What is the mean deviation about the mean for the data 4, 7, 8, 9, 10, 12, 13, 17?

Options

A. 2.5
B. 3
C. 3.5
D. 4

Explanation

To find the mean deviation about the mean for the data set \(4, 7, 8, 9, 10, 12, 13, 17\), we will follow these steps:

### Step 1: Calculate the Mean
The mean is calculated by taking the sum of all observations and dividing it by the number of observations.

\[ \text{Mean} = \frac{4 + 7 + 8 + 9 + 10 + 12 + 13 + 17}{8} \]

Calculating the sum:

\[ 4 + 7 = 11 \]
\[ 11 + 8 = 19 \]
\[ 19 + 9 = 28 \]
\[ 28 + 10 = 38 \]
\[ 38 + 12 = 50 \]
\[ 50 + 13 = 63 \]
\[ 63 + 17 = 80 \]

Now, divide by the number of observations (8):

\[ \text{Mean} = \frac{80}{8} = 10 \]

### Step 2: Calculate the Absolute Deviations from the Mean
Next, we will calculate the absolute deviations of each observation from the mean (10):

\[ |4 - 10| = 6 \]
\[ |7 - 10| = 3 \]
\[ |8 - 10| = 2 \]
\[ |9 - 10| = 1 \]
\[ |10 - 10| = 0 \]
\[ |12 - 10| = 2 \]
\[ |13 - 10| = 3 \]
\[ |17 - 10| = 7 \]

### Step 3: Sum of Absolute Deviations
Now, we sum all the absolute deviations:

\[ 6 + 3 + 2 + 1 + 0 + 2 + 3 + 7 = 24 \]

### Step 4: Calculate the Mean Deviation
Finally, we divide the sum of absolute deviations by the number of observations (8) to find the mean deviation:

\[ \text{Mean Deviation} = \frac{24}{8} = 3 \]

### Final Answer
The mean deviation about the mean for the data set \(4, 7, 8, 9, 10, 12, 13, 17\) is **3**.

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