at £600
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Either paper 2 marksIsla records the age and the value of seven cars of the same make and model. The table shows the data.
Isla draws this scatter diagram and line of best fit for the data. Write down two mistakes she has made. Mistake 1 Mistake 2Age (years) 1 2 3 5 6 7 8 Value (£000) 20 20 20 16 14 12 8 Show answer
The point \((2, 20)\) from the table has not been plotted: there is no cross for it. The line of best fit is drawn too low, so it is not close to the crosses.Check each table value against the graph, reading across for the first number and up for the second.
Every other table value has a cross in the right place, but the point \((2, 20)\) from the table has not been plotted: there is no cross for it.
Now compare the line with the crosses: The line of best fit is drawn too low, so it is not close to the crosses.
Not mistakes: a line of best fit does not have to pass through every cross or start at \((0, 0)\), and the point \((5, 16)\) is plotted correctly.
- B1the point \((2, 20)\) from the table has not been plotted: there is no cross for it.
- B1The line of best fit is drawn too low, so it is not close to the crosses.
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Either paper 2 marks
The scatter graph shows information about ten students. It shows the number of hours of television each student watched in a week and the mark each student got in a test. The line of best fit has been drawn.
Ben says, “There is strong negative correlation.”
Is Ben correct? Give a reason for your answer.
Show answer
Yes. The crosses are close to the line and y goes down as x goes up, so there is strong negative correlation.Two things decide this claim: how close the crosses are to the line of best fit, and which way the crosses slope. Every cross is close to the line, so the correlation is strong. The crosses slope down as x increases, so the correlation is negative.
So Ben is correct.
- B1Yes: Ben is correct
- B1reason names the feature of the graph: every cross is close to the line
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Either paper 3 marksThe table shows the number of pupils on the roll of a school in each of eight years.
A time series graph is drawn to show the data. The first four points have been plotted.Year 2014 2015 2016 2017 2018 2019 2020 2021 Number of pupils 470 475 465 455 450 420 385 350 (a)Complete the graph by plotting the other four points.
(b)Use the graph to estimate the number of pupils on the roll in 2022, by continuing the trend of the last three years.
You cannot draw on this page. Draw (a) on paper, then show the answer and check your drawing against the model.
Show answer
(a) Points plotted at \((2018, 450)\), \((2019, 420)\), \((2020, 385)\) and \((2021, 350)\)
(b) \(317\)(a)Read each missing year from the table and go up to its value.
Each square up the vertical axis is 10 pupils, so go to the line for each value.
In 2018 the value is 450 pupils, in 2019 it is 420, in 2020 it is 385 and in 2021 it is 350.
Put a cross at each of these four heights above its year. The graph now shows all eight years.
The yearly changes are uneven, so base the estimate on the most recent years rather than the first, as the question asks.
From 2018 to 2021 the number of pupils on the roll went from 450 to 350, which is a change of 100 pupils in 3 years.
The average yearly decrease is \(100 \div 3 \approx 33\) pupils.
Continue the pattern for one more year: \(350 - 33 = 317\).
The estimate for 2022 is about 317 pupils. Any value from 313 to 321 pupils is acceptable.
- (a) B1at least two of the four missing points plotted correctly
- (a) B1all four missing points plotted correctly: \((2018, 450)\), \((2019, 420)\), \((2020, 385)\), \((2021, 350)\)
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Either paper 3 marks
Nine students recorded how long they spent revising for a test and the mark they scored. The test was marked out of \(100\).
The scatter graph shows the results, with a line of best fit.
(a)Use the line of best fit to estimate the mark of a student who revised for 13 hours.
(b)Anya says the graph can be used to estimate the mark of a student who revised for 40 hours.
Explain why she would not be able to rely on that estimate.
Show answer
(a) \(56\)
(b) 40 hours is far outside the range of the data (2 hours to 18 hours), so there is no evidence that the pattern carries on that far; the line would give a mark of 164, which is impossible out of 100.\(13\) hours is inside the range of the data, which runs from \(2\) hours to \(18\) hours, so the line can be trusted here. Estimating *between* the data like this is called interpolation.
Go up from \(13\) hours to the line, then across to the mark axis.
The line gains \(4\) marks for every extra hour, so from \(10\) hours (where it passes through \(44\)) to \(13\) hours it changes by \(4 \times 3 = 56 - 44\).
The estimate is about \(56\) marks.
Reading the line *outside* the data is called extrapolation, and it assumes something nobody measured: that the pattern keeps going.
The students in this sample revised for between \(2\) and \(18\) hours. Nothing in the data says what happens at \(40\) hours.
Continuing the line that far gives \(164\) marks, and the test is only out of \(100\) — so the line has already stopped making sense.
In real life the marks would flatten off somewhere near the top, because there is a limit to how much revision can help.
- (a) M1reads the line at \(13\) hours (change of \(4 \times 3\) from \(44\) at \(10\) hours)
- (a) A1estimate \(56\) marks