Download Lesson 9: The Mean Absolute Deviation (MAD)

Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
6•6
Lesson 9: The Mean Absolute Deviation (MAD)
Student Outcomes

Students calculate the mean absolute deviation (MAD) for a given data set.

Students interpret the MAD of a data set as the average distance of the data values from the mean.
Lesson Notes
Variability was discussed informally in Lesson 8. This lesson focuses on developing a more formal measure of variability
in a data distribution called the mean absolute deviation, denoted by MAD. The concept of distance from the mean
should be clear to students by now since previous lessons used distances to the right (above) and distances to the left
(below) of the mean to develop the idea of the mean as a balance point. This lesson introduces the idea of a deviation
as a distance and direction (to the right or to the left) from the mean and challenges students to think about why
deviations provide information about variability in a data set.
Mean absolute deviation is the measure of variability used in the middle school curriculum. At the high school level,
students see other measures of variability called the variance and the standard deviation that are also based on
deviations from the mean.
Classwork
Example 1 (5 minutes): Variability
Example 1: Variability
In Lesson 8, Robert wanted to decide where he would rather move (New York City or San Francisco). He planned to make
his decision by comparing the average monthly temperatures for the two cities. Since the mean of the average monthly
temperatures for New York City and the mean for San Francisco turned out to be about the same, he decided instead to
compare the cities based on the variability in their monthly average temperatures. 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.
Read through Example 1 as a class, and recall the main idea of Lesson 8.



Variability is when not all the values in a data set are the same. When
the values in a data set are all very similar, there isn’t much variability,
but if the values differ a lot, there is more variability.
What does a distribution that has no variability look like?


Scaffolding:
What is variability?
Show students different dot
plots, and have them discuss
which one has no variability,
little variability, and a lot of
variability.
All of the data values are the same.
What does a distribution that has a lot of variability look like?

The data points are spread far apart.
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
6•6
Remind students that variability is another word for “spread” and that whether there is a lot or a little variability can be
seen visually in a dot plot of a data set. Suggest that students can visually order data sets, based on the spread in the dot
plots. Consider showing several dot plots of data and asking students to order them from least variability to most
variability.
Exercises 1–3 (8 minutes)
Students work in small groups on these exercises. Then, confirm answers as a class. The discussion of Exercise 3 leads
into Example 2.
Students are asked to order the seven data sets from least variability to most variability. Students will no doubt suggest
different orderings. Several orderings are reasonable—focus on students’ explanations for ordering the distributions.
MP.3
What is important is not their suggested orderings but rather their arguments to support their orderings. Also, the goal
for these exercises is for students to realize that they need to have a more formal way of determining the best ordering.
Sabina suggests that a formula is needed to measure variability, and she proceeds in this lesson to develop one.
Exercises 1–3
The following temperature distributions for seven other cities all have a mean monthly temperature of approximately 𝟔𝟑
degrees Fahrenheit. They do not have the same variability.
City B
City A
30
35
40
45
50
55
60
65
70
75
80
85
90
30
Temperature (degrees F)
35
40
45
50
55
35
40
45
50
55
60
65
70
75
80
85
90
30
35
40
45
50
55
City E
65
70
75
80
85
90
30
Temperature (degrees F)
35
40
45
50
55
60
60
65
70
75
80
85
90
75
80
85
90
Temperature (degrees F)
Temperature (degrees F)
City D
30
60
City C
City F
65
70
Temperature (degrees F)
75
80
85
90
30
35
40
45
50
55
60
65
70
Temperature (degrees F)
City G
30
35
40
45
50
55
60
65
70
75
80
85
90
Temperature (degrees F)
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
1.
6•6
Which distribution has the smallest variability? Explain your answer.
City A has the smallest variability because all the data points are the same.
2.
Which distribution or distributions seem to have the most variability? Explain your answer.
One or more of the following is acceptable: Cities D, E, and F. They appear to have data points that are the most
spread out.
3.
Order the seven distributions from least variability to most variability. Explain why you listed the distributions in
the order that you chose.
Several orderings are reasonable. Focus on students’ explanations for choosing the order, making sure that the
ordering is consistent with an understanding of spread. There are some that will be hard for students to order, and
if students have trouble, use this opportunity to point out that it would be useful to have a more formal way to
measure variability in a data set. Such a measure is developed in Example 2.
Example 2 (5 minutes): Measuring Variability
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 produce a number that measures the variability in a data
distribution. She would then use the formula to measure the variability in each data set and use these values to order the
distributions from smallest variability to largest variability. She proposes beginning by looking at how far the values in a
data set are from the mean of the data set.
No doubt students had different orderings of variability for the seven cities in Exercise 2. Sabina suggests that, in this
example, a formula is needed to measure variability and that this could be used to establish an order. She proposes
looking at distances from the mean to develop a formula for a measure of variability.

