Rescaling for desired standard deviation

Question

Apologies for what is probably a very basic question. I have looked around both here and in the usual places and haven't had any luck.

I have read that there are at least two methods for linearly transforming data so that you can give your distribution a certain desired standard deviation. What are they and are there cases where you'd want to use one method rather than another?

Just for concreteness's sake, let's say you have test scores from 0-50, a mean of 35 and sd of 10, and you wanted to rescale so the sd is 15.

Answer 1

The SD is directly proportional to the data. Therefore, to change it from 10 to 15 = 1.5 * 10, multiply all scores by 1.5. The other way is to multiply all scores by -1.5, because negating all values does not change the SD. Of course you can also add an arbitrary constant to all the scores, too, without changing the SD. That is an exhaustive description of the linear transformations of the data that change the SD to the desired value.

You would use the negative multiple when you want to reverse the order of the data.

Answer 2

If you have a random variable (or observed data) $X$ with mean $\mu_x$ and standard deviation $\sigma_x$, and then apply any linear transformation $$Y=a+bX$$ then you will find the mean of $Y$ is $$\mu_y = a + b \mu_x$$ and the standard deviation of $Y$ is $$\sigma_y = |b|\; \sigma_x.$$

So for example, as whuber says, to multiply the standard deviation by 1.5, the two possibilities are $b=1.5$ or $b=-1.5$, while $a$ can have any value.