Does Standard Deviation Change When You Add or Multiply by a Constant?

The standard deviation measures how far values sit from their mean. Whether it changes depends on whether the transformation changes those distances.

The rule

For any constants $a$ and $b$:

$$\bar{y} = a\bar{x} + b, \qquad s_y = |a|\, s_x, \qquad s_y^2 = a^2 s_x^2 \qquad \text{where } y_i = a x_i + b.$$

The same holds for random variables: $\operatorname{SD}(aX + b) = |a| \operatorname{SD}(X)$ and $\operatorname{Var}(aX+b) = a^2 \operatorname{Var}(X)$.

Why: look at the deviations

Adding a constant. If $y_i = x_i + b$, then $\bar y = \bar x + b$, so

$$y_i - \bar y = (x_i + b) - (\bar x + b) = x_i - \bar x.$$

The deviations are identical, so the standard deviation is unchanged. Shifting all the data moves it along the axis without spreading it out.

Multiplying by a constant. If $y_i = a x_i$, then $\bar y = a \bar x$ and $y_i - \bar y = a(x_i - \bar x)$. Squaring multiplies each squared deviation by $a^2$:

$$s_y^2 = \frac{1}{n-1}\sum (y_i - \bar y)^2 = \frac{a^2}{n-1} \sum (x_i - \bar x)^2 = a^2 s_x^2,$$

and taking the square root gives $s_y = |a|\, s_x$. The absolute value is needed because a standard deviation is never negative. Multiplying by $-2$ flips the data and doubles its spread.

Worked example

Data: $2, 4, 4, 4, 5, 5, 7, 9$. Mean $5$, population standard deviation $2$.

Transformation	New data	Mean	SD
none	2, 4, 4, 4, 5, 5, 7, 9	5	2
$+10$	12, 14, 14, 14, 15, 15, 17, 19	15	2
$\times 3$	6, 12, 12, 12, 15, 15, 21, 27	15	6
$\times(-1)$	−2, −4, …, −9	−5	2
$3x + 10$	16, 22, …, 37	25	6

You can confirm any row with the calculator above.

Where this comes up

• Unit conversion. Converting °C to °F uses $F = 1.8C + 32$. A temperature SD of $5\ ^\circ\text{C}$ becomes $1.8 \times 5 = 9\ ^\circ\text{F}$. The $+32$ has no effect on the spread.
• z-scores. $z = (x - \bar x)/s$ is the transformation with $a = 1/s$ and $b = -\bar x/s$. That is why z-scores always have mean $0$ and SD $1$.
• Other measures of spread behave the same way: the range, the IQR and the mean absolute deviation are all unchanged by adding a constant and scaled by $|a|$ when multiplying.

Common mistakes

• Scaling the variance by $a$ instead of $a^2$.
• Writing $\operatorname{SD}(-2X) = -2\operatorname{SD}(X)$. A standard deviation can’t be negative.
• Confusing $aX$ with the sum of $a$ independent copies of $X$. $\operatorname{SD}(2X) = 2\operatorname{SD}(X)$, but $\operatorname{SD}(X_1 + X_2) = \sqrt{2}\operatorname{SD}(X)$ for independent copies, as with rolling two dice.