Describe Distributions Using the Mean and MAD

Videos and solutions to help grade 6 students learn how describe the data distribution in terms of center and variability and use the mean and MAD to describe similarities and differences between two distributions.

New York State Common Core Math Grade 6, Module 6, Lesson 11

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Lesson 11 Student Outcomes

• Students use the mean and MAD to describe a data distribution in terms of center and variability.
• Students use the mean and MAD to describe similarities and differences between two distributions.

Lesson 11 Summary

This lesson focused on comparing two data distributions based on center and variability. It is important to consider the context when comparing distributions. In decision-making, drawing dot plots and calculating means and MADs can help you make informed decisions.

Lesson 11 Classwork

Example 1: Comparing Distributions with the Same Mean
In Lesson 10, a data distribution was characterized mainly by its center (mean) and variability (MAD). How these measures help us make a decision often depends on the context of the situation. For example, suppose that two classes of students took the same test and their grades (based on 100 points) are shown in the following dot plots. The mean score for each distribution is 79 points. Would you rather be in Class A or Class B if you had a score of 79?

Exercises 1–6

1. Looking at the dot plots, which class has the greater MAD? Explain without actually calculating the MAD.
2. If Liz had one of the highest scores in her class, in which class would she rather be? Explain your reasoning.
3. If Logan scored below average, in which class would he rather be? Explain your reasoning.

Exercises 4–6
Your little brother asks you to replace the battery in his favorite remote control car. The car is constructed so that it is difficult to replace its battery. Your research of the lifetimes (in hours) of two different battery brands (A and B) shows the following data for 20 batteries from each brand:
4. To help you decide which battery to purchase, start by drawing a dot plot for each brand.
5. Find the mean battery life for each brand and compare them.
6. Looking at the variability of each data set shown in its dot plot, give one reason you would choose Brand A. What is one reason you would choose Brand B? Explain your reasoning.

Example 2: Comparing Distributions with Different Means
You have been comparing distributions that have the same mean, but different variability. As you have seen, deciding whether large variability or small variability is best depends on the context and on what is being asked. For example, in Exercise 2, Liz preferred to be in the distribution with more variability because she had one of the highest scores in the class. Thus, her score would have been higher had she been in Class A than had she been in Class B. Logan, on the other hand, preferred the class with lesser variability (i.e., Class B), since his score was below average. If two data distributions have different means, how does a measure of variability play a part in making decisions?

Exercises 7–9
Suppose that you wanted to answer the following question: Are field crickets better predictors of atmospheric temperature than katydids are? Both species of insect make chirping sounds by rubbing their front wings together. The following data are the number of chirps (per minute) for 10 insects each. All the data were taken on the same evening at the same time.
7. Draw dot plots for these two data distributions using the same scale, going from 30 to 70. Visually, what conclusions can you draw from the dot plots?
8. Calculate the mean and MAD for each distribution.
9. The outside temperature can be predicted by counting the number of chirps made by these insects.
a. For crickets, T is found by adding 40 to its mean number of chirps per minute. What value of T is being predicted by the crickets?
b. For katydids, T is found by adding to 161 its mean number of chirps per minute and then dividing the sum by 3. What value of is being predicted by the katydids?
c. The temperature was 75 degrees when these data were recorded, so using the mean from each data set gave an accurate prediction of temperature. If you were going to use the number of chirps from a single cricket or a single katydid to predict the temperature, would you use a cricket or a katydid? Explain how variability in the distributions of number of chirps played a role in your decision.

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Solution Set

1. Two classes took the same mathematics test. Summary measures for the two classes are as follows:
   a. Suppose that you received the highest score in your class. Would your score have been higher if you were in Class A or Class B? Explain your reasoning.
   b. Suppose that your score was below the mean score. In which class would you prefer to have been? Explain your reasoning.
2. Eight of each of two varieties of tomato plants, LoveEm and Wonderful, are grown under the same conditions. The numbers of tomatoes produced from each plant of each variety are shown:
   a. Draw dot plots to help you decide which variety is more productive.
   b. Calculate the mean number of tomatoes produced for each variety. Which one produces more tomatoes on average?
   c. If you want to be able to accurately predict the number of tomatoes a plant is going to produce, which variety should you choose—the one with the smaller MAD or the one with the larger MAD? Explain your reasoning.
   d. Calculate the MAD of each plant variety.

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