Find the mean deviation about the median for the following data.
`{:(x,1,2,3,4,5,6),(f,3,7,5,8,2,5):}`

Options

A. `1`
B. `0.bar(7)`
C. `3`
D. `1.bar(3)`

Explanation

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

### Step 1: Organize the Data
We have the following data:
- \( x = \{1, 2, 3, 4, 5, 6\} \)
- \( f = \{3, 7, 5, 8, 2, 5\} \)

### Step 2: Create a Frequency Table with Cumulative Frequency
We will create a table to find the cumulative frequency.

| \( x \) | \( f \) | Cumulative Frequency |
|---------|---------|----------------------|
| 1 | 3 | 3 |
| 2 | 7 | 10 |
| 3 | 5 | 15 |
| 4 | 8 | 23 |
| 5 | 2 | 25 |
| 6 | 5 | 30 |

### Step 3: Calculate the Total Frequency (n)
The total frequency \( n \) is the sum of all frequencies:
\[ n = 3 + 7 + 5 + 8 + 2 + 5 = 30 \]

### Step 4: Find \( n/2 \)
Calculate \( n/2 \):
\[ n/2 = 30/2 = 15 \]

### Step 5: Determine the Median
To find the median, we look for the cumulative frequency that is greater than or equal to \( n/2 \):
- The cumulative frequency just before 15 is 10 (for \( x = 2 \)).
- The cumulative frequency at 15 is 15 (for \( x = 3 \)).
Thus, the median \( m \) is:
\[ m = 3 \]

### Step 6: Calculate Deviations from the Median
Now, we calculate the absolute deviations from the median:
- \( |x_i - m| \)

| \( x \) | \( f \) | \( |x_i - m| \) | \( f \cdot |x_i - m| \) |
|---------|---------|------------------|------------------------|
| 1 | 3 | \( |1 - 3| = 2 \) | \( 3 \cdot 2 = 6 \) |
| 2 | 7 | \( |2 - 3| = 1 \) | \( 7 \cdot 1 = 7 \) |
| 3 | 5 | \( |3 - 3| = 0 \) | \( 5 \cdot 0 = 0 \) |
| 4 | 8 | \( |4 - 3| = 1 \) | \( 8 \cdot 1 = 8 \) |
| 5 | 2 | \( |5 - 3| = 2 \) | \( 2 \cdot 2 = 4 \) |
| 6 | 5 | \( |6 - 3| = 3 \) | \( 5 \cdot 3 = 15 \) |

### Step 7: Sum of the Products
Now, we sum the last column:
\[ 6 + 7 + 0 + 8 + 4 + 15 = 40 \]

### Step 8: Calculate the Mean Deviation
Finally, we calculate the mean deviation using the formula:
\[ \text{Mean Deviation} = \frac{\sum f_i |x_i - m|}{n} = \frac{40}{30} = \frac{4}{3} \approx 1.33 \]

### Final Answer
The mean deviation about the median is approximately \( 1.33 \).

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