This question checks some key vocabulary used with cumulative frequency.
(Total for Question 1 is 1 mark)
2
60 people took part in a puzzle challenge. The table shows the time, t minutes, each person took to complete a puzzle.
Time, t (minutes)
Frequency
0 < t ≤ 5
6
5 < t ≤ 10
14
10 < t ≤ 15
22
15 < t ≤ 20
12
20 < t ≤ 25
6
(Total for Question 2 is 2 marks)
3
Using the completed cumulative frequency table from Question 2 (60 people; t ≤ 15 has cumulative frequency 42), answer the following.
(Total for Question 3 is 2 marks)
4
A cumulative frequency graph for the delivery times, in minutes, of 200 parcels gives these estimates: lower quartile = 14 minutes, median = 21 minutes, upper quartile = 33 minutes.
(a)Write down the interquartile range of the delivery times.(1)
(b)By comparing the distances of the median from each quartile, state whether the distribution of delivery times is positively skewed or negatively skewed. Give a reason.(2)
(Total for Question 4 is 3 marks)
5
A set of exam marks has minimum 22, lower quartile 48, median 61, upper quartile 77 and maximum 95.
(a)Write down the interquartile range.(1)
(b)Write down the range.(1)
(c)The range will always be greater than or equal to the interquartile range for any data set. Explain why.(1)
(Total for Question 5 is 3 marks)
6
The five-number summary for the length, in minutes, of 90 phone calls is:
Minimum = 1, Lower quartile = 4, Median = 7, Upper quartile = 12, Maximum = 20
(Total for Question 6 is 4 marks)
7
The box plot below shows the number of goals scored by a football team in each of its last 32 matches.
(a)Write down the median number of goals scored.(1)
(b)Work out the interquartile range of the number of goals scored.(1)
(c)The team is said to have had an "attacking game" if it scored more goals than the upper quartile. Estimate the number of matches, out of the 32, in which the team had an attacking game.(2)
(Total for Question 7 is 4 marks)
8
Zara works for a courier company. The table shows the mass, m kg, of each of 80 parcels she delivered in a week.
Mass, m (kg)
Frequency
0 < m ≤ 2
8
2 < m ≤ 4
18
4 < m ≤ 6
26
6 < m ≤ 8
20
8 < m ≤ 10
8
(a)Complete the cumulative frequency table below.
Mass, m (kg)
Cumulative frequency
m ≤ 2
8
m ≤ 4
...
m ≤ 6
...
m ≤ 8
...
m ≤ 10
80
(2)
(b)On the grid provided, draw a cumulative frequency graph for this data.(2)
(c)Use your graph to estimate the median mass of the parcels.(1)
(Total for Question 8 is 5 marks)
9
The cumulative frequency graph for Zara's 80 parcels (from Question 8) is shown below. Parcels with a mass greater than 7 kg require a large-parcel label.
(Total for Question 9 is 4 marks)
10
For a sample of house prices, the lower quartile is £180,000 and the upper quartile is £340,000. An estate agent identifies outliers using the rule: a value is an outlier if it is more than 1.5 x IQR below the lower quartile, or more than 1.5 x IQR above the upper quartile.
(a)Work out the interquartile range.(1)
(b)Work out the lower and upper boundaries for outliers.(2)
(Total for Question 10 is 3 marks)
11
Two classes, 10A and 10B, sat the same maths test out of 100. Their results are summarised in the box plots below.
10A: minimum = 35, lower quartile = 52, median = 68, upper quartile = 79, maximum = 95 10B: minimum = 40, lower quartile = 60, median = 63, upper quartile = 70, maximum = 100
(a)Write down the range of test scores for each class.(2)
(b)Since the two classes have the same range, a student concludes that they must have the same spread of scores. Compare the distributions of test scores for the two classes to show whether this conclusion is correct. You must use appropriate statistics and refer to the context.(3)
(Total for Question 11 is 5 marks)
12
The cumulative frequency graph below shows the waiting times, w minutes, for 120 patients at a clinic.
