Which of the following statements about a cumulative frequency graph is correct?
A) It shows the frequency of each class interval only, not a running total.
B) It can be used to find estimates for the median and the quartiles of grouped data.
C) It can only be drawn for ungrouped, discrete data.
D) The cumulative frequency always decreases as the value of the variable increases.
(Total for Question 1 is 1 mark)
2
A charity organises a fun run. The times, t minutes, taken by 60 runners to complete the course are recorded below.
Time, t (minutes)
Frequency
30 < t ≤ 40
4
40 < t ≤ 50
9
50 < t ≤ 60
15
60 < t ≤ 70
18
70 < t ≤ 80
10
80 < t ≤ 90
4
(a)Complete the cumulative frequency table below. Some values have already been filled in for you.
Time, t (minutes)
Cumulative frequency
t ≤ 40
4
t ≤ 50
13
t ≤ 60
...
t ≤ 70
46
t ≤ 80
...
t ≤ 90
60
(2)
(b)Write down the coordinates you would plot on a cumulative frequency graph to represent (i) the class 40 < t ≤ 50, and (ii) the class 80 < t ≤ 90.(2)
(Total for Question 2 is 4 marks)
3
The cumulative frequency graph below has been drawn for the fun run data (60 runners).
(a)Use the graph to estimate the number of runners who took at most 55 minutes to complete the run.(2)
(b)Show that an estimate for the median completion time is greater than 60 minutes.(2)
(Total for Question 3 is 4 marks)
4
The same cumulative frequency graph for the 60 fun run times is shown again below.
(a)Use the graph to find estimates for (i) the lower quartile and (ii) the upper quartile of the completion times.(4)
(b)Hence write down an estimate for the interquartile range of the completion times.(1)
(c)Runners who complete the course in 45 minutes or less receive a 'fast finisher' certificate. Use the graph to estimate the number of runners who receive a certificate.(1)
(d)Use the graph to estimate the percentage of runners who took more than 70 minutes to complete the run.(1)
(Total for Question 4 is 7 marks)
5
The fastest runner in the fun run completed the course in 32 minutes, and the slowest runner took 88 minutes. Using your estimates for the median, lower quartile and upper quartile from Question 4, draw a box plot to represent the fun run data. Use a scale of 2 cm to represent 10 minutes, from 30 to 90 minutes.
(Total for Question 5 is 3 marks)
6
A delivery depot records the weights, w kg, of 80 parcels sent in one day.
Weight, w (kg)
Frequency
0 < w ≤ 2
5
2 < w ≤ 4
13
4 < w ≤ 6
23
6 < w ≤ 8
21
8 < w ≤ 10
12
10 < w ≤ 12
6
(a)Complete the cumulative frequency table below.
Weight, w (kg)
Cumulative frequency
w ≤ 2
5
w ≤ 4
18
w ≤ 6
...
w ≤ 8
62
w ≤ 10
...
w ≤ 12
80
(2)
(b)Write down the coordinates you would plot to represent the class 6 < w ≤ 8.(1)
(c)The cumulative frequency graph for this data is shown above. Use the graph to estimate the median weight of a parcel.(2)
(Total for Question 6 is 5 marks)
7
Using the cumulative frequency graph for the parcel weights (shown again below), find an estimate for the interquartile range of the parcel weights.
(Total for Question 7 is 3 marks)
8
Parcels weighing more than 7 kg attract a delivery surcharge. Use the cumulative frequency graph for the parcel weights to estimate the percentage of parcels that attract the surcharge.
(Total for Question 8 is 2 marks)
9
The box plot below shows the heights, in cm, of a sample of sunflowers grown in a greenhouse.
Minimum = 10 cm, Lower quartile = 18 cm, Median = 24 cm, Upper quartile = 30 cm, Maximum = 42 cm
(a)Find the range of the sunflower heights.(1)
(b)Find the interquartile range of the sunflower heights.(1)
(Total for Question 9 is 2 marks)
10
A commuter can travel to work using Route A or Route B. Box plots summarising the journey times (minutes) recorded over several weeks for each route are shown below.
