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7.05 Box plots

Worksheet
Five number summary
1

For each of the the following sets of data:

i

Sort the data in ascending order.

ii

Find the maximum value.

iii

Find the minimum value.

iv

Find the median value.

v

Find Q_1 for this data set.

vi

Find Q_3 for this data set.

a

The data set shows number of points scored by a basketball team in each game of their previous season:

75,\, 53,\, 84,\, 66,\, 89,\, 55,\, 63,\, 70,\, 92,\, 51,\, 90,\, 55,\, 81,\, 87,\, 68
b

The data set shows marks in an end-of-year exam for a class of students:

59,\, 53,\, 75,\, 80,\, 82,\, 96,\, 81,\, 79,\, 64,\, 58,\, 77,\, 62,\, 62,\, 86
c

The data set shows finishing times (in minutes) of the competitors in a 1500-meter swimming race:

24.41, \, 22.95,\, 21.88,\, 24.19,\, 16.12,\, 25.64,\, 16.83,\, 23.62,\, 24.52,\, 23.74,\, 19.44
2

There is a test to measure the Emotional Quotient (EQ) of an individual. Here are the EQ results for 21 people, listed in ascending order:

92,\, 94,\, 100,\, 103,\, 103,\, 105,\, 105,\, 109,\, 110,\, 113,\, 114,\\ 114,\, 116,\, 118,\, 118,\, 119,\, 120,\, 125,\, 125,\, 126,\, 130

a

Determine the median EQ score.

b

Determine Q_1 for this data set.

c

Determine Q_3 for this data set.

3

Below is the luggage weight of 30 passengers:

a

What is the mean check in weight? Round your answer to two decimal places.

b

Find the:

i

Median

ii

Lower quartile

iii

Upper quartile

c

In which quartile does the mean lie?

WeightFrequency
165
175
182
194
206
214
224
4

Consider the grouped data displayed in the following column graph:

a

Complete a frequency table for this data.

b

Use the class centres to estimate:

i

The range

ii

The interquartile range

c

Use technology to construct a boxplot for the above data, then write down the five number summary.

5

The histogram shows the heights of tomato plants in a greenhouse:

a

Complete the given frequency table.

b

Use the class centres to estimate:

i

The range

ii

The interquartile range

c

Use technology to construct a boxplot for the above data, then write down the five number summary.

\text{Height (cm)}\text{Class}\\ \text{centre}\text{Frequency}
95 \leq x \lt 10097.51
100 \leq x \lt 105
105 \leq x \lt 110
110 \leq x \lt 115
115 \leq x \lt 120
120 \leq x \lt 125
125 \leq x \lt 130
130 \leq x \lt 135
6

In a survey the mass (in grams) of 30 individual apples from an orchard were noted and recorded in the following list:

86,\quad 87,\quad 91,\quad 93,\quad 94,\quad 95,\quad 96,\quad 96,\quad 98,\quad99

100,\quad 101,\quad 102,\quad 103,\quad 103,\quad 104,\quad 104,\quad 105,\quad 106,\quad 106

106,\quad 107,\quad 107,\quad 107,\quad 108,\quad 108,\quad 109,\quad 109,\quad 109,\quad 109

Use technology to construct a boxplot for the above data, then write down the five number summary.

7

For each of the following sets of data:

i
Find the five number summary using technology.
ii
Calculate the interquartile range.
a

4,\, 27,\, 16,\, 29,\, 27,\, 10,\, 21,\, 12,\, 23,\, 8,\, 3,\, 1,\, 23,\, 9,\, 22

b
47.1,\, 42.3,\, 41.8,\, 47.8,\, 27.1,\, 36.2,\, 31.8,\, 24,\, 48.2,\, 45.1,\, 47.2,\, 45.3,\, 43,\, 43,\, 39,\, 34.4
c
Leaf
70\ 2\ 3\ 7\ 8\ 9
81\ 5\ 6\ 9
90\ 1\ 2\ 2\ 6\ 6\ 6\ 6\ 7\ 8

Key: 7 \vert 0 = 70

d
e
ScoreFrequency
1513
169
1723
1819
198
2013
f
8

Consider the following grouped frequency table and use the class centres to find approximate values for:

a

The five number summary.

b

The interquartile range.

