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GCSE · Statistics

4.20D Scatter Graphs: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A scatter graph is drawn for two variables. As the value of one variable increases, the value of the other variable tends to decrease. State the type of correlation shown.

(Total for Question 1 is 1 mark)

A market trader sells sun hats. State the type of correlation you would expect between the number of hours of sunshine in a day and the number of sun hats sold that day.

(Total for Question 2 is 1 mark)

A seaside kiosk sells ice creams. State the type of correlation you would expect between the number of rainy days in a month and the number of ice creams sold that month.

(Total for Question 3 is 1 mark)

For each pair of quantities below, state whether you would expect positive correlation, negative correlation or no correlation.

(a)The age of an oak tree, in years, and its height, in metres.(1)

(b)The distance a runner has already covered in a marathon and the distance remaining to the finish line.(1)

(c)A student's favourite colour and their score in a maths test.(1)

(Total for Question 4 is 3 marks)

A hairdressing salon records the number of customers seen each day and the total tips (in GBP) the stylists collected that day, for 10 days.
Number of customers (x): 4, 6, 8, 9, 11, 13, 15, 17, 19, 22
Tips collected in GBP (y): 12, 18, 25, 30, 38, 45, 52, 60, 68, 78

(a)Plot all 10 points on the scatter graph grid.(2)

(b)Describe the correlation shown by the scatter graph.(1)

(Total for Question 5 is 3 marks)

Using the scatter graph you completed in Question 5, write down the tip total collected on the day when there were 11 customers.

(Total for Question 6 is 1 mark)

State one feature of a good line of best fit drawn on a scatter graph.

(Total for Question 7 is 1 mark)

A clothes shop records the outdoor temperature (in deg C) and the number of woolly hats it sells each day, for 8 days. The table shows the first 6 days; the points for these 6 days have already been plotted on the grid.
Temperature in deg C (x): 2, 5, 7, 9, 12, 14
Woolly hats sold (y): 42, 35, 30, 24, 18, 14
The remaining two days had a temperature of 16 deg C with 10 hats sold, and a temperature of 19 deg C with 5 hats sold.

(a)Plot the remaining two points on the grid.(2)

(b)State the type of correlation shown.(1)

(Total for Question 8 is 3 marks)

Using the completed scatter graph from Question 8, describe what the correlation shows about the relationship between temperature and the number of woolly hats sold.

(Total for Question 9 is 2 marks)

Eight players record their number of hours of sleep the night before a computer game tournament (x) and their reaction time in milliseconds in a test round (y); a lower reaction time means a faster response.
Hours of sleep (x): 4, 5, 6, 7, 8, 9, 10, 11
Reaction time in ms (y): 340, 310, 290, 265, 105, 230, 200, 180

(a)Write down the coordinates of the outlier.(1)

(b)Suggest a possible reason for this outlier.(1)

(Total for Question 10 is 2 marks)

A scatter graph shows the number of gym sessions attended per month (x) and the kilograms of weight lost that month (y), for data collected between 1 and 12 sessions per month. Aisha uses the line of best fit to estimate the weight lost by someone who attends 40 sessions in a month. Explain why this estimate would not be reliable.

(Total for Question 11 is 2 marks)

Six children record the number of pages they read each day and the number of minutes they spend reading that day.
Pages read per day (x): 3, 5, 6, 8, 9, 11
Minutes spent reading (y): 15, 25, 30, 42, 47, 57

(a)Calculate the mean number of pages read per day.(1)

(b)Calculate the mean number of minutes spent reading.(1)

(c)Write down the coordinates of the point that must lie on the line of best fit.(1)

(Total for Question 12 is 3 marks)

Graph A shows outdoor temperature plotted against sun cream sales at a chemist's, with points lying close to a straight line. Graph B shows shoe size plotted against spelling test score for the same group of people, with points scattered across the grid with no clear pattern.

(a)State which graph shows the stronger correlation.(1)

(b)Give a reason for your answer.(1)

(Total for Question 13 is 2 marks)

For each statement, write True or False.

