Standard Deviation and Changing Scale

Definition Let $X$ be a random variable with a finite second moment $E[X^2]$, so that its mean $\mu = E[X]$ and its variance are finite. The standard deviation of $X$ is the nonnegative square root of its variance,

It is often written $\sigma$. It is also called the root-mean-square deviation, a name that lists the operations from the outside in. The computation runs the other way. Take each deviation $X - \mu$, square it, average the squares with the probability weights, and then take the root. The standard deviation has the same units as $X$, and it equals $0$ exactly when $P(X = \mu) = 1$.

Definition For a random variable $X$ with finite mean $\mu$, the mean absolute deviation is the probability-weighted average distance from the mean,

On this page it is always measured from the mean. When $X$ also has finite variance,

because $\lvert X - \mu \rvert^2 = (X - \mu)^2$ gives $0 \le \operatorname{Var}(\lvert X - \mu \rvert) = \operatorname{Var}(X) - \big(E[\lvert X - \mu \rvert]\big)^2$. Equality holds exactly when $\lvert X - \mu \rvert$ is constant with probability $1$, that is, when $P(X = \mu) = 1$ or when $X$ equals $\mu - c$ and $\mu + c$ with probability $\tfrac12$ each for some $c > 0$.

Definition Let $X$ have finite mean and let $b$ be a constant. The shifted variable $X + b$ adds $b$ to every value of $X$ and keeps every probability. Linearity gives $E[X + b] = E[X] + b$, so the deviations do not change,

If $X$ has finite variance, it follows that

Definition Let $X$ have finite variance and let $a$ be a constant. A change of scale multiplies $X$ by $a$. Every deviation is multiplied by $a$, since $aX - E[aX] = a\big(X - E[X]\big)$, and every squared deviation by $a^2$. Hence

A change of units writes the same quantity on another scale. Often it multiplies by a positive conversion factor, such as $a = 60$ from minutes to seconds, which is a change of scale. Some changes of units also add a constant, as in the conversion from degrees Celsius to degrees Fahrenheit, and those are affine transformations, described in the next card.

Definition An affine transformation of a random variable $X$ is a random variable $aX + b$ for constants $a$ and $b$, a change of scale followed by a shift. If $X$ has finite variance, then

The reason is that the shift cancels from the deviation, $(aX + b) - E[aX + b] = a\big(X - E[X]\big)$. Squaring and taking expectations gives the variance, and the nonnegative square root of $a^2$ is $\lvert a \rvert$. When $a = 0$, the variable is the constant $b$ and its variance and standard deviation are both $0$. When $a < 0$, the order of the values is reversed, but the spread is not negative.