Question

If the standard deviation of the series \( x_1, x_2, \dots, x_n \) is \( \sigma \), then the standard deviation of the series \( \frac{6x_1 - 7}{3}, \frac{6x_2 - 7}{3}, \dots, \frac{6x_n - 7}{3} \) is:

• A. \( 2\sigma \)
• B. \( \sigma \)
• C. \( 6\sigma - 7 \)
• D. \( 2\sigma - \frac{7}{3} \)

Hint:

When applying linear transformations to data, the standard deviation is multiplied by the absolute value of the scaling factor (here, 2).

Answer 1

The Correct Option is A

We are given that the standard deviation of the series \( x_1, x_2, \dots, x_n \) is \( \sigma \). The general formula for transforming the standard deviation is: If the transformation is of the form \( y_i = a x_i + b \), the standard deviation of the transformed series is \( |a| \sigma \), where \( \sigma \) is the standard deviation of the original series. In this case, the transformation is \( \frac{6x_i - 7}{3} \), so \( a = \frac{6}{3} = 2 \). Therefore, the standard deviation of the transformed series is \( 2 \times \sigma = 2\sigma \).