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: Create a Frequency Table
We will create a table with three columns: \(x_i\) (the values), \(f_i\) (the frequencies), and \(c_f\) (the cumulative frequencies).

| \(x_i\) | \(f_i\) | \(c_f\) |
|---------|---------|---------|
| 1 | 8 | |
| 2 | 10 | |
| 3 | 11 | |
| 4 | 16 | |
| 5 | 20 | |
| 6 | 25 | |
| 7 | 15 | |
| 8 | 9 | |
| 9 | 6 | |

### Step 2: Calculate Cumulative Frequency
We will calculate the cumulative frequency by adding the frequencies sequentially.

- \(c_f(1) = 8\)
- \(c_f(2) = 8 + 10 = 18\)
- \(c_f(3) = 18 + 11 = 29\)
- \(c_f(4) = 29 + 16 = 45\)
- \(c_f(5) = 45 + 20 = 65\)
- \(c_f(6) = 65 + 25 = 90\)
- \(c_f(7) = 90 + 15 = 105\)
- \(c_f(8) = 105 + 9 = 114\)
- \(c_f(9) = 114 + 6 = 120\)

Now the table looks like this:

| \(x_i\) | \(f_i\) | \(c_f\) |
|---------|---------|---------|
| 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: Calculate Total Frequency (n)
The total frequency \(n\) is the sum of all frequencies \(f_i\).

\[ n = 8 + 10 + 11 + 16 + 20 + 25 + 15 + 9 + 6 = 120 \]

### Step 4: Find \(n/2\)
Since \(n = 120\), we calculate:

\[ \frac{n}{2} = \frac{120}{2} = 60 \]

### Step 5: Locate the Median Class
We need to find the cumulative frequency that is just greater than or equal to 60. From our cumulative frequency table, we see:

- \(c_f(5) = 65\) (which is the first cumulative frequency greater than 60)

### Step 6: Identify the Median
The value of \(x_i\) corresponding to \(c_f = 65\) is 5. Therefore, the median of the frequency distribution is:

\[ \text{Median} = 5 \]

### Final Answer
The median for the given frequency distribution is **5**.

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