 # 10.08 Dot plots and stem and leaf plots

Lesson

Dot plots and stem-and-leaf plots are used to represent numerical data. They are suitable for small to moderate sized data sets.

Both plots are very good for highlighting outliers and clusters in the data. This will be discussed more in the next chapter.

### Dot plots

A dot plot is used for organising and displaying numerical data. It is good for small sets of discrete numerical data, but can also be used for data that is categorical.

In a dot plot, each individual value, or score, is represented by a single dot, displayed above a horizontal number line. When data values are identical, the dots are stacked vertically. • To correctly display the distribution of the data, the dots must be evenly spaced in columns above the number line
• The scale on the number line must be evenly spaced
• A dot plot does not have a vertical axis
• The dot plot should be appropriately labelled

#### Practice questions

##### Question 1

Here is a dot plot of the number of goals scored in each of Bob’s soccer games. 1. How many times were five goals scored?

2. Which number of goals were scored equally and most often?

$1$1

A

$0$0

B

$4$4

C

$3$3

D

$2$2

E

$5$5

F

$1$1

A

$0$0

B

$4$4

C

$3$3

D

$2$2

E

$5$5

F
3. How many games were played in total?

##### Question 2

The goals scored by a football team in their matches are represented in the following dot plot. 1. Complete the following frequency distribution table.

Goals scored Frequency
$0$0 $\editable{}$
$1$1 $\editable{}$
$2$2 $\editable{}$
$3$3 $\editable{}$
$4$4 $\editable{}$
$5$5 $\editable{}$

### Stem-and-leaf plot

A stem-and-leaf plot, or stemplot, is used for organising and displaying numerical data. It is appropriate for small to moderately sized data sets.

In a stem-and-leaf plot, the right-most digit in each data value is split from the other digits, to become the 'leaf'. The remaining digits become the 'stem'.

All of the values in a stem-and-leaf plot are arranged in ascending order (from lowest to highest). For this reason, it is often called an ordered stem-and-leaf plot.

The data values $10,13,16,21,26,27,28,35,35,36,41,41,45,46,49,50,53,56,58$10,13,16,21,26,27,28,35,35,36,41,41,45,46,49,50,53,56,58 are displayed in the stem-and-leaf plot below. • The stems are arranged in ascending order, to form a column, with the lowest value at the top
• The leaf values are arranged in ascending order, in rows, next to their corresponding stem
• A single vertical line separates the stem and leaf values
• There are no commas or other symbols between the leaves, only a space between them
• In order to correctly display the distribution of the data, the leaves must line up in imaginary columns, with each data value directly below the one above
• A stem-and-leaf plot includes a key that describes the way in which the stem and the leaf combine to form the data value  #### Practice questions

##### Question 1

Which of the following is true of a stem-and-leaf plot?

Stem Leaf
$0$0 $7$7
$1$1
$2$2
$3$3 $1$1 $3$3 $3$3 $3$3
$4$4 $1$1 $2$2 $3$3 $4$4 $9$9
$5$5 $1$1 $2$2 $4$4 $5$5 $5$5
$6$6 $0$0

 Key: $1$1$\mid$∣$2$2$=$=$12$12
1. The scores are ordered.

A

A stem-and-leaf plot does not give an idea of outliers and clusters.

B

It is only appropriate for data where scores have high frequencies.

C

The individual scores cannot be read on a stem-and-leaf plot.

D

The scores are ordered.

A

A stem-and-leaf plot does not give an idea of outliers and clusters.

B

It is only appropriate for data where scores have high frequencies.

C

The individual scores cannot be read on a stem-and-leaf plot.

D

##### QUESTION 2

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

Stem Leaf
$1$1 $1$1 $2$2 $4$4 $5$5 $6$6 $6$6 $7$7 $9$9 $9$9
$2$2 $2$2 $3$3 $5$5 $5$5 $7$7
$3$3 $1$1 $3$3 $8$8 $9$9
$4$4
$5$5 $8$8

 Key: $1$1$\mid$∣$2$2$=$=$12$12
years old
1. How many people passed through the gates in the first $5$5 seconds?

2. What was the age of the youngest person?

The youngest person was $\editable{}$ years old.

3. What was the age of the oldest person?

The oldest person was $\editable{}$ years old.

4. What proportion of the concert-goers were under $20$20 years old?

### Back-to-back stem-and-leaf plots

Two sets of data can be displayed side-by-side using a back-to-back stem-and-leaf plot.

In the example below, the pulse rates of $18$18 students were recorded before and after exercise. Reading a back-to-back stem-and-leaf plot is very similar to reading a regular stem-and-leaf plot.

Referring to the example above:

• The central column displays the stems, with the leaf values on each side.
• The values on the left are the pulse rates of the students before exercise, while the values on the right are their pulse rates after exercise.
• In this example, the fourth row of the plot, $4$4 $3$3 $0$0 | $8$8 | $2$2 $2$2 $6$6, displays pulse rates of $80$80, $83$83 and $84$84 before exercise and pulse rates of $82$82, $82$82 and $86$86 after exercise. They are not necessarily the pulse rates of the same students.
• On both sides of the stem column, the leafs are displayed in ascending order with the lowest value closest to the stem.

Remember!

To create a stem-and-leaf plot, it is usually easier to arrange all of the data values in ascending order, before ordering them in the plot.

#### Practice questions

##### Question 3

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

Local International
Stem Leaf
$6$6 $0$0 $5$5 $5$5 $8$8
$7$7 $0$0 $5$5 $6$6 $7$7 $9$9
$8$8 $0$0 $1$1 $1$1 $4$4 $7$7
$9$9 $0$0 $2$2 $3$3 $3$3 $3$3
$10$10 $9$9

 Stem Leaf $6$6 $3$3 $9$9 $7$7 $2$2 $4$4 $7$7 $7$7 $8$8 $0$0 $3$3 $4$4 $8$8 $9$9 $1$1 $1$1 $1$1 $2$2 $3$3 $10$10 $0$0 $1$1 $1$1 $3$3 $9$9
 Key: $1$1$\mid$∣$2$2$=$=$12$12
1. What was the most expensive ticket price at the international venue?

$\editable{}$ dollars

2. At the international venue, what percentage of tickets cost between $\$90$$90 and \110$$110 (inclusive)?

3. At the local venue, what percentage of tickets cost between $\$90$$90 and \100$$100 (inclusive)?

##### QUESTION 4

The back-to-back stem plots show the number of pieces of paper used over several days by Maximilian’s and Charlie’s students.

Maximilian Stem Charlie
$7$7 $0$0 $7$7
$3$3 $1$1 $1$1 $2$2 $3$3
$8$8 $2$2 $8$8
$4$4 $3$3 $3$3 $2$2 $3$3 $4$4
$7$7 $6$6 $5$5 $4$4 $9$9
$3$3 $2$2 $5$5 $2$2
 Key: $6\mid1\mid2$6∣1∣2 $=$= $16$16 and $12$12
1. Which of the following statements are true?

I. Maximilian's students did not use $7$7 pieces of paper on any day.

II. Charlie's median is higher than Maximilian’s median.

III. The median is greater than the mean in both groups.

I and II

A

II and III

B

None of the statements are correct.

C

III only

D

II only

E

I only

F

I and II

A

II and III

B

None of the statements are correct.

C

III only

D

II only

E

I only

F

### Outcomes

#### MS11-2

represents information in symbolic, graphical and tabular form

#### MS11-7

develops and carries out simple statistical processes to answer questions posed