Question

If the standard deviation of \( n \) elements of the series \( x_1, x_2, x_3, \dots, x_n \) is \( \sigma \), then find the variance of the series \( ax_1, ax_2, ax_3, \dots, ax_n \) is:

• A. \( a^2\sigma \)
• B. \( a^2 n\sigma \)
• C. \( a\sigma \)
• D. \( a^2\sigma^2 \)

Hint:

Remember the scaling property of dispersion parameters:
- Standard Deviation scales linearly: \( \text{S.D.}(aX) = |a|\text{S.D.}(X) \)
- Variance scales quadratically: \( \text{Var}(aX) = a^2\text{Var}(X) \)

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem asks for the variance of a new series obtained by multiplying every term of a given series by a constant \( a \).

Step 2: Key Formula or Approach:
If a series of observations is multiplied by a constant \( a \), the new standard deviation \( \sigma' \) is:
\[ \sigma' = |a| \sigma \] The variance is the square of the standard deviation:
\[ \text{Variance} = \sigma^2 \]

Step 3: Detailed Explanation:
Given that the standard deviation of the series \( x_1, x_2, \dots, x_n \) is \( \sigma \).
So, the variance of the original series is:
\[ \text{Var}(X) = \sigma^2 \] When each element of the series is multiplied by \( a \), the new elements are \( ax_1, ax_2, \dots, ax_n \).
Using the properties of variance:
\[ \text{Var}(aX) = a^2 \text{Var}(X) = a^2 \sigma^2 \] Therefore, the variance of the scaled series is \( a^2 \sigma^2 \).

Step 4: Final Answer:
(D) \( a^2\sigma^2 \)