(a)Use the graph to estimate the median waiting time.(2)
(b)The clinic manager wants to know the waiting time that only the slowest 10% of patients exceeded (the 90th percentile). Estimate this waiting time, and interpret your answer in context.(2)
(Total for Question 12 is 4 marks)
13
The distances, d km, travelled to work by 100 employees of a company were recorded.
Distance, d (km)
Frequency
0 < d ≤ 5
14
5 < d ≤ 10
28
10 < d ≤ 15
32
15 < d ≤ 20
18
20 < d ≤ 25
8
(a)Complete the cumulative frequency table below.
Distance, d (km)
Cumulative frequency
d ≤ 5
14
d ≤ 10
...
d ≤ 15
...
d ≤ 20
...
d ≤ 25
100
(2)
(b)State the x-coordinate you would use to plot the cumulative frequency for the class 10 < d ≤ 15.(1)
(c)Use the graph to estimate the median distance travelled.(1)
(d)Use the graph to estimate the interquartile range of the distances travelled.(2)
(Total for Question 13 is 6 marks)
14
A distribution of delivery-driver ages has median 40, lower quartile 25 and upper quartile 44.
(a)Find the interquartile range.(1)
(b)By comparing the distance from the median to each quartile, comment on the skew of the distribution.(2)
(Total for Question 14 is 3 marks)
15
A box plot for the annual salaries, in £, of 40 employees at a small company has: minimum (excluding outliers) = £18,000, lower quartile = £24,000, median = £29,000, upper quartile = £38,000, maximum (excluding outliers) = £52,000, with one additional outlier plotted at £95,000.
(a)Using the rule that a value is an outlier if it is more than 1.5 x IQR above the upper quartile, show that £95,000 is an outlier.(3)
(b)Write down the range of the data, including the outlier.(1)
(Total for Question 15 is 4 marks)
16
Kwame, a horticultural researcher, grew 120 sunflowers as part of a trial. The table shows their heights, h cm, at the end of the trial.
Height, h (cm)
Frequency
100 < h ≤ 120
10
120 < h ≤ 140
24
140 < h ≤ 160
46
160 < h ≤ 180
30
180 < h ≤ 200
10
(a)Complete the cumulative frequency table below.
Height, h (cm)
Cumulative frequency
h ≤ 120
10
h ≤ 140
...
h ≤ 160
...
h ≤ 180
...
h ≤ 200
120
(2)
(b)Use the graph to estimate the median height.(1)
(c)Use the graph to estimate the interquartile range of the heights.(2)
(d)A sunflower is classed as exceptionally tall if its height is more than 1.5 x IQR above the upper quartile. Determine the height threshold for this, and state what this tells you about the sunflowers in this trial.(1)
(Total for Question 16 is 6 marks)
17
Priya, a delivery analyst, believes that Courier A's delivery times are more consistent than Courier B's. Cumulative frequency graphs for 50 deliveries by each courier gave these estimates:
(a)Calculate the interquartile range for each courier.(2)
(b)Does the data support Priya's claim? Justify your answer with reference to the interquartile ranges.(2)
(c)State one limitation of using the interquartile range alone to compare the consistency of the two couriers.(1)
(Total for Question 17 is 5 marks)
18
Kwame, a quality controller at a battery factory, tested the battery life, in hours, of a sample of 200 batteries. He found: the median battery life is 18 hours; the interquartile range is 6 hours; 25% of batteries lasted less than 15 hours; the range of the data is 20 hours; the shortest-lasting battery lasted 6 hours.
(a)Write down the five-number summary for this data.(3)
(b)The manufacturer claims: "At least 75% of our batteries last longer than 14 hours." Using your five-number summary, determine whether this claim is definitely true, definitely false, or cannot be determined. Justify your answer fully.(4)
(Total for Question 18 is 7 marks)
Mark scheme · 6.11D Cumulative Frequency and Box Plots: Fluency and Exam Drill
Question 1
B1 oe: the running total of frequencies up to (and including) that class interval/value
Answer: The running total of all the frequencies up to and including that class interval.