Do you think using distances from the mean is a good basis for a formula to measure variability?

Yes. If there is not much variability in a data set, the values will all be close together, so the distances
from the mean will be small. If there is a lot of variability in a data set, the data values will be spread
out, and some points will be far away from the mean.
Introduce the term deviation at this point. In Grade 6, a deviation is defined as a distance and a direction (to the left or
to the right of the mean). In later grades, after students have more experience with signed numbers, a deviation is
defined as a signed distance from the mean and is calculated by subtracting the mean from a data value. If the data
value is to the left of (below) the mean, the deviation is negative. If the data value is to the right of (above) the mean,
the deviation is positive. Here, just focus on distance, and then attach a direction to the distance, as is done in
Exercise 4.
The name mean absolute deviation comes from the fact that this measure of variability is calculated as the mean of the
unsigned distances from the mean (the absolute values of the deviations).
Exercises 4–5 (5 minutes)
Guide students to work in pairs during the following exercises.
In these exercises, students use City G to calculate deviations and verify that the total of the distances from the mean on
either side are equal.
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
6•6
Exercises 4–5
The dot plot for the monthly temperatures in City G is shown below. Use the dot plot and the mean monthly
temperature of 𝟔𝟑 degrees Fahrenheit to answer the following questions.
City G
30
35
40
45
50
55
60
65
70
75
80
85
90
Temperature (degrees F)
4.
Fill in the following table for City G’s temperature deviations.
Temperature
(in degrees Fahrenheit)
𝟓𝟑
𝟓𝟕
𝟔𝟎
𝟔𝟎
𝟔𝟒
𝟔𝟒
𝟔𝟒
𝟔𝟒
𝟔𝟒
𝟔𝟖
𝟔𝟖
𝟕𝟎
5.
Distance
(in degrees Fahrenheit)
from the Mean of 𝟔𝟑℉
𝟏𝟎
𝟔
𝟑
𝟑
𝟏
𝟏
𝟏
𝟏
𝟏
𝟓
𝟓
𝟕
Deviation from the
Mean (distance and
direction)
𝟏𝟎 to the left
𝟔 to the left
𝟑 to the left
𝟑 to the left
𝟏 to the right
𝟏 to the right
𝟏 to the right
𝟏 to the right
𝟏 to the right
𝟓 to the right
𝟓 to the right
𝟕 to the right
What is the sum of the distances to the left of the mean? What is the sum of the distances to the right of the mean?
The sum of the distances to the left of the mean is 𝟏𝟎 + 𝟔 + 𝟑 + 𝟑 = 𝟐𝟐. The sum of the distances to the right of the
mean is 𝟏 + 𝟏 + 𝟏 + 𝟏 + 𝟏 + 𝟓 + 𝟓 + 𝟕 = 𝟐𝟐.
Example 3 (6 minutes): Finding the Mean Absolute Deviation (MAD)
Example 3: Finding the Mean Absolute Deviation (MAD)
Sabina notices that when there is not much variability in a data set, the distances from the mean are small and that when
there is a lot of variability in a data set, the data values are spread out and at least some of the distances from the mean
are large. She wonders how she can use the distances from the mean to help her develop a formula to measure
variability.
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
6•6
One way to measure variability in a data set is to look at the mean (average) distance of data values from the mean of
the data set. To help students understand the name mean absolute deviation, define the term absolute deviation as
distance from the mean. In later grades, students see deviations as signed numbers, and then absolute deviation is just
the absolute value of the deviation. Because this is equivalent to distance, in Grade 6, it is appropriate to just define
absolute deviation as the distance, ignoring the direction.
Introduce the MAD as the mean of the absolute deviations, meaning the mean of the distances. Consider showing a dot
plot with little variability, and ask students if the distances from the mean would be large or small and if the average