(a)A strong positive correlation between two variables always means that increasing one variable directly causes the other to increase.(1)

(b)The point given by the mean of the x-values and the mean of the y-values always lies on the line of best fit.(1)

(c)A scatter graph in which the points are spread randomly across the grid, with no visible pattern, shows no correlation.(1)

(d)A strong correlation between the number of people visiting a beach and the number of ice creams sold proves that buying ice cream makes people want to visit the beach.(1)

(Total for Question 14 is 4 marks)

A market stall records the outdoor temperature (in deg C) and the number of bottles of water it sells on 9 days. The first 7 days have already been plotted on the grid.
Temperature in deg C (x): 14, 16, 18, 19, 21, 22, 24
Bottles of water sold (y): 20, 24, 28, 30, 34, 36, 40
On the remaining two days, the temperature was 26 deg C with 44 bottles sold, and 28 deg C with 48 bottles sold.

(a)Plot the remaining two points on the grid.(2)

(b)Draw a line of best fit on the scatter graph.(1)

(c)Use your line of best fit to estimate the number of bottles of water sold when the temperature is 23 deg C.(1)

(Total for Question 15 is 4 marks)

A teacher records, for 8 pupils, the number of hours each pupil spent using a revision app in the week before a test (x) and their test score out of 50 (y).
Hours on app (x): 1, 2, 3, 4, 5, 6, 7, 8
Test score out of 50 (y): 22, 25, 8, 30, 34, 38, 41, 45

(a)Write down the coordinates of the outlier.(1)

(b)The teacher removes this outlier before drawing a line of best fit for the remaining 7 pupils. Explain why this is likely to make the line of best fit a more useful model of the relationship between hours on the app and test score.(2)

(c)State one reason why it would not be appropriate to use this line of best fit to predict the test score of a pupil who spends 20 hours a week on the app.(1)

(Total for Question 16 is 4 marks)

A streaming company records, for a sample of online videos, the number of adverts shown before each video starts and the percentage of viewers who watch to the end. The scatter graph for this data shows strong negative correlation. Explain what this tells you about the relationship between the number of adverts shown and the percentage of viewers who watch to the end, and discuss whether this data alone proves that showing more adverts causes viewers to stop watching.

(Total for Question 17 is 3 marks)

A car magazine investigates the relationship between a car's engine size, in litres, and its fuel economy, in miles per gallon, for 40 cars. The scatter graph shows moderate negative correlation, but the points are fairly scattered rather than lying close to a line.

(a)State what 'moderate negative correlation, but fairly scattered' tells you about the strength of the relationship between engine size and fuel economy.(1)

(b)A second magazine collects data on the same 40 cars, also recording each car's weight. They find that heavier cars tend to have both bigger engines and worse fuel economy. Explain how this new information affects the conclusion that a bigger engine directly causes worse fuel economy.(2)

(c)Suggest one other variable, apart from car weight, that a researcher could investigate to help explain a car's fuel economy.(1)

(Total for Question 18 is 4 marks)

A scientist measures the wing length, in mm, and the flying speed, in mph, of 12 swifts. The wing lengths recorded were between 165 mm and 210 mm.

(a)A birdwatcher uses the line of best fit to estimate the flying speed of a swift with a wing length of 195 mm. State whether this estimate is likely to be reliable, giving a reason.(2)

(b)The birdwatcher then uses the same line of best fit to estimate the flying speed of a swift with a wing length of 260 mm. State whether this estimate is likely to be reliable, giving a reason.(2)

(Total for Question 19 is 4 marks)

A researcher wants to investigate the relationship between the number of hours of daylight and the number of people using a park each day. She only collects data on sunny weekend days between May and August.

(a)Explain one way in which this sampling method could affect the reliability of any conclusions drawn from the scatter graph.(2)

(b)Suggest one improvement the researcher could make to her data collection to give a more reliable picture of the relationship across the whole year.(1)

(Total for Question 20 is 3 marks)

A sports scientist records the resting heart rate, in beats per minute, and the VO2 max score (a measure of aerobic fitness) for 15 club athletes. The scatter graph shows strong negative correlation. The mean resting heart rate for the group is 58 bpm and the mean VO2 max score is 52.

