This question checks some key vocabulary used with box plots.
(a)Which of the following best describes the interquartile range (IQR) of a data set?
A) The highest value minus the lowest value B) The upper quartile minus the lower quartile C) The median minus the lower quartile D) The mean of all the data values(1)
A) The highest value minus the lowest value
B) The upper quartile minus the lower quartile
C) The median minus the lower quartile
D) The mean of all the data values
(b)State what is meant by the median of a data set.(1)
(Total for Question 1 is 2 marks)
2
A hockey team's goal totals in each of their 9 matches this season were:
3, 5, 5, 7, 9, 10, 11, 13, 15
(a)Find the median number of goals.(1)
(b)Find the range of the number of goals.(1)
(c)Find the lower quartile.(1)
(d)Find the upper quartile.(1)
(e)Find the interquartile range.(1)
(Total for Question 2 is 5 marks)
3
The box plot below shows the time, in minutes, that a sample of Year 11 students spent on their maths homework one evening.
Minimum = 10, Lower quartile = 25, Median = 35, Upper quartile = 50, Maximum = 65
(a)Write down the median homework time.(1)
(b)Work out the range of the homework times.(1)
(c)Work out the interquartile range of the homework times.(2)
(Total for Question 3 is 4 marks)
4
In a data set, the median is 18 and the maximum value is 30.
A) 15
B) 20
C) 24
D) 29
(Total for Question 4 is 1 mark)
5
During a charity walking challenge, one participant's five-number summary for the distances, in km, walked on each day of the challenge is:
Minimum = 4 km, Lower quartile = 9 km, Median = 14 km, Upper quartile = 19 km, Maximum = 23 km
(Total for Question 5 is 3 marks)
6
Amelia recorded the number of text messages she sent on each of 13 days:
2, 5, 6, 6, 7, 8, 9, 9, 10, 12, 14, 15, 20
(a)Find the median number of texts sent.(1)
(b)Find the lower quartile and the upper quartile of the number of texts sent.(2)
(c)Hence find the interquartile range of the number of texts sent.(1)
(d)Draw a box plot to represent Amelia's data on the grid provided.(3)
(Total for Question 6 is 7 marks)
7
Box plots for the percentage exam marks of two classes are summarised below.
Class A: minimum = 35, lower quartile = 50, median = 62, upper quartile = 75, maximum = 95 Class B: minimum = 40, lower quartile = 58, median = 70, upper quartile = 80, maximum = 90
(a)Work out the interquartile range for Class A and for Class B.(2)
(b)Make one comparison about the average mark, and one comparison about the consistency of marks, between the two classes. You must refer to the context in both comparisons.(2)
(Total for Question 7 is 4 marks)
8
A call centre records the time, in minutes, taken for calls to be answered. The box plot for a large sample of calls has the following five-number summary.
Minimum = 5, Lower quartile = 15, Median = 22, Upper quartile = 34, Maximum = 50
For each statement below, write True, False or Cannot Tell, giving a brief reason.
(a)"More than half of the calls were answered in less than 20 minutes."(1)
(b)"The middle 50% of call answering times span 19 minutes."(1)
(c)"No call took longer than 50 minutes to be answered."(1)
(Total for Question 8 is 3 marks)
9
A box plot has been drawn for a set of data.
(Total for Question 9 is 1 mark)
10
The box plot below shows the ages, in years, of members at a running club.
Minimum = 10, Lower quartile = 20, Median = 22, Upper quartile = 40, Maximum = 44
(Total for Question 10 is 2 marks)
11
A list of 8 numbers, written in ascending order, is:
12, 15, 18, x, 24, 27, 30, 33
The median of the 8 numbers is 21.
(a)Find the value of x.(2)
(b)Using your value of x, find the interquartile range of the 8 numbers.(2)
(Total for Question 11 is 4 marks)
12
In a marathon, the finishing times of a large group of runners are shown on a box plot. The upper quartile finishing time is 4 hours 15 minutes, and 45 runners finished in a time longer than this.
(a)Work out the total number of runners in the group.(2)
(b)Hence work out the number of runners who finished in less than the median time.(1)
(Total for Question 12 is 3 marks)
13
As part of a fitness survey, the distances, in km, cycled in one week by 20 members of a cycling club were recorded, and are shown below in ascending order.
(a)Find the median, the lower quartile, the upper quartile and the interquartile range of these distances.(4)
(b)Draw a box plot to represent this data on the grid provided.(3)
(c)Describe the skewness of the distribution of distances cycled, giving a reason for your answer.(2)
(Total for Question 13 is 9 marks)
14
A gym recorded the number of visits made in a month by a sample of 11 members of Gym A:
3, 4, 4, 6, 7, 8, 8, 9, 11, 13, 18
The box plot for a sample of members of Gym B has the following five-number summary: minimum = 2, lower quartile = 6, median = 9, upper quartile = 12, maximum = 20.
