Question

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

• A. 10
• B. 20
• C. 40
• D. 15

Hint:

Multiplication affects spread: If you multiply by $k$, Mean changes by $k$, Standard Deviation changes by $k$, but Variance changes by $k^2$!

Answer 1

The Correct Option is B

Concept: Variance measures the "squared" spread of data. When data points are modified by a constant factor, the variance changes according to the square of that factor.

Step 1: Recall the property of Variance.
If every observation $x_i$ in a set is multiplied by a constant $k$, the new standard deviation ($\sigma'$) and new variance ($Var'$) are:
• $\sigma' = |k| \cdot \sigma$
• $Var' = k^2 \cdot Var$

Step 2: Apply the values from the problem.
Given:
• Original Variance ($Var$) = 5
• Multiplication factor ($k$) = 2 Applying the formula: \[ \text{New Variance} = 2^2 \times 5 = 4 \times 5 = 20 \]

Step 3: Final Result.
The resulting variance after multiplying each observation by 2 is 20.