Answer on paper first, then reveal the mark scheme one mark at a time.
1Grade 1–3FoundationNon-calculator3 marks
The scatter graph shows the number of hours some students spent revising and their test scores.
(a) One of the points does not follow the trend. Write down its coordinates. [1]
(b) Write down the type of correlation shown by the other points. [1]
(c) Describe, in words, the relationship between the number of hours spent revising and the test score. [1]
(a) (8, 25) (accept 8 hours and 25%)B1
(b) positive (ignore ‘strong’ or ‘weak’)B1
(c) the more hours spent revising, the higher the test score oe in context, e.g. ‘students who revise longer tend to score higher’ (do not accept ‘positive correlation’ alone)C1
Answer: (a) (8, 25) (b) positive (c) students who revise for longer tend to get higher scores
Examiner tip: In (c) you must use the context (hours of revision and test scores) – just writing ‘positive correlation’ again earns nothing.
2Grade 4–5Non-calculator4 marks
The scatter graph shows the midday temperature and the number of cups of hot chocolate sold by a café on 10 days.
(a) Describe the correlation. [1]
(b) On another day the midday temperature was 9 °C. Estimate the number of cups of hot chocolate sold. [2]
(c) Explain why it may not be reliable to use the graph to estimate the number of cups sold on a day when the temperature is 30 °C. [1]
(a) negative (ignore ‘strong’ or ‘weak’)B1
(b) a single straight line of best fit drawn between the points (or implied by a correct reading), with a reading taken at 9 °CM1
(b) answer in the range 31 to 34A1
(c) 30 °C is outside the range of the data (extrapolation), so the trend may not continue (accept: the line would give a negative number of cups, which is impossible)C1
Answer: (a) negative (b) about 32 (accept 31–34) (c) 30 °C is outside the range of the data, so the trend may not continue
Examiner tip: A line of best fit should follow the trend with roughly equal points on each side – it does not have to go through the origin or through the first point.
3Grade 1–3FoundationNon-calculator3 marks
For each pair of variables, state whether you would expect positive correlation, negative correlation or no correlation.
(a) The height of an adult and their shoe size. [1]
(b) The age of a car and its value. [1]
(c) The number on a person's front door and their weight. [1]
(a) positive (ignore ‘strong’ or ‘weak’)B1
(b) negative (ignore ‘strong’ or ‘weak’)B1
(c) no correlation (accept ‘none’)B1
Answer: (a) positive (b) negative (c) no correlation
Examiner tip: Ask: as one goes up, does the other tend to go up (positive), go down (negative), or do neither (none)?
4Grade 4–5Non-calculator2 marks
A council records the number of ice creams sold and the number of people treated for sunburn at a beach each day during the summer.
A scatter graph of the data shows strong positive correlation.
Leah says, “This shows that eating ice cream causes sunburn.”
(a) Is Leah correct? Give a reason for your answer. [1]
(b) Give a likely reason for the correlation. [1]
(a) No, with a reason, e.g. correlation does not show that one variable causes the other / another factor (e.g. hot weather) could cause both. ‘No’ with no reason scores 0C1
(b) both are likely to increase on hot, sunny days (a third factor, e.g. temperature or hours of sunshine, affects both) oeC1
Answer: (a) No, correlation does not prove causation (b) both increase on hot, sunny days
Examiner tip: ‘Correlation does not mean causation’ – look for a third variable that drives both quantities.
5Grade 4–5Calculator3 marks
The scatter graph shows the ages and values of 11 cars of the same model. A line of best fit has been drawn.
(a) Use the line of best fit to estimate the age of a car of this model that is worth £8000. [1]
(b) Abdul extends the line of best fit and says, “A 15-year-old car of this model will be worth nothing.”
Comment on Abdul's statement. [2]
(a) answer in the range 5.5 to 6 years (line gives 5.67)B1
(b) states the prediction is unreliable because 15 years is outside the range of the data (the oldest car is 9 years old) / it is extrapolationC1
(b) second valid comment, e.g. the line reaches £0 at about 11 years and gives a negative value (about −£6000) at 15 years, which is impossible / cars do not usually lose value at a constant rate, so the car is likely to still be worth something. (Two comments that both only say ‘outside the data’ score C1 only)C1
Answer: (a) about 5.7 years (accept 5.5–6) (b) not reliable: 15 years is outside the range of the data (extrapolation), and the line would give a negative value at 15 years, which is impossible
Examiner tip: In (a) read from the line, not from the cross at (5, 8); in (b) remember predictions outside the range of the data are unreliable.
