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GCSE · Statistics

S29 Cumulative Frequency

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

For each statement about cumulative frequency graphs, write down whether it is True or False.

(i)A cumulative frequency graph can be used to find estimates for the median and the quartiles of grouped data.(1)

(ii)As the value of the variable increases, the cumulative frequency can decrease.(1)

(iii)Each point on a cumulative frequency graph is plotted at the upper boundary of its class interval.(1)

(Total for Question 1 is 3 marks)

70 students were asked how many minutes, m, they spent on a revision app one evening. The grouped frequency table shows the results.

Time, m (minutes)	Frequency
0 < m ≤ 20	6
20 < m ≤ 40	12
40 < m ≤ 60	18
60 < m ≤ 80	20
80 < m ≤ 100	10
100 < m ≤ 120	4

(a)Complete the cumulative frequency table below. Some values have already been filled in.

Time, m (minutes)	Cumulative frequency
m ≤ 20	6
m ≤ 40	...
m ≤ 60	36
m ≤ 80	...
m ≤ 100	66
m ≤ 120	70

(2)

(b)Write down the coordinates you would plot on a cumulative frequency graph to represent (i) the class 60 < m ≤ 80, and (ii) the class 100 < m ≤ 120.(2)

(Total for Question 2 is 4 marks)

Using your completed table from Question 2, draw a cumulative frequency graph on the grid below.

(Total for Question 3 is 3 marks)

The cumulative frequency graph shows the heights, h cm, of 100 sunflower plants grown in a school garden.

(a)Use the graph to find an estimate for the median height of the sunflower plants.(2)

(b)Use the graph to estimate the number of plants with a height of at most 130 cm.(2)

(Total for Question 4 is 4 marks)

The same cumulative frequency graph for the 100 sunflower plants is shown again below.

(a)Use the graph to estimate the number of plants with a height of more than 170 cm.(2)

(b)Use the graph to estimate the number of plants with a height between 110 cm and 150 cm.(3)

(Total for Question 5 is 5 marks)

120 employees at a company were asked their annual salary, s (GBP). The grouped frequency table shows the results.

Salary, s (GBP)	Frequency
15000 < s ≤ 25000	15
25000 < s ≤ 35000	25
35000 < s ≤ 45000	40
45000 < s ≤ 55000	25
55000 < s ≤ 65000	10
65000 < s ≤ 75000	5

(a)Complete the cumulative frequency table below. Some values have already been filled in.

Salary, s (GBP)	Cumulative frequency
s ≤ 25000	15
s ≤ 35000	...
s ≤ 45000	80
s ≤ 55000	...
s ≤ 65000	115
s ≤ 75000	120

(2)

(b)Write down the coordinates you would plot on a cumulative frequency graph to represent (i) the class 45000 < s ≤ 55000, and (ii) the class 65000 < s ≤ 75000.(2)

(Total for Question 6 is 4 marks)

The cumulative frequency graph for the 100 sunflower plants is shown again below.

(a)Use the graph to find an estimate for the lower quartile of the heights.(1)

(b)Use the graph to find an estimate for the upper quartile of the heights.(1)

(c)Hence work out an estimate for the interquartile range of the heights.(2)

(Total for Question 7 is 4 marks)

The shortest plant measured had a height of 85 cm and the tallest plant measured had a height of 198 cm. Using your estimates from Question 7 (lower quartile 120 cm, median 137 cm, upper quartile 156 cm), draw a box plot for the heights of the sunflower plants on the grid below.

(Total for Question 8 is 3 marks)

The box plots show the delivery times, in minutes, for a sample of parcels delivered by Company A and by Company B.

(a)Work out the interquartile range of the delivery times for (i) Company A and (ii) Company B.(2)

(b)Compare the delivery times of the two companies. Give your answer in the context of the question.(2)

(Total for Question 9 is 4 marks)

The cumulative frequency graph shows the annual salaries, s (GBP), of the 120 employees from Question 6.

(a)Use the graph to find an estimate for the median annual salary.(2)

(b)Use the graph to find estimates for the lower quartile and the upper quartile of the salaries.(2)

(c)Hence write down an estimate for the interquartile range of the salaries.(1)

(Total for Question 10 is 5 marks)

A value is considered an outlier if it is more than 1.5 x IQR above the upper quartile, or more than 1.5 x IQR below the lower quartile. Use your estimates from Question 7 (lower quartile 120 cm, upper quartile 156 cm, interquartile range 36 cm).

(a)Work out the upper and lower outlier fences for the heights.(2)

(b)Show that neither the shortest plant (85 cm) nor the tallest plant (198 cm) is an outlier.(2)

(Total for Question 11 is 4 marks)

The lowest salary recorded was GBP 18,000 and the highest salary recorded was GBP 92,000. Using your estimates from Question 10 (lower quartile GBP 31,000, median GBP 40,000, upper quartile GBP 49,000), draw a box plot for the salaries on the grid below.