\text{Class}\text{Class}\\ \text{centre}\text{Frequency}
40\leq x \lt 4542.53
45\leq x \lt 5047.54
50\leq x \lt 5552.57
55\leq x \lt 6057.53
60\leq x \lt 6562.53
65\leq x \lt 7067.59
70\leq x \lt 7572.54
75\leq x \lt 8077.55
Box plots
9

For the box plot shown, find the following:

a

Lowest score

b

Highest score

c

Range

d

Median

e

Interquartile range

0
2
4
6
8
10
12
14
16
18
20
10

Construct a box plot for each five number summary:

a
  • Median = 47

  • Lower Quartile = 33

  • Upper Quartile = 61

  • Lowest score = 16

  • Highest score = 71

b
  • Median = 36

  • Lower Quartile = 28

  • Upper Quartile = 42

  • Lowest score = 20

  • Highest score = 52

c
  • Median = 35

  • Lower Quartile = 25

  • Upper Quartile = 60

  • Lowest score = 5

  • Highest score = 75

11

A geography teacher has marked a set of tests. She wants to represent the results in a box plot. She has already sorted her data and created the table shown. Create a box plot to match the data in the table:

Minimum8
Lower Quartile10
Median16
Upper Quartile24
Maximum28
12

Consider the following data set:

20,\, 36,\, 52,\, 56,\, 24,\, 16,\, 40,\, 4,\, 28

a

Find the five number summary.

b

Construct a box plot for the data.

13

The box plot shows the age at which a group of people got their driving licences:

a

What is the oldest age?

b

What is the youngest age?

c

What percentage of people were aged from 18 to 22?

d

The middle 50\% of responders were within how many years of one another?

e

In which quartile are the ages least spread out?

f

The bottom 50\% of responders were within how many years of one another?

Age
15
20
25
30
35
14

Consider the box plot shown:

a

State the percentage of scores that lie between each of the following values:

i

7 and 15

ii

1 and 7

iii

19 and 9

iv

7 and 19

v

1 and 15

b

In which quartile is the data the least spread out?

0
5
10
15
20
15

The glass windows for an airplane are cut to a certain thickness, but machine production means there is some variation. The thickness of each pane of glass produced is measured (in millimetres), and the dot plot shows the results:

a

Find the median thickness, to two decimal places.

b

Find the interquartile range.

c

Construct a box plot to represent the data.

d

What percentage of thicknesses were between 10.8 \text{ mm} and 11.2 \text{ mm} inclusive? Round your answer to two decimal places.

e

According to the box plot, in which quartile are the results the most spread out?

f

Which statistics cannot be found from a box plot?

16

The marks in an end-of-year exam for a class of students is given below:

52,\, 95,\, 80,\, 56,\, 59,\, 80,\, 86,\, 77,\, 80,\, 81,\, 78,\, 64,\, 84,\, 66,\, 96,\, 90

a

Construct a box plot for the data.

b

Calculate the interquartile range.

c

What percentage of marks lie in the range 85 to 96?

d

Which values do the lowest 75\% of scores lie between?

17

In training, a fighter pilot measures the number of seconds he blacks out over a number of flights. He constructs the following box and whisker plot for his data:

As long as the pilot is not unconscious for more than 7 seconds, he will be safe to fly.

The pilot concludes that he is safe to fly all the time. Is his conclusion correct? Explain your answer.

0
1
2
3
4
5
6
7
8
9
10
18

Salaries earned by employees at a software company is given in the histogram below:

a

Using the class centres and technology construct a box plot.

b

Using the class centres find an estimate for the interquartile range.

c

Using the box plot, approximately what percentage of salaries lie in the range \$90\,000 to \$100\,000?

d

Using the box plot, what can be said about the highest 25\% of salaries? Would this conclusion be different from using the histogram? Explain your answer.