6Grade 6–7HigherNon-calculator4 marks
The scatter graph shows the number of hours some students practised typing and their typing speeds. A line of best fit has been drawn. It passes through (0, 18) and (40, 48).
(a) Find an equation of the line of best fit. Give your answer in the form $y = mx + c$. [3]
(b) Interpret the gradient of the line in this context. [1]
(a) method for the gradient, e.g. $\frac{48 - 18}{40 - 0}$M1
(b) for each extra hour of practice, typing speed increases by about 0.75 words per minute (accept 3 words per minute for every 4 hours oe; must be in context)C1
Answer: (a) $y = 0.75x + 18$ (b) each extra hour of practice increases typing speed by about 0.75 words per minute
Examiner tip: For ‘interpret’, give the number WITH its units and context – ‘it is the rate of change’ alone gets no credit.
7Grade 1–3FoundationNon-calculator3 marks
Here are four scatter graphs, A, B, C and D.
(a) Which graph shows strong negative correlation? [1]
(b) Which graph shows no correlation? [1]
(c) Describe fully the correlation shown in graph A. [1]
(a) CB1
(b) DB1
(c) weak negative (both words needed; accept ‘fairly weak negative’ or ‘moderate negative’; ‘negative’ alone or ‘strong negative’ scores 0)B1
Answer: (a) C (b) D (c) weak negative correlation
Examiner tip: ‘Describe fully’ means give the strength (strong or weak) as well as the type (positive or negative); one word is not enough.
8Grade 1–3FoundationNon-calculator4 marks
The table shows the number of pages and the mass of eight books.
Book
A
B
C
D
E
F
G
H
Number of pages
120
180
250
300
360
420
480
550
Mass (g)
190
260
330
400
440
540
590
680
Charlie plots the information from the table on a scatter graph. He makes one mistake.
(a) Which book has Charlie plotted incorrectly? [1]
(b) Write down the type of correlation shown by the data. [1]
(c) Another book has 400 pages. Estimate the mass of this book. [2]
(a) F (accept the book with 420 pages; it is plotted at 450 g instead of 540 g)B1
(b) positive (ignore ‘strong’ or ‘weak’)B1
(c) a single straight line of best fit drawn through the trend of the points (using F at 540 g), or a reading taken from such a line at 400 pages (may be implied by an answer in range)M1
(c) answer in the range 480 to 530 gA1
Answer: (a) F (b) positive (c) about 505 g (accept 480–530 g)
Examiner tip: Check every point against the table before using the graph: a misplotted point drags your line of best fit away from the real trend.
9Grade 4–5Non-calculator4 marks
The scatter graph shows the distance ten runners ran in training each week and their time for a 5 km race.
(a) Describe the correlation shown by the scatter graph. [1]
Sam has drawn a line on the graph and says it is a line of best fit.
(b) Give one reason why Sam’s line is not a good line of best fit. [1]
(c) Another runner trains 25 km each week. Use a suitable line of best fit to estimate this runner’s time for the 5 km race. [2]
(a) negative (ignore ‘strong’ or ‘weak’; accept a description in context such as ‘the further they train, the faster (shorter) their race time’)B1
(b) a valid reason, e.g. all (or most) of the points are below the line / there are not roughly equal numbers of points on each side / the line is not steep enough (gradient too shallow) / it does not follow the trend of the pointsC1
(c) a single straight line of best fit with negative gradient passing between the points (not Sam’s line), or a reading taken from such a line at 25 kmM1
(c) answer in the range 28 to 29.5 minutesA1
Answer: (a) negative (b) e.g. all the points are below Sam’s line (c) about 28.7 minutes (accept 28–29.5)
Examiner tip: A line of best fit should have roughly the same number of points on each side; it does not have to start at the axis or join the first and last points.
10Grade 4–5Non-calculator3 marks
A gardener gives six tomato plants different amounts of fertiliser each week. The table shows the amount of fertiliser and the mass of tomatoes from each plant.
Fertiliser (ml per week)
10
20
30
40
50
60
Mass of tomatoes (kg)
1.9
2.4
2.9
3.4
3.8
4.3
The gardener draws a scatter graph and a line of best fit for the data.
Priya uses the line of best fit to estimate the mass of tomatoes from a plant given 35 ml of fertiliser per week. Tom uses the line of best fit to estimate the mass of tomatoes from a plant given 150 ml of fertiliser per week.
