# 11.6: Measures of Central Tendency and Dispersion

## 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:

Answer:

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

Answer:

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

Answer:

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:

Answer:

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

Answer:

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

Answer:

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:

Answer: and .

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

Answer: None.

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

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:

Answer:

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

Answer:

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

Answer:

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.