Find the mean deviation from the mean for the following data :
4,7,8,9,10,12,13,17

Answer

To find the mean deviation from the mean for the given 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 using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \( n \) is the number of observations.

**Calculating the sum of the data:**
\[ \sum x_i = 4 + 7 + 8 + 9 + 10 + 12 + 13 + 17 = 70 \]
**Finding the number of observations:**
\[ n = 8 \]
**Calculating the mean:**
\[ \bar{x} = \frac{70}{8} = 8.75 \]

### Step 2: Calculate the Absolute Deviations from the Mean
Next, we calculate the absolute deviations from the mean for each observation:
\[ |x_i - \bar{x}| \]
Calculating each deviation:
- For \( x_1 = 4 \): \( |4 - 8.75| = 4.75 \)
- For \( x_2 = 7 \): \( |7 - 8.75| = 1.75 \)
- For \( x_3 = 8 \): \( |8 - 8.75| = 0.75 \)
- For \( x_4 = 9 \): \( |9 - 8.75| = 0.25 \)
- For \( x_5 = 10 \): \( |10 - 8.75| = 1.25 \)
- For \( x_6 = 12 \): \( |12 - 8.75| = 3.25 \)
- For \( x_7 = 13 \): \( |13 - 8.75| = 4.25 \)
- For \( x_8 = 17 \): \( |17 - 8.75| = 8.25 \)

### Step 3: Calculate the Mean Deviation
Now, we find the mean deviation using the formula:
\[ \text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{n} \]
**Calculating the sum of absolute deviations:**
\[ \sum |x_i - \bar{x}| = 4.75 + 1.75 + 0.75 + 0.25 + 1.25 + 3.25 + 4.25 + 8.25 = 24.5 \]
**Calculating the mean deviation:**
\[ \text{Mean Deviation} = \frac{24.5}{8} = 3.0625 \]

### Final Result
The mean deviation from the mean for the given data is approximately \( 3.06 \).

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