distance would be large or small. Then, show a dot plot with more variability, and ask students if the average distance of
data values from the mean would be smaller than or greater than the average for the first dot plot. This helps students
understand how average distance from the mean works as a measure of variability.
Exercises 6–7 (9 minutes)
Students continue to work in pairs or small groups. If students have trouble getting started, consider working through
Exercise 6 parts (a)–(c) as a class.
Exercises 6–7
6.
Use the data on monthly temperatures for City G given in Exercise 4 to answer the following questions.
a.
b.
Fill in the following table.
Temperature
(in degrees Fahrenheit)
𝟓𝟑
𝟓𝟕
Distance from the Mean
(absolute deviation)
𝟏𝟎
𝟔
𝟔𝟎
𝟑
𝟔𝟎
𝟔𝟒
𝟑
𝟏
𝟔𝟒
𝟔𝟒
𝟏
𝟏
𝟔𝟒
𝟏
𝟔𝟒
𝟔𝟖
𝟏
𝟓
𝟔𝟖
𝟕𝟎
𝟓
𝟕
The absolute deviation for a data value is its distance from the mean of the data set. For example, for the
first temperature value for City G (𝟓𝟑 degrees), the absolute deviation is 𝟏𝟎. What is the sum of the absolute
deviations?
The sum of the absolute deviations is 𝟏𝟎 + 𝟔 + 𝟑 + 𝟑 + 𝟏 + 𝟏 + 𝟏 + 𝟏 + 𝟏 + 𝟓 + 𝟓 + 𝟕 = 𝟒𝟒.
c.
Sabina suggests that the mean of the absolute deviations (the mean of the distances) could be a measure of
the variability in a data set. Its value is the average distance of the data values from the mean of the monthly
temperatures. It is called the mean absolute deviation and is denoted by the letters MAD. Find the MAD for
this data set of City G’s temperatures. Round to the nearest tenth.
The MAD (mean absolute deviation) is
Lesson 9:
𝟒𝟒
, or 𝟑. 𝟕 degrees to the nearest tenth of a degree.
𝟏𝟐
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
d.
6•6
Find the MAD values in degrees Fahrenheit for each of the seven city temperature distributions, and use the
values to order the distributions from least variability to most variability. Recall that the mean for each data
set is 𝟔𝟑 degrees Fahrenheit. Looking only at the distributions, does the list that you made in Exercise 2
match the list made by ordering MAD values?
If time is a factor in completing this lesson, assign cities to individual students. After each student has
calculated the mean deviation, organize the results for the whole class. Direct students to calculate the MAD
to the nearest tenth of a degree.
MAD values (in ℉):
City A = 𝟎
City B = 𝟓. 𝟑
City C = 𝟑. 𝟐
City D = 𝟏𝟎. 𝟓
e.
City E = 𝟏𝟎. 𝟓
City F = 𝟏𝟎. 𝟓
City G = 𝟑. 𝟕
The order from least to greatest is
A, C, G, B, and
D, E, and F (all tied).
Which of the following is a correct interpretation of the MAD?
i.
The monthly temperatures in City G are all within 𝟑. 𝟕 degrees from the approximate mean of 𝟔𝟑
degrees.
ii.
The monthly temperatures in City G are, on average, 𝟑. 𝟕 degrees from the approximate mean
temperature of 𝟔𝟑 degrees.
iii.
All of the monthly temperatures in City G differ from the approximate mean temperature of 𝟔𝟑
degrees by 𝟑. 𝟕 degrees.
The answer is (ii). Remind students that the MAD is an average of the distances from the mean, so
some distances may be smaller and some larger than the value of the MAD. Point out that the
distances from the mean for City G were not all equal to 𝟑. 𝟕 and that some were smaller (for
example, the distances of 𝟏 and 𝟑) and that some were larger (for example, the distances of 𝟓 and
𝟏𝟎).
7.
The dot plot for City A’s temperatures follows.
City A
30
35
40
45
50
55
60
65
70
75
80
85
90
Temperature (degrees F)
a.
How much variability is there in City A’s temperatures? Why?
There is no variability in City A’s temperatures. The absolute deviations (distances from the mean) are all 𝟎.
b.
Does the MAD agree with your answer in part (a)?
The MAD does agree with my answer from part (a). The value of the MAD is 𝟎.
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
6•6
Closing (2 minutes)

