Box Plots (GCSE Statistics) - Worksheets, Questions and Revision

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GCSE · Statistics

S30 Box Plots

EDEXCEL 1ST0 · Calculator allowed · about 85 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Here is a list of five statistical terms used with box plots:

median, lower quartile, upper quartile, interquartile range, range

For each description (i) to (v), write down the term from the list that matches it.

(i)The value exactly in the middle of an ordered data set.(1)

(ii)The value below which the bottom quarter of the data lies.(1)

(iii)The value below which the top quarter of the data lies (i.e. 75% of the data is below it).(1)

(iv)The difference between the upper quartile and the lower quartile.(1)

(v)The difference between the maximum and the minimum values.(1)

(Total for Question 1 is 5 marks)

For each statement about box plots, write down whether it is True or False.

(i)The line drawn inside the box represents the median.(1)

(ii)The two ends of the box mark the minimum and the maximum values in the data set.(1)

(iii)The interquartile range is not affected by a few very small or very large extreme values in the data.(1)

(Total for Question 2 is 3 marks)

The box plot shows the times, in minutes, taken by a group of 40 students to complete an exam paper.

(a)Write down the median time.(1)

(b)Write down the upper quartile time.(1)

(c)Work out the range of the times.(2)

(d)Work out the interquartile range of the times.(2)

(Total for Question 3 is 6 marks)

The box plot shows the annual salaries, in £1000s, of the 60 employees at a small company.

(a)Write down the median annual salary.(1)

(b)Work out the range of annual salaries, giving your answer in £.(2)

(c)Work out the interquartile range of annual salaries, giving your answer in £.(2)

(d)Write down the percentage of employees who earn a salary between the lower quartile and the upper quartile.(1)

(Total for Question 4 is 6 marks)

The box plots show the journey times, in minutes, of two bus routes, Bus A and Bus B, recorded over several weeks.

(a)Write down the median journey time for Bus B.(1)

(b)Work out the interquartile range for Bus A.(2)

(c)Compare the two bus routes, using the median and the interquartile range.(3)

(Total for Question 5 is 6 marks)

The box plots show the monthly rainfall, in mm, recorded in City P and City Q over a year.

(a)Write down the median monthly rainfall for City P.(1)

(b)Work out the range of monthly rainfall for City Q.(2)

(c)Compare the typical rainfall and the spread of rainfall in the two cities, using the median and the range.(2)

(Total for Question 6 is 5 marks)

The five-number summary for the heights, in cm, of 20 sunflowers grown by a gardening club is given in the table.

Minimum	Lower quartile	Median	Upper quartile	Maximum
40	55	68	80	95

On the grid, draw a box plot to show this information.

(Total for Question 7 is 3 marks)

The weights, in kg, of 11 parcels sent by a courier are shown below, in order:

2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 12

(a)Find the median, the lower quartile and the upper quartile of the weights.(3)

(b)On the grid provided, draw a box plot to show this data.(3)

(Total for Question 8 is 6 marks)

A student wants to investigate whether Year 11 students spend more time than Year 10 students on homework each week.

(a)Write a suitable hypothesis for the student's investigation.(1)

(b)Give one thing the student should keep the same for both year groups when collecting the data, so that the comparison is fair.(1)

(c)The student collects the data and summarises it using the two box plots shown, for time spent on homework (hours per week).

Compare the amount of time spent on homework by Year 10 and Year 11, using the median and the interquartile range.(3)

(d)State whether the student's data supports the hypothesis from part (a).(1)

(Total for Question 9 is 6 marks)

The box plot shows the amount of time, in minutes, that 50 customers spent in a shop.

(Total for Question 10 is 3 marks)

The box plot shows the finishing times, in seconds, of 30 runners in a sprint race. One runner's time, 52 seconds, is much slower than the rest and is plotted as a separate circle beyond the whisker, because it is an outlier.

A value is classed as 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.

(a)Show that 52 seconds is an outlier.(2)

(b)Write down the value that the upper whisker of the box plot is drawn to.(1)

(c)Write down the least finishing time recorded in the race.(1)

(Total for Question 11 is 4 marks)

The times, in minutes, taken by 13 people to complete a charity fun run are shown below, in order:

32, 35, 36, 38, 39, 40, 41, 42, 44, 45, 47, 48, 71

(a)Write down the median time.(1)

(b)Work out the lower quartile.(2)

(c)Work out the upper quartile.(2)

(d)Work out the interquartile range.(1)

(e)A value is an outlier if it is more than 1.5 x IQR above the upper quartile. Show that 71 minutes is an outlier.(2)

(f)State the value that the upper whisker of a box plot for this data would be drawn to.(1)

(Total for Question 12 is 9 marks)

The box plots show two distributions, X and Y, plotted on the same scale.

