Question

If standard deviation of the data \[ 1,15,35,53,72,64 \] is \(x\), then the variance of the data \[ 62,70,51,33,13,-1 \] is

• A. \(x\)
• B. \(2x\)
• C. \(x+2\)
• D. \(x^2\)

Hint:

Adding or subtracting a constant from all observations does not change variance. Also remember that variance is the square of the standard deviation.

Answer 1

The Correct Option is D

Concept: Variance remains unchanged when the same constant is added to or subtracted from every observation. Also, \[ \text{Variance}=(\text{Standard Deviation})^2. \]

Step 1: Relate the two data sets. The first data set is \[ 1,\;15,\;35,\;53,\;72,\;64. \] The second data set is \[ 62,\;70,\;51,\;33,\;13,\;-1. \] Observe that \[ 62=63-1,\qquad 70=85-15, \] \[ 51=86-35,\qquad 33=86-53, \] \[ 13=85-72,\qquad -1=63-64. \] Thus every observation of the second data set is obtained from the first by the transformation \[ y=K-x, \] where \(K\) is a constant.

Step 2: Use the property of variance. For the transformation \[ y=K-x, \] the variance remains unchanged because multiplication by \(-1\) changes only the sign and not the spread. Hence, \[ \text{Variance of second data set} = \text{Variance of first data set}. \]

Step 3: Express the variance in terms of \(x\). Given that the standard deviation of the first data set is \[ x. \] Therefore, \[ \text{Variance of first data set} = x^2. \] Hence, \[ \text{Variance of second data set} = x^2. \] \[ \boxed{x^2} \] \[ \boxed{\text{Answer = (D)}} \]