The Median of the following discrete series is
`{:(x_(i),3,6,5,8,12,7),(f_(i)"",5,2,4,6,7,6):}`

Options

A. 7
B. 8
C. 9
D. 6

Explanation

To find the median of the given discrete series, we will follow these steps:

### Step 1: Organize the Data
First, we need to organize the data in increasing order along with their corresponding frequencies.

Given data:
- \( x_i = \{3, 5, 6, 7, 8, 12\} \)
- \( f_i = \{5, 4, 2, 6, 7, 6\} \)

Arranging the data in increasing order:
- \( 3 \) (frequency \( 5 \))
- \( 5 \) (frequency \( 4 \))
- \( 6 \) (frequency \( 2 \))
- \( 7 \) (frequency \( 6 \))
- \( 8 \) (frequency \( 7 \))
- \( 12 \) (frequency \( 6 \))

### Step 2: Calculate Cumulative Frequency
Next, we calculate the cumulative frequency for the organized data.

- Cumulative frequency for \( 3 \): \( 5 \)
- Cumulative frequency for \( 5 \): \( 5 + 4 = 9 \)
- Cumulative frequency for \( 6 \): \( 9 + 2 = 11 \)
- Cumulative frequency for \( 7 \): \( 11 + 6 = 17 \)
- Cumulative frequency for \( 8 \): \( 17 + 7 = 24 \)
- Cumulative frequency for \( 12 \): \( 24 + 6 = 30 \)

So, the cumulative frequency table looks like this:

| \( x_i \) | \( f_i \) | Cumulative Frequency |
|-----------|-----------|----------------------|
| 3 | 5 | 5 |
| 5 | 4 | 9 |
| 6 | 2 | 11 |
| 7 | 6 | 17 |
| 8 | 7 | 24 |
| 12 | 6 | 30 |

### Step 3: Find \( N \) and \( \frac{N}{2} \)
Now, we find \( N \), which is the total frequency, and then calculate \( \frac{N}{2} \).

- Total \( N = 30 \)
- \( \frac{N}{2} = \frac{30}{2} = 15 \)

### Step 4: Determine the Median Class
Next, we need to find the cumulative frequency that is greater than or equal to \( 15 \).

From the cumulative frequency table:
- The cumulative frequency just greater than \( 15 \) is \( 17 \), which corresponds to the value \( 7 \).

### Step 5: Conclusion
Since we are dealing with a discrete series and we have found the median class, the median is the value corresponding to this class.

Thus, the median of the given discrete series is \( 7 \).

### Final Answer
The median is \( 7 \).

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