19

Minimum temperatures are recorded on a sample of days throughout the year and is given in the histogram below:

a

Using the class centres and technology construct a box plot.

b

Using the class centres find an estimate for the range.

c

Using the box plot, approximately what percentage of temperatures lie in the range 2.5 \degree \text{C} to 12.5 \degree \text{C}?

Parallel box plots
20

The box plots below show the monthly profits (in thousands of dollars) of two derivatives traders over a year:

Ned
10
15
20
25
30
35
40
45
50
55
Tobias
10
15
20
25
30
35
40
45
50
55
a

Who made a higher median monthly profit?

b

Whose profits had a higher interquartile range?

c

Whose profits had a higher range?

d

How much more did Ned make in his most profitable month than Tobias did in his most profitable month?

21

The parallel box plots shows the distances, in centimetres, jumped by two high jumpers:

a

Who had a higher median jump?

b

Who made the highest jump?

c

Who made the lowest jump?

22

The parallel box plots shows the prices, in dollars, of the items on the menu of an upmarket restaurant and the menu of a fast food restaurant:

a

Which restaurant has the higher median price for the items they sell?

b

What is the difference between the median prices of the items sold by each restaurant?

c

Which restaurant has a greater price range for the items on the their menu?

d

What is the price difference between the most expensive items sold by each restaurant?

e

What amount of the cheapest item at the fast food restaurant could be bought for the same price of the most expensive item at the upmarket restaurant?

23

The parallel box plots show the number of goals scored by two football players in each season:

a

Who scored the most goals in a season?

b

How many more goals did Holly score in her best season compared to Sophie in her best season?

c

What is the difference between the median number of goals scored in a season by each player?

d

What is the difference between the interquartile range for both players?

24

Two groups of people, athletes and non-athletes, had their resting heart rate measured. The results are displayed in the given pair of box plots:

a

What is the median heart rate of athletes?

b

What is the median heart rate of the non-athletes?

c

Using this measure, which group has the lower heart rates?

d

What is the interquartile range of the athletes' heart rates?

e

What is the interquartile range of the non-athletes' heart rates?

f

Using this measure, which group has more consistent heart rate measures?

Athletes
40
50
60
70
80
90
Non-athletes
40
50
60
70
80
90
25

The parallel box plots below shows the data collected by the manufacturers on the life-span of light bulbs, measured in thousands of hours:

a

Complete the following table. Write each answer in terms of hours.

Manufacturer AManufacturer B
Median
Lower quartile
Upper quartile
Range
Interquartile range
b

Hence, which manufacturer produces light bulbs with the best lifespan? Explain your answer.

26

A mathematics test is given to two classes. The marks out of 20 received by students in each class are represented in the box plots below:

Class 9P
0
2
4
6
8
10
12
14
16
18
20
Class 9Q
0
2
4
6
8
10
12
14
16
18
20
a

Complete the following table:

Class 9PClass 9Q
Median
Lower quartile
Upper quartile
Range
Interquartile range
b

Which class tended to score better marks? Explain your answer.

27

A class took an English test and a Mathematics test. Both tests had a maximum possible mark of 25. The results are illustrated below.

a

Complete the following table using the two box plots:

EnglishMathematics
Median
Lower Quartile
Upper Quartile
Range
Interquartile Range
b

In which test did the class tend to score better marks? Explain your answer.

28

A builder can choose between two different types of brick that are coloured red or yellow. The parallel box plots below shows the results of tests on the strength of the bricks:

a

Using the box plot, explain why a builder might to prefer to use the red bricks.

b

Using the box plot, explain why a builder might to prefer to use yellow bricks.

29

The box plots below represent the daily sales made by Carl and Angelina over the course of one month:

a

What is the range in Angelina's sales?

b

What is the range in Carl's sales?

c

By how much did Carl's median sales exceed Angelina's?

d

Considering the middle 50\% of sales for both sales people, whose sales were more consistent?

e

Which salesperson had a more successful sales month?

