Topics
12. Univariate Data
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Standard Level

12.09 Recognising the shape of data

Lesson
Worksheet
Practice
Worksheet
Symmetry and skew
1

Describe the shape of the data in the following graphs:

a
Leaf
16\ 7\ 7
22\ 2\ 2\ 2\ 3\ 3\ 3
33\ 3\ 3\ 6\ 6\ 6\ 7\ 7\ 7\ 7\ 7
44\ 4\ 4\ 4\ 4\ 4
57\ 7

Key: 2 \vert 3 = 23

b
c
d
e
f
g
h
i
j
1
2
3
4
5
6
7
\text{Score}
5
10
15
20
\text{Frequency}
k
1
2
3
4
5
6
7
\text{Score}
5
10
15
20
25
30
\text{Frequency}
l
1
2
3
4
5
6
7
\text{Score}
5
10
15
20
25
30
\text{Frequency}
2

Determine whether the following graphs are bimodal:

a
b
c
Leaf
10\ 0\ 2\ 7
22\ 2\ 3\ 3\ 5\ 8
31\ 6\ 4
47
50\ 1\ 6
65\ 7\ 7\ 8\ 8
74\ 4\ 4
84\ 4

Key: 2 \vert 3 = 23

d
Leaf
12\ 3\ 4\ 5\ 5\ 6\ 6\ 7\ 7\ 9
21\ 2\ 4\ 4
30\ 2\ 9\ 9
4
55

Key: 1\vert 2=12

e
f
g
h
i
1
2
3
4
5
6
7
8
\text{Score}
5
10
15
20
25
\text{Frequency}
j
1
2
3
4
5
6
7
\text{Score}
5
10
15
20
25
30
\text{Frequency}
3

How would you describe the modality of the following dot plot? Explain your answer.

4

The table shows the number of crime novels in a bookshop for different price ranges rounded off to the nearest \$ 5:

a

Graph this data as a histogram.

b

Describe the shape of the distribution of the data.

Price of crime novelFrequency
\$55
\$1010
\$1517
\$208
\$2517
\$3010
\$355
Clusters and outliers
5

The number of hours worked per week by a group of people is represented in the following stem-and-leaf plot:

a

State the value of any outliers.

b

Is there any clustering of data? If so, in what interval?

c

State the mode.

Leaf
02
1
20\ 3\ 6\ 6
31\ 4\ 5\ 6\ 6\ 7
40\ 4\ 6\ 7\ 9
50

Key: 2 \vert 3 = 23

6

Consider the stem plot given:

a

State the value of any outliers.

b

Is there any clustering of data? If so, in what interval?

c

State the mode.

d

Describe the shape of the distribution.

Leaf
05
17\ 8
20\ 8
31\ 3\ 3\ 7\ 8\ 9
41\ 3\ 5\ 8\ 8\ 8
5
6
7
8
92

Key: 2 \vert 3 = 23

7

Temperatures were recorded over a period of time and presented as a dot plot:

a

Are there any outliers?

b

Is there any clustering of data? If so, in what interval?

c

What is the modal temperature?

d

Describe the shape of the distribution.

8

The number of peanuts in mixed nut packets were sampled and recorded in the following stem plot:

a

Complete the frequency distribution table:

ScoreFrequency
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110-119
Leaf
43\ 6\ 8
51\ 2\ 2\ 6\ 7\ 7\ 8
60\ 0\ 2
73\ 3\ 4\ 5\ 9
81\ 1\ 1\ 4\ 6\ 8\ 8\ 9
90\ 2\ 5\ 6
101\ 2\ 3\ 5\ 5\ 6\ 7\ 8
110\ 4\ 5\ 7

Key: 5 | 2 = 52

b

Describe the modaility of the distribution.

c

State the modal class or classes of the data.

9

The percentage of faulty computer chips in 42 batches were recorded in the given histogram:

a

Describe the modaility of the distribution.

b

State the modal class or classes of the data.

10

Consider the dot plot below:

a

Are there any outliers?

b

Is there any clustering of data?

c

State the modal score(s).

d

Describe the shape of the distribution.

11

The reaction time of drivers was tested and recorded in the dot plot below:

a

Construct a frequency distribution table for the individual data values.

b

Describe the modaility of the distribution.

c

State the mode(s).

