For a symmetrical distribution `Q_(1) =20 and Q_(3) =40. ` The value of `50^(th)` percentile, is

Options

A. 20
B. 30
C. 40
D. none of these

Explanation

To find the value of the 50th percentile (which is also the median) for a symmetrical distribution where \( Q_1 = 20 \) and \( Q_3 = 40 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Quartiles**: We are given that \( Q_1 = 20 \) and \( Q_3 = 40 \).

2. **Use the Formula for the Median**: In a symmetrical distribution, the median (or the 50th percentile, \( P_{50} \)) can be calculated using the formula:
\[ P_{50} = \frac{Q_1 + Q_3}{2} \]

3. **Substitute the Values**: Plug in the values of \( Q_1 \) and \( Q_3 \) into the formula:
\[ P_{50} = \frac{20 + 40}{2} \]

4. **Calculate the Sum**: First, calculate the sum of \( Q_1 \) and \( Q_3 \):
\[ 20 + 40 = 60 \]

5. **Divide by 2**: Now, divide the sum by 2 to find the median:
\[ P_{50} = \frac{60}{2} = 30 \]

6. **Conclusion**: Therefore, the value of the 50th percentile is:
\[ P_{50} = 30 \]

### Final Answer:
The value of the 50th percentile is **30**.