Route A: minimum = 12, lower quartile = 18, median = 25, upper quartile = 33, maximum = 45 Route B: minimum = 15, lower quartile = 22, median = 27, upper quartile = 30, maximum = 40
(a)Calculate the interquartile range for Route A and for Route B.(2)
(b)Compare the two routes using the median journey time.(1)
(c)Compare the two routes using the interquartile range.(1)
(d)A commuter wants the most reliable, predictable journey time each day. Which route should they choose? Give a reason for your answer.(1)
(Total for Question 10 is 5 marks)
11
Box plots showing the number of hours spent revising in the two weeks before a mock exam are shown below for a sample of Year 10 students and a sample of Year 11 students.
Year 10: minimum = 2, lower quartile = 5, median = 8, upper quartile = 11, maximum = 16 Year 11: minimum = 4, lower quartile = 9, median = 13, upper quartile = 19, maximum = 20
(a)Find the interquartile range for Year 10 and for Year 11.(2)
(b)Compare the revision times of the Year 10 and Year 11 students, using the median and the interquartile range.(2)
(Total for Question 11 is 4 marks)
12
The box plot below shows the percentage scores of students in a Year 11 mock exam.
Minimum = 22, Lower quartile = 48, Median = 52, Upper quartile = 80, Maximum = 95
State, giving a reason, whether the distribution of scores is skewed, and if so in which direction.
(Total for Question 12 is 2 marks)
13
A box plot has been drawn to represent the delivery times of 200 parcels sent by a courier company. Using the definition of the upper quartile, estimate how many of the 200 parcels had a delivery time longer than the upper quartile.
(Total for Question 13 is 2 marks)
14
Using the cumulative frequency graph for the fun run times (60 runners, shown in Question 3), the organisers decide to award a medal to the fastest 20% of runners. Estimate the completion time below which a runner receives a medal.
(Total for Question 14 is 3 marks)
15
Two schools, School X and School Y, each entered 100 students for the same mock exam. Cumulative frequency graphs have been drawn using the following points.
School X: (0,0), (20,5), (40,20), (60,55), (80,85), (100,100) School Y: (0,0), (20,10), (40,30), (60,55), (80,80), (100,100)
(a)Use the graphs to find an estimate for the median score at each school.(2)
(b)Use the graphs to find an estimate for the interquartile range of scores at each school.(2)
(c)Using your answers to parts (a) and (b), compare the performance of the students at the two schools, and comment on which school's students performed more consistently.(2)
(Total for Question 15 is 6 marks)
16
A student has recorded the following five-number summary for a data set, and claims it represents a valid box plot.
Minimum = 8, Lower quartile = 15, Median = 12, Upper quartile = 20, Maximum = 25
Explain why this five-number summary cannot be correct, and state what must be true instead.
(Total for Question 16 is 3 marks)
17
The battery life, in hours, of 50 tablets was tested. A cumulative frequency graph was drawn using the points (0,0), (4,3), (8,11), (12,27), (16,42), (20,48), (24,50).
(a)Use the graph to estimate the number of tablets with a battery life between 10 and 18 hours.(3)
(b)Express your answer to part (a) as a percentage of the 50 tablets tested.(1)
(Total for Question 17 is 4 marks)
18
Two data sets have the same range but different interquartile ranges. Explain why the interquartile range might be considered a better measure of spread than the range for comparing these data sets.
(Total for Question 18 is 2 marks)
Mark scheme · 6.11 Cumulative Frequency
Question 1
B1 B cao
Answer: B
Question 2
(a) B1 t ≤ 60 : 28 cao
(a) B1 t ≤ 80 : 56 cao
(a) Answer: 28 and 56
(b) M1 using the upper class boundary and the cumulative frequency oe
(b) A1 (50, 13) and (90, 60) cao
(b) Answer: (50, 13) and (90, 60)
Question 3
(a) M1 reading cumulative frequency at t = 55 from the curve
(a) A1 awrt 21 (accept 19-22)
(a) Answer: awrt 21 runners (accept 19-22)
(b) M1 identifying the median position as the 30th value (n/2 = 30) and using correct interpolation between (60,28) and (70,46)
(b) A1 cso: value = awrt 61.1 (accept 60-63), which is greater than 60
(b) Answer: 61.1 minutes (greater than 60, as required)
Question 4
(a) M1 identifying the lower quartile position as the 15th value (n/4 = 15) and reading/interpolating
(b) B1 correct comparison in context, e.g. Route B has the higher median (27 minutes) than Route A (25 minutes), so journeys on Route B tend to take slightly longer on average oe
(b) Answer: Route B has the higher median, so on average its journeys take slightly longer than Route A's.
