Let $x_1, x_2, ..., x_n$ be a sample data and let $c$ be a non-zero constant.
Suppose $y_i = x_i + c$, $i = 1, 2, ..., n$. Prove that $ar{y} = ar{x} + c$, and that $s_y = s_x$, where $s_y$ and $s_x$ are the standard deviations of $y$ and $x$ respectively.
Suppose $y_i = cx_i$, $i = 1, 2, ..., n$. Prove that $ar{y} = car{x}$, and that $s_y = |c|s_x$.