Find the mean deviation about the mean for the mean for the following data :
15, 17, 10, 13, 7, 18, 9, 6, 14, 11

Options

A. `3.4`
B. `4.5`
C. `3.9`
D. `5.4`

Explanation

To find the mean deviation about the mean for the given data set \( 15, 17, 10, 13, 7, 18, 9, 6, 14, 11 \), we will follow these steps:

### Step 1: Calculate the Mean
First, we need to find the mean (\( \bar{x} \)) of the data.

\[ \text{Mean} (\bar{x}) = \frac{\sum x_i}{n} \]

Where:
- \( \sum x_i \) is the sum of all observations.
- \( n \) is the number of observations.

Calculating the sum of the data:

\[ \sum x_i = 15 + 17 + 10 + 13 + 7 + 18 + 9 + 6 + 14 + 11 = 120 \]

Now, the number of observations \( n = 10 \).

Now, substituting the values into the mean formula:

\[ \bar{x} = \frac{120}{10} = 12 \]

### Step 2: Calculate the Absolute Deviations
Next, we calculate the absolute deviations from the mean for each observation.

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

Calculating each deviation:

- For \( 15 \): \( |15 - 12| = 3 \)
- For \( 17 \): \( |17 - 12| = 5 \)
- For \( 10 \): \( |10 - 12| = 2 \)
- For \( 13 \): \( |13 - 12| = 1 \)
- For \( 7 \): \( |7 - 12| = 5 \)
- For \( 18 \): \( |18 - 12| = 6 \)
- For \( 9 \): \( |9 - 12| = 3 \)
- For \( 6 \): \( |6 - 12| = 6 \)
- For \( 14 \): \( |14 - 12| = 2 \)
- For \( 11 \): \( |11 - 12| = 1 \)

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

\[ \sum |x_i - \bar{x}| = 3 + 5 + 2 + 1 + 5 + 6 + 3 + 6 + 2 + 1 = 34 \]

### Step 4: Calculate the Mean Deviation
Finally, we calculate the mean deviation using the formula:

\[ \text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{n} \]

Substituting the values:

\[ \text{Mean Deviation} = \frac{34}{10} = 3.4 \]

### Final Answer
The mean deviation about the mean for the given data is **3.4**.

---