(a) Whose estimate is likely to be more reliable? Give a reason for your answer. [1]
(b) Priya says, “If you double the amount of fertiliser, you double the mass of tomatoes.” Use values from the table to show that Priya is wrong. [1]
(c) Give one way the gardener could make her conclusions more reliable. [1]
(a) Priya’s, with a reason, e.g. 35 ml is within the range of the data (10 to 60 ml) but 150 ml is outside it / Tom’s estimate is extrapolation and the trend may not continue. (‘Priya’ with no reason scores 0)C1
(b) a correct pair of values from the table, e.g. 20 ml gives 2.4 kg but 40 ml gives 3.4 kg (not 4.8 kg) / 10 ml gives 1.9 kg but 20 ml gives 2.4 kg (not 3.8 kg) / 30 ml gives 2.9 kg but 60 ml gives 4.3 kg (not 5.8 kg)C1
(c) e.g. use more plants (a bigger sample) / repeat the experiment / keep other conditions such as water, sunlight and variety of plant the same (ignore vague answers such as ‘be more accurate’)C1
Answer: (a) Priya’s: 35 ml is inside the range of the data, 150 ml is outside it (b) e.g. 20 ml gives 2.4 kg but 40 ml gives 3.4 kg, not 4.8 kg (c) e.g. use more plants
Examiner tip: When asked to ‘show’ someone is wrong, quote actual numbers from the table and say what doubling would have given.
11Grade 4–5Non-calculator4 marks
Twelve students sat a Maths test, a Physics test and an Art test.
Graph 1 shows each student’s Maths mark and Physics mark. Graph 2 shows each student’s Maths mark and Art mark.
(a) Describe the relationship between the students’ Maths marks and their Art marks. [1]
Ben scored 64 in the Maths test. He missed the Physics test and the Art test. His teacher wants to use his Maths mark to estimate his mark in one of the other tests.
(b) Should she estimate his Physics mark or his Art mark? Give a reason for your answer. [1]
(c) Estimate Ben’s Physics mark. [2]
(a) there is no correlation / no relationship oe, e.g. ‘a student’s Maths mark tells you nothing about their Art mark’B1
(b) Physics, with a reason, e.g. graph 1 shows (strong) positive correlation / the points in graph 1 lie close to a straight line, but graph 2 shows no correlation. (‘Physics’ with no reason scores 0)C1
(c) a single straight line of best fit drawn on graph 1 between the points, or a reading taken from such a line at a Maths mark of 64M1
(c) answer in the range 58 to 62A1
Answer: (a) no correlation (b) Physics, because Maths and Physics marks show strong positive correlation (c) about 61 (accept 58–62)
Examiner tip: You can only make a sensible prediction from a scatter graph that shows correlation; with no correlation, a line of best fit is meaningless.
12Grade 4–5Calculator4 marks
The scatter graph shows the floor area and the monthly rent of 12 flats in a town. A line of best fit has been drawn.
(a) Use the line of best fit to estimate the monthly rent of a flat in this town with a floor area of 70 m². [1]
Flat Q is bigger than flat P, but the rent for flat Q is less than the rent for flat P.
(b) Use the line of best fit to explain which of flats P and Q is better value for money. [2]
(c) Explain why this line of best fit should not be used to estimate the monthly rent of a 70 m² flat in a different city. [1]
(a) answer in the range £1120 to £1160B1
(b) compares each flat with the line, e.g. P is above the line (its rent is more than expected for its size) and Q is below the line (its rent is less than expected for its size); may be shown with values, e.g. line gives about £1020 for P and about £1320 for QM1
(b) conclusion: Q is better value for money, with the comparison to the line correctly supporting it (do not accept ‘Q because it is cheaper and bigger’ without reference to the line or expected rent)C1
(c) the data only come from flats in one town, so they may not represent rents in the other city oe (e.g. rents in a city may be much higher)C1
Answer: (a) about £1140 (accept £1120–£1160) (b) Q: it lies below the line, so its rent is less than expected for its size, while P lies above the line (c) the data are from one town only, so may not represent the other city
Examiner tip: A point below a line of best fit has a smaller value than the trend predicts for its x-value; use this to judge ‘good value’.
13Grade 6–7HigherNon-calculator4 marks
A car was driven at different constant speeds. The scatter graph shows the fuel economy of the car, in miles per gallon, at each speed.
Ali says, “There is no correlation on this graph, so the speed of the car makes no difference to its fuel economy.”
