Find the mean deviation from mean for the following data :
`{:(x_i:, 3,5,7,9,11,13),(f_i: , 2,7,10,9,5,1):}`

Answer

To find the mean deviation from the mean for the given data, we will follow these steps:

### Step 1: Calculate the Mean (X̄)
We will first calculate the mean (X̄) using the formula:
\[ X̄ = \frac{\sum (f_i \cdot x_i)}{\sum f_i} \]

#### Given Data:
- \( x_i = \{3, 5, 7, 9, 11, 13\} \)
- \( f_i = \{2, 7, 10, 9, 5, 1\} \)

#### Calculation:
1. Calculate \( f_i \cdot x_i \):
- \( 3 \cdot 2 = 6 \)
- \( 5 \cdot 7 = 35 \)
- \( 7 \cdot 10 = 70 \)
- \( 9 \cdot 9 = 81 \)
- \( 11 \cdot 5 = 55 \)
- \( 13 \cdot 1 = 13 \)

2. Sum of \( f_i \cdot x_i \):
\[ \sum (f_i \cdot x_i) = 6 + 35 + 70 + 81 + 55 + 13 = 260 \]

3. Sum of \( f_i \):
\[ \sum f_i = 2 + 7 + 10 + 9 + 5 + 1 = 34 \]

4. Calculate the mean:
\[ X̄ = \frac{260}{34} \approx 7.64 \]

### Step 2: Calculate the Mean Deviation
The mean deviation is calculated using the formula:
\[ \text{Mean Deviation} = \frac{\sum f_i \cdot |x_i - X̄|}{\sum f_i} \]

#### Calculation:
1. Calculate \( |x_i - X̄| \):
- \( |3 - 7.64| = 4.64 \)
- \( |5 - 7.64| = 2.64 \)
- \( |7 - 7.64| = 0.64 \)
- \( |9 - 7.64| = 1.36 \)
- \( |11 - 7.64| = 3.36 \)
- \( |13 - 7.64| = 5.36 \)

2. Calculate \( f_i \cdot |x_i - X̄| \):
- \( 2 \cdot 4.64 = 9.28 \)
- \( 7 \cdot 2.64 = 18.48 \)
- \( 10 \cdot 0.64 = 6.4 \)
- \( 9 \cdot 1.36 = 12.24 \)
- \( 5 \cdot 3.36 = 16.8 \)
- \( 1 \cdot 5.36 = 5.36 \)

3. Sum of \( f_i \cdot |x_i - X̄| \):
\[ \sum (f_i \cdot |x_i - X̄|) = 9.28 + 18.48 + 6.4 + 12.24 + 16.8 + 5.36 = 68.56 \]

4. Calculate the mean deviation:
\[ \text{Mean Deviation} = \frac{68.56}{34} \approx 2.01 \]

### Final Answer:
The mean deviation from the mean for the given data is approximately **2.01**.

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