(a)State whether Distribution X is positively skewed, negatively skewed, or symmetrical. Give a reason based on the position of the median within the box.(2)

(b)State whether Distribution Y is positively skewed, negatively skewed, or symmetrical. Give a reason based on the position of the median within the box.(2)

(Total for Question 13 is 4 marks)

60 students took a puzzle-solving test. The grouped frequency table shows their completion times.

Time, t (minutes)	Frequency
0 < t ≤ 20	5
20 < t ≤ 40	15
40 < t ≤ 60	25
60 < t ≤ 80	10
80 < t ≤ 100	5

The cumulative frequency graph for this data is shown below.

(a)Use the graph to find an estimate for the median completion time.(1)

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

(c)Work out an estimate for the interquartile range.(1)

(d)Given that the fastest student finished in 10 minutes and the slowest student finished in 95 minutes, draw a box plot to show this information, using your estimates from parts (a) and (b).(3)

(Total for Question 14 is 7 marks)

The box plots show the distances thrown, in metres, by three javelin throwers, A, B and C, over a season of competitions.

(a)Work out the interquartile range for each thrower.(3)

(b)Which thrower was the most consistent? Give a reason.(1)

(c)Describe the skewness of Thrower B's distribution, giving a reason based on the box plot.(2)

(Total for Question 15 is 6 marks)

Mark scheme · S30 Box Plots

Question 1

(i) B1 median cao
(i) Answer: Median
(ii) B1 lower quartile cao
(ii) Answer: Lower quartile
(iii) B1 upper quartile cao
(iii) Answer: Upper quartile
(iv) B1 interquartile range cao
(iv) Answer: Interquartile range
(v) B1 range cao
(v) Answer: Range

Question 2

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

Question 3

(a) B1 25 cao
(a) Answer: 25 minutes
(b) B1 32 cao
(b) Answer: 32 minutes
(c) M1 45 - 10, oe
(c) A1 35 cao
(c) Answer: 35 minutes
(d) M1 32 - 18, oe
(d) A1 14 cao
(d) Answer: 14 minutes

Question 4

(a) B1 £30 000 cao
(a) Answer: £30,000
(b) M1 65 - 18, oe
(b) A1 £47 000 cao
(b) Answer: £47,000
(c) M1 42 - 24, oe
(c) A1 £18 000 cao
(c) Answer: £18,000
(d) B1 50% oe (half)
(d) Answer: 50%

Question 5

(a) B1 25 cao
(a) Answer: 25 minutes
(b) M1 35 - 20, oe
(b) A1 15 cao
(b) Answer: 15 minutes
(c) B1 Bus B has the lower median (25 < 28), so Bus B is generally faster
(c) B1 Bus B has the smaller IQR (30 - 22 = 8 compared with 15 for Bus A), so Bus B's journey times are more consistent
(c) B1 valid overall conclusion in context, e.g. Bus B is both faster on average and more reliable than Bus A
(c) Answer: Bus B is faster on average (lower median) and more consistent (smaller IQR of 8 compared with 15 for Bus A).

Question 6

(a) B1 55 cao
(a) Answer: 55 mm
(b) M1 100 - 15, oe
(b) A1 85 cao
(b) Answer: 85 mm
(c) B1 City Q has the lower median (48 < 55), so City Q typically has less rainfall per month
(c) B1 City Q has the larger range (85 > 75), so City Q's monthly rainfall is more variable
(c) Answer: City Q has lower typical rainfall (median 48 mm vs 55 mm) but more variable rainfall (range 85 mm vs 75 mm).

Question 7

B1 whiskers drawn correctly to the minimum (40) and maximum (95)
B1 box drawn from the lower quartile (55) to the upper quartile (80)
B1 median (68) marked correctly inside the box, box plot drawn accurately to scale
Answer: Box plot: min 40, LQ 55, median 68, UQ 80, max 95 (cm).

