Parameters of Random Variables

A discrete random variable's variance is $\sigma_X^2 = \sum (x - \mu_X)^2 P(x)$, and the standard deviation, its square root, is the version carrying the variable's units. A linear transformation $Y = aX + b$ gives $\mu_Y = a\mu_X + b$ and $\sigma_Y = |a|\sigma_X$: a shift moves the center and leaves the spread alone, while a scale factor moves both. For sums and differences, $E(X \pm Y) = E(X) \pm E(Y)$, and for independent variables the variances ADD in both cases, so $\sigma_{X-Y} = \sqrt{\sigma_X^2 + \sigma_Y^2}$: with standard deviations 100 and 60 that is $\sqrt{13600} \approx 116.6$.

The errors are all in the spread. Standard deviations get added directly ($100 + 60 = 160$), or variances get subtracted for a difference ($\sqrt{10000 - 3600} = 80$, and sometimes a negative variance), when a difference of independent variables is more variable than either piece. A shift is treated as changing the spread, or a scale factor as leaving it alone. And $2X$ is treated as $X_1 + X_2$, though doubling one observation gives $2\sigma_X = 200$ while summing two independent ones gives $\sqrt{2}\,\sigma_X \approx 141.4$, since two draws partly cancel each other.