Effect of a Linear Transformation on Mean and Standard Deviation

For the mean: $\bar y = \frac{1}{n}\sum_{i=1}^{n}y_i = \frac{1}{n}\sum_{i=1}^{n}(ax_i+b) = \frac{a}{n}\sum_{i=1}^{n}x_i + \frac{1}{n}\sum_{i=1}^{n}b = a\bar x + b$, using that $\frac{1}{n}\sum x_i = \bar x$ and $\frac{1}{n}\sum_{i=1}^{n}b = b$.

For the spread, first note that $y_i-\bar y = (ax_i+b)-(a\bar x+b) = a(x_i-\bar x)$: the constant shift $b$ cancels out completely, leaving only the scaled deviation.

Squaring and averaging, $s_y^2 = \frac{1}{n}\sum_{i=1}^{n}(y_i-\bar y)^2 = \frac{1}{n}\sum_{i=1}^{n}a^2(x_i-\bar x)^2 = a^2 s_x^2$. Taking the square root of both sides gives $s_y = |a|s_x$, since a square root is always non-negative.