How do you calculate the mean absolute deviation (MAD)?

The MAD is computed by finding the sum of the absolute deviations and then dividing the sum by the
number of data values.
Lesson 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 the distance that data point is from the mean.

The mean absolute deviation (MAD) is computed by finding the mean of the absolute
deviations (distances from the mean) for the data set.

The value of MAD is the average distance that the data values are from the mean.

A small MAD indicates that the data distribution has very little variability.

A large MAD indicates that the data points are spread out and that at least some are far
away from the mean.
Exit Ticket (5 minutes)
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
6•6
Date
Lesson 9: The Mean Absolute Deviation (MAD)
Exit Ticket
1.
The mean absolute deviation (MAD) is a measure of variability for a data set. What does a data distribution look like
if its MAD equals zero? Explain.
2.
Is it possible to have a negative value for the MAD of a data set?
3.
Suppose that seven students have the following numbers of pets: 1, 1, 1, 2, 4, 4, 8.
a.
The mean number of pets for these seven students is 3 pets. Use the following table to find the MAD for this
distribution of number of pets.
Student
Number of Pets
1
1
2
1
3
1
4
2
5
4
6
4
7
8
Deviation from
the Mean
(distance and
direction)
Absolute Deviation
(distance from the
mean)
Sum
b.
Explain in words what the MAD means for this data set.
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
6•6
Exit Ticket Sample Solutions
1.
The mean absolute deviation (MAD) is a measure of variability for a data set. What does a data distribution look
like if its MAD equals zero? Explain.
If the MAD is zero, then all of the absolute deviations are zero. The MAD measures the average distance from the
mean, and distance is never negative. The only way the MAD could average to zero is if all the absolute deviations
are zero. For example, City A had a dot plot where all of the temperatures were the same. Because all of the
temperatures were the same, all of the absolute deviations were zero, which indicates that there was no variability
in the temperatures.
2.
Is it possible to have a negative value for the MAD of a data set?
Because a MAD is the average of distances, which can never be negative, the MAD is always zero or a positive
number.
3.
Suppose that seven students have the following numbers of pets: 𝟏, 𝟏, 𝟏, 𝟐, 𝟒, 𝟒, 𝟖.
a.
The mean number of pets for these seven students is 𝟑 pets. Use the following table to find the MAD for this
distribution of number of pets.
𝟏𝟒
𝟕
=𝟐
The MAD number of pets is 𝟐.
Student
Number of Pets
Deviation from
the Mean
(distance and direction)
Absolute Deviation
(distance from the
mean)
𝟏
𝟏
𝟐 to the left
𝟐
𝟐
𝟏
𝟐 to the left
𝟐
𝟑
𝟏
𝟐 to the left
𝟐
𝟒
𝟐
𝟏 to the left
𝟏
𝟓
𝟒
𝟏 to the right
𝟏
𝟔
𝟒
𝟏 to the right
𝟏
𝟕
𝟖
𝟓 to the right
𝟓
𝟏𝟒
Sum
b.
Explain in words what the MAD means for this data set.
On average, the number of pets for these students differs by 𝟐 from the mean of 𝟑 pets.
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
6•6
Problem Set Sample Solutions
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.
Dot Plot of Number of Goals Scored by Girls’ Team
CREATED
Dot Plot of Number of Goals Scored by Boys’ Team
a.
Before doing any calculations, which dot plot has the larger MAD? Explain how you know.
