# 11.6: Measures of Central Tendency and Dispersion

**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.

Additional Examples:

*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.

Additional Examples:

*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.

Additional Examples:

*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*}

Answer: None.

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:

Use Examples 6-8 to introduce measures of dispersion.

Additional Examples:

*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*}

Answer:

\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*}

Answer:

\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*}

Answer:

\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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