(a) Is Ali correct? Explain your answer. [2]
(b) Explain why a straight line of best fit would not be suitable for estimating the fuel economy of the car at a given speed. [1]
(c) The car has a fuel economy of 48 miles per gallon. Use the graph to estimate two possible speeds of the car. [1]
(a) No – there is (little or) no linear correlation, but there is still a clear relationship between speed and fuel economy (accept ‘the points follow a curve’, ‘the relationship is not a straight line’)C1
(a) describes the relationship, e.g. fuel economy increases as speed increases up to about 50 mph, then decreases as speed increases further (accept the peak anywhere from 45 to 55 mph)C1
(b) the points follow a curve (rise and then fall), so a straight line would not follow the trend / would give poor estimates oeC1
(c) one speed in the range 27 to 33 mph and one speed in the range 67 to 73 mph (both needed)B1
Answer: (a) No: there is no linear correlation, but fuel economy rises up to about 50 mph and then falls (b) the points follow a curve, not a straight line (c) about 30 mph and about 70 mph
Examiner tip: ‘No correlation’ only means no straight-line trend; always look for a curved pattern before saying the variables are unrelated.
14Grade 6–7HigherCalculator5 marks
A scientist records the height above sea level, $h$ metres, and the mean July temperature, $T$ °C, at 15 weather stations. The weather stations are between 10 m and 1200 m above sea level.
She draws a scatter graph and a line of best fit. The equation of her line of best fit is
$$T = 21.5 - 0.0065h$$
(a) Use the equation to estimate the mean July temperature at a weather station 750 m above sea level. [1]
(b) Give an interpretation of the number 21.5 in the equation. [1]
(c) Use the equation to estimate the height above sea level at which the mean July temperature would be 0 °C. [2]
(d) Comment on the reliability of your answer to part (c). [1]
(a) 16.6 °C (accept 16.625, 16.63 or 17 if 16.625 or 16.6 seen)B1
(b) the (estimated) mean July temperature at sea level / at a height of 0 m is 21.5 °C oeC1
(c) answer in the range 3300 to 3310 m (exact 3307.69…)A1
(d) not reliable, because about 3300 m is far outside the range of the data (the highest station is 1200 m) / it is extrapolation, so the trend may not continueC1
Answer: (a) 16.6 °C (b) estimated mean July temperature at sea level is 21.5 °C (c) about 3310 m (accept 3300–3310) (d) unreliable: extrapolation far beyond the 1200 m maximum of the data
Examiner tip: The constant in a line-of-best-fit equation is the predicted value when the x-variable is zero; say this in the context of the question.
15Grade 6–7HigherCalculator7 marks
The table shows the number of hours of sunshine and the number of visitors to a garden on nine days.
Day
A
B
C
D
E
F
G
H
I
Hours of sunshine
4
1
7
5
9
2
6
3
8
Number of visitors
230
110
340
62
420
150
300
195
375
One of the numbers of visitors was recorded incorrectly.
(a) Which day is most likely to have been recorded incorrectly? Give a reason for your answer. [1]
The incorrect day is ignored. The line of best fit for the other eight days passes through the mean point of their data.
(b) Work out the coordinates of this mean point. [2]
The gradient of the line of best fit is 40.
(c) Find an equation of the line of best fit. Give your answer in the form $y = mx + c$, where $x$ is the number of hours of sunshine and $y$ is the number of visitors. [2]
The garden is full when it has 450 visitors.
(d) Use your equation to estimate the number of hours of sunshine for the garden to be full. [2]
(a) day D, with a reason, e.g. 62 visitors is far fewer than the trend suggests for 5 hours of sunshine (days with 4 and 6 hours had 230 and 300) / it does not fit the positive correlation. (‘D’ with no reason scores 0)C1
(b) method to find either mean for the eight days without day D, e.g. $(1 + 2 + 3 + 4 + 6 + 7 + 8 + 9) \div 8$ or $2120 \div 8$ (do not award for $45 \div 9$ or other means that include day D)M1
(b) (5, 265)A1
(c) substitutes the mean point into $y = 40x + c$, e.g. $265 = 40 \times 5 + c$ (ft their mean point)M1
(c) $y = 40x + 65$A1
(d) $40x + 65 = 450$ oe (ft their equation)M1
(d) 9.6 hours (accept 9.625, 9.63 or 9 hours 37.5 minutes; ft their equation)A1
Answer: (a) day D: 62 visitors does not fit the trend (b) (5, 265) (c) $y = 40x + 65$ (d) about 9.6 hours
Examiner tip: Leave out an obvious outlier before finding the mean point, then use $y = mx + c$ with the mean point to find $c$.
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