(a)Write down the coordinates of the point that must lie on the line of best fit.(1)

(b)One athlete has a resting heart rate of 50 bpm (below the group mean) but a VO2 max score of only 38 (also below the group mean), placing them well away from the line of best fit. Explain what this tells you about this athlete compared with the general trend shown by the rest of the group.(2)

(c)A sports journalist writes: 'This data proves that lowering your resting heart rate directly increases your VO2 max fitness score.' Evaluate this claim.(3)

(Total for Question 21 is 6 marks)

Mark scheme · 4.20D Scatter Graphs: Fluency and Exam Drill

Question 1

B1 negative correlation (oe)
Answer: Negative correlation

Question 2

B1 positive correlation (oe)
Answer: Positive correlation

Question 3

B1 negative correlation (oe)
Answer: Negative correlation

Question 4

(a) B1 positive correlation
(a) Answer: Positive correlation
(b) B1 negative correlation
(b) Answer: Negative correlation
(c) B1 no correlation
(c) Answer: No correlation

Question 5

(a) B2 all 10 points plotted correctly (allow small tolerance); award B1 for at least 7 points plotted correctly
(a) Answer: Points at (4,12), (6,18), (8,25), (9,30), (11,38), (13,45), (15,52), (17,60), (19,68), (22,78)
(b) B1 positive correlation
(b) Answer: Positive correlation

Question 6

B1 GBP 38 (or 38) cao
Answer: GBP 38

Question 7

B1 any correct feature, e.g. it follows the trend of the points with roughly equal numbers of points above and below it, or it passes through the mean point (oe)
Answer: It should follow the trend of the data with roughly equal numbers of points above and below the line.

Question 8

(a) B1 point (16, 10) plotted correctly
(a) B1 point (19, 5) plotted correctly
(a) Answer: (16, 10) and (19, 5)
(b) B1 negative correlation
(b) Answer: Negative correlation

Question 9

B1 identifies negative correlation
B1 correct statement in context, e.g. as the temperature increases, the number of woolly hats sold tends to decrease
Answer: There is negative correlation: as the temperature increases, the number of woolly hats sold tends to decrease.

Question 10

(a) B1 (8, 105) cao
(a) Answer: (8, 105)
(b) B1 any sensible reason, e.g. the reaction time may have been recorded incorrectly, or this player may have had an unusually fast practice attempt (oe)
(b) Answer: The reaction time may have been recorded incorrectly, or this player had an unusually fast response on that attempt.

Question 11

B1 40 sessions is outside the range of data collected (1 to 12), so this is extrapolation
B1 the same trend/rate of weight loss may not continue that far beyond the data, e.g. there could be a natural limit to how much weight can be lost, or attending 40 sessions a month may not be realistic (oe)
Answer: The estimate is unreliable because 40 sessions is far outside the data collected (1 to 12 sessions), so it involves extrapolation and the same trend may not continue.

Question 12

(a) B1 7 cao
(a) Answer: 7 pages
(b) B1 36 (minutes) cao
(b) Answer: 36 minutes
(c) B1 (7, 36) ft from parts (a) and (b)
(c) Answer: (7, 36)

Question 13

(a) B1 Graph A
(a) Answer: Graph A
(b) B1 the points in Graph A lie closer to a straight line / are less scattered than the points in Graph B (oe)
(b) Answer: The points in Graph A lie closer to a straight line than the scattered points in Graph B.

Question 14

(a) B1 False
(a) Answer: False
(b) B1 True
(b) Answer: True
(c) B1 True
(c) Answer: True
(d) B1 False
(d) Answer: False

Question 15

(a) B1 point (26, 44) plotted correctly
(a) B1 point (28, 48) plotted correctly
(a) Answer: (26, 44) and (28, 48)
(b) B1 a single straight line, drawn by eye, that follows the trend of all 9 points with a roughly equal number of points on each side
(b) Answer: A straight line following the trend of the 9 points, e.g. close to the line y = 2x - 8.
(c) B1 38 (bottles), accept 36 to 40, ft from candidate's line
(c) Answer: 38 bottles (accept 36 to 40)

Question 16

(a) B1 (3, 8) cao
(a) Answer: (3, 8)
(b) B1 the outlier does not follow the trend/pattern shown by the rest of the data
(b) B1 removing it means the line is not pulled away from the trend by this single unusual result, so it better represents the general relationship for the remaining pupils (oe)
(b) Answer: The outlier does not fit the pattern of the rest of the data, so removing it stops the line of best fit being distorted, giving a line that better represents the true relationship for the other 7 pupils.
(c) B1 20 hours is far outside the range of data collected (1 to 8 hours), so this would be extrapolation and the trend may not continue (oe)
(c) Answer: 20 hours is well outside the data collected (1 to 8 hours), so predicting from the line would be extrapolation and may not be reliable.

