Lesson 4The Mean

Learning Goal

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

Learning Targets

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

Lesson Terms

average
mean
measure of center

Warm Up: Which One Doesn’t Belong: Division

Which expression does not belong? Be prepared to explain your reasoning.

$\frac{8 + 8 + 4 + 4}{4}$
$\frac{10 + 10 + 4}{4}$
$\frac{9 + 9 + 5 + 5}{4}$
$\frac{6 + 6 + 6 + 6 + 6}{5}$

Activity 1: Spread Out and Share

Problem 1

The kittens in a room at an animal shelter are placed in 5 crates, as shown.

A large rectangle that is divided into 5 equal squares that represent crates. There are 2 cats are in the first square, 1 cat is in the second square, 4 cats are in the third square and 3 cats in the fourth square. There are no cats are in the fifth square.

The manager of the shelter wants the kittens distributed equally among the crates. How might that be done? How many kittens will end up in each crate?
The number of kittens in each crate after they are equally distributed is called the mean number of kittens per crate, or the average number of kittens per crate.

Explain how the expression $10 \div 5$ is related to the average.
A different room in the shelter has 6 crates. No two crates contain the same number of kittens, and there is an average of 3 kittens per crate.
Draw or describe at least two different arrangements of kittens that match this description. You may choose to use the applet to help.
open applet in presentation mode

Print Version

The kittens in a room at an animal shelter are placed in 5 crates, as shown.

The manager of the shelter wants the kittens distributed equally among the crates. How might that be done? How many kittens will end up in each crate?
The number of kittens in each crate after they are equally distributed is called the mean number of kittens per crate, or the average number of kittens per crate. Explain how the expression $10 \div 5$ is related to the average.
Another room in the shelter has 6 crates. No two crates has the same number of kittens, and there is an average of 3 kittens per crate. Draw or describe at least two different arrangements of kittens that match this description.

Problem 2

Five servers were scheduled to work the number of hours shown. They decided to share the workload, so each one would work equal hours.

Server A: 3
Server B: 6
Server C: 11
Server D: 7
Server E: 4

On the grid, draw 5 bars whose heights represent the hours worked by servers A, B, C, D, and E.
Think about how you would rearrange the hours so that each server gets a fair share. Then, draw a new graph to represent the rearranged hours. Be prepared to explain your reasoning.
Based on your second drawing, what is the average or mean number of hours that the servers will work?
Explain why we can also find the mean by finding the value of $31 \div 5$ .
Which server will see the biggest change to work hours? Which server will see the least change?

Are you ready for more?

Server F, working 7 hours, offers to join the group of five servers, sharing their workload. If server F joins, will the mean number of hours worked increase or decrease? Explain how you know.

Activity 2: Travel Times (Part 2)

Problem 1

Here are dot plots showing how long Diego’s trips to school took in minutes—which you studied earlier—and how long Andre’s trips to school took in minutes. The dot plots include the means for each data set, marked by triangles.

Two dot plots labeled “Diego’s travel time in minutes” and “Andre’s travel time in minutes” are indicated. Each dot plot has the numbers 7 through 22 indicated. The dot plot “Diego’s travel time in minutes” has the following data: 7 minutes, 1 dot. 9 minutes, 1 dot. 11 minutes, 1 triangle. 12 minutes, 1 dot. 13 minutes, 1 dot. 14 minutes, 1 dot. The dot plot “Andre’s travel time in minutes” has the following data: 12 minutes, 2 dots. 13 minutes, 1 dot. 14 minutes, 1 triangle. 16 minutes, 1 dot. 17 minutes, 1 dot.

Which of the two data sets has a larger mean? In this context, what does a larger mean tell us?
Which of the two data sets has larger sums of distances to the left and right of the mean? What do these sums tell us about the variability in Diego’s and Andre’s travel times?

Problem 2

Here is a dot plot showing lengths of Lin’s trips to school.

A dot plot for “travel time in minutes.” The numbers 7 through 22, are indicated. The data are as follows: 8 minutes, 1 dot. 11 minutes, 2 dots. 18 minutes, 1 dot. 22 minutes, 1 dot.

Calculate the mean of Lin’s travel times.
Complete the table with the distance between each point and the mean as well as whether the point is to the left or right of the mean.
time in minutes
distance from the mean
left or right of the mean?
$22$
$18$
$11$
$8$
$11$
Find the sum of distances to the left of the mean and the sum of distances to the right of the mean.
Use your work to compare Lin’s travel times to Andre’s. What can you say about their average travel times? What about the variability in their travel times?

Lesson Summary

Sometimes a general description of a distribution does not give enough information, and a more precise way to talk about center or spread would be more useful. The mean, or average, is a number we can use to summarize a distribution.

We can think about the mean in terms of “fair share” or “leveling out.” That is, a mean can be thought of as a number that each member of a group would have if all the data values were combined and distributed equally among the members.

For example, suppose there are 5 bottles which have the following amounts of water: 1 liter, 4 liters, 2 liters, 3 liters, and 0 liters.

There are 5 identical tape diagrams that are each partitioned into 4 equal parts. The first diagram has 1 part shaded. The second diagram has 4 parts shaded. The third diagram has 2 parts shaded. The fourth diagram has 3 parts shaded. The fifth diagram has no parts shaded.

To find the mean, first we add up all of the values. We can think of as putting all of the water together: $1 + 4 + 2 + 3 + 0 = 10$

A tape diagram partitioned into 10 equal parts. All 10 parts are shaded.

To find the “fair share,” we divide the 10 liters equally into the 5 containers: $10 \div 5 = 2$ .

There are 5 identical tape diagrams each partitioned into 4 equal parts. Each diagram has 2 parts shaded.

Suppose the quiz scores of a student are 70, 90, 86, and 94. We can find the mean (or average) score by finding the sum of the scores $(70 + 90 + 86 + 94 = 340)$ and dividing the sum by four $(340 \div 4 = 85)$ . We can then say that the student scored, on average, 85 points on the quizzes.

In general, to find the mean of a data set with values, we add all of the values and divide the sum by .

The mean is often used as a measure of center of a distribution. This is because the mean of a distribution can be seen as the “balance point” for the distribution.

The sum of the distances for the data points to the left of the mean is equal to the sum of the distances for the data points to the right of the mean. So, the mean is often near the middle of the distribution, especially when the data is symmetric.

time in minutes	distance from the mean	left or right of the mean?
$22$
$18$
$11$
$8$
$11$