Given the that variance of 50 observations is 18. If each of the 50 observations is increased by 2, then variance of new data is

Options

A. 50
B. 52
C. 18
D. 25

Answer

To solve the problem, we need to determine the variance of a new set of observations obtained by increasing each of the original observations by 2.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We have 50 observations with a variance of 18.
- Let the original observations be represented as \( x_1, x_2, x_3, \ldots, x_{50} \).
- The variance of these observations is given by \( \sigma^2_x = 18 \).

2. **Define the New Observations:**
- The new observations after increasing each original observation by 2 can be represented as:
\[ y_i = x_i + 2 \quad \text{for } i = 1, 2, \ldots, 50 \]

3. **Calculate the Mean of the New Observations:**
- The mean of the new observations \( \bar{y} \) can be calculated as follows:
\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{1}{50} \sum_{i=1}^{50} (x_i + 2) \]
- This can be simplified to:
\[ \bar{y} = \frac{1}{50} \left( \sum_{i=1}^{50} x_i + \sum_{i=1}^{50} 2 \right) = \frac{1}{50} \left( \sum_{i=1}^{50} x_i + 100 \right) \]
- Thus, we can express it as:
\[ \bar{y} = \bar{x} + 2 \]
- where \( \bar{x} \) is the mean of the original observations.

4. **Calculate the Variance of the New Observations:**
- The variance of the new observations \( \sigma^2_y \) is given by:
\[ \sigma^2_y = \frac{1}{n} \sum_{i=1}^{n} (y_i - \bar{y})^2 \]
- Substituting \( y_i = x_i + 2 \):
\[ \sigma^2_y = \frac{1}{n} \sum_{i=1}^{n} ((x_i + 2) - (\bar{x} + 2))^2 \]
- This simplifies to:
\[ \sigma^2_y = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
- Therefore, we find that:
\[ \sigma^2_y = \sigma^2_x \]

5. **Conclusion:**
- Since the variance of the original observations \( \sigma^2_x = 18 \), we conclude that:
\[ \sigma^2_y = 18 \]

Thus, the variance of the new data is **18**.