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

Options

A. 2.5
B. 3
C. 3.25
D. 4

Answer

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 (\( \bar{x} \)) is calculated by summing all the data points and dividing by the number of data points.

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

Calculating the sum:

\[ 4 + 7 + 8 + 9 + 10 + 12 + 13 + 17 = 80 \]

Now, divide by the number of terms (which is 8):

\[ \bar{x} = \frac{80}{8} = 10 \]

### Step 2: Calculate the Absolute Deviations
Next, we will calculate the absolute deviations of each data point from the mean:

\[ |x_i - \bar{x}| \]

Calculating for each data point:

- For \(4\): \(|4 - 10| = 6\)
- For \(7\): \(|7 - 10| = 3\)
- For \(8\): \(|8 - 10| = 2\)
- For \(9\): \(|9 - 10| = 1\)
- For \(10\): \(|10 - 10| = 0\)
- For \(12\): \(|12 - 10| = 2\)
- For \(13\): \(|13 - 10| = 3\)
- For \(17\): \(|17 - 10| = 7\)

### Step 3: Sum of Absolute Deviations
Now, we will sum all the absolute deviations calculated in the previous step:

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

### Step 4: Calculate the Mean Deviation
Finally, we will calculate the mean deviation by dividing the total absolute deviation by the number of data points:

\[ \text{Mean Deviation} = \frac{\text{Sum of Absolute Deviations}}{n} = \frac{24}{8} = 3 \]

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

---