Find the mean deviation from the mean for the following data :
6,7,10,12,13,4,8,12

Answer

To find the mean deviation from the mean for the given data set \(6, 7, 10, 12, 13, 4, 8, 12\), we will follow these steps:

### Step 1: Calculate the Mean
The mean (\( \bar{x} \)) is calculated using the formula:

\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]

Where \( n \) is the number of observations and \( x_i \) are the data points.

**Calculating the sum:**
\[ 6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 = 72 \]

**Number of observations:**
\[ n = 8 \]

**Calculating the mean:**
\[ \bar{x} = \frac{72}{8} = 9 \]

### Step 2: Calculate the Mean Deviation
The mean deviation is calculated using the formula:

\[ \text{Mean Deviation} = \frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n} \]

Where \( |x_i - \bar{x}| \) is the absolute deviation of each observation from the mean.

**Calculating the absolute deviations:**
- For \( x_1 = 6 \): \( |6 - 9| = 3 \)
- For \( x_2 = 7 \): \( |7 - 9| = 2 \)
- For \( x_3 = 10 \): \( |10 - 9| = 1 \)
- For \( x_4 = 12 \): \( |12 - 9| = 3 \)
- For \( x_5 = 13 \): \( |13 - 9| = 4 \)
- For \( x_6 = 4 \): \( |4 - 9| = 5 \)
- For \( x_7 = 8 \): \( |8 - 9| = 1 \)
- For \( x_8 = 12 \): \( |12 - 9| = 3 \)

**Sum of absolute deviations:**
\[ 3 + 2 + 1 + 3 + 4 + 5 + 1 + 3 = 22 \]

**Calculating the mean deviation:**
\[ \text{Mean Deviation} = \frac{22}{8} = 2.75 \]

### Final Answer
The mean deviation from the mean for the data set \(6, 7, 10, 12, 13, 4, 8, 12\) is **2.75**.

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