Angelina's Sales
0
10
20
30
40
50
60
70
Carl's Sales
0
10
20
30
40
50
60
70
30

Two bookstores recorded the selling price of all their books. The results are presented in the parallel box plots:

a

Which bookstore had the more consistent prices?

b

Comparing the most expensive books in each store, how much more expensive is the one in store B?

c

True or False: 25\% of the books in Bookstore B are at least as expensive than the most expensive book in Bookstore A.

d

True or False: 25\% of the books in Bookstore B are cheaper than the cheapest book in Bookstore A.

31

A cinema is showing three films, labelled A, B and C. The ages of people watching each of the films are illustrated in the parallel box plots:

a

Which film do you think has an adults only rating, restricting it to viewers 18 years of age and older? Explain your answer.

b

Which film would you recommend for a group of 15 year olds to watch? Explain your answer.

c

Which film would you recommend to a family of two parents in their 40's and two teenagers? Explain your answer.

32

The heights (in metres) of the boys and girls in a class of 30 students were recorded. The results are given below:

Boy's heights: 1.65, 1.66, 1.67, 1.68, 1.63, 1.62, 1.61, 1.60, 1.75, 1.76, 1.77, 1.78, 1.73, 1.72, 1.71

Girl's heights: 1.55, 1.56, 1.57, 1.58, 1.53, 1.52, 1.51, 1.50, 1.69, 1.70, 1.71, 1.72, 1.67, 1.66, 1.65

a

Find the five number summary for the heights of boys in the class.

b

Find the five number summary for the heights of girls in the class.

c

Draw a parallel box plots for this data.

33

The number of vehicles sold by two companies each week from a dealership over three months was recorded in the back-to-back stem plot:

a

Find the five number summary for the weekly number of vehicles sold over these three months by company A.

b

Find the five number summary for the weekly number of vehicles sold over these three months by company B.

c

Draw parallel box plots for this data.

Company ACompany B
5\ 003\ 9
8\ 7\ 4\ 1\ 1\ 010\ 2\ 2\ 2\ 3\ 7
9\ 2\ 1\ 020\ 1\ 1\ 7
931

Key: 2 \vert 1 \vert 0 = 12 \text{ and }10

34

The data below shows the results of a survey conducted on the price of concert tickets locally and the price of the same concert at an international venue:

a

Find the five number summary for the price of concert tickets at local venues.

b

Find the five number summary for the price of concert tickets at international venues.

c

Draw parallel box plots for this data.

LocalInternational
7\ 6\ 3\ 061\ 8
8\ 6\ 4\ 3\ 273\ 5\ 5\ 9
9\ 6\ 5\ 1\ 181\ 5\ 7\ 9
8\ 7\ 5\ 2\ 091\ 3\ 4\ 6\ 8
1101\ 2\ 4\ 7\ 8

Key: 2 \vert 6 \vert 0 = 62 \text{ and }60

35

The batting scores of two cricket teams, A and B, are recorded in the back-to-back stem plot below:

a

Find the five number summary for the batting scores of team A.

b

Find the five number summary for the batting scores of team B.

c

Draw parallel box plots for this data.

Team ATeam B
9\ 532\ 3\ 6\ 6\ 8
8\ 8\ 5\ 5\ 4\ 142\ 9
9\ 550\ 8
62

Key: 2 \vert 3 \vert 0 = 32 \text{ and }30

36

The back-to-back stem plot below shows the number of pieces of paper used over several days by two classes, A and B:

a

Find the five number summary for the number of pieces of paper used in class A.

b

Find the five number summary for the number of pieces of paper used in class B.

c

Draw parallel box plots for this data.

Class AClass B
707
311\ 2\ 3
828
4\ 332\ 3\ 4
7\ 6\ 549
3\ 252

Key: 2 \vert 1 \vert 0 = 12 \text{ and }10

37

The back-to-back stem plot shows the test scores of two classes, A and B:

Draw parallel box plots for this data.

