The median for the following frequency distribution is :
`{:(x_(i) " :",1,2,3,4,5,6,7,8,9),(f_(i)" :",8,10,11,16,20,25,15,9,6):}`

Options

A. 6
B. 5
C. 7
D. 4

Explanation

To find the median for the given frequency distribution, we will follow these steps:

### Step 1: Organize the Data
We have the following data:
- \( x_i \) (values): 1, 2, 3, 4, 5, 6, 7, 8, 9
- \( f_i \) (frequencies): 8, 10, 11, 16, 20, 25, 15, 9, 6

### Step 2: Calculate the Cumulative Frequency
We will calculate the cumulative frequency by adding the frequencies sequentially:
- For \( x_1 = 1 \): Cumulative frequency = 8
- For \( x_2 = 2 \): Cumulative frequency = 8 + 10 = 18
- For \( x_3 = 3 \): Cumulative frequency = 18 + 11 = 29
- For \( x_4 = 4 \): Cumulative frequency = 29 + 16 = 45
- For \( x_5 = 5 \): Cumulative frequency = 45 + 20 = 65
- For \( x_6 = 6 \): Cumulative frequency = 65 + 25 = 90
- For \( x_7 = 7 \): Cumulative frequency = 90 + 15 = 105
- For \( x_8 = 8 \): Cumulative frequency = 105 + 9 = 114
- For \( x_9 = 9 \): Cumulative frequency = 114 + 6 = 120

The cumulative frequency table is as follows:

| \( x_i \) | \( f_i \) | Cumulative Frequency |
|-----------|-----------|----------------------|
| 1 | 8 | 8 |
| 2 | 10 | 18 |
| 3 | 11 | 29 |
| 4 | 16 | 45 |
| 5 | 20 | 65 |
| 6 | 25 | 90 |
| 7 | 15 | 105 |
| 8 | 9 | 114 |
| 9 | 6 | 120 |

### Step 3: Find \( n \) and \( n/2 \)
The total frequency \( n \) is the last cumulative frequency, which is 120. Now, we calculate \( n/2 \):
\[ n/2 = 120/2 = 60 \]

### Step 4: Identify the Median Class
We need to find the cumulative frequency that is just greater than \( n/2 \) (which is 60). From the cumulative frequency table, we see:
- The cumulative frequency for \( x = 5 \) is 65, which is the first cumulative frequency greater than 60.

Thus, the median class is the class corresponding to \( x = 5 \).

### Step 5: Conclusion
Since the median class is identified as 5, we can conclude that the median for the given frequency distribution is:
\[ \text{Median} = 5 \]