Using the following data, calculate the mean absolute deviation:7 2 8 5 9 9 5 6 8 1What is the mean absolute deviation for these data? (4 points)2.266.58

1. Break Down the Problem

To calculate the Mean Absolute Deviation (MAD), we need to follow these steps:

1. Calculate the mean of the data set.
2. Find the absolute deviations from the mean for each data point.
3. Calculate the mean of these absolute deviations.

2. Relevant Concepts

The formulas involved are:

• Mean: $\text{Mean} = \frac{\sum {x_i}}{n}$
• Absolute Deviation: $|d_i| = |x_i - \text{Mean}|$
• Mean Absolute Deviation: $\text{MAD} = \frac{\sum |d_i|}{n}$

3. Analysis and Detail

1. Calculate the Mean:

   • Data set: $7, 2, 8, 5, 9, 9, 5, 6, 8, 1$
   • Number of elements, $n = 10$. $\text{Mean} = \frac{7 + 2 + 8 + 5 + 9 + 9 + 5 + 6 + 8 + 1}{10} = \frac{60}{10} = 6$

2. Calculate Absolute Deviations:

   • Deviations from the mean:

     • For $7$: $|7 - 6| = 1$
     • For $2$: $|2 - 6| = 4$
     • For $8$: $|8 - 6| = 2$
     • For $5$: $|5 - 6| = 1$
     • For $9$: $|9 - 6| = 3$
     • For $9$: $|9 - 6| = 3$
     • For $5$: $|5 - 6| = 1$
     • For $6$: $|6 - 6| = 0$
     • For $8$: $|8 - 6| = 2$
     • For $1$: $|1 - 6| = 5$

   • Absolute deviations: $1, 4, 2, 1, 3, 3, 1, 0, 2, 5$

3. Calculate the Mean Absolute Deviation: $\text{MAD} = \frac{1 + 4 + 2 + 1 + 3 + 3 + 1 + 0 + 2 + 5}{10} = \frac{22}{10} = 2.2$

4. Verify and Summarize

The calculations have been verified, and all steps confirm that the Mean Absolute Deviation is correct.

Final Answer

The Mean Absolute Deviation for the data set is 2.2.