If the variance of the data $2, 4, 5, 6, 8, 17$ is $23.33$, then the variance of $4, 8, 10, 12, 16, 34$ will be
Show Hint
Remember that variance is a measure of squared deviations. Thus, any addition or subtraction of a constant does not alter the variance at all, while multiplying by a constant multiplies the variance by that constant squared. Multiplying the dataset by 2 means the variance must scale up by 4!
Step 1: Understanding the Question:
We are given the numerical variance of an initial set of data observations. A second, transformed data set is presented, and we need to determine its new variance. Step 2: Key Formula or Approach:
Analyze how the data has been altered. If every observation in a dataset $X$ is multiplied by a constant factor $c$, the new dataset becomes $cX$.
The fundamental statistical theorem for the change of scale states:
$$\text{Var}(cX) = c^2 \cdot \text{Var}(X)$$
Step 3: Detailed Explanation:
Let's compare the original data set $X$ with the new data set $Y$:
$$X = \{2, 4, 5, 6, 8, 17\}$$
$$Y = \{4, 8, 10, 12, 16, 34\}$$
Notice that each element in $Y$ is exactly double its corresponding element in $X$:
$$4 = 2(2), \quad 8 = 2(4), \quad 10 = 2(5), \dots, \quad 34 = 2(17)$$
Therefore, the scale factor is $c = 2$, which means $Y = 2X$.
Using our change of scale property:
$$\text{Var}(Y) = \text{Var}(2X) = 2^2 \cdot \text{Var}(X)$$
$$\text{Var}(Y) = 4 \cdot \text{Var}(X)$$
Substitute the given variance value of $23.33$:
$$\text{Var}(Y) = 4 \times 23.33 = 93.32$$
Step 4: Final Answer:
The variance of the modified dataset is $93.32$, matching option (A).