Question 17

B1 correctly interprets the negative correlation in context, e.g. as the number of adverts shown increases, the percentage of viewers who watch to the end tends to decrease
B1 states that correlation, however strong, does not prove causation
B1 identifies a plausible alternative explanation/confounding factor, e.g. longer videos may naturally have more adverts and viewers may stop watching because of the video's length rather than the adverts themselves (oe)
Answer: As the number of adverts increases, the percentage of viewers watching to the end tends to decrease, but this does not prove adverts cause viewers to leave; another factor, such as longer videos having both more adverts and lower completion for unrelated reasons, could explain the pattern.

Question 18

(a) B1 there is a genuine negative relationship (bigger engines tend to go with worse fuel economy), but it is not a strong or highly reliable one, since the points are not tightly clustered around a line (oe)
(a) Answer: There is a real negative link between engine size and fuel economy, but it is not very strong, since the points are quite spread out rather than close to a line.
(b) B1 identifies car weight as a third factor (confounding variable) linked to both engine size and fuel economy
(b) B1 explains that this means engine size may not be the only, or even the main, direct cause of worse fuel economy, since the apparent link could partly or wholly be explained by weight (oe)
(b) Answer: Car weight could be causing both a bigger engine and worse fuel economy, so it is a confounding variable; this means the original conclusion that engine size directly causes worse fuel economy is weakened, since weight might explain some or all of the pattern instead.
(c) B1 any sensible variable other than weight, e.g. driving style/average speed, tyre type, aerodynamic design, or fuel type (oe)
(c) Answer: For example, the driving style or average speed of the car's usual driver.

Question 19

(a) B1 reliable / yes
(a) B1 195 mm lies within the range of data collected (165 mm to 210 mm), so this is interpolation, which tends to be reliable (oe)
(a) Answer: Reliable, because 195 mm lies within the range of wing lengths collected (165 mm to 210 mm), so this is interpolation.
(b) B1 not reliable / no
(b) B1 260 mm is well outside the range of data collected (165 mm to 210 mm), so this is extrapolation and the relationship may not continue in the same way beyond the observed data (oe)
(b) Answer: Not reliable, because 260 mm is well beyond the range of wing lengths collected (165 mm to 210 mm), so this is extrapolation and the trend may not continue.

Question 20

(a) B1 identifies that only sunny weekend days in May to August were sampled, excluding weekdays, poor-weather days and other seasons
(a) B1 explains a consequence, e.g. the relationship seen may not represent the true pattern across all days/seasons, since weekday attendance or attendance in other weather/seasons could behave differently, so conclusions may not generalise (oe)
(a) Answer: Only sunny weekend days in May to August were sampled, so weekdays, poor weather and other seasons are missing; the relationship found may not hold for the park's attendance the rest of the year, so conclusions drawn from it may not be reliable more generally.
(b) B1 any sensible improvement, e.g. collect data across weekdays and weekends, throughout the year, and in a range of weather conditions (oe)
(b) Answer: Collect data on a mix of weekdays and weekends, across all seasons and a range of weather conditions, not just sunny summer weekends.

Question 21

(a) B1 (58, 52) cao
(a) Answer: (58, 52)
(b) B1 recognises this athlete is an outlier/anomalous point that does not follow the trend shown by the rest of the group
(b) B1 explains the implication, e.g. a lower resting heart rate is generally linked with a higher VO2 max in this data, but this athlete has a low heart rate and a low fitness score, so some other factor must be affecting their result (oe)
(b) Answer: This athlete does not follow the general trend: in the rest of the group, a lower resting heart rate goes with a higher VO2 max, but this athlete has a low heart rate and a low VO2 max score, so some other factor must be affecting their fitness score.
(c) B1 states that correlation, even a strong one, does not prove causation
(c) B1 identifies a plausible confounding factor, e.g. overall training/fitness level could improve both resting heart rate and VO2 max at the same time, rather than one directly causing the other (oe)
(c) B1 states that establishing causation would require further evidence, such as a controlled study, not just an observed correlation from a scatter graph (oe)
(c) Answer: The claim is not proven: correlation does not establish causation. Both resting heart rate and VO2 max could be improved together by a third factor, such as overall training level, rather than a lower heart rate directly causing higher fitness. A controlled study, not just this scatter graph, would be needed to establish a genuine causal link.

Scatter Graphs: Fluency and Exam Drill - Worksheets, Questions and Revision