12

Consider the data shown in the histogram:

a

Are there any outliers?

b

Is there any clustering of data? If so, in what interval?

c

State the mode.

d

Describe the shape of the distribution.

13

The shoe sizes of all the students in a class were measured and the data was presented in a graph:

a

Are there any outliers?

b

Is there any clustering of data? If so, in what interval?

c

What is the modal shoe size?

d

Describe the shape of the distribution.

14

Estimate the value of the mean of the following data set correct to one decimal place:

15

Consider the histogram representing students' heights in centimetres:

a

Does the histogram most likely represent grouped data or individual scores?

b

Estimate the value of the mean to one decimal place.

c

Describe the shape of the distribution.

16

Consider the given column graph:

a

Describe the shape of the distribution.

b

Find the following:

i

Lower quartile

ii

Upper quartile

c

Hence, calculate the interquartile range.

d

Are there any outliers? If so, state the value.

17

Consider the dot plot given:

a

Describe the shape of the distribution.

b

Find the following:

i

Lower quartile

ii

Upper quartile

c

Hence, calculate the interquartile range.

d

Are there any outliers? If so, state the value.

18

The stem-and-leaf plot below shows the age of people to enter through the gates of a concert in the first 5 seconds:

a

Find the median age.

b

Find the difference between the lowest age and the median.

c

Find the difference between the highest age and the median.

d

Calculate the mean age, correct to two decimal places.

e

Is the data positively or negatively skewed?

Leaf
10\ 1\ 2\ 2\ 3\ 3\ 4\ 4\ 4\ 8\ 8\ 8
21\ 7
34\ 5\ 5
40
54

Key: 1 | 2 \ = \ 12 years old

19

\text{VO}_2 \text{Max} is a measure of how efficiently your body uses oxygen during exercise. The more physically fit you are, the higher your \text{VO}_2 \text{Max}. Here are some people's results, listed in ascending order, when their \text{VO}_2 \text{Max} was measured:

21,\, 21,\, 23,\, 25,\, 26,\, 27,\, 28,\, 29,\, 29,\, 29,\, 30,\, 30,\, 32,\, 38,\, 38,\, 42,\, 43,\, 44,\, 48,\, 50,\, 76

a

Determine the median \text{VO}_2 \text{Max}.

b

Determine the upper quartile value.

c

Determine the lower quartile value.

d

Consider the box plot for this data set and state whether the results are positively or negatively skewed.

e

State the value of the outlier.

f

An average untrained healthy person has a \text{VO}_2 \text{Max} between 30 and 40. What can we say about the exercise habits of the majority of this group of people?

20
30
40
50
60
70
80
20

A die is rolled for a large number of trials and the number appearing is noted.

a

Which histogram would you expect to match the data? Explain your answer.

A
B
b

Describe the shape of the distribution of the data chosen above.

21

A pair of dice are rolled and the numbers appearing on the uppermost face are added to create a score.

a

How many combinations would result in a score of:

i

4

ii

7

iii

10

b

A pair of dice are rolled for a large number of trials and the numbers appearing are added to create a score. Draw a sketch of a histogram you would expect to match the data.

Connect histograms and box plots
22

Match the histograms on the left to the corresponding box plots on the right:

\text{}\\
\text{}\\
Box Plot 1
10
20
30
40
50
60
70
80
90
\text{}\\\text{}\\\text{}\\\text{}\\
Box Plot 2
0
1
2
3
4
5
6
7
8
9
10
\text{}\\\text{}\\\text{}\\
Box Plot 3
0
1
2
3
4
5
6
7
8
9
10

Histogram A

Histogram B

Histogram C

\text{}\\
Box Plot 4
1
2
3
4
5
6
7
8
9
\text{}\\\text{}\\\text{}\\\text{}\\
Box Plot 5
0
10
20
30
40
50
60
70
80
90
100
\text{}\\\text{}\\\text{}\\
Box Plot 6
0
10
20
30
40
50
60
70
80
90
100

Histogram D

Histogram E

Histogram F

23

Construct a box plot for the following histograms:

a
b
c
d
e
f
24

State whether the following pairs of histograms and box plots are correctly matched with respect to their shape:

a
b
c
d
e
25

Explain why the following pairs of histograms and box plots do not match:

a
b
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