The value of `Q_(3)` for the following distribution is
`{:("Marks group:",5-10,10-15,15-20,20-25,25-30,30-35,35-40,40-45),("No of Student:",5,6,15,10,5,4,2,1):}`

Options

A. 15
B. 21.5
C. 25
D. 25.5

Explanation

To find the value of \( Q_3 \) (the third quartile) for the given distribution, we will follow these steps:

### Step 1: Create a Cumulative Frequency Table
We start by calculating the cumulative frequency for the given data.

| Marks Group | No of Students (f) | Cumulative Frequency (cf) |
|-------------|---------------------|----------------------------|
| 5 - 10 | 5 | 5 |
| 10 - 15 | 6 | 5 + 6 = 11 |
| 15 - 20 | 15 | 11 + 15 = 26 |
| 20 - 25 | 10 | 26 + 10 = 36 |
| 25 - 30 | 5 | 36 + 5 = 41 |
| 30 - 35 | 4 | 41 + 4 = 45 |
| 35 - 40 | 2 | 45 + 2 = 47 |
| 40 - 45 | 1 | 47 + 1 = 48 |

### Step 2: Find Total Frequency (n)
Now, we find the total number of students (n) by adding all the frequencies:
\[ n = 5 + 6 + 15 + 10 + 5 + 4 + 2 + 1 = 48 \]

### Step 3: Calculate \( \frac{3n}{4} \)
To find \( Q_3 \), we calculate \( \frac{3n}{4} \):
\[ \frac{3n}{4} = \frac{3 \times 48}{4} = 36 \]

### Step 4: Identify the Median Class
Now, we need to find the cumulative frequency just greater than \( 36 \). From the cumulative frequency table, we see that:
- The cumulative frequency just greater than \( 36 \) is \( 41 \), which corresponds to the class interval \( 25 - 30 \).

### Step 5: Identify Values for the Formula
For the class \( 25 - 30 \):
- Lower limit \( l = 25 \)
- Cumulative frequency of the preceding class \( cf = 36 \)
- Frequency of the median class \( f = 5 \)
- Class width \( h = 5 \) (since \( 30 - 25 = 5 \))

### Step 6: Apply the Formula for \( Q_3 \)
The formula for the third quartile \( Q_3 \) is given by:
\[ Q_3 = l + \left( \frac{\frac{3n}{4} - cf}{f} \right) \times h \]
Substituting the values we have:
\[ Q_3 = 25 + \left( \frac{36 - 36}{5} \right) \times 5 \]
\[ Q_3 = 25 + \left( \frac{0}{5} \right) \times 5 \]
\[ Q_3 = 25 + 0 = 25 \]

### Final Answer
Thus, the value of \( Q_3 \) is \( 25 \).

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