# ﻿Finding and Interpreting the Mean as the Balance Point

Warmup

Activity #1

Interpreting the Mean.

Here is the data set showing how long it takes for Diego to walk to school, in minutes, over 5 days.

12, 7, 13, 9, 14.

The mean number of minutes is 11.

• Represent Diego’s data on a dot plot.
• Mark the location of the mean by using red triangle.

The mean can also be seen as a measure of center that balances the points in a data set. If we find the distance between every point and the mean, add the distances on each side of the mean, and compare the two sums, we can see this balancing.

• Record the distance between each point and 11 and its location relative to 11.

(1.) Sum of distances left of 11: ___________ Sum of distances right of 11: ___________

(2.) What do you notice about the two sums?

(3.) Can another point that is not the mean produce similar sums of distances? Let’s investigate whether 10 can produce similar sums as those of 11.

• Complete the table with the distance of each data point from 10.

(4.) Sum of distances left of 10: ___________ Sum of distances right of 10: ___________

(5.) What do you notice about the two sums?

(6.) Based on your work so far, explain why the mean can be considered a balance point for the data set.

Activity #2

`Interpret Diagrams that Represent Mean as a "Balance Point".`

• Study the diagrams below carefully and answer the questions that follow.

Here are dot plots showing how long Diego’s trips to school took in minutes—which you studied in the previous activity—and how long Andre’s trips to school took in minutes.

The dot plots include the means for each data set, marked by triangles.

(1.) Which of the two data sets has a larger mean? In this context, what does a larger mean tell us?

(2.) Which of the two data sets has larger sums of distances to the left and right of the mean?

(3.) What do these sums tell us about the variation in Diego’s and Andre’s travel times?

Activity #3

`Exercise Questions.`

• Study the data plot below and answer the questions that follow.

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

(1.) Calculate the mean of Lin’s travel times.

(2.) Complete the table with the distance between each point and the mean as well whether the point is to the left or right of the mean.

(3.) Find the sum of distances to the left of the mean and the sum of distances to the right of the mean.

(4.) Use your work to compare Lin’s travel times to Andre’s. What can you say about their average travel times?

(5.) What about the variability in their travel times?

Challenge #1

Below is a distribution of 11 data points.

• Move the blue point to see how the mean responds to changing this single value within the distribution.​

(1.) At what point will the blue point be in order to have the same value as the mean?

(2.) Which point on the line will affect the mean the most? Use the blue point to make the determination.

Challenge #2

On school days, Kiran walks to school. Here are the lengths of time, in minutes, for Kiran’s walks on 5 school days: 16, 11, 18, 12, 13. Create a dot plot for Kiran’s data.

(1.) Without calculating, decide if 15 minutes would be a good estimate of the mean. If you think it is a good estimate, explain your reasoning. If not, give a better estimate and explain your reasoning.

(2.) Calculate the mean for Kiran’s data.

Challenge #3

In the table, record the distance of each data point from the mean you found above, and its location relative to the mean.

(1.) Calculate the sum of all distances to the left of the mean, then calculate the sum of distances to the right of the mean. Explain how these sums show that the mean is a balance point for the values in the data set.

(2.) Noah scored 20 points in a game. Mai’s score was 30 points. The mean score for Noah, Mai, and Clare was 40 points. What was Clare’s score? Explain or show your reasoning.