Question 8

(a) B1 median = 6 cao
(a) B1 lower quartile = 3 cao
(a) B1 upper quartile = 8 cao
(a) Answer: Median = 6 kg, lower quartile = 3 kg, upper quartile = 8 kg
(b) B1 whiskers drawn to the minimum (2) and maximum (12)
(b) B1 box drawn from the lower quartile (3) to the upper quartile (8)
(b) B1 median (6) marked correctly inside the box, ft from (a), drawn accurately to scale
(b) Answer: Box plot: min 2 kg, LQ 3 kg, median 6 kg, UQ 8 kg, max 12 kg.

Question 9

(a) B1 a testable statement comparing the two groups, e.g. 'Year 11 students spend more time on homework per week, on average, than Year 10 students', oe
(a) Answer: Year 11 students spend more time on homework per week, on average, than Year 10 students.
(b) B1 any valid control, e.g. ask about the same week/time period, use the same question wording, use the same sample size, oe
(b) Answer: Ask both year groups about the same week (the same time period).
(c) B1 Year 11 has the higher median (10 > 7), so Year 11 students typically spend more time on homework
(c) B1 Year 11 has the larger IQR (13 - 7 = 6 compared with 9 - 5 = 4 for Year 10), so the amount of time Year 11 students spend is more spread out/less consistent
(c) B1 valid overall statement in context linking both measures
(c) Answer: Year 11 spends more time on homework on average (median 10 hours vs 7 hours) but with more variation between students (IQR 6 hours vs 4 hours).
(d) B1 Yes, ft from (a)/(c): Year 11's median (10 hours) is higher than Year 10's (7 hours), so the data supports the hypothesis
(d) Answer: Yes, the data supports the hypothesis, since Year 11's median homework time is higher than Year 10's.

Question 10

B1 compares median - LQ (9 - 8 = 1) with UQ - median (20 - 9 = 11), noting UQ - median is much larger
B1 correctly states the distribution is positively skewed (skewed to the right)
B1 valid contextual interpretation, e.g. most customers left quite quickly, but a few stayed much longer, stretching the upper part of the distribution
Answer: Positively skewed (skewed to the right).

Question 11

(a) M1 IQR = 30 - 20 = 10, upper boundary = 30 + 1.5 x 10 (= 45)
(a) A1 52 > 45, so 52 seconds is an outlier
(a) Answer: 52 > 45, so 52 seconds is an outlier.
(b) B1 44 cao
(b) Answer: 44 seconds
(c) B1 15 cao
(c) Answer: 15 seconds

Question 12

(a) B1 41 cao
(a) Answer: 41 minutes
(b) M1 identifies 36 and 38 as the two middle values of the lower 6 values
(b) A1 37 cao
(b) Answer: 37 minutes
(c) M1 identifies 45 and 47 as the two middle values of the upper 6 values
(c) A1 46 cao
(c) Answer: 46 minutes
(d) B1 9 cao, ft from (b) and (c)
(d) Answer: 9 minutes
(e) M1 upper boundary = 46 + 1.5 x 9 (= 59.5)
(e) A1 71 > 59.5, so 71 minutes is an outlier
(e) Answer: 71 > 59.5, so 71 minutes is an outlier.
(f) B1 48 cao, ft
(f) Answer: 48 minutes

Question 13

(a) B1 median is much closer to the lower quartile (17-15=2) than to the upper quartile (30-17=13)
(a) B1 positively skewed (skewed to the right)
(a) Answer: Positively skewed (skewed to the right).
(b) B1 median is much closer to the upper quartile (27-25=2) than to the lower quartile (25-10=15)
(b) B1 negatively skewed (skewed to the left)
(b) Answer: Negatively skewed (skewed to the left).

Question 14

(a) B1 awrt 46-50
(a) Answer: 48 minutes (accept 46-50)
(b) B1 lower quartile awrt 31-35
(b) B1 upper quartile awrt 58-62
(b) Answer: Lower quartile approximately 33 minutes; upper quartile approximately 60 minutes.
(c) B1 27 (ft from (b))
(c) Answer: 27 minutes
(d) B1 whiskers drawn to the minimum (10) and maximum (95)
(d) B1 box drawn from the lower quartile (awrt 33) to the upper quartile (awrt 60), ft from (b)
(d) B1 median (awrt 48) marked correctly inside the box, ft from (a), drawn accurately to scale
(d) Answer: Box plot: min 10, LQ approx. 33, median approx. 48, UQ approx. 60, max 95 (minutes).