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11.05 Describing data distributions

Worksheet
Shape of data distributions
1

Describe the shape of each of the following data sets as positively skewed, negatively skewed or symmetrical:

a
b
c
d
e
1
2
3
4
5
6
7
8
x
3
6
9
12
15
18
21
y
f
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: 1 \vert 6 = 16

g
h
1
2
3
4
5
6
7
8
5
10
15
20
25
i
1
2
3
4
5
6
7
8
5
10
15
20
25
2

How many peaks are there on the following dot plot?

3

Describe the distribution of the following graphs:

a
b
4

The table shows the number of crime novels in a bookshop for different price ranges.

\text{Price of crime novel, to the nearest } \$55101520253035
\text{Frequency}391961993
a

Plot this data as a bar graph.

b

Describe the shape of the data in the graph.

Measures of centre
5

Consider the stem plot below:

a

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

b

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

c

What is the mode?

d

Describe the shape of the data.

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

6

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

a

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

b

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

c

What is 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

7

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 of the data.

8

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 of the data.

9

Consider the data shown in the histogram:

a

Are there any outliers? If so, what is the value?

b

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

c

What is the mode?

d

Describe the shape of the distribution of the data.

10

Estimate the value of the mean of the following data set:

11

If a set of data is strongly positively skewed and the median is 70, what can we conclude about the mean?

12

State whether each of the following statements are true or false:

a

If two sets of data have the same median then the data sets must be the same.

b

If two sets of data have different modes then the highest values cannot be the same.

c

Two sets of data have the same highest and lowest values. This means they must have the same median.

d

Two sets of data that have the same highest and lowest values must have the same mean.

13

Which of the following dot plots has the highest median?

A
B
14

Determine whether the following sets of data have equal median and mean:

a
b
c
d
15

Consider the dot plot given:

From which score can a dot be removed so that the mean, median and mode remain unchanged?

16

Consider the frequency distribution table below:

a

Complete the table.

b

Calculate the mean, correct to two decimal places.

c

State the mode.

d

Find the range.

e

Determine the number of scores that are less than the mode.

\text{Score } (x)\text{Frequency } (f)fx
411
535
16
14
\text{Total}43365
17

For the following scenarios, determine the value of the x:

a

A rating system of 1 - 4 was used in a survey to determine the usefulness of a new feature. The 14 scores shown below are known to be bi-modal with values 2 \text{ and }4.

2,\, 4,\, 2,\, 4,\, 3,\, 2,\, 3,\, 4,\, 4,\, 1,\, 1,\, 2,\, 3,\, x
b

A rating system of 1 - 3 was used in a survey to determine the usefulness of a new feature. The 10 scores shown below are known to have a mode of 1.

3,\, 2,\, 3,\, 2,\, 1,\, 3,\, 1,\, 1,\, 2,\, x
18

The six numbers 6, 2, 7, 18, 17 and an unknown number x have a median of 8.5. Find the value of x.

19

Five numbers have a range of 16, a mode of 2, a median of 7 and a mean of 8. The minimum number in the set is 2. Calculate:

a

The minimum

b

The median

c

The maximum

20

Three numbers have a mode of 10 and a mean of 10. Write the three numbers of the data set.

21

Four numbers have a range of 5, a median of 9 and a mode of 11. Write the four numbers of this data set.

22

State the measure of centre which more accurately describes the centre of each of the following data sets:

a

12,\, 15,\, 16,\, 21,\, 22,\, 25

b

15,\,13,\,16,\,17,\,15,\,15,\,15

c
8,\, 10,\, 14,\, 18,\, 19,\, 91
23

Consider the histogram below:

Determine the measure of centre that would be most appropriate to use to represent the data in this graph. Explain your answer.

24

Find the most appropriate measure of centre for the following data sets:

a
b
c
25

What measure of center would be most appropriate to use to represent the data in this graph?

1
2
3
4
5
6
7
8
x
5
10
15
20
25
y
26

Carl has been recording his spelling test scores for the past semester:14,\, 16,\, 2,\, 15,\, 15,\, 16,\, 15

a

Calculate the median of Carl's scores.

b

Calculate the mean of Carl's scores. Round your answer to two decimal places.

c

Which measure of centre most accurately describes the centre of this data set? Explain your answer.

Applications
27

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

What was the median age?

b

What was the difference between the lowest age and the median?

c

What is the difference between the highest age and the median?

d

What was the mean age? Round your answer 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

28

The following stem plot shows the ages of 20 employees in a company:

a

How many of the employees are in their 30s?

b

What is the age of the oldest employee?

c

What is the age of the youngest employee?

d

What is the median age of the employees?

e

What is the modal age group?

Leaf
20\ 3\ 5\ 6\ 7\ 7\ 9
30\ 2\ 2\ 2\ 7
44\ 4\ 5\ 7\ 8
52\ 3\ 7

Key: 2\vert 0=20

29

\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

Find the median.

b

Find the upper quartile.

c

Find the lower quartile.

d

Consider the box plot for this data set:

Are the results positively or negatively skewed?

e

Determine 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 majority of this group of people?

20
30
40
50
60
70
80
30

Susanah has been growing watermelons. The weight of the watermelons (in kilograms) are shown below:15,\, 6,\, 5,\, 2,\, 4,\, 4,\, 5

a

Calculate the median weight of Oprah's watermelons.

b

Calculate the mean weight of Oprah's watermelons. Round your answer to two decimal places if necessary.

c

Find the most appropriate measure of centre of this data set.

31

A timed quiz consists of 6 puzzles. The data below shows the times (in seconds) that it took Sophia to complete each question on her first and second attempts of the quiz:

  • Times in first attempt: \, 11,\, 15,\, 17,\, 20,\, 23,\, 34

  • Times in second attempt: \, 20,\, 20,\, 20,\, 19,\, 21,\, 20

a

Calculate the mean time spent on each puzzle in the first attempt.

b

Calculate the mean time spent on each puzzle in the second attempt.

c

For which of Sophia's attempts would the mean number of minutes spent per question be a better indicator of her performance than the median number of minutes spent per question? Explain your answer.

32

The price of petrol at a petrol station was recorded each day for two weeks. The results are presented in the table below:

MondayTuesdayWednesdayThursdayFridaySaturdaySunday
Week 1\$1.70\$1.50\$1.62\$1.46\$1.49\$1.46\$1.55
Week 2\$1.25\$1.36\$1.25\$1.21\$1.21\$1.20\$3.30

The mean petrol price across the 14 days of records is \$1.54 per litre. For which week is this mean a better indication of the price of petrol? Explain your answer.

33

Every week over 45 weeks, a kayaking club ran social sessions and the number of people who attended each session was recorded in the table:

Number of people attending121314151617181920
Number of weeks656565656

Explain why the mean and the median are equally accurate indicators of the typical number of people who attended each session.

34

The histograms below represents the luggage weight of each passenger on board an airline's morning and afternoon flight:

The airline wants to get an indication of how much luggage each passenger is checking in. For which flight's data would the median be a more precise indicator of a typical passenger's luggage weight than the mean? Explain your answer.

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MS11-7

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