Question

If the standard deviation of \( x_1, x_2, x_3, \dots, x_n \) is \( k \), then what will be the standard deviation of \( \frac{10x_1 - 7}{2} \), \( \frac{10x_2 - 7}{2} \), ......., \( \frac{10x_n - 7}{2} \)?

• A. \( k \)
• B. \( \frac{10k - 7}{2} \)
• C. \( 5k \)
• D. \( \frac{5k - 7}{2} \)

Hint:

When scaling data by a constant factor, the standard deviation is multiplied by the same constant. Shifting the data does not affect the standard deviation.

Answer 1

The Correct Option is C

The standard deviation of \( x_1, x_2, x_3, \dots, x_n \) is \( k \).
When a constant multiplier is applied to each data point, the standard deviation is scaled by the same factor.
In this case, the transformation \( \frac{10x_i - 7}{2} \) can be broken down into two operations: multiplying each \( x_i \) by \( 5 \) (since \( \frac{10}{2} = 5 \)) and subtracting \( 7 \), which does not affect the standard deviation.
Thus, the standard deviation of the transformed data will be \( 5k \).