(a)Find the median and the interquartile range for Gym A's 11 members.(3)
(b)Compare the number of gym visits made by members of Gym A and Gym B, making two comparisons in context.(2)
(Total for Question 14 is 5 marks)
15
A company's weekly sales figures, in GBP, for a product across all of its shops are shown on a box plot. The upper quartile is GBP 1200 and the maximum is GBP 2000. It is known that 18 shops had weekly sales between the upper quartile and the maximum.
(a)Work out the total number of shops in the sample.(2)
(b)The lower quartile of weekly sales is GBP 700. A sales manager claims that 9 shops had weekly sales below this figure. State, with a reason, whether this claim is consistent with the box plot.(1)
(Total for Question 15 is 3 marks)
16
A list of 10 numbers, written in ascending order, is:
4, 7, 10, 13, x, 20, 23, 26, 29, y
The median of the 10 numbers is 17, and the range of the 10 numbers is 35.
(a)Find the value of x.(2)
(b)Find the value of y.(2)
(c)Hence find the interquartile range of the 10 numbers.(2)
(Total for Question 16 is 6 marks)
17
Six positive integers, a, b, c, d, e, f, are written in ascending order. The minimum value is 6, the range is 22, the median is 15, the interquartile range is 10, and the mean of all six numbers is 17.
(a)Find the value of f (the maximum).(1)
(b)Find the values of the lower quartile (b) and the upper quartile (e).(3)
(Total for Question 17 is 4 marks)
18
Two box plots, one for Data Set P and one for Data Set Q, have exactly the same five-number summary (minimum, lower quartile, median, upper quartile and maximum).
(Total for Question 18 is 2 marks)
Mark scheme · 6.12 Box Plots
Question 1
(a) B1 B cao
(a) Answer: B
(b) B1 oe: the middle value when the data are arranged in order of size
(b) Answer: The middle value when the data are arranged in ascending order.
Question 2
(a) B1 9 cao
(a) Answer: 9 goals
(b) B1 12 cao
(b) Answer: 12 goals
(c) B1 5 cao
(c) Answer: 5 goals
(d) B1 12 cao
(d) Answer: 12 goals
(e) B1 ft: 7 cao
(e) Answer: 7 goals
Question 3
(a) B1 35 minutes cao
(a) Answer: 35 minutes
(b) B1 55 minutes cao
(b) Answer: 55 minutes
(c) M1 50 - 25 oe
(c) A1 25 minutes cao
(c) Answer: 25 minutes
Question 4
B1 A cao, with the reason that the upper quartile must be greater than or equal to the median (18)
Answer: A
Question 5
B1 box drawn correctly from the lower quartile (9) to the upper quartile (19), with the median (14) marked inside the box
B1 whiskers drawn correctly from the box to the minimum (4) and the maximum (23)
B1 correct linear scale used and axis labelled (distance, km)
Answer: Box plot with min 4, LQ 9, median 14, UQ 19, max 23 (km)
Question 6
(a) B1 9 cao
(a) Answer: 9 texts
(b) M1 splitting the data into a lower half (2,5,6,6,7,8) and an upper half (9,10,12,14,15,20) and finding the median of each half
(b) A1 LQ = 6 and UQ = 13, both cao
(b) Answer: LQ = 6 texts, UQ = 13 texts
(c) B1 ft: 7 cao
(c) Answer: 7 texts
(d) B1 ft: whiskers drawn correctly from the minimum (2) to the box, and from the box to the maximum (20)
(d) B1 ft: box drawn correctly from the lower quartile (6) to the upper quartile (13)
(d) B1 ft: median (9) correctly marked inside the box
(d) Answer: Box plot with min 2, LQ 6, median 9, UQ 13, max 20
Question 7
(a) B1 Class A IQR = 25 cao
(a) B1 Class B IQR = 22 cao
(a) Answer: Class A: 25; Class B: 22
(b) B1 median comparison in context, e.g. Class B's median (70) is higher than Class A's median (62), so Class B typically scored higher marks oe
(b) B1 ft: IQR comparison in context, e.g. Class A's IQR (25) is greater than Class B's (22), so Class A's marks were more spread out/less consistent than Class B's oe
(b) Answer: Class B typically scored higher (higher median); Class A's marks were more varied/less consistent (larger IQR).
Question 8
(a) B1 Cannot tell, with reason: the median (22) tells us 50% of calls were answered in less than 22 minutes, but 20 is not one of the known five-number summary values, so the proportion below 20 cannot be found oe
(a) Answer: Cannot tell
(b) B1 True, since IQR = 34 - 15 = 19
(b) Answer: True
(c) B1 True, since the maximum value recorded was 50 minutes
(c) Answer: True
Question 9
B1 oe: a box plot only shows the minimum, lower quartile, median, upper quartile and maximum, not the individual data values (or their total), so the exact mean cannot be calculated from it
Answer: The mean cannot be found because a box plot only shows the five-number summary, not the actual data values or their total.
