Question

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

• A. 5
• B. 10
• C. 4
• D. 20

Hint:

Statistics Tip: If $Y = aX + b$, then $Var(Y) = a^2 \times Var(X)$.The addition or subtraction of a constant does not affect the variance.

Answer 1

The Correct Option is D

Concept:
Statistics - Properties of Variance. When each observation is multiplied by a constant $k$, the new variance becomes $k^2$ times the original variance.

Step 1: Define the original variance and its formula.
Let the original set of observations be denoted as $X=\{x_1,x_2,...,x_{20}\}$. The variance of these observations, $Var(X)$, is given as 5. The formula is $Var(X)=\frac{\sum_{i=1}^{20}(x_{i}-\mu)^{2}}{20}$, where $\mu$ is the mean of the original observations.

Step 2: Define the new set of observations.
If each observation is multiplied by 2, we get a new set of observations $Y=\{2x_{1},2x_{2},...,2x_{20}\}$.

Step 3: Determine the new mean.
The new variance is $Var(Y)=\frac{1}{20}\sum_{i=1}^{20}(2x_{i}-\mu^{\prime})^{2}$, where $\mu^{\prime}$ is the new mean. Since every term is doubled, the new mean is also doubled: $\mu^{\prime}=2\mu$.

Step 4: Substitute and simplify the new variance expression.
Substitute $\mu^{\prime}$ into the equation: $(2x_{i}-2\mu)^{2} = 4(x_{i}-\mu)^{2}$. Factoring out the 4 gives us: $Var(Y) = \frac{1}{20}\sum_{i=1}^{20}4(x_{i}-\mu)^{2} = 4\times\frac{1}{20}\sum_{i=1}^{20}(x_{i}-\mu)^{2}$.

Step 5: Calculate the final numerical value.
Recognize that the remaining sum is the original variance. Therefore, $Var(Y) = 4\times Var(X)$. Since $Var(X)=5$, we calculate $Var(Y) = 4\times 5 = 20$. $$ \therefore \text{The correct answer is Option D: 20.} $$