Question

For each student in a certain class, a teacher adjusted the student's test score using the formula \( y = 0.8x + 20 \), where \( x \) is the student's original test score and \( y \) is the student's adjusted test score. If the standard deviation of the original test scores of the students in the class was 20, what was the standard deviation of the adjusted test scores of the students in the class? [Official GMAT-2018]

Hint:

When applying a linear transformation to a data set (multiplying by a constant and adding a constant), only the multiplication by the constant affects the standard deviation.

Answer 1

Step 1: Understand the effect of the adjustment.
The formula for the adjusted score is: \[ y = 0.8x + 20 \] Here, the transformation involves multiplying the original score by 0.8 and adding a constant 20. The multiplication by 0.8 affects the standard deviation, but the addition of 20 does not.
Step 2: Effect of multiplication on standard deviation.
Multiplying all values by a constant (0.8 in this case) scales the standard deviation by the same factor. Thus, the standard deviation of the adjusted scores is: \[ \text{Standard deviation of adjusted scores} = 0.8 \times \text{Standard deviation of original scores} \] \[ = 0.8 \times 20 = 16 \] Step 3: Conclusion.
The standard deviation of the adjusted test scores is 16.