Question 10
B1 correct conclusion: the distribution is positively skewed (skewed to the right, towards older ages) oe
B1 valid reason, e.g. the gap between the upper quartile and the median (40 - 22 = 18) is much bigger than the gap between the median and the lower quartile (22 - 20 = 2), showing greater spread among the older members oe
Answer: Positively (right) skewed
Question 11
(a) M1 (x + 24)/2 = 21 oe
(a) A1 x = 18 cao
(a) Answer: x = 18
(b) M1 ft: identifying LQ from the lower half (12,15,18,18) and UQ from the upper half (24,27,30,33)
(b) A1 ft: IQR = 12 cao
(b) Answer: IQR = 12
Question 12
(a) M1 0.25 x n = 45 oe, recognising that 25% of runners finish after the upper quartile
(a) A1 n = 180 cao
(a) Answer: 180 runners
(b) B1 ft: 90 cao
(b) Answer: 90 runners
Question 13
(a) M1 correct method for the median with n = 20, using the mean of the 10th and 11th values
(a) A1 median = 32 km cao
(a) M1 correct method for the quartiles, splitting the data into two halves of 10 values and finding the mean of the 5th and 6th value in each half
(a) A1 LQ = 23 km, UQ = 42 km and IQR = 19 km, all cao
(a) Answer: Median = 32 km, LQ = 23 km, UQ = 42 km, IQR = 19 km
(b) B1 ft: box drawn correctly from the lower quartile (23) to the upper quartile (42)
(b) B1 ft: whiskers drawn correctly from the box to the minimum (12) and the maximum (60)
(b) B1 ft: median (32) correctly marked inside the box, with correct scale and axis labelled
(b) Answer: Box plot with min 12, LQ 23, median 32, UQ 42, max 60 (km)
(c) B1 ft: correct conclusion, e.g. the distribution is approximately symmetric (only very slight positive skew) oe
(c) B1 ft: valid reason comparing the two gaps, e.g. median - LQ = 32 - 23 = 9 and UQ - median = 42 - 32 = 10, which are very close in size, so the data are roughly evenly spread either side of the median oe
(c) Answer: Approximately symmetric (only a very slight positive skew)
Question 14
(a) B1 median = 8 cao
(a) M1 identifying LQ from the lower half (3,4,4,6,7) and UQ from the upper half (8,9,11,13,18)
(a) A1 IQR = 7 cao
(a) Answer: Median = 8 visits, IQR = 7 visits
(b) B1 ft: median comparison in context, e.g. Gym B's median (9) is slightly higher than Gym A's median (8), so Gym B members typically visit slightly more often oe
(b) B1 ft: IQR comparison in context, e.g. Gym A's IQR (7) is greater than Gym B's IQR (6), so the number of visits made by Gym A members is slightly more variable/less consistent than Gym B's oe
(b) Answer: Gym B members typically visit slightly more often (higher median); Gym A's visit numbers are slightly more variable (larger IQR).
Question 15
(a) M1 0.25 x n = 18 oe, recognising 18 shops represents the top 25% of the sample
(a) A1 n = 72 cao
(a) Answer: 72 shops
(b) B1 ft: not consistent, since 25% of 72 shops = 18 shops should lie below the lower quartile, not 9
(b) Answer: Not consistent: 18 shops (25% of 72), not 9, should have sales below the lower quartile.
Question 16
(a) M1 (x + 20)/2 = 17 oe
(a) A1 x = 14 cao
(a) Answer: x = 14
(b) M1 y - 4 = 35 oe
(b) A1 y = 39 cao
(b) Answer: y = 39
(c) M1 ft: identifying LQ from the lower half (4,7,10,13,14) and UQ from the upper half (20,23,26,29,39)
(c) A1 ft: IQR = 16 cao
(c) Answer: IQR = 16
Question 17
(a) B1 f = 28 cao
(a) Answer: f = 28
(b) M1 using the mean to form total sum = 6 x 17 = 102, and the median to give c + d = 30, to obtain b + e = 102 - 6 - 28 - 30 = 38
(b) dM1 combining b + e = 38 with e - b = 10 (the interquartile range) to solve simultaneously
B1 correct conclusion: No, the data sets do not have to be identical
B1 valid justification, e.g. a box plot only summarises five key values; the individual data values between the minimum and lower quartile, between the quartiles, and between the upper quartile and maximum could differ between the two sets while still producing the same five-number summary oe
Answer: No; a box plot only shows five summary values, so two different data sets can share the same five-number summary without containing identical data.