Estimating Centers and Interpreting the Mean as a Balance Point

Examples, videos, lessons, and solutions to help Algebra I students learn to estimate the mean and median of a distribution represented by a dot plot or a histogram.

New York State Common Core Math Algebra I, Module 2, Lesson 3

Worksheets for Algebra 1

Lesson 3 Student Outcomes

• Students learn how to estimate the mean and median of a distribution represented by a dot plot or a histogram.
• Students indicate that the mean is a reasonable description of a typical value for a distribution that is symmetrical but that the median is a better description of a typical value for a distribution that is skewed. • Students interpret the mean as a balance point of a distribution. • Students indicate that for a distribution in which neither the mean nor the median is a good description of a typical value, the mean still provides a description of the center of a distribution in terms of the balance point.

Lesson 3 Summary

The mean of a data distribution represents a balance point for the distribution. The sum of the distances to the right of the mean is equal to the sum of the distances to the left of the mean.

Exit Ticket

1. Draw a dot plot of a data distribution representing the ages of twenty people for which the median and the mean would be approximately the same.
2. Draw a dot plot of a data distribution representing the ages of twenty people for which the median is noticeably less than the mean.
3. An estimate of the balance point for a distribution of ages represented on a number line resulted in a greater sum of the distances to the right than the sum of the distances to the left. In which direction should you move your estimate of the balance point? Explain.

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Exercise

Twenty-two students from the junior class and twenty-six students from the senior class at River City High School participated in a walkathon to raise money for the school’s band. Dot plots indicating the distances in miles students from each class walked are shown below:

8. Estimate the mean number of miles walked by a junior, and mark it with an “X” on the junior class dot plot. How did you estimate this position?
9. What is the median of the junior data distribution?
10. Is the mean number of miles walked by a junior less than, approximately equal to, or greater than the median number of miles? If they are different, explain why. If they are approximately the same, explain why.
11. How would you describe the typical number of miles walked by a junior in this walkathon?
12. Estimate the mean number of miles walked by a senior, and mark it with an “X” on the senior class dot plot. How did you estimate this position?
13. What is the median of the senior data distribution?
14. Estimate the mean and the median of the miles walked by the seniors. Is your estimate of the mean number of miles less than, approximately equal to, or greater than the median number of miles walked by a senior? If they are different, explain why. If they are approximately the same, explain why.
15. How would you describe the typical number of miles walked by a senior in this walkathon?
16. A junior from River City High School indicated that the number of miles walked by a typical junior was better than the number of miles walked by a typical senior. Do you agree? Explain your answer.
    Finally, the twenty-five sophomores who participated in the walkathon reported their results. A dot plot is shown below.
17. What is different about the sophomore data distribution compared to the data distributions for juniors and for seniors?
18. Estimate the balance point of the sophomore data distribution.
19. What is the median number of miles walked by a sophomore?
20. How would you describe the sophomore data distribution?

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