The graph of the boys’ team has a larger MAD because the data are more spread out and have the larger
distances from the mean.
b.
Use the following tables to find the MAD for each distribution. Round your calculations to the nearest
hundredth.
Boys’ Team
Number of Goals
Absolute Deviation
𝟎
𝟑
𝟎
𝟑
𝟑
𝟎
𝟑
𝟎
𝟓
𝟐
𝟕
𝟒
Sum
𝟏𝟐
The MAD for the boys’ team is 𝟐 goals because
𝟒
𝟔
c.
Girls’ Team
Number of Goals
Absolute Deviation
𝟐
𝟏
𝟐
𝟏
𝟑
𝟎
𝟑
𝟎
𝟑
𝟎
𝟓
𝟐
Sum
𝟒
𝟏𝟐
𝟔
= 𝟐. The MAD for the girls’ team is 𝟎. 𝟔𝟕 goal because
≈ 𝟎. 𝟔𝟕.
Based on the computed MAD values, for which distribution is the mean a better indication of a typical value?
Explain your answer.
The mean is a better indicator of a typical value for the girls’ team because the measure of variability given by
the MAD is lower (𝟎. 𝟔𝟕 goal) than the boys’ MAD (𝟐 goals).
Lesson 9:
The Mean Absolute Deviation (MAD)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
6•6
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:
Month
Temperature
Absolute Deviation
a.
Jan.
𝟑𝟗
𝟐𝟒
Feb.
𝟒𝟐
𝟐𝟏
Average Temperature in New York City
Mar. Apr. May June July Aug.
𝟓𝟎
𝟔𝟏
𝟕𝟏
𝟖𝟏
𝟖𝟓
𝟖𝟒
𝟏𝟑
𝟐
𝟖
𝟏𝟖
𝟐𝟐
𝟐𝟏
Sep.
𝟕𝟔
𝟏𝟑
Oct.
𝟔𝟓
𝟐
Nov.
𝟓𝟓
𝟖
Dec.
𝟒𝟕
𝟏𝟔
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.
The sum of the absolute deviations is 𝟏𝟔𝟖. The MAD is the average of the absolute deviations. The MAD is
𝟏𝟒 degrees because
𝟏𝟔𝟖
𝟏𝟐
= 𝟏𝟒. On average, the monthly temperatures in New York City differ from the
mean of 𝟔𝟑 degrees Fahrenheit by 𝟏𝟒 degrees.
b.
Complete the following table, and then use the values to calculate the MAD for the San Francisco data
distribution.
The sum of the absolute deviations is 𝟒𝟐. The MAD is the mean of the absolute deviations. The MAD is 𝟑. 𝟓
degrees because
Month
Temperature
Absolute Deviation
c.
𝟒𝟐
𝟏𝟐
= 𝟑. 𝟓.
Jan.
𝟓𝟕
𝟕
Feb.
𝟔𝟎
𝟒
Average Temperature in San Francisco
Mar. Apr. May June July
𝟔𝟐
𝟔𝟑
𝟔𝟒
𝟔𝟕
𝟔𝟕
𝟐
𝟏
𝟎
𝟑
𝟑
Aug.
𝟔𝟖
𝟒
Sep.
𝟕𝟎
𝟔
Oct.
𝟔𝟗
𝟓
Nov.
𝟔𝟑
𝟏
Dec.
𝟓𝟖
𝟔
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.
New York City has a MAD of 𝟏𝟒 degrees, as compared to 𝟑. 𝟓 degrees in San Francisco. Robert should choose
New York City if he wants to have more variability in monthly temperatures.
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.
Awesome
Delight
Finest
Sweeties
YumYum
a.
Bag 1
𝟒𝟎
𝟐𝟐
𝟐𝟔
𝟑𝟔
𝟑𝟑
Bag 2
𝟒𝟎
𝟑𝟏
𝟑𝟔
𝟑𝟗
𝟑𝟔
Bag 3
𝟒𝟏
𝟑𝟔
𝟒𝟎
𝟒𝟐
𝟒𝟐
Bag 4
𝟒𝟐
𝟒𝟐
𝟒𝟑
𝟒𝟐
𝟒𝟐
Bag 5
𝟒𝟐
𝟒𝟖
𝟒𝟕
𝟒𝟐
𝟒𝟓
Bag 6
𝟒𝟑
𝟓𝟑
𝟓𝟎
𝟒𝟒
𝟒𝟖
Bag 7
𝟒𝟔
𝟔𝟐
𝟓𝟐
𝟒𝟗
𝟒𝟖
Complete the following table of the absolute deviations for the seven bags for each candy manufacturer.
Awesome
Delight
Finest
Sweeties
YumYum
Lesson 9:
Bag 1
𝟐
𝟐𝟎
𝟏𝟔
𝟔
𝟗
Bag 2
𝟐
𝟏𝟏
𝟔
𝟑
𝟔
Absolute Deviations
Bag 3
Bag 4
𝟏
𝟎
𝟔
𝟎
𝟐
𝟏
𝟎
𝟎
𝟎
𝟎
The Mean Absolute Deviation (MAD)
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from G6-M6-TE-1.3.0-10.2015
Bag 5
𝟎
𝟔
𝟓
𝟎
𝟑
Bag 6
𝟏
𝟏𝟏
𝟖
𝟐
𝟔
Bag 7
𝟒
𝟐𝟎
𝟏𝟎
𝟕
𝟔
108
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
b.
6•6
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.
Use the MAD for each manufacturer to evaluate students’ responses.
Awesome
Delight
Finest
Sweeties
YumYum
Bag 1
𝟐
𝟐𝟎
𝟏𝟔
𝟔
𝟗
Lesson 9:
Bag 2
𝟐
𝟏𝟏
𝟔
𝟑
𝟔
Bag 3
𝟏
𝟔
𝟐
𝟎
𝟎
Bag 4
𝟎
𝟎
𝟏
𝟎
𝟎
Bag 5
𝟎
𝟔
𝟓
𝟎
𝟑
The Mean Absolute Deviation (MAD)
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from G6-M6-TE-1.3.0-10.2015
Bag 6
𝟏
𝟏𝟏
𝟖
𝟐
𝟔
Bag 7
𝟒
𝟐𝟎
𝟏𝟎
𝟕
𝟔
SUM
𝟏𝟎
𝟕𝟒
𝟒𝟖
𝟏𝟖
𝟑𝟎
MAD
𝟏. 𝟒
𝟏𝟎. 𝟔
𝟔. 𝟗
𝟐. 𝟔
𝟒. 𝟑
109
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.