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Answer :
- Calculate the mean of the data set: $\frac{10 + 15 + 5 + 10 + 20}{5} = 12$.
- Find the absolute deviation of each data point from the mean: 2, 3, 7, 2, 8.
- Calculate the mean of the absolute deviations: $\frac{2 + 3 + 7 + 2 + 8}{5} = 4.4$.
- The mean absolute deviation is $\boxed{4.4}$.
### Explanation
1. Understanding the Problem
We are given the data set representing the number of books checked out by five students: 10, 15, 5, 10, 20. Our goal is to find the mean absolute deviation (MAD) of this data set. The mean absolute deviation measures the average distance between each data point and the mean of the data set.
2. Calculating the Mean
First, we need to calculate the mean of the data set. The mean is the sum of all the data points divided by the number of data points. In this case, the mean is: $$\frac{10 + 15 + 5 + 10 + 20}{5} = \frac{60}{5} = 12$$
3. Finding Absolute Deviations
Next, we need to find the absolute deviation of each data point from the mean. The absolute deviation is the absolute value of the difference between each data point and the mean. The absolute deviations are:
* $|10 - 12| = |-2| = 2$
* $|15 - 12| = |3| = 3$
* $|5 - 12| = |-7| = 7$
* $|10 - 12| = |-2| = 2$
* $|20 - 12| = |8| = 8$
4. Calculating the Mean Absolute Deviation
Now, we need to calculate the mean of the absolute deviations. This is the sum of the absolute deviations divided by the number of data points: $$\frac{2 + 3 + 7 + 2 + 8}{5} = \frac{22}{5} = 4.4$$
5. Final Answer
Therefore, the mean absolute deviation of the given data set is 4.4.
### Examples
The mean absolute deviation is useful in many real-world scenarios. For example, in quality control, it can be used to measure the variability in the dimensions of manufactured parts. In finance, it can be used to measure the volatility of stock prices. In education, it can be used to measure the spread of student test scores. Understanding the mean absolute deviation helps in making informed decisions based on the variability of data.
- Find the absolute deviation of each data point from the mean: 2, 3, 7, 2, 8.
- Calculate the mean of the absolute deviations: $\frac{2 + 3 + 7 + 2 + 8}{5} = 4.4$.
- The mean absolute deviation is $\boxed{4.4}$.
### Explanation
1. Understanding the Problem
We are given the data set representing the number of books checked out by five students: 10, 15, 5, 10, 20. Our goal is to find the mean absolute deviation (MAD) of this data set. The mean absolute deviation measures the average distance between each data point and the mean of the data set.
2. Calculating the Mean
First, we need to calculate the mean of the data set. The mean is the sum of all the data points divided by the number of data points. In this case, the mean is: $$\frac{10 + 15 + 5 + 10 + 20}{5} = \frac{60}{5} = 12$$
3. Finding Absolute Deviations
Next, we need to find the absolute deviation of each data point from the mean. The absolute deviation is the absolute value of the difference between each data point and the mean. The absolute deviations are:
* $|10 - 12| = |-2| = 2$
* $|15 - 12| = |3| = 3$
* $|5 - 12| = |-7| = 7$
* $|10 - 12| = |-2| = 2$
* $|20 - 12| = |8| = 8$
4. Calculating the Mean Absolute Deviation
Now, we need to calculate the mean of the absolute deviations. This is the sum of the absolute deviations divided by the number of data points: $$\frac{2 + 3 + 7 + 2 + 8}{5} = \frac{22}{5} = 4.4$$
5. Final Answer
Therefore, the mean absolute deviation of the given data set is 4.4.
### Examples
The mean absolute deviation is useful in many real-world scenarios. For example, in quality control, it can be used to measure the variability in the dimensions of manufactured parts. In finance, it can be used to measure the volatility of stock prices. In education, it can be used to measure the spread of student test scores. Understanding the mean absolute deviation helps in making informed decisions based on the variability of data.
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