Calculate mean deviation about mean from the following data: `x_i :` 3 9 17 23 27 `f_i :` 8 10 12 9 5

Options

A. `5.09`
B. `6.09`
C. `7.09`
D. `8.09`

Explanation

To calculate the mean deviation about the mean from the given data, we will follow these steps:

### Step 1: Create a table with \(x_i\) and \(f_i\)

We have the following data:

| \(x_i\) | \(f_i\) |
|---------|---------|
| 3 | 8 |
| 9 | 10 |
| 17 | 12 |
| 23 | 9 |
| 27 | 5 |

### Step 2: Calculate \(f_i \cdot x_i\)

Now, we will calculate the product of \(f_i\) and \(x_i\) for each row:

\[ \begin{align*} f_i \cdot x_i \text{ for } x_1 (3) & : 3 \cdot 8 = 24 \\ f_i \cdot x_i \text{ for } x_2 (9) & : 9 \cdot 10 = 90 \\ f_i \cdot x_i \text{ for } x_3 (17) & : 17 \cdot 12 = 204 \\ f_i \cdot x_i \text{ for } x_4 (23) & : 23 \cdot 9 = 207 \\ f_i \cdot x_i \text{ for } x_5 (27) & : 27 \cdot 5 = 135 \\ \end{align*} \]

Now, we will sum these products:

\[ \Sigma (f_i \cdot x_i) = 24 + 90 + 204 + 207 + 135 = 660 \]

### Step 3: Calculate \(\Sigma f_i\)

Now, we will sum the frequencies \(f_i\):

\[ \Sigma f_i = 8 + 10 + 12 + 9 + 5 = 44 \]

### Step 4: Calculate the mean (\(\bar{x}\))

The mean \(\bar{x}\) is given by:

\[ \bar{x} = \frac{\Sigma (f_i \cdot x_i)}{\Sigma f_i} = \frac{660}{44} = 15 \]

### Step 5: Calculate the deviations \(|d_i|\)

Now we will calculate the deviations from the mean for each \(x_i\):

\[ \begin{align*} d_1 & = 3 - 15 = -12 \\ d_2 & = 9 - 15 = -6 \\ d_3 & = 17 - 15 = 2 \\ d_4 & = 23 - 15 = 8 \\ d_5 & = 27 - 15 = 12 \\ \end{align*} \]

Now, we take the absolute values:

\[ |d_i| = \{12, 6, 2, 8, 12\} \]

### Step 6: Calculate \(f_i \cdot |d_i|\)

Now, we will calculate the product of \(f_i\) and \(|d_i|\):

\[ \begin{align*} f_1 \cdot |d_1| & : 8 \cdot 12 = 96 \\ f_2 \cdot |d_2| & : 10 \cdot 6 = 60 \\ f_3 \cdot |d_3| & : 12 \cdot 2 = 24 \\ f_4 \cdot |d_4| & : 9 \cdot 8 = 72 \\ f_5 \cdot |d_5| & : 5 \cdot 12 = 60 \\ \end{align*} \]

Now, we will sum these products:

\[ \Sigma (f_i \cdot |d_i|) = 96 + 60 + 24 + 72 + 60 = 312 \]

### Step 7: Calculate the mean deviation

The mean deviation is given by:

\[ \text{Mean Deviation} = \frac{\Sigma (f_i \cdot |d_i|)}{\Sigma f_i} = \frac{312}{44} \approx 7.09 \]

### Final Answer

The mean deviation about the mean is approximately **7.09**.

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