Question

Mean and variance of six observations are 6 and 12 respectively. If each observation is multiplied by 3, then new variance of the resulting observations is

• A. 288
• B. 36
• C. 18
• D. 108

Answer 1

Answer: (D)

108

Answer 2

Let each observation be $x_i$ with variance $\sigma^2 = 12$. When each observation is multiplied by 3, the new observation is $y_i = 3x_i$. The property of variance for a constant multiplier $a$ is:

$$\text{Var}(aX) = a^2 \times \text{Var}(X)$$

Here, $a = 3$. So, the new variance becomes:

$$\text{Var}(y_i) = 3^2 \times 12 = 9 \times 12 = 108.$$

Therefore, the new variance is 108. Multiplying each observation by 3 scales the variance by $3^2$.