Math Academy

Interpreting Shapes of Distributions Using Dot Plots

Introduction

Identifying Outliers From a Dot Plot

Tails, Skew, and Symmetry

Identifying Symmetry and Skew From a Dot Plot

The Mean of a Symmetric Distribution

The Effect of Skew on the Mean

Estimating the Location of the Mean From a Dot Plot

Prerequisites

Measuring Range and MAD From Dot Plots

Outliers

Introduction

Remember that an outlier of a data set is a value much higher or much lower than the other values.

When a data set is represented as a dot plot, it's easy to pick out the outliers: we have to look for values much further to the left or right than the main cluster of values.

To demonstrate, consider the dot plot below:

Notice that most values are concentrated between and but one value is separated from the main group.

Since this value is separated from the main group, we conclude that is an outlier of the distribution.

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Example: Identifying Outliers From a Dot Plot

How many outliers are there for the data set whose dot plot is shown above?

EXPLANATION

Notice that most values are concentrated between and but one value is separated from the main group.

The value is the outlier of the distribution. Therefore, there is outlier.

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Practice: Identifying Outliers From a Dot Plot

How many outliers are there for the data set whose dot plot is shown above?

a	$3$
b	$4$
c	$1$
d	$2$
e	$5$

EXPLANATION

Notice that most values are concentrated between $40$ and $50,$ but two values are separated from the main group.

The values $\color{blue}10$ and $\color{blue}80$ are the outliers of the distribution. Therefore, there are $2$ outliers.

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Practice: Identifying Outliers From a Dot Plot

What are the outliers for the data set whose dot plot is shown above?

a	$95$ and $110$ only
b	$110$ only
c	$75$ and $80$ only
d	$80$ only
e	$95$ only

EXPLANATION

Notice that most values are concentrated between $95$ and $110,$ but two values are separated from the main group.

The values $\color{blue}75$ and ${\color{blue}80}$ are the outliers of the distribution.

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Tails, Skew, and Symmetry

A tail of a distribution is a part that extends away from the main cluster.

If the left tail of the distribution is longer than the right tail, then we say that the distribution is left-skewed (or negatively skewed). An example of a left-skewed distribution is shown below.

Similarly, if the right tail of the distribution is longer than the left tail, then we say that the distribution is right-skewed (or positively skewed). An example of a right-skewed distribution is shown below.

If the tails of the distribution are the same, we say that the distribution is symmetric. An example of a symmetric distribution is shown below.

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Example: Identifying Symmetry and Skew From a Dot Plot

The dot plot above shows the working hours of a small group of employees. Each dot represents a single employee. What is the shape of the distribution?

EXPLANATION

From the dot plot, we see that the distribution is left-skewed. It has a long tail on the left-hand side.

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Practice: Identifying Symmetry and Skew From a Dot Plot

The dot plot above shows the daily high temperatures of several US cities on a specific day. Each dot represents a single city. What is the shape of the distribution?

a	the distribution is left-skewed
b	the distribution is symmetric
c	the distribution is right-skewed
d	the distribution is left-skewed and symmetric
e	the distribution is right-skewed and symmetric

Practice: Identifying Symmetry and Skew From a Dot Plot

The dot plot above shows the heights of a small group of children, where each child's height has been rounded to the nearest ten centimeters. Each dot represents a different child. What is the shape of the distribution?

a	the distribution is both left-skewed and symmetric
b	the distribution is right-skewed
c	none of the options describes the shape
d	the distribution is symmetric
e	the distribution is left-skewed

The Mean of a Symmetric Distribution

When a distribution is perfectly symmetric, the mean lies at the midpoint of the distribution.

To demonstrate, consider the symmetric distribution below:

To calculate this distribution's mean (midpoint), we find the mean of the two most extreme values. The largest value is , and the smallest value is Therefore,

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The Effect of Skew on the Mean

Let's now consider the right-skewed distribution below.

Notice that the midpoint (i.e., the mean of the two most extreme values) is also

Since the distribution is right-skewed, most of the data points lie to the left of the midpoint. Consequently, the mean also lies to the left of the midpoint.

The situation is reversed when the distribution is left-skewed.

In this case, most of the data points lie to the right of the midpoint. Consequently, the mean also lies to the right of the midpoint.

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Example: Estimating the Location of the Mean From a Dot Plot

The dot plot above shows how much time each student in Mr. Johnson's class spent studying math on a particular day. Each dot represents a single student.

Which of the following statements are true?

The distribution is approximately symmetric.
The mean of the distribution equals
The mean of the distribution exceeds
The mean of the distribution is less than

EXPLANATION

Let's examine the statements one by one.

Statement I is false. The distribution is left-skewed (since we have a longer tail on the left). It is not symmetric.

Statement III is true, while statements II and IV are false. All of the data is situated between and minutes. If the data were symmetric, then the mean would be given by the midpoint, which is However, since the distribution is left-skewed, most of the data points are concentrated on the right-hand side. This indicates that the mean of our distribution lies to the right of the midpoint.
So, the mean is greater than minutes.

Therefore, the correct answer is "III only."

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Practice: Estimating the Location of the Mean From a Dot Plot

The dot plot above shows the height distribution for a particular group of people, where each person's height has been rounded to the nearest centimeter. Each dot represents a different person. Which of the following statements are true?

The distribution is approximately symmetric.
The mean of the distribution is less than $155 \, \text{cm}.$
The mean of the distribution equals $155 \, \text{cm}.$

a	I and III only
b	III only
c	I only
d	I and II only
e	II only

EXPLANATION

Let's examine the statements one by one.

Statement I is false. The distribution is right-skewed (since we have a longer tail on the right). It is not symmetric.

Statement II is true, while statement III is false. All of the data is situated between $150\,\textrm{cm}$ and $160\,\textrm{cm}.$ If the data were symmetric, then the mean would be given by the midpoint, which is $\dfrac{150+160}{2} = 155\,\textrm{cm}.$ However, since the distribution is right-skewed, most of the data points are concentrated on the left-hand side. This indicates that the mean of the distribution lies to the left of the midpoint.
So, the mean is less than $155\,\textrm{cm}.$

Therefore, the correct answer is "II only."

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Practice: Estimating the Location of the Mean From a Dot Plot

The dot plot above shows the number of riders in each team that completed the Tour de France final stage in a particular year. Each dot represents a different team. Which of the following statements are true?

The distribution is symmetric.
The mean of the distribution equals $5.$
The mean of the distribution exceeds $5.$
The mean of the distribution is less than $5.$

a	III only
b	I and II only
c	IV only
d	I and III only
e	II only

EXPLANATION

Let's examine the statements one by one:

Statement I is true. The distribution is symmetric (since the left-hand side is a mirror image of the right-hand side).

Statement II is true, while statements III and IV are false. All of the data is situated between $2$ and $8$ riders. Since the data is symmetric, the mean is given by the midpoint. Therefore, $\textrm{mean} = \dfrac{2+8}{2} = 5.$ The location of the mean (midpoint) is shown below.

Therefore, the correct answer is "I and II only."

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