Lesson 5Variability and MAD

Learning Goal

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

Learning Targets

  • I can use means and MADs to compare groups.

  • I know what the mean absolute deviation (MAD) measures and what information it provides.

Lesson Terms

  • average
  • mean
  • mean absolute deviation (MAD)
  • measure of center

Warm Up: Shooting Hoops (Part 1)

Problem 1

Elena, Jada, and Lin enjoy playing basketball during recess. Lately, they have been practicing free throws. They record the number of baskets they make out of 10 attempts. Here are their data sets for 12 school days.

  • Elena

  • 4

  • 5

  • 1

  • 6

  • 9

  • 7

  • 2

  • 8

  • 3

  • 3

  • 5

  • 7

  • Jada

  • 2

  • 4

  • 5

  • 4

  • 6

  • 6

  • 4

  • 7

  • 3

  • 4

  • 8

  • 7

  • Lin

  • 3

  • 6

  • 6

  • 4

  • 5

  • 5

  • 3

  • 5

  • 4

  • 6

  • 6

  • 7

  1. Calculate the mean number of baskets each player made, and compare the means. What do you notice?

  2. What do the means tell us in this context?

Activity 1: Shooting Hoops (Part 3)

The tables show Elena, Jada, and Lin’s basketball data from an earlier activity. Recall that the mean of Elena’s data, as well as that of Jada and Lin’s data, was 5. 

Problem 1

  1. Record the distance between each of Elena’s scores and the mean. 

    Elena

    distance from 5

  2. Now find the average of the distances in the table. Show your reasoning and round your answer to the nearest tenth.

    This value is the mean absolute deviation (MAD) of Elena’s data.
    Elena’s MAD:

Problem 2

Find the mean absolute deviation of Jada’s data. Round it to the nearest tenth.

Jada

distance from 5

Jada’s MAD:

Problem 3

Find the mean absolute deviation of Lin’s data. Round it to the nearest tenth.

Lin

distance from 5

Lin’s MAD:

Problem 4

Compare the MADs and dot plots of the three students’ data. Do you see a relationship between each student’s MAD and the distribution on her dot plot? Explain your reasoning.

Three dot plots labeled “number of baskets Elena made,” “number of baskets Jada made,” and “number of baskets Lin made” are indicated. Each dot plot has the numbers 0 through 10 indicated with a triangle at the number 5. The dot plot “number of baskets Elena made,” has the following data: 1 basket, 1 dot. 2 baskets, 1 dot. 3 baskets, 2 dots. 4 baskets, 1 dot. 5 baskets, 2 dots. 6 baskets, 1 dot. 7 baskets, 2 dots. 8 baskets, 1 dot. 9 baskets, 1 dot. The dot plot “number of baskets Jada made,” has the following data: 2 baskets, 1 dot. 3 baskets, 1 dot. 4 baskets, 4 dots. 5 baskets, 1 dot. 6 baskets, 2 dots. 7 baskets, 2 dots. 8 baskets, 1 dot. The dot plot “number of baskets Lin made,” has the following data: 3 baskets, 2 dots. 4 baskets, 2 dots. 5 baskets, 3 dots. 6 baskets, 4 dots. 7 baskets, 1 dot.

Are you ready for more?

Problem 1

Invent another data set that also has a mean of 5 but has a MAD greater than 2. Remember, the values in the data set must be whole numbers from 0 to 10.

Activity 2: Which Player Would You Choose?

Problem 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 - data set A and B. The mean is noted at approximately 5.2 in both plots. A has data from 3 to 8 and B has data from 1 to 9.

  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.

Problem 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.

  1. Complete the table.

    eighth‐grade student

    distance from 6

  2. Calculate the MAD. Show your reasoning.

  3. Draw a dot plot to represent his data and mark the location of the mean with a triangle ().

  4. Compare the eighth-grade student’s mean and MAD to Noah’s mean and MAD. What do you notice?

  5. Compare their dot plots. What do you notice about the distributions?

  6. What can you say about the two players’ shooting accuracy and consistency?

Are you ready for more?

Problem 1

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

Activity 3: Swimmers Over the Years

Problem 1

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 Summary

We use the mean of a data set as a measure of center of its distribution, but two data sets with the same mean could have very different distributions.

This dot plot shows the weights, in grams, of 22 cookies.

A dot plot for “cookie weights in grams.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 grams. The data are as follows: 18 grams, 1 dot; 19 grams, 3 dots; 20 grams, 4 dots; 21 grams, 5 dots; 22 grams, 6 dots; 23 grams, 2 dots, 24 grams, 1 dot.

The mean weight is 21 grams. All the weights are within 3 grams of the mean, and most of them are even closer. These cookies are all fairly close in weight.

This dot plot shows the weights, in grams, of a different set of 30 cookies.

A dot plot for “cookie weights in grams.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 grams. The data are as follows: 9 grams, 1 dot; 10 grams, 1 dot; 11 grams, 2 dots; 12 grams, 1 dot; 14 grams, 1 dot; 16 grams, 2 dots; 17 grams, 1 dot; 18 grams, 2 dots; 19 grams, 1 dot; 20 grams, 3 dots; 21 grams, 1 dot; 22 grams, 3 dots; 23 grams, 1 dot; 24 grams, 2 dots; 26 grams, 2 dots; 28 grams, 1 dot; 30 grams, 1 dot; 32 grams, 2 dots; 33 grams, 1 dot; 34 grams, 1 dot.

The mean weight for this set of cookies is also 21 grams, but some cookies are half that weight and others are one-and-a-half times that weight. There is a lot more variability in the weight.

There is a number we can use to describe how far away, or how spread out, data points generally are from the mean. This measure of spread is called the mean absolute deviation (MAD).

Here the MAD tells us how far cookie weights typically are from 21 grams. To find the MAD, we find the distance between each data value and the mean, and then calculate the mean of those distances.

For instance, the point that represents 18 grams is 3 units away from the mean of 21 grams. We can find the distance between each point and the mean of 21 grams and organize the distances into a table, as shown.

A dot plot for “cookie weights in grams.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 grams. There are two perpendicular lines drawn, one at 18 grams and the other at 21 grams. A horizontal line between the two lines is labeled 3. The data are as follows: 18 grams, 1 dot; 19 grams, 3 dots; 20 grams, 4 dots; 21 grams, 5 dots; 22 grams, 6 dots; 23 grams, 2 dots, 24 grams, 1 dot.

in grams

18

19

19

19

20

20

20

20

21

21

21

distance from mean

3

2

2

2

1

1

1

1

0

0

0

in grams

21

21

22

22

22

22

22

22

23

23

24

distance from mean

0

0

1

1

1

1

1

1

2

2

3

The values in the first row of the table are the cookie weights for the first set of cookies. Their mean, 21 grams, is the mean of the cookie weights.

The values in the second row of the table are the distances between the values in the first row and 21. The mean of these distances is the MAD of the cookie weights.

What can we learn from the averages of these distances once they are calculated?

  • In the first set of cookies, the distances are all between 0 and 3. The MAD is 1.2 grams, which tells us that the cookie weights are typically within 1.2 grams of 21 grams. We could say that a typical cookie weighs between 19.8 and 22.2 grams.

  • In the second set of cookies, the distances are all between 0 and 13. The MAD is 5.6 grams, which tells us that the cookie weights are typically within 5.6 grams of 21 grams. We could say a typical cookie weighs between 15.4 and 26.6 grams.

The MAD is also called a measure of the variability of the distribution. In these examples, it is easy to see that a higher MAD suggests a distribution that is more spread out, showing more variability.