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18.04 Stem-and-leaf and dot plots

Lesson

Introduction

Representing data can be tricky because we want to make it easy to interpret without losing any information. Most of the time we are forced to compromise, either making the data simpler to express in a simpler manner or instead having more complex ways to express our data.

This lesson looks at a couple of the ways in which we can present our data visually to express information, with the trade-off that we need to learn how to read them.

Dot plots

The dot plot is a useful way to express discrete data in a visually simple manner. The main advantages of the dot plot are that we can find the mode and range very easily, as well as quickly see how the data is distributed. The main disadvantages are that we need to count each dot when finding the median and it is often easier to convert to a table to find the mean. The dot plot is particularly suited to discrete data where the frequency of results are often greater than one.

In a dot plot, each dot represents one data point belonging to the result that it is placed above. The modes of a dot plot will be the results with the most dots. Since a dot plot stacks vertically, the greatest columns will belong to the modes.

Examples

Example 1

A group of adults is asked: "How old were you when you passed your driving test?". The responses were: 22,\,17,\,17,\,17,\,19,\,21,\,17,\,22,\,21,\,18,\,18,\,17,\,18,\,22,\,18

The following dot plot represents the responses.

A dot plot titled Responses. Ranging from 17 to 22 in steps of 1. Ask your teacher for more information.
a

What is the range of this data set?

Worked Solution
Create a strategy

To find the range, use the formula: \text{Range}=\text{Highest score}-\text{Lowest score}

Apply the idea
\displaystyle \text{Range}\displaystyle =\displaystyle 22-17Find the difference
\displaystyle =\displaystyle 5Evaluate the subtraction
b

What is the mode of this data set?

Worked Solution
Create a strategy

Choose the score which has the most dots above it.

Apply the idea

\text{Mode}=17

c

What is the median of this data set?

Worked Solution
Create a strategy

Order the scores and find the middle score.

Apply the idea

Arrange the scores in order:17,\,17,\,17,\,17,\,17,\,18,\,18,\,18,\,18,\,19,\,21,\,21,\,22,\,22,\,22

The middle score is: 18

\text{Median}=18

d

How many people passed their driving test on or after their 19\text{th} birthday?

Worked Solution
Create a strategy

Count the number of dots that are greater or equal to 19.

Apply the idea
\displaystyle \text{No. People}\displaystyle =\displaystyle 6Count the scores
Idea summary

In a dot plot, each dot represents one data point belonging to the result that it is placed above.

The mode(s) of a dot plot will be the result(s) with the most dots. Since a dot plot stacks vertically, the highest column(s) will belong to the mode(s).

Stem-and-leaf plot

The stem-and-leaf plot is an example of a way to express data in a more complicated way so that we can express more information visually. In particular, the stem-and-leaf plot is used when we have lots of numerical data points.

A stem-and-leaf plot is made up of two components, the stem and the leaf. The stem is usually used to represent the tens part of a score while the leaf is used to represent the ones part of the score.

StemLeaf
52
Key 5\vert 2 = 52

The score 52 is expressed on this stem-and-leaf plot. The ones digit is in the row corresponding to its tens digit. In other words, we attached the leaf, 2, to its stem, 5, to make the score 52.

What is useful about the stem-and-leaf plot is that we can record as many scores as we like by writing the leaves in the appropriate rows.

StemLeaf
30\ 1
46\ 9
51\ 2\ 2\ 7
Key 5\vert 2 = 52

We can express the data set: 52,\,46,\,31,\,57,\,49,\,51,\,52,\,30with this stem-and-leaf plot.

As we can see, each score has been expressed on the plot as a ones digit written in its tens row.

Notice that the leaves have been arranged in ascending order from left to right. We need to do this so that we can find the median without jumping back and forth across our rows.

It is also worth noting that if there is more than one of the same score, in this case 52 appears twice, each score should have its own leaf.

While stem-and-leaf plots are used primarily to store data of two digit numbers, there are some cases where the stem and leaf might mean something different. It is for this reason that we should always check the key before translating the leaves into scores.

StemLeaf
13\ 6\ 7
20\ 2\ 2\ 7
38\ 9
Key 2\vert 7 = 2.7 km

For example, in this stem-and-leaf plot the stem represents the whole number of kilometres while the leaf represents tenths of a kilometre.

There are also cases of the stem representing the number of tens as usual, except it uses two digit numbers in the stem to express three digit numbers.

StemLeaf
92\ 5
100\ 3\ 3\ 9
117\ 8
123\ 4\ 6\ 8
131\ 1\ 4
Key 12\vert 8 = 128

In this case, the score 12 is represented by the leaf "8" attached to the "12" stem.

In both cases, we need the key to tell us how to interpret the stem-and-leaf plot since the data is different from our usual two digit scores.

Examples

Example 2

A city council selected a number of houses at random. They determined the fastest travel time (in minutes) from each house to the nearest hospital, and recorded the following results:25, \, 37, \, 16, \, 27, \, 27, \, 35, \, 21, \, 18, \, 19, \, 49, \, 14, \, 19, \, 31, \, 42, \, 18

Represent this data as an ordered stem-and-leaf plot.

Worked Solution
Create a strategy

Put the scores in a stem and leave plot where the tens digit is the stem and the leaves are in increasing order.

Apply the idea
StemLeaf
14\ 6\ 8\ 8\ 9\ 9
21\ 5\ 7\ 7
31\ 5\ 7
42\ 9
Key 2\vert 5 = 25

Example 3

The following stem-and-leaf plot shows the ages of 20 employees in a company.

StemLeaf
20\ 1\ 1\ 2\ 8\ 8\ 9
30\ 2\ 4\ 8\ 8
41\ 1\ 1\ 2\ 5
53\ 4\ 8
Key 1\vert 2 = 12
a

How many of the employees are in their 30s?

Worked Solution
Create a strategy

Count how many leaves the stem 3 has.

Apply the idea

\text{Employees} = 5

b

What is the age of the oldest employee?

Worked Solution
Create a strategy

Find the largest leaf of the largest stem, and put them together to find the oldest age.

Apply the idea

\text{Age} = 58

c

What is the age of the youngest employee?

Worked Solution
Create a strategy

Find the smallest leaf of the smallest stem, and put them together to find the youngest age.

Apply the idea

\text{Age} = 20

d

What is the median age of the employees?

Worked Solution
Create a strategy

Since there are 20 scores, find the average of the 10th and 11th score.

Apply the idea

By counting the leaves, the 10th score is 34 and the 11th score is 38.

\displaystyle \text{Median}\displaystyle =\displaystyle \dfrac{34+38}{2}Find the average of the scores
\displaystyle =\displaystyle \dfrac{72}{2}Perform the addition
\displaystyle =\displaystyle 36Perform the division
e

What is the modal age group?

A
20s
B
30s
C
40s
D
50s
Worked Solution
Create a strategy

The modal age group is the most common age group.

Apply the idea

Stem 2, which represents the 20s, has the most leaves. So the correct answer is option A.

Idea summary

A stem-and-leaf plot is made up of two components, the stem and the leaf. The stem is usually used to represent the tens part of a score while the leaf is used to represent the ones part of the score.

Outcomes

MA4-19SP

collects, represents and interprets single sets of data, using appropriate statistical displays

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