Class AClass B
5\ 5\ 061\ 5\ 9
9\ 8\ 4\ 3\ 271\ 4\ 7
5\ 580\ 4\ 7\ 9
92\ 7

Key: 2 \vert 7 \vert 0 = 72 \text{ and }70

38

Ten participants had their pulse measured in beats per minute before and after exercise with results shown in the back-to-back stem plot on the right.

a

Find the five number summary for the participants' pulse before exercise.

b

Find the five number summary for the participants' pulse after exercise.

c

Draw a parallel box plots for this data.

Pulse beforePulse after
5\ 5\ 05
9\ 9\ 7\ 46
4\ 37
084
95\ 7\ 8
103
113\ 5\ 5
120\ 1

Key: 2 \vert 8 \vert 0 = 82 \text{ and }80

39

The test scores of 12 students in English and Music are listed below:

  • English:\ 55,\, 57,\, 63,\, 69,\, 71,\, 74,\, 77,\, 81,\, 84,\, 88,\, 91,\, 98

  • Music:\ 55,\, 61,\, 66,\, 69,\, 72,\, 74,\, 76,\, 81,\, 84,\, 86,\, 89,\, 93

a

Find the five number summary for the test scores for English.

b

Find the five number summary for the test scores for Music.

c

Draw parallel box plots for this data.

40

The data below represents how long each person in two different groups could hold their breath for, measured to the nearest second:

  • Group A: \,13,\, 17,\, 22,\, 28,\, 32,\, 34,\, 44,\, 49,\, 55,\, 56,\, 63,\, 64,\, 66,\, 51,\, 39

  • Group B: \,13,\, 18,\, 23,\, 28,\, 32,\, 36,\, 42,\, 44,\, 53,\, 56,\, 64,\, 67,\, 13,\, 45,\, 47

a

Find the five number summary for the length of time people in group A could hold their breath for.

b

Find the five number summary for the length of time people in group B could hold their breath for.

c

Draw a parallel box plots for this data.

41

Two friends, Rhonda and Michael, have been growing sunflowers. They have each measured the heights of their sunflowers to the nearest centimetre. The data from these measurements is shown below:

  • Rhonda:\ 3,\, 7,\, 7,\, 11,\, 16,\, 21,\, 24,\, 27,\, 28,\, 31,\, 41,\, 46,\, 49

  • Michael:\ 13,\, 18,\, 19,\, 23,\, 28,\, 35,\, 39,\, 46

a

Find the five number summary for the heights of Rhonda's sunflowers.

b

Find the five number summary for the heights of Michael's sunflowers.

c

Draw parallel box plots for this data.

42

At every training session of the season, a cyclist measured her pulse rate before a sprint and after a sprint. The before and after rates, measured in beats per minute \left(\text{bpm}\right), recorded throughout the season are presented in the box plots below:

a

How much greater was her median pulse rate after the sprint than before the sprint?

b

Find the interquartile range of her pulse rate before the sprint.

c

Find the interquartile range of her pulse rate after the sprint.

d

Find the range of her pulse rate before the sprint.

e

Find the range of her pulse rate after the sprint.

f

Are her pulse rate readings more consistent before or after the sprint?

g

In the last session of the season, the cyclist recorded her highest pulse rate of the season both before and after the sprint. By how much did her pulse rate increase during this particular training session?

Pulse Rate Before (bpm)
60
70
80
90
100
110
120
130
140
Pulse Rate After (bpm)
60
70
80
90
100
110
120
130
140
43

Eileen competed in the high beam gymnastics event at both the 2006 and 2010 Olympics. Her judges' scores in both years are presented in the parallel box plots:

a

What was the difference between the minimum scores she was awarded?

b

What was the difference between the maximum scores she was awarded?

c

One particular judge at the 2010 games gave Eileen score of 8.3. In which quartile of her 2006 scores would this lie?

d

In which year did the judges score Eileen most consistently?

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Outcomes

ACMGM031

construct and use parallel box plots (including the use of the ‘Q1 – 1.5 x IQR’ and ‘Q3 + 1.5 x IQR’ criteria for identifying possible outliers) to compare groups in terms of location (median), spread (IQR and range) and outliers and to interpret and communicate the differences observed in the context of the data

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