(Total for Question 12 is 3 marks)

Using the outlier rule from Question 11 (a value is an outlier if it is more than 1.5 x IQR above the upper quartile or below the lower quartile), and your estimates from Question 10 (lower quartile GBP 31,000, upper quartile GBP 49,000, interquartile range GBP 18,000):

(a)Work out the upper outlier fence for the salaries.(2)

(b)One employee earns GBP 92,000 a year. Determine, giving a reason, whether this salary is an outlier.(1)

(Total for Question 13 is 3 marks)

A second company, Company Y, publishes a summary of its employees' annual salaries: median GBP 42,000 and interquartile range GBP 12,000. Using your estimates for Company X from Question 10 (median GBP 40,000, interquartile range GBP 18,000), compare the two companies' salaries. Give your answer in the context of the question.

(Total for Question 14 is 3 marks)

A gardener wants to investigate whether a new fertiliser increases the height of sunflower plants. She grows 100 sunflowers without the fertiliser (the data used in Questions 4, 5, 7, 8 and 11) and 100 sunflowers with the fertiliser, then compares the two groups using the statistical enquiry cycle.

(a)PLAN: Write down a suitable hypothesis for the gardener's investigation.(1)

(b)COLLECT: Give one step the gardener should take when growing the two groups of plants so that the comparison is fair, and explain why it matters.(2)

(c)PROCESS: Explain why the gardener grouped the raw height measurements into class intervals before drawing a cumulative frequency graph.(1)

(d)INTERPRET: For the fertilised group, the median height was 152 cm, the lower quartile was 132 cm and the upper quartile was 171 cm. Compare this group with the group grown without fertiliser (Question 7: median 137 cm, lower quartile 120 cm, upper quartile 156 cm), and use your comparison to evaluate the gardener's hypothesis.(3)

(Total for Question 15 is 7 marks)

The cumulative frequency graph for the 100 sunflower plants (Question 4) is shown again below.

(a)Explain why the value you found for the median height in Question 4 is only an estimate.(1)

(b)Use the graph to estimate the number of plants with a height in the class 140 < h ≤ 160.(2)

(Total for Question 16 is 3 marks)

Mark scheme · S29 Cumulative Frequency

Question 1

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 True cao
(iii) Answer: True

Question 2

(a) B1 m ≤ 40 : 18 cao
(a) B1 m ≤ 80 : 56 cao
(a) Answer: 18 and 56
(b) M1 using the upper class boundary and the cumulative frequency oe
(b) A1 (80, 56) and (120, 70) cao
(b) Answer: (80, 56) and (120, 70)

Question 3

B1 a suitable linear scale used on the cumulative frequency axis, matching the grid provided
B1 all 7 points plotted correctly: (0,0), (20,6), (40,18), (60,36), (80,56), (100,66), (120,70), ± half a small square
B1 points joined with a single smooth rising curve (or line segments), starting at (0,0)
Answer: Cumulative frequency curve rising smoothly through (0,0), (20,6), (40,18), (60,36), (80,56), (100,66), (120,70).

Question 4

(a) M1 identifying the median position as the 50th value (n/2 = 50) and reading/interpolating from the curve
(a) A1 awrt 137 (accept 135-139)
(a) Answer: awrt 137 cm (accept 135-139 cm)
(b) M1 reading cumulative frequency at h = 130 from the curve
(b) A1 awrt 40 (accept 37-43)
(b) Answer: awrt 40 plants (accept 37-43)

Question 5

(a) M1 reading cumulative frequency at h = 170 from the curve and subtracting from 100
(a) A1 awrt 13 (accept 10-16)
(a) Answer: awrt 13 plants (accept 10-16)
(b) M1 reading cumulative frequency at h = 110 (awrt 16-17) and at h = 150 (awrt 67-68) from the curve
(b) M1 subtracting the two cumulative frequencies oe
(b) A1 awrt 51 (accept 47-55)
(b) Answer: awrt 51 plants (accept 47-55)

Question 6

(a) B1 s ≤ 35000 : 40 cao
(a) B1 s ≤ 55000 : 105 cao
(a) Answer: 40 and 105
(b) M1 using the upper class boundary and the cumulative frequency oe
(b) A1 (55000, 105) and (75000, 120) cao
(b) Answer: (55000, 105) and (75000, 120)

Question 7

(a) B1 120 cao (accept 118-122)
(a) Answer: 120 cm
(b) B1 156 cao (accept 154-158)
(b) Answer: 156 cm
(c) M1 ft: upper quartile minus lower quartile
(c) A1 36 cao (accept 32-40)
(c) Answer: 36 cm

Question 8

B1 ft: whiskers drawn from 85 to 120 and from 156 to 198
B1 ft: box drawn from 120 to 156
B1 ft: median marked at 137 inside the box
Answer: Box plot: minimum 85, LQ 120, median 137, UQ 156, maximum 198.

