Question

The variance of a set of 20 observations is \(16\). If \(7\) is added to each observation, and then \(5\) is subtracted from each resulting observation, what will be the new standard deviation?

• A. \(4\)
• B. \(18\)
• C. \(9\)
• D. \(2\)

Hint:

Adding or subtracting the same constant from every observation never changes the variance or standard deviation.

Answer 1

The Correct Option is A

Understanding the Concept: Variance and standard deviation measure the spread or dispersion of data. A very important property is:
Adding or subtracting a constant from every observation changes the mean but does not change the variance or standard deviation.
Multiplying every observation by a constant changes the variance and standard deviation accordingly.
Thus, whenever only addition or subtraction is involved, the spread of the data remains unchanged.
Step 1: Write the given variance. We are given: \[ \text{Variance}=16 \] We know: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} \] Therefore, \[ \text{Standard Deviation} = \sqrt{16} \] \[ =4 \]
Step 2: Understand the effect of adding and subtracting constants. According to the question:
First, \(7\) is added to each observation.
Then, \(5\) is subtracted from each resulting observation.
Net effect: \[ x \to x+7-5 \] \[ x \to x+2 \] Thus, every observation increases by a constant value \(2\). Adding a constant affects only the mean, not the variance or standard deviation. Hence, the variance remains: \[ 16 \] and therefore the standard deviation remains: \[ \sqrt{16}=4 \] Thus, the new standard deviation is: \[ \boxed{4} \]