 # The Mean Absolute Deviation (MAD)

Videos and solutions to help grade 6 students learn how to calculate the mean absolute deviation (MAD) for a given data set and interpret the MAD as the average distance of data values from the mean.

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Common Core For Grade 6

### New York State Common Core Math Grade 6, Module 6, Lesson 9

Lesson 9 Student Outcomes

• Students calculate the mean absolute deviation (MAD) for a given data set.
• Students interpret the MAD as the average distance of data values from the mean.

Lesson 9 Summary

In this lesson, a formula was developed that measures the amount of variability in a data distribution.
• The absolute deviation of a data point is how far away that data point is from the mean.
• The Mean Absolute Deviation (MAD) is computed by finding the mean of the absolute deviations in the distribution.
• The value of MAD is the average distance that all the data values are from the mean.
• A small MAD indicates that the distribution has very little variability.
• A large MAD indicates that the data points are spread far away from the mean.

Lesson 9 Classwork

Example 1: Variability
In Lesson 8, Robert tried to decide to which of two cities he would rather move, based on comparing their mean annual temperatures. Since the mean yearly temperature for New York City and San Francisco turned out to be about the same, he decided instead to compare the cities based on the variability in their monthly temperatures from the overall mean. He looked at the two distributions and decided that the New York City temperatures were more spread out from their mean than were the San Francisco temperatures from their mean.

Exercises 1–3
The following temperature distributions for seven other cities all have a mean temperature of approximately 63 degrees. They do not have the same variability. Consider the following dot plots of the mean yearly temperatures of the seven cities in degrees Fahrenheit.
1. Which distribution has the smallest variability of the temperatures from its mean of 63 degrees? Explain your answer.

2. Which distribution(s) seems to have the most variability of the temperatures from the mean of 63 degrees? Explain your answer.

3. Order the seven distributions from least variability to most variability. Explain why you listed the distributions in the order that you chose.

Example 2: Measuring Variability
Based on just looking at the distributions, there are different orderings of variability that seem to make some sense. Sabina is interested in developing a formula that will give a number that measures the variability in a data distribution. She would then use the formula for each data set and order the distributions from lowest to highest. She remembers from a previous lesson that a deviation is found by subtracting the mean from a data point. The formula was summarized as: deviation = data point – mean. Using deviations to develop a formula measuring variability is a good idea to consider.

Exercises 4–6
The dot plot for the temperatures in City is shown below. Use the dot plot and the mean temperature of degrees to answer the following questions.
4. Fill in the following table for City temperature deviations.

5. Why should the sum of your deviations column be equal to zero? (Hint: Recall the balance interpretation of the mean of a data set.)

6. Another way to graph the deviations is to write them on a number line as follows. What is the sum of the positive deviations (the deviations to the right of the mean)? What is the sum of the negative deviations (the deviations to the left of the mean)? What is the total sum of the deviations?

Example 3: Finding the Mean Absolute Deviation (MAD)
By the balance interpretation of the mean, the sum of the deviations for any data set will always be zero. Sabina is disappointed that her idea of developing a measure of variability using deviations isn’t working. She still likes the concept of using deviations to measure variability, but the problem is that the sum of the positive deviations is cancelling out the sum of the negative deviations. What would you suggest she do to keep the deviations as the basis for a formula but to avoid the deviations cancelling out each other?

Exercises 7–8
7. One suggestion to possibly help Sabina is to take the absolute value of the deviations.
a. Fill in the following table.
b. From the following graph, what is the sum of the absolute deviations?
c. Sabina suggests that the mean of the absolute deviations could be a measure of the variability in a data set. Its value is the average distance that all the data values are from the mean temperature. It is called the Mean Absolute Deviation and is denoted by the letters, MAD. Find the MAD for this data set of City G temperatures. Round to the nearest tenth.
d. Find the MAD for each of the temperature distributions in all seven cities, and use the values to order the distributions from least variability to most variability. Recall that the mean for each data set is degrees. Does the list that you made in Exercise 2 by just looking at the distributions match this list made by ordering MAD values?
e. Which of the following is a correct interpretation of the MAD?
i. The monthly temperatures in City G are spread degrees from the approximate mean of degrees.
ii. The monthly temperatures in City G are, on average, degrees from the approximate mean temperature of degrees.
iii. The monthly temperatures in City G differ from the approximate mean temperature of degrees by degrees.

8. The dot plot for City A temperatures follows.
a. How much variability is there in City A’s temperatures? Why?
b. Does the MAD agree with your answer in part (a)? Problem Set
1. Suppose the dot plot on the left shows the number of goals a boys’ soccer team has scored in six games so far this season, and the dot plot on the right shows the number of goals a girls’ soccer team has scored in six games so far this season. The mean for both of these teams is 3.
a. Before doing any calculations, which dot plot has the larger MAD? Explain how you know.
b. Use the following tables to find the MAD for each distribution. Round your calculations to the nearest hundredth.
c. Based on the computed MAD values, for which distribution is the mean a better indication of a typical value? Explain your answer.

2. Recall Robert’s problem of deciding whether to move to New York City or to San Francisco. A table of temperatures (in degrees Fahrenheit) and absolute deviations for New York City follows:
a. The absolute deviations for the monthly temperatures are shown in the above table. Use this information to calculate the MAD. Explain what the MAD means in words.
b. Complete the following table, and then use the values to calculate the MAD for the San Francisco data distribution.
c. Comparing the MAD values for New York City and San Francisco, which city would Robert choose to move to if he is interested in having a lot of variability in monthly temperatures? Explain using the MAD.

3. Consider the following data of the number of green jelly beans in seven bags sampled from each of five different candy manufacturers (Awesome, Delight, Finest, Sweeties, YumYum). Note that the mean of each distribution is 42 green jelly beans.
a. Complete the following table of the absolute deviations for the seven bags for each candy manufacturer.
b. Based on what you learned about MAD, which manufacturer do you think will have the lowest MAD? Calculate the MAD for the manufacturer you selected.

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