What Happens to the Mean and Standard Deviation When You Add or Multiply All Data by a Constant?

Adding a constant to every data value changes the mean by that constant but leaves the standard deviation unchanged. Multiplying every value by a constant multiplies the mean by that constant and the standard deviation by its absolute value.

This is part of SL 4.3: Mean, median, and mode, studied in both SL and HL Mathematics: Applications and Interpretation.

Transformation Rules

Suppose the original data has mean $\bar{x}$, standard deviation $\sigma$, and variance $\sigma^2$. If each value $x$ is transformed into $y=ax+b$, then

$$\bar{y}=a\bar{x}+b$$

and

$$\sigma_y=|a|\sigma, \qquad \operatorname{Var}(Y)=a^2\sigma^2.$$

Adding a constant shifts the whole distribution without changing the distances between values, so the spread is unchanged. Multiplication scales every distance from the mean, so the standard deviation is scaled too.

More formally, each deviation from the new mean is $y-\bar{y}=a(x-\bar{x})$. The translation $b$ cancels, while multiplication changes every deviation by factor . Squaring these deviations explains why variance uses , and taking the square root gives (|a|) for standard deviation.

This rule applies whether the original data are positive, negative, or mixed. For IB Math AI questions, use the population standard deviation $\sigma$ when the listed data represent the whole population, unless the question states that they are a sample.

For example, the data $2,4,6$ has mean $4$ and population standard deviation

$$\sigma=\sqrt{\frac{(2-4)^2+(4-4)^2+(6-4)^2}{3}}=\sqrt{\frac{8}{3}}.$$

Adding $3$ gives $5,7,9$, with mean $7$ and the same standard deviation. Multiplying the original data by $-2$ gives $-4,-8,-12$, with mean and standard deviation .

Exam Technique

A common misconception is that a negative multiplier produces a negative standard deviation. This is impossible: standard deviation is non-negative, so use (|a|). In an IB response, state the transformation rule and distinguish carefully between standard deviation, which scales by (|a|), and variance, which scales by $a^2$.