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### Key Concepts

• Find the mean absolute deviation to describe variability.
• Find the interquartile range(IQR) to describe variability.
• Use the mean absolute deviation(MAD) to find the variability of a data set.

## 8.5 Summarize Data Using Measures of Variability

### Mean Absolute Deviation:

The mean absolute deviation of a set of data is the average distance between each data value and the mean.

Steps to find the Mean Absolute Deviation:

1. Find the mean.
2. Find the difference/distance between each data value and the mean. It means, find the absolute value of the difference between each data value and the mean.
3. Find the average of those differences obtained.

### Interquartile range (IQR):

The interquartile range (IQR) is the distance between the first and third quartiles of the data set. To find the IQR, subtract the first quartile from the third quartile.

To find the interquartile range (IQR):

1. Find the median (middle value) of the lower and upper half of the data. These values are quartile 1 (Q1) and quartile 3 (Q3).
2. The IQR is the difference between Q3 and Q1.

### Example 1:

Mr. Larry works at a shoe store. He measured the feet of seven customers and found their shoe sizes as below:

5, 5, 6, 7, 8, 8, and 10.

He knows that the average (mean) size is 7. How can Mr. Larry determine how much the shoe sizes varied for the seven customers?

### Solution: Find the differences between each of the shoe size and the mean (average) size. Show all the differences as positive integers. 5 – 7 = 2

5 – 7 = 2

6 – 7 = 1

7 – 7 = 0

8 – 7 = 1

8 – 7 = 1

10 – 7 = 3 Find the mean absolute deviation (MAD) by finding the mean of all of the differences, or absolute deviations.

2+2+1+0+1+1+372+2+1+0+1+1+3 / 7

= 10/7  or 1.43 Mr. Larry can find the mean absolute deviation (MAD) to determine how much the shoe sizes varied for the seven customers. ### Example 2:

The dot plot shows the distribution of Kayla’s health quiz scores. How can Kayla determine the variability in her health quiz scores? ### Solution: Read the data set from the dot plot,

84, 86, 88, 90, 92, 92, 94, 96, 96 Find the minimum, median, and maximum values as well as the first and third quartiles,  Draw a box plot to determine the interquartile range,  ### Example 3:

The data set shows prices for concert tickets in 10 different cities in Washington. Find the IQR of the data set.

Solution:

Find the minimum, median, and maximum values as well as the first and third quartiles. Draw a box plot to determine the interquartile range, ### Example 4:

Larry recorded scores of his last six science exams. The mean number of marks scored was 60 and the MAD was 35. How can Larry use these measures to describe the variability of the marks scored during the last six science exams?

Would you say that Larry is a consistent or inconsistent test taker? Explain.

Science Exam Scores: 20, 90, 25, 100, 95, 30

### Solution:

The MAD shows that the marks generally varied greatly from the mean. The marks were mostly less than 25 (60 – 35) or greater than 95 (60 + 35).

Larry is an inconsistent test taker. Ha has a high MAD. This means, there is more variation from the mean. His marks tend to be farther from his average.

## Exercise:

1. Find the mean absolute deviation (MAD) of the following data set.
52, 48, 60, 55, 59, 54, 58, 62
2. Find the IQR of the following data set.
0, 0, 1, 1, 2, 2, 2, 3, 4, 5, 6, 6, 7, 7, 7, 8
3. Find the Mean Absolute Deviation of the following data set.
2, 5, 7, 13, 18
4. Larry scored the following percentages on 10 quizzes in a science class:
55, 65, 70, 70, 72, 85, 90, 90, 93, 100
Find the MAD of the quiz scores.
5. John surveyed the approximate maximum speeds, in miles per hour, of different animals in a zoo and noted in a table.

Find the interquartile range to describe how the data vary.

6. The table below shows the number of hours different animals spend sleeping per day in a zoo. Use the interquartile range to describe how the data vary.

7. Bob surveyed his friends about the number of apps they use in their mobile phones. The responses were noted as 15, 16, 18, 9, 18, 4, 19, 20, 17, and 36 apps. Use the interquartile range to describe how the data vary.

8.The following table shows the maximum speeds of roller coasters at an amusement park. Find the mean absolute deviation of the given set of data of eight roller coasters.

9. The following table shows the maximum flying speeds of the ten fastest birds. Find the mean absolute deviation of the following set of data.

10. Suppose that seven students have the following numbers of pets: 𝟏, 𝟏, 𝟏, 𝟐, 𝟒, 𝟒, 𝟖.
The mean number of pets for these seven students is 𝟑 pets. The MAD number of
pets is 𝟐.
Explain in words what the MAD means for this data set.

### Concept Map ### What have we learned:

• Calculate the mean absolute deviation and describe variability of a data set.
• Calculate the interquartile range(IQR) and describe variability of a data set.
• Find the variability of a data set by using the mean absolute deviation(MAD)

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