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 find the new variance after each of the 50 observations is increased by 2. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We have 50 observations (n = 50).
- The variance of these observations is given as 18.

2. **Define the Observations:**
- Let the original observations be represented as \( x_1, x_2, x_3, \ldots, x_{50} \).

3. **Transformation of Observations:**
- Each observation is increased by 2. Therefore, the new observations can be defined as:
\[ y_i = x_i + 2 \quad \text{for } i = 1, 2, \ldots, 50 \]

4. **Calculate the Mean of New Observations:**
- The mean of the original observations is denoted as \( \bar{x} \).
- The mean of the new observations \( \bar{y} \) can be calculated as follows:
\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{1}{n} \sum_{i=1}^{n} (x_i + 2) = \frac{1}{n} \sum_{i=1}^{n} x_i + \frac{1}{n} \sum_{i=1}^{n} 2 \]
- This simplifies to:
\[ \bar{y} = \bar{x} + 2 \]

5. **Variance Formula:**
- The variance of a set of observations is defined as:
\[ \text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
- For the new observations, the variance is given by:
\[ \text{Variance of } y = \frac{1}{n} \sum_{i=1}^{n} (y_i - \bar{y})^2 \]

6. **Substituting the New Observations:**
- Substitute \( y_i = x_i + 2 \) and \( \bar{y} = \bar{x} + 2 \):
\[ \text{Variance of } y = \frac{1}{n} \sum_{i=1}^{n} ((x_i + 2) - (\bar{x} + 2))^2 \]
- This simplifies to:
\[ \text{Variance of } y = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
- Thus, we see that the variance of \( y \) is the same as the variance of \( x \).

7. **Conclusion:**
- Since the variance of the original observations \( x \) is 18, the variance of the new observations \( y \) remains:
\[ \text{Variance of } y = 18 \]

### Final Answer:
The variance of the new data is **18**.