Linear transformation of a random variable

A linear transformation replaces $X$ with $Y = a + bX$ for fixed numbers $a$ and $b$. The mean follows the rule the values follow, $\mu_Y = a + b\mu_X$ (mu-Y, the mean of Y), but the standard deviation answers only to the multiplier: $\sigma_Y = |b|\sigma_X$ (sigma-Y). Adding $a$ slides every value the same distance, which cannot change how far apart any two of them are, and the absolute value keeps $\sigma_Y$ positive when $b$ is negative. In variance terms $\sigma_Y^2 = b^2\sigma_X^2$.

Daily high temperatures in Celsius have mean 20 and standard deviation 4. Converting with $F = 32 + 1.8C$ gives a mean of $32 + 1.8(20) = 68$ degrees Fahrenheit and a standard deviation of $1.8(4) = 7.2$ degrees.

"So the standard deviation is $32 + 1.8(4) = 39.2$" is the wrong sentence, and it is the one this topic produces most. A standard deviation measures distances between values. Lifting the whole distribution by 32 moves both ends of every distance, so every distance survives untouched. Only the stretch by 1.8 does anything at all, and a spread of 39.2 degrees on daily temperatures should have failed a sanity check before the algebra did.

Standardizing is the case worth recognizing on sight: $Z = \frac{X - \mu_X}{\sigma_X}$ is $Y = a + bX$ with $b = \frac{1}{\sigma_X}$, which is why a z-score has mean 0 and standard deviation 1.

Linear is the entire condition. Squaring is not linear and neither rule survives it: for $X$ taking 0, 1 and 2 with probabilities 0.5, 0.3 and 0.2, $\mu_X = 0.7$, but $X^2$ takes 0, 1 and 4 with those same probabilities, so its mean is $0(0.5) + 1(0.3) + 4(0.2) = 1.1$, not $0.7^2 = 0.49$.

Transforming one variable is a different operation from combining two. Adding or subtracting random variables, where variances add under independence, is among the topics College Board removed from the course with the Fall 2026 revision, and that removal says nothing either way about rescaling a single variable.