(c) B1 ft: correct comparison in context, e.g. Route A has a much larger IQR (15) than Route B (8), so journey times on Route A are more variable/less consistent than on Route B oe
(c) Answer: Route A's journey times are more spread out/variable than Route B's, since its IQR (15) is greater than Route B's (8).
(d) B1 ft: Route B, because it has the smaller interquartile range (less variation in journey times) oe
(d) Answer: Route B, because its journey times are more consistent (smaller IQR).
Question 11
(a) B1 Year 10 IQR = 6 cao
(a) B1 Year 11 IQR = 10 cao
(a) Answer: Year 10: 6 hours; Year 11: 10 hours
(b) B1 median comparison in context ft, e.g. Year 11 students typically revised for longer than Year 10 students, since their median (13 hours) is higher than Year 10's (8 hours)
(b) B1 IQR/spread comparison in context ft, e.g. the amount of time Year 11 students spent revising was more variable/spread out than Year 10's, since their IQR (10 hours) is greater than Year 10's (6 hours)
(b) Answer: Year 11 students revised for longer on average (higher median) and their revision times were more variable (higher IQR) than Year 10 students.
Question 12
B1 correct conclusion: the distribution is positively skewed (skewed to the right/towards higher scores) oe
B1 valid reason, e.g. the gap between the upper quartile and the median (80 - 52 = 28) is much bigger than the gap between the median and the lower quartile (52 - 48 = 4), showing greater spread among the higher-scoring students oe
Answer: Positively (right) skewed
Question 13
M1 recognising that the upper quartile is the value below which approximately 75% of the data lies, so 25% lies above it oe
A1 50 parcels cao
Answer: 50 parcels
Question 14
M1 identifying the required position as the 12th value (20% of 60 = 12)
M1 correct interpolation method between (40,4) and (50,13)
A1 awrt 49 minutes (accept 47-51)
Answer: awrt 49 minutes
Question 15
(a) B1 School X median = awrt 57 (accept 55-59)
(a) B1 School Y median = awrt 56 (accept 54-58)
(a) Answer: School X: awrt 57; School Y: awrt 56
(b) B1 School X IQR = awrt 30 (accept 27-34)
(b) B1 School Y IQR = awrt 41 (accept 37-45)
(b) Answer: School X: awrt 30; School Y: awrt 41
(c) B1 ft: the medians are similar (awrt 57 and awrt 56), so the overall level of performance at the two schools was about the same oe
(c) B1 ft: School X has the smaller IQR (awrt 30 compared with awrt 41), so School X's scores were less spread out/more consistent than School Y's oe
(c) Answer: Similar median performance overall, but School X's students performed more consistently (smaller IQR).
Question 16
B1 identify the contradiction: the median (12) is less than the lower quartile (15)
B1 valid explanation, e.g. the lower quartile is the value below which approximately 25% of the data lies, and the median is the middle value with 50% of the data below it, so the median must always be greater than or equal to the lower quartile oe
B1 correct statement of what must change, e.g. the median value must be at least 15 (and no more than 20); the data must have been recorded or plotted incorrectly oe
Answer: The summary is invalid because the median (12) is less than the lower quartile (15); the median must be at least 15.
Question 17
(a) M1 cumulative frequency at 10 hours, awrt 17-21 (e.g. 19)
(a) M1 cumulative frequency at 18 hours, awrt 43-47 (e.g. 45)
(a) A1 ft: subtract to give awrt 26 tablets (accept 23-29)
(a) Answer: awrt 26 tablets
(b) B1 ft: awrt 52% (accept range consistent with part (a), e.g. 46-58%)
(b) Answer: awrt 52%
Question 18
B1 the range only uses the two extreme values (the minimum and the maximum), so it can be greatly affected by an unusually small or large value (an outlier/anomaly) oe
B1 the interquartile range is based on the middle 50% of the data, so it is not affected by extreme values/outliers and gives a more reliable/representative measure of spread oe
Answer: The IQR is less affected by outliers/extreme values than the range, since it only considers the middle 50% of the data.