Question

The variance of 20 observations is 5. If each observation is multiplied by 2, then the new variance of the resulting observation is

• A. \(2^3 \times 5\)
• B. \(2^2 \times 5\)
• C. \(2 \times 5\)
• D. \(2^4 \times 5\)

Hint:

Remember: Adding a constant to all observations does NOT change the variance, but multiplying by a constant \(k\) changes it by \(k^2\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept: 
Variance measures the spread of a set of data. If every observation in a dataset is scaled (multiplied) by a constant factor, the variance is scaled by the square of that constant. 
Step 2: Key Formula or Approach: 
For a random variable $X$ or a set of observations, the property of variance under a linear transformation is: \[ \text{Var}(aX) = a^2 \text{Var}(X) \] where $a$ is a constant multiplier. 
Step 3: Detailed Explanation: 
Let the original variance of the observations be $\text{Var}(X) = 5$. 
Each observation is multiplied by the constant $a = 2$. 
The new variance will be: \[ \text{Var}(2X) = 2^2 \times \text{Var}(X) \] \[ \text{Var}(2X) = 2^2 \times 5 = 4 \times 5 = 20 \] Matching this with the given options, the result is explicitly written as $2^2 \times 5$. Note that the number of observations ($n = 20$) is irrelevant to the scaling property of the variance itself. 
Step 4: Final Answer: 
The new variance is $2^2 \times 5$.