Question 9

(a) A1 Company A: 25 - 12 = 13 minutes
(a) A1 Company B: 21 - 15 = 6 minutes
(a) Answer: Company A: 13 minutes; Company B: 6 minutes
(b) B1 comparison of median: Company A has a slightly higher median (18 minutes) than Company B (17 minutes) oe
(b) B1 comparison of spread in context: Company B has a smaller interquartile range, so Company B's delivery times are more consistent (less varied) than Company A's, despite similar medians
(b) Answer: Similar median delivery times, but Company B is more consistent (smaller IQR).

Question 10

(a) M1 identifying the median position as the 60th value (n/2 = 60) and reading/interpolating from the curve
(a) A1 GBP 40000 cao (accept 39000-41000)
(a) Answer: GBP 40,000
(b) B1 lower quartile = GBP 31000 (accept 30000-32000)
(b) B1 upper quartile = GBP 49000 (accept 48000-50000)
(b) Answer: LQ = GBP 31,000, UQ = GBP 49,000
(c) B1 ft: upper quartile minus lower quartile, GBP 18000 (accept 16000-20000)
(c) Answer: GBP 18,000

Question 11

(a) M1 ft: 156 + 1.5x36 and 120 - 1.5x36
(a) A1 upper fence = 210 cm, lower fence = 66 cm
(a) Answer: Upper fence 210 cm, lower fence 66 cm
(b) B1 ft ao: 85 > 66 (the lower fence), so the shortest plant is not an outlier
(b) B1 ft ao: 198 < 210 (the upper fence), so the tallest plant is not an outlier
(b) Answer: 85 > 66 and 198 < 210, so neither is an outlier.

Question 12

B1 ft: whiskers drawn from 18000 to 31000 and from 49000 to 92000
B1 ft: box drawn from 31000 to 49000
B1 ft: median marked at 40000 inside the box
Answer: Box plot: minimum 18000, LQ 31000, median 40000, UQ 49000, maximum 92000.

Question 13

(a) M1 ft: 49000 + 1.5x18000
(a) A1 GBP 76000 cao
(a) Answer: GBP 76,000
(b) B1 ft: yes, since 92000 > 76000 (the upper fence)
(b) Answer: Yes, an outlier (92000 > 76000)

Question 14

B1 ft: comparison of medians: Company Y has a slightly higher median salary (GBP 42000) than Company X (GBP 40000)
B1 ft: comparison of spread: Company Y has a smaller interquartile range (GBP 12000) than Company X (GBP 18000)
B1 conclusion in context: Company Y's salaries are slightly higher on average and much more consistent (less variation) than Company X's
Answer: Company Y pays slightly more on average and its salaries are more consistent (smaller IQR).

Question 15

(a) B1 a sensible hypothesis comparing height with and without fertiliser, e.g. 'sunflowers grown with the fertiliser will tend to be taller than sunflowers grown without it'
(a) Answer: Sunflowers grown with the fertiliser will tend to be taller than those grown without it.
(b) B1 a valid control, e.g. use the same variety of seed, the same amount of water/sunlight, or plant both groups at the same time
(b) B1 explanation: this ensures fertiliser is the only variable that differs between the groups, so any height difference can be attributed to the fertiliser
(b) Answer: Use the same seed variety and growing conditions for both groups, so fertiliser is the only variable that differs.
(c) B1 height is continuous data with many different individual values, so grouping into class intervals makes the data manageable to record and display
(c) Answer: Height is continuous data with many different values, so grouping it makes it easier to record, summarise and display.
(d) M1 working out the interquartile range of the fertilised group: 171 - 132 = 39 cm
(d) M1 comparing medians: 152 cm > 137 cm, so the fertilised group has a higher median height
(d) A1 conclusion in context: the fertilised group tends to be taller (higher median), and the spread is similar (IQR 39 cm vs 36 cm), so this supports the hypothesis that the fertiliser increases height
(d) Answer: Fertilised group IQR = 39 cm; median is higher (152 vs 137), similar spread, so the hypothesis is supported.

Question 16

(a) B1 the original heights were grouped into class intervals, so the exact individual data values are unknown/lost, and the median is found by interpolation from the graph oe
(a) Answer: The raw data was grouped, so the exact individual heights are unknown; only an estimate can be read from the graph.
(b) M1 reading the cumulative frequency at h = 140 (awrt 55) and at h = 160 (awrt 80) from the curve
(b) A1 awrt 25 (accept 22-28)
(b) Answer: awrt 25 plants (accept 22-28)

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