Question 2
B1 at least one of t ≤ 15 : 42 or t ≤ 20 : 54 correct, with a correct running-total method shown
B1 all three values correct: t ≤ 10 : 20, t ≤ 15 : 42, t ≤ 20 : 54
Answer: t ≤ 10 : 20, t ≤ 15 : 42, t ≤ 20 : 54 (t ≤ 5 : 6 and t ≤ 25 : 60 were already given)
(b) B1 ft: positively skewed, since the upper quartile is further from the median than the lower quartile is oe
(b) Answer: Positively skewed
Question 5
(a) B1 29 cao
(a) Answer: 29
(b) B1 73 cao
(b) Answer: 73
(c) B1 oe: the IQR only measures the spread of the middle 50% of the data, so it excludes the most extreme values, which can only make the range bigger (or equal), never smaller
(c) Answer: The IQR measures the spread of only the middle 50% of the data, excluding the most extreme values, so it can never be more than the full range from minimum to maximum.
Question 6
B1 box drawn correctly from the lower quartile (4) to the upper quartile (12)
B1 median (7) correctly marked inside the box
B1 whiskers drawn correctly from the box to the minimum (1) and to the maximum (20)
B1 correct linear scale used with the axis labelled (length of call, minutes)
Answer: Box plot with min 1, LQ 4, median 7, UQ 12, max 20 (minutes)
Question 7
(a) B1 2 cao
(a) Answer: 2 goals
(b) B1 3 cao
(b) Answer: 3 goals
(c) M1 recognising that exactly 25% of matches lie above the upper quartile, i.e. 1/4 x 32 oe
(c) A1 8 cao
(c) Answer: 8 matches
Question 8
(a) B1 at least two of m ≤ 4 : 26, m ≤ 6 : 52, m ≤ 8 : 72 correct
(a) B1 all three values correct: m ≤ 4 : 26, m ≤ 6 : 52, m ≤ 8 : 72
(a) Answer: m ≤ 4 : 26, m ≤ 6 : 52, m ≤ 8 : 72
(b) M1 ft: points plotted at the upper class boundary for each class, using their table
(b) A1 ft: a correct smooth increasing curve (or line segments) drawn through all six points, including (0,0)
(b) Answer: Curve through (0,0), (2,8), (4,26), (6,52), (8,72), (10,80).
(c) B1 ft: awrt 5.1 kg (accept 4.9 - 5.3 kg read from a correctly drawn graph)
(c) Answer: approx 5.1 kg (accept 4.9 - 5.3 kg)
Question 9
M1 using the graph (or interpolation) to find the cumulative frequency at m = 7, e.g. 52 + 1/2 x 20 oe
A1 CF(7) = awrt 62 (accept 58-64 read from the graph)
M1 dep: 80 - their CF(7)
A1 ft: 18 cao (accept 16-20 read from the graph)
Answer: approx 18 parcels (accept 16-20)
Question 10
(a) B1 £160,000 cao
(a) Answer: £160,000
(b) M1 1.5 x their IQR (= 240,000) used correctly in at least one boundary calculation
(b) A1 lower boundary = -£60,000 and upper boundary = £580,000, both cao (ft)
(b) Answer: Lower boundary = -£60,000 (so no house price could be a low outlier); upper boundary = £580,000
Question 11
(a) B1 10A range = 60 cao
(a) B1 10B range = 60 cao
(a) Answer: 10A: 60; 10B: 60 (the ranges are equal)
(b) B1 IQR(10A) = 79 - 52 = 27 and IQR(10B) = 70 - 60 = 10, both correct
(b) B1 conclusion in context: 10A's IQR is much greater than 10B's, so 10A's scores (the middle 50%) were far more spread out/varied than 10B's, even though the ranges are equal
(b) B1 comparison of medians in context: 10A's median (68) is higher than 10B's (63), so 10A typically scored higher/better
(b) Answer: The conclusion is wrong: although the ranges are equal, 10A's IQR (27) is much larger than 10B's (10), so 10A's scores were far more spread out/less consistent in the middle 50%; 10A also had the higher median (68 vs 63), so typically scored higher.
