# 11.6: Measures of Central Tendency and Dispersion

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

• Compare measures of central tendency.
• Measure the dispersion of a collection of data.
• Calculate and interpret measures of central tendency and dispersion for real-world situations.

## Vocabulary

Terms introduced in this lesson:

measure of central tendency
average
data set
mean
median
mode
sum of values, number of values
ordered list
outlier
dispersion
variance
standard deviation

## Teaching Strategies and Tips

Students learn that “average” has a general meaning:

• Averages describe the general characteristics of a group.
• Mean, median, and mode are examples of averages.

Use Examples 1 and 2 to introduce the mean.

Find the mean of the given data.

a. The students in Mr. Peterson’s math class took the AP Statistics exam. Their math scores are:

\begin{align*}3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3\end{align*}

Answer: \begin{align*}3.18\end{align*}

b. The weights, in ounces, of several cookies taken from the same package are:

\begin{align*}0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91\end{align*}

Answer: \begin{align*}0.92\end{align*}

c. The precipitation, in mm, for the city of Townville in the month of October is:

\begin{align*}142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00, 0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 15.20, 12.22, 0.00, 0.00, 0.00, 0.00, 0.02, 0.03, 0.01, 14.16, 18.13\end{align*}

Answer: \begin{align*}38.37\end{align*}

Use Examples 3 and 4 to introduce the median.

Find the median of the given data.

a. The students in Mr. Peterson’s math class took the AP Statistics exam. Their math scores are:

\begin{align*}3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3\end{align*}

Answer: \begin{align*}3\end{align*}

b. The weights, in ounces, of several cookies taken from the same package are:

\begin{align*}0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91\end{align*}

Answer: \begin{align*}0.92\end{align*}

c. The precipitation, in mm, for the city of Townville in the month of October is:

\begin{align*}142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00, 0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 15.20, 12.22, 0.00, 0.00, 0.00, 0.00, 0.02, 0.03, 0.01, 14.16, 18.13\end{align*}

Answer: \begin{align*}12.185\end{align*}

Use Example 5 to introduce the mode.

Find the mode of the given data.

a. The students in Mr. Peterson’s math class took the AP Statistics exam. Their math scores are:

\begin{align*}3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3\end{align*}

Answer: \begin{align*}3\end{align*} and \begin{align*}4\end{align*}.

b. The weights, in ounces, of several cookies taken from the same package are:

\begin{align*}0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91\end{align*}

c. The precipitation, in mm, for the city of Townville in the month of October is:

\begin{align*}142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00, 0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 15.20, 12.22, 0.00, 0.00, 0.00, 0.00, 0.02, 0.03, 0.01, 14.16, 18.13\end{align*}

Use Examples 6-8 to introduce measures of dispersion.

Find the range, variance, and standard deviation of the given data.

a. The students in Mr. Peterson’s math class took the AP Statistics exam. Their math scores are:

\begin{align*}3, 2, 2, 3, 4, 4, 3, 1, 2, 3, 4, 5, 4, 2, 2, 3, 4, 3, 4, 5, 4, 3\end{align*}

\begin{align*}\text{range} & = 4 \\ s^2 & = 1.109\\ s & = 1.053\end{align*}

b. The weights, in ounces, of several cookies taken from the same package are:

\begin{align*}0.95, 0.85, 0.93, 0.90, 0.97, 0.96, 0.87, 0.91\end{align*}

\begin{align*}\text{range} & = 0.12\\ s^2 & = 0.0018\\ s & = 0.0430\end{align*}

c. The precipitation, in mm, for the city of Townville in the month of October is:

\begin{align*}142.20, 0.02, 0.01, 12.15, 92.72, 103.21, 138.90, 102.92, 12.07, 1.00,0.00, 0.00, 12.01, 21.02, 22.02, 87.91, 89.60, 132.72, 120.82, 15.20, 12.22, 0.00, 0.00, 0.00, 0.00, 0.02, 0.03, 0.01, 14.16, 18.13\end{align*}

\begin{align*}\text{range} & = 142.2\\ s^2 & = 2601\\ s & = 51\end{align*}

In Example 7, students learn to separate the components of the standard deviation formula into manageable pieces.

• Suggest that students reconstruct such a multi-column table for Review Question 2.
• Suggest that students use a calculator to only to check their calculations.

Encourage students to provide the kind of explanations shown in Example 9 for Review Questions 4 and 5. Ask:

• What does a larger mean imply?
• What does a larger standard deviation imply?

Emphasize that the appropriateness of an average depends on the shape of the distribution of the data.

• Use the mean for tightly clustered data sets.
• Use the median for data sets with a lot of spread.
• Use the mode when there is no apparent pattern in the distribution.

## Error Troubleshooting

General Tip: Remind students to order the data set when computing the median and the range.

General Tip: Suggest that students transfer data to calculator carefully.

• Have students put a line through the data points as they get transferred.
• Have students count the number of data points in both lists to prevent omissions.

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