# Lesson 4

The Mean

### Lesson Narrative

In this lesson, students find and interpret the mean of a distribution (MP2) as the amount each member of the group would get if everything is distributed equally and as a balance point of a numerical distribution. The first technique is sometimes called the “leveling out” or the “fair share” interpretation of the mean. For a quantity that cannot actually be redistributed, like the weights of the dogs in a group, this interpretation translates into a thought experiment.

Suppose all of the dogs in a group had different weights and their combined weight was 200 pounds. The mean would be the weight of the dogs if all the dogs were replaced with the same number of identical dogs and the total weight was still 200 pounds.

Then, students use the structure of the data (MP7) to interpret the mean as the balance point of a numerical distribution. They calculate how far away each data point is from the mean and study how the distances on either side of the mean compare.

Students connect this interpretation to why we call the mean a measure of the center of a distribution and, through this interpretation, begin to see how the mean is useful in characterizing a “typical” value for the group. Students continue to practice calculating the mean of a data set (MP8) and interpreting it in context (MP2).

### Learning Goals

Teacher Facing

• Comprehend the words “mean” and “average” as a measure of center that summarizes the data using a single number.
• Explain (using words and other representations) how to calculate the mean for a numerical data set.
• Represent the mean of a data set on a dot plot and interpret it in the context of the situation.

### Student Facing

Let’s explore the mean of a data set and what it tells us.

### Student Facing

• I can describe what the mean tells us in the context of the data.
• I can find the mean for a numerical data set.

### Glossary Entries

• average

The average is another name for the mean of a data set.

For the data set 3, 5, 6, 8, 11, 12, the average is 7.5.

$$3+5+6+8+11+12=45$$

$$45 \div 6 = 7.5$$

• mean

The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11.

To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. $$7+9+12+13+14=55$$ and $$55 \div 5 = 11$$.

• measure of center

A measure of center is a value that seems typical for a data distribution.

Mean and median are both measures of center.