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
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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: Understand how variance responds to linear transformations of a dataset—specifically scaling by a constant factor. Step 2: Key Formula or Approach: Variance measures squared deviations from the mean. For a transformation $Y = aX + b$: $\text{Var}(Y) = a^2\text{Var}(X)$. Addition of a constant $b$ shifts all values equally without affecting spread, so it contributes zero to variance. Step 3: Detailed Explanation: When multiplying every data point by 2, the deviations from the mean also double, and their squares quadruple. Therefore, the new variance becomes $2^2 = 4$ times the original variance. The additive constant merely translates the entire distribution, leaving the dispersion unchanged. Step 4: Final Answer: Multiplying the dataset by 2 scales the variance by a factor of 4.