Question 12
(a) M1 identifying the median position as the 60th value (n/2 = 60) and interpolating between (20,36) and (30,80)
(a) A1 awrt 25.5 minutes (accept 25.0-26.0)
(a) Answer: approx 25.5 minutes
(b) M1 identifying the 90th percentile position as 0.9 x 120 = 108th value
(b) A1 40 minutes cao, with interpretation, e.g. 90% of patients waited 40 minutes or less oe
(b) Answer: 40 minutes; so 90% of patients waited 40 minutes or less (only 10% waited longer than 40 minutes).
Question 13
(a) B1 at least two of d ≤ 10 : 42, d ≤ 15 : 74, d ≤ 20 : 92 correct
(a) B1 all three values correct: d ≤ 10 : 42, d ≤ 15 : 74, d ≤ 20 : 92
(a) Answer: d ≤ 10 : 42, d ≤ 15 : 74, d ≤ 20 : 92
(b) B1 15 cao
(b) Answer: 15
(c) B1 ft: awrt 11.3 km (accept 11.0-11.5)
(c) Answer: approx 11.3 km
(d) M1 ft: correct interpolation (or graph reading) for both the lower quartile (25th value, awrt 7.0 km) and the upper quartile (75th value, awrt 15.3 km)
(a) A1 cso: 95,000 > 59,000, so £95,000 is an outlier
(a) Answer: Upper boundary = £59,000; since £95,000 > £59,000, it is an outlier.
(b) B1 £77,000 cao
(b) Answer: £77,000
Question 16
(a) B1 at least two of h ≤ 140 : 34, h ≤ 160 : 80, h ≤ 180 : 110 correct
(a) B1 all three values correct: h ≤ 140 : 34, h ≤ 160 : 80, h ≤ 180 : 110
(a) Answer: h ≤ 140 : 34, h ≤ 160 : 80, h ≤ 180 : 110
(b) B1 ft: awrt 151.3 cm (accept 150.8-151.8)
(b) Answer: approx 151.3 cm
(c) M1 ft: correct interpolation (or graph reading) for both the lower quartile (30th value, awrt 136.7 cm) and the upper quartile (90th value, awrt 166.7 cm)
(c) A1 ft: 30.0 cm cao (accept 29.0-31.0)
(c) Answer: 30.0 cm
(d) B1 ft: awrt 211.7 cm (accept 209-215), with the observation that since the tallest sunflower recorded was only 200 cm, no sunflower in this trial reached the exceptionally-tall threshold
(d) Answer: approx 211.7 cm; since no sunflower in the trial was taller than 200 cm, none reached this threshold.
(b) B1 ft: recognising that a smaller IQR means the middle 50% of times are less spread out, i.e. more consistent
(b) B1 ft: yes, the claim is supported, since Courier A's IQR (11) is much smaller than Courier B's (28), so Courier A's delivery times are more consistent
(b) Answer: Yes, the data supports the claim: Courier A's IQR (11 minutes) is much smaller than Courier B's (28 minutes), so Courier A's delivery times are more consistent.
(c) B1 oe: the IQR ignores the top and bottom 25% of deliveries, so it does not account for any unusually early or late deliveries (outliers) in the full data set
(c) Answer: The IQR only describes the middle 50% of deliveries, so it ignores any very fast or very slow deliveries in the top and bottom 25%.
Question 18
(a) M1 maximum = 6 + 20 (= 26), using range = maximum - minimum
(a) A1 all five values correct: minimum 6, lower quartile 15, median 18, upper quartile 21, maximum 26
(a) Answer: Minimum = 6, lower quartile = 15, median = 18, upper quartile = 21, maximum = 26 (all in hours)
(b) M1 recognising that the lower quartile of 15 hours means exactly 25% of batteries lasted less than 15 hours, so the remaining 75% lasted 15 hours or more
(b) M1 recognising that 15 hours is itself greater than 14 hours
(b) A1 concluding that every battery in that 75% (lasting 15 hours or more) therefore also lasted longer than 14 hours
(b) A1 cso: the claim is definitely TRUE, since at least 75% (in fact exactly the top 75%) of batteries lasted 15 hours or more, which is more than 14 hours