When you **add or subtract** a constant from the outcomes of a random variable, the mean will increase or decrease by that amount, and the standard deviation will stay the same.

Example: This distribution is called X.

\begin{align*}\mu_X &=3 \\ \sigma_X &=1\end{align*}

If we add 2 to the distribution, it becomes X + 2. Now the distribution has a mean of 5, but the standard deviation is still 1:

\begin{align*}\mu_{X+2} &=5 \\ \sigma_{X+2} &=1\end{align*}

When you **multiply or divide** the outcomes of a random variable by a constant, the mean *and the standard deviation* will also be multiplied or divided by that constant.

Let’s see what happens if we multiply each outcome by 3, and find \begin{align*}\mu_{3Y}\end{align*} and \begin{align*}\sigma_{3Y}\end{align*} :

\begin{align*}\mu_{3Y} = 3 \times 3=9\end{align*}

\begin{align*}\sigma_{3Y}=1 \times 3=3\end{align*}