Find the variance of $x$ when the standard deviation is given in the form of $ax+b$.

Question

The standard deviation of $3x+7$ is $4$; then what is the variance of $x$? I know that variance is the square of the standard deviation but how does $3x+7$ and $x$ relate?

Answer 1

Hints: Adding a constant doesn't change the standard deviation. What does multiplying by a constant do?

Answer 2

If the standard deviation of $3x+7$ is $4$, then the standard deviation of $x$ is $4/3$. So the variance must be $16/9$.

When you multiply data by a constant, you multiply the standard deviation by that same constant. You can prove this to yourself by using the formula defining standard deviation. Left as an exercise for the student, of course.