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Lesson 12Using Mean and MAD to Make Comparisons

Let's use mean and MAD to describe and compare distributions.

Learning Targets:

  • I can say what the MAD tells us in a given context.
  • I can use means and MADs to compare groups.

12.1 Number Talk: Decimal Division

Find the value of each expression mentally.

42\div12

2.4\div12

44.4\div12

46.8\div12

12.2 Which Player Would You Choose?

  1. Andre and Noah joined Elena, Jada, and Lin in recording their basketball scores. They all recorded their scores in the same way: the number of baskets made out of 10 attempts. Each collected 12 data points.

    Andre’s mean number of baskets was 5.25, and his MAD was 2.6. Noah’s mean number of baskets was also 5.25, but his MAD was 1.

    Here are two dot plots that represent the two data sets. The triangle indicates the location of the mean.

    Two dot plots for “number of baskets made.” The numbers 0 through 10 are indicated. On each dot plot there is a red triangle located between 5 and 6 baskets made. The data for “data set A” are as follows: 3 baskets, 1 dot. 4 baskets, 2 dots. 5 baskets, 5 dots. 6 baskets, 2 dots. 7 baskets, 1 dot. 8 baskets, 1 dot. The data for “data set B” are as follows: 1 basket, 1 dot. 2 baskets, 2 dots. 3 baskets, 1 dot. 4 baskets, 2 dots. 6 baskets, 1 dot. 7 baskets, 1 dot. 8 baskets, 2 dots. 9 baskets, 2 dots.
    1. Without calculating, decide which dot plot represents Andre’s data and which represents Noah’s. Explain how you know.
    2. If you were the captain of a basketball team and could use one more player on your team, would you choose Andre or Noah? Explain your reasoning.
  2. An eighth-grade student decided to join Andre and Noah and kept track of his scores. His data set is shown here. The mean number of baskets he made is 6.
    eighth‐grade student 6 5 4 7 6 5 7 8 5 6 5 8
    distance from 6
    1. Calculate the MAD. Show your reasoning.
    2. Draw a dot plot to represent his data and mark the location of the mean with a triangle ( \Delta ).
    3. Compare the eighth-grade student’s mean and MAD to Noah’s mean and MAD. What do you notice?
    4. Compare their dot plots. What do you notice about the distributions?
    5. What can you say about the two players’ shooting accuracy and consistency?

Are you ready for more?

Invent a data set with a mean of 7 and a MAD of 1.

12.3 Swimmers Over the Years

In 1984, the mean age of swimmers on the U.S. women’s swimming team was 18.2 years and the MAD was 2.2 years. In 2016, the mean age of the swimmers was 22.8 years, and the MAD was 3 years.

  1. How has the average age of the women on the U.S. swimming team changed from 1984 to 2016? Explain your reasoning.
  2. Are the swimmers on the 1984 team closer in age to one another than the swimmers on the 2016 team are to one another? Explain your reasoning.

  3. Here are dot plots showing the ages of the women on the U.S. swimming team in 1984 and in 2016. Use them to make two other comments about how the women’s swimming team has changed over the years.

    Two dot plots labeled “age of swimmers, in years" each have the numbers 14 through 30, in increments of 2, indicated. The top dot plot is for 1984 and the bottom dot plot is for 2016. The data for 1984 is as follows. 14 years, 0 dots 15 years, 2 dots 16 years, 2 dots 17 years,1 dot 18 years, 2 dots 19 years, 4 dots 20 years, 1 dot 21 years, 1 dot 22 years, 0 dots 23 years, 1 dot 24 years, 0 dots 25 years, 0 dots 26 years, 0 dots 27 years, 0 dots 28 years, 0 dots 29 years, 0 dots 30 years, 0 dots The data for 2016 is as follows. 14 years, 0 dots 15 years, 0 dots 16 years, 0 dots 17 years, 0 dots 18 years, 0 dots 19 years, 3 dots 20 years, 3 dots 21 years, 3 dots 22 years, 3 dots 23 years, 1 dot 24 years, 2 dots 25 years, 4 dots 26 years, 1 dot 27 years, 0 dots 28 years, 0 dots 29 years, 1 dot 30 years, 1 dot

Lesson 12 Summary

Sometimes two distributions have different means but the same MAD.

Pugs and beagles are two different dog breeds. The dot plot shows two sets of weight data—one for pugs and the other for beagles.

A dot plot for two sets of data: "pug weights in kilograms" and "beagle weights in kilograms". The numbers 6 through 11 are indicated and there are tick marks midway between each indicated number. There are also two triangles indicated. The first triangle is at 7 kilograms with a horizontal line drawn below the triangle that begins at 6.5 and ends at 7.5 kilograms. The second triangle is indicated at 10 kilograms with a horizontal line drawn below the triangle that begins at 9.5 and ends at 10.5 kilograms. The data for "pug weights in kilograms" are as follows: 6 kilograms, 1 dot. 6.2 kilograms, 2 dots. 6.4 kilograms, 2 dots. 6.6 kilograms, 2 dots. 6.8 kilograms, 2 dots. 7 kilograms, 3 dots. 7.2 kilograms, 3 dots. 7.4 kilograms, 1 dot. 7.6 kilograms, 2 dots. 7.8 kilograms, 1 dot. 8 kilograms, 1 dot. The data for "beagle weights in kilograms" are as follows: 9 kilograms, 1 x. 9.2 kilograms, 2 x's. 9.4 kilograms, 1 x. 9.6 kilograms, 3 x's. 9.8 kilograms, 1 x. 10 kilograms, 3 x's. 10.2 kilograms, 3 x's. 10.4 kilograms, 1 x. 10.6 kilograms, 2 x's. 10.8 kilograms, 2 x's. 11 kilograms, 1 x.
  • The mean weight for pugs is 7 kilograms, and the MAD is 0.5 kilogram.
  • The mean weight for beagles is 10 kilograms, and the MAD is 0.5 kilogram.

We can say that, in general, the beagles are heavier than the pugs. A typical weight for the beagles in this group is about 3 kilograms heavier than a typical weight for the pugs.

The variability of pug weights, however, is about the same as the variability of beagle weights. In other words, the weights of pugs and the weights of beagles are equally spread out.

Lesson 12 Practice Problems

  1. The dot plots show the amounts of time that ten U.S. students and ten Australian students took to get to school. Which statement is true about the MAD of the Australian data set?

    Two dot plots for “travel time in minutes”. The upper dot plot is labeled “U.S.” and the lower dot plot is labeled “Australia.” On each dot plot, the numbers 0 through 60, in increments of 10, are indicated. On the “U.S.” dot plot the data are as follows: 2 minutes, 2 dots. 6 minutes, 2 dots. 7 minutes, 3 dots. 11 minutes, 1 dot. 17 minutes, 1 dot. 20 minutes, 1 dot. On the “Australia” dot plot the data are as follows: 5 minutes, 1 dot. 7 minutes, 1 dot. 9 minutes, 1 dot. 15 minutes, 2 dots. 20 minutes, 3 dots. 25 minutes, 1 dot. 45 minutes, 1 dot.
    1. It is significantly less than the MAD of the U.S. data set.
    2. It is exactly equal to the MAD of the U.S. data set.
    3. It is approximately equal to the MAD of the U.S. data set.
    4. It is significantly greater than the MAD of the U.S. data set.

  2. The dot plots show the amounts of time that ten South African students and ten Australian students took to get to school. Without calculating, answer the questions.

    Two dot plots for "travel time in minutes" with the top labeled "South Africa" and the bottom labeled "Australia". The numbers 0 through 60 are indicated in increments of 10, and there are also tick marks midway between. The 10 data values for "South Africa" are as follows: 5 minutes, 2 dots. 10 minutes, 2 dots. 15 minutes, 2 dots. 30 minutes, 1 dot. 40 minutes, 1 dot. 45 minutes, 1 dot. 60 minutes, 1 dot. The 10 data values for "Australia" are as follows: 5 minutes, 1 dot. 7 minutes, 1 dot. 9 minutes, 1 dot. 15 minutes, 2 dots. 20 minutes, 3 dots. 25 minutes, 1 dot. 45 minutes, 1 dot.
    1. Which data set has the smaller mean? Explain your reasoning.
    2. Which data set has the smaller MAD? Explain your reasoning.
    1. What does a smaller mean tell us in this context?
    2. What does a smaller MAD tell us in this context?

  3. Two high school basketball teams have identical records of 15 wins and 2 losses. Sunnyside High School's mean score is 50 points and its MAD is 4 points. Shadyside High School's mean score is 60 points and its MAD is 15 points.

    Lin read the records of each team’s score. She likes the team that had nearly the same score for every game it played. Which team do you think Lin likes? Explain your reasoning.

  4. Jada thinks the perimeter of this rectangle can be represented with the expression a+a+b+b . Andre thinks it can be represented with 2a+2b .

    Do you agree with either, both, or neither of them? Explain your reasoning.

    a rectangle that has the side lengths of a and b

  5. Draw a number line.

    1. Plot and label three numbers between -2 and -8 (not including -2 and -8).
    2. Use the numbers you plotted and the symbols < and > to write three inequality statements.

  6. Adult elephant seals generally weigh about 5,500 pounds. If you weighed 5 elephant seals, would you expect each seal to weigh